Showing papers on "Measure (mathematics) published in 1993"

Abundance of strange attractors

Abstract: In 1976 [He] H~non performed a numerical study of the family of diffeomorphisms of the plane ha,b(X, y)=(1-ax2+y, bx) and detected for parameter values a=l.4, b=0.3, what seemed to be a non-trivial attractor with a highly intricate geometric structure. This family has since then been the subject of intense research, both numerical and theoretical, but its dynamics is still far from being completely understood. In particular one could not exclude the possibility that the attractor observed by H6non were just a periodic orbit with a very high period. Recently, in a remarkable paper [BC2], Benedicks and Carleson were able to show that this is not the case, at least for a positive Lebesgue measure set of parameter values near a=2, b=O. More precisely, they showed that if b>0 is small enough then for a positive measure set of a-values near a=2 the corresponding diffeomorphism ha,b exhibits a strange attractor. Their argument is a very creative extension of the techniques they had previously developed in [BUll for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated, nonhyperbolic dynamics. When acquainted in 1985 with the work by Benedicks and Carleson, then in progress, Palls suggested that one should in this context think of the H6non family as a particular, although important, model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies. He proposed that the correct setting for Benedicks-Carleson's results is within this more general framework of homoclinic bifurcations and stated the following

Polynomial diffeomorphisms of C2. IV: The measure of maximal entropy and laminar currents.

Abstract: The simplest holomorphic dynamical systems which display interesting behavior are the polynomial maps of C The dynamical study of these maps began with Fatou and Julia in the s and is currently a very active area of research If we are interested in studying invertible holomorphic dynamical systems then the simplest examples with interesting behavior are probably the polynomial di eomorphisms ofC These are maps f C C such that the coordinate functions of f and f are holomorphic polynomials For polynomial maps of C the algebraic degree of the polynomial is a useful dynamical invariant In particular the only dynamically interesting maps are those with degree d greater than one For polynomial di eomorphisms we can de ne the algebraic degree to be the maximum of the degrees of the coordinate functions This is not however a conjugacy invariant Friedland and Milnor FM gave an alternative de nition of a positive integer deg f which is more natural from a dynamical point of view If deg f then deg f coincides with the minimal algebraic degree of a di eomorphism in the conjugacy class of f As in the case of polynomial maps of C the polynomial di eomorphisms f with deg f are rather uninteresting We will make the standing assumption that deg f For a polynomial map of C the point at in nity is an attractor Thus the recurrent dynamics can take place only on the set K consisting of bounded orbits A normal families argument shows that there is no expansion on the interior of K so chaotic dynamics can occur only on J K This set is called the Julia set and plays a major role in the study of polynomial maps For di eomorphisms of C each of the objectsK and J has three analogs Correspond ing to the set K in one dimension we have the sets K resp K consisting of the points whose orbits are bounded in forward resp backward time and the set K K K consisting of points with bounded total orbits Each of these sets is invariant and K is compact As is in the one dimensional case recurrence can occur only on the set K Corresponding to the set J in dimension one we have the sets J K and the set J J J Each of these sets is invariant and J is compact A normal families argu ment shows that there is no forward instability in the interior of K and no backward instability in the interior of K Thus chaotic dynamics that is recurrent dynamics with instability in both forward and backward time can occur only on the set J The techniques that Fatou and Julia used in one dimension are based on Montel s theory of normal families and do not readily generalize to higher dimensions A di erent tool appears in the work of Brolin Br who made use of the theory of the logarithmic

Ergodic theory for Markov fibred systems and parabolic rational maps

Abstract: A paraboIic rational map of the Riemann sphere admits a non-atomic h-conformal measure on its Julia set where h = the Hausdorff dimension of the Julia set and satisfies 1/2 < h < 2. With respect to this measure the rational map is conservative, exact and there is an equivalent a-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework

Steady-state electrical conduction in the periodic Lorentz gas

Abstract: We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an electric external field is applied and the particle kinetic energy is held fixed by a “thermostat” constructed according to Gauss’ principle of least constraint (a model problem previously studied numerically by Moran and Hoover). The resulting dynamics is reversible and deterministic, but does not preserve Liouville measure. For a sufficiently small field, we prove the following results: (1) existence of a unique stationary, ergodic measure obtained by forward evolution of initial absolutely continuous distributions, for which the Pesin entropy formula and Young's expression for the fractal dimension are valid; (2) exact identity of the steady-state thermodynamic entropy production, the asymptotic decay of the Gibbs entropy for the time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an explicit expression for the full nonlinear current response (Kawasaki formula); and (4) validity of linear response theory and Ohm's transport law, including the Einstein relation between conductivity and diffusion matrices. Results (2) and (4) yield also a direct relation between Lyapunov exponents and zero-field transport (=diffusion) coefficients. Although we restrict ourselves here to dimensiond=2, the results carry over to higher dimensions and to some other physical situations: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical systems and the method of Markov sieves, an approximation of Markov partitions.

Dilation for Sets of Probabilities

Abstract: Suppose that a probability measure $P$ is known to lie in a set of probability measures $M$. Upper and lower bounds on the probability of any event may then be computed. Sometimes, the bounds on the probability of an event $A$ conditional on an event $B$ may strictly contain the bounds on the unconditional probability of $A$. Surprisingly, this might happen for every $B$ in a partition $\mathscr{B}$. If so, we say that dilation has occurred. In addition to being an interesting statistical curiosity, this counterintuitive phenomenon has important implications in robust Bayesian inference and in the theory of upper and lower probabilities. We investigate conditions under which dilation occurs and we study some of its implications. We characterize dilation immune neighborhoods of the uniform measure.

Validation of self-ideal body size discrepancy as a measure of body dissatisfaction

Abstract: Recently, body dissatisfaction has been conceptualized as the discrepancy between self and ideal body size estimates. This study evaluated the validity of this conceptualization using three methods for estimating actual and ideal body size: (a) the Body Image Assessment, (b) the Body Image Testing System, and (c) the Body Image Detection Device. The three body image assessment procedures were concurrently administered to a sample of 110 women diagnosed: bulimia nervosa (n=18),obese (n=34),and non-eating disorder (n=58).The Eating Disorder Inventory Body Dissatisfaction scale was also used to measure body dissatisfaction. Measures of self-ideal body size discrepancy were found to correlate more highly with measures of body dissatisfaction than were measures of current body size perception, ideal body size, body size estimation accuracy, or indices based on actual body size. Estimation of both current and ideal body size were found to significantly predict overall body dissatisfaction; thus, both self and ideal body size measures were found to be significant components in determining body size dissatisfaction. These data were interpreted as supportive of the conceptualization of body dissatisfaction as the discrepancy between self and ideal body size estimates.

Skewness and Kurtosis in Large-Scale Cosmic Fields

Abstract: In this paper, I present the calculation of the third and fourth moments of both the distribution function of the large--scale density and the large--scale divergence of the velocity field, $\theta$. These calculations are made by the mean of perturbative calculations assuming Gaussian initial conditions and are expected to be valid in the linear or quasi linear regime. The moments are derived for a top--hat window function and for any cosmological parameters $\Omega$ and $\Lambda$. It turns out that the dependence with $\Lambda$ is always very weak whereas the moments of the distribution function of the divergence are strongly dependent on $\Omega$. A method to measure $\Omega$ using the skewness of this field has already been presented by Bernardeau et al. (1993). I show here that the simultaneous measurement of the skewness and the kurtosis allows to test the validity of the gravitational instability scenario hypothesis. Indeed there is a combination of the first three moments of $\theta$ that is almost independent of the cosmological parameters $\Omega$ and $\Lambda$, $${( -3 ^2) \over ^2}\approx 1.5,$$ (the value quoted is valid when the index of the power spectrum at the filtering scale is close to -1) so that any cosmic velocity field created by gravitational instabilities should verify such a property.

Discrepancy and approximations for bounded VC-dimension

Abstract: Let (X, ℛ) be a set system on ann-point setX. For a two-coloring onX, itsdiscrepancy is defined as the maximum number by which the occurrences of the two colors differ in any set in ℛ. We show that if for anym-point subset $$Y \subseteq X$$ the number of distinct subsets induced by ℛ onY is bounded byO(m d) for a fixed integerd, then there is a coloring with discrepancy bounded byO(n 1/2−1/2d(logn)1+1/2d ). Also if any subcollection ofm sets of ℛ partitions the points into at mostO(m d) classes, then there is a coloring with discrepancy at mostO(n 1/2−1/2dlogn). These bounds imply improved upper bounds on the size of e-approximations for (X, ℛ). All the bounds are tight up to polylogarithmic factors in the worst case. Our results allow to generalize several results of Beck bounding the discrepancy in certain geometric settings to the case when the discrepancy is taken relative to an arbitrary measure.

Sharing of electrons in molecules

Abstract: An index which gives a quantitative measure of the degree of sharing of an electron between two points in space in systems containing many electrons is introduced. This sharing index, denoted by I(ξ;ξ), is defined as the absolute value squared of the matrix element of a sharing amplitude, (ξ;ξ), which in turn is the square root of the first-order density matrix. These quantities are invariant under transformations of the orbitals in terms of which the wavefunction is typically expressed and are independent of the basis set providod it is sufficiently complete. The sharing amplitude has many of the characteristics of a wavefunction. By integration of the sharing index over volumes assigned to atoms, indices which measure the degree of sharing of an electron between atoms in molecules are found

The dynamics of perturbations of the contracting Lorenz attractor

Abstract: We show here that by modifying the eigenvalues λ2 0 by acontracting condition λ3+λ1 < 0, we can obtain vector fields exhibiting transitive non-hyperbolic attractors which are persistent in the following measure theoretical sense: They correspond to a positive Lebesgue measure set in a twoparameter space Actually, there is a codimension-two submanifold in the space of all vector fields, whose elements are full density points for the set of vector fields that exhibit a contracting Lorenz-like attractor in generic two parameter families through them On the other hand, for an open and dense set of perturbations, the attractor breaks into one or at most two attracting periodic orbits, the singularity, a hyperbolic set and a set of wandering orbits linking these objects

The uniform random tree in a Brownian excursion

Abstract: To any Brownian excursione with duration σ(e) and anyt 1, ...,t p ∈[0,σ(e)], we associate a branching tree withp branches denoted byT p (e, t 1,...,t p ), which is closely related to the structure of the minima ofe. Our main theorem states that, ife is chosen according to the Ito measure and (t 1, ...,t p ) according to Lebesgue measure on [0,σ(e)] p , the treeT p (e, t 1, ...,t p ) is distributed according to the uniform measure on the set of trees withp branches. The proof of this result yields additional information about the “subexcursions” ofe corresponding to the different branches of the tree, thus generalizing a well-known representation theorem of Bismut. If we replace the Ito measure by the law of the normalized excursion, a simple conditioning argument leads to another remarkable result originally proved by Aldous with a very different method.

A class of path-valued Markov processes and its applications to superprocesses

Abstract: Let (ξ s ) be a continuous Markov process satisfying certain regularity assumptions. We introduce a path-valued strong Markov process associated with (ξ s ), which is closely related to the so-called superprocess with spatial motion (ξ s ). In particular, a subsetH of the state space of (ξ s ) intersects the range of the superprocess if and only if the set of paths that hitH is not polar for the path-valued process. The latter property can be investigated using the tools of the potential theory of symmetric Markov processes: A set is not polar if and only if it supports a measure of finite energy. The same approach can be applied to study sets that are polar for the graph of the superprocess. In the special case when (ξ s ) is a diffusion process, we recover certain results recently obtained by Dynkin.

Space Syntax: Standardised Integration Measures and Some Simulations:

Abstract: Central to the space syntax method (a technique for morphological analysis of urban areas) is the concept of integration. Unfortunately, the integration values are not independent of the size of urban areas. Consequently it is difficult to compare areas of different size, implying a need for standardisation of integration measures.In this paper three such standardised integration measures are discussed. The properties of these measures are simulated for differently sized urban areas, with the assumption of a spatial structure with morphological constants. The results of these simulations suggest that a measure based on a comparison with an ‘axial grid’ has most desirable properties. Comments regarding the interpretation and use of the chosen standardised integration measure are provided.

Correlation functions in the Itzykson-Zuber model

Abstract: Then-point function for the integral over unitary matrices with Itzykson-Zuber measure is reduced to the integral over the Gelfand-Tzetlin table; the integrand (for genericn) is given by linear exponential times the rational function. Forn=2 and in some cases forn>2 later in fast is the polynomial and this allows to give an explicit and simple expression for all 2-point and a set ofn-point functions. For the most generaln-point function a simple linear differential equation is constructed.

Polynomial averages in the Kontsevich model

Abstract: We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic potential) measure, as derivatives of the partition function with respect to traces of inverse odd powers of the external argument. The proofs are based on elementary algebraic identities involving a new set of invariant polynomials of the linear group, closely related to the general Schur functions.

New ergodic billiards: exact results

Abstract: The author presents an improved version of work on the fundamental theorem of Sinai and Chernov (1987). It can be used to study the ergodicity of dynamical systems with singularities if an increasing nondegenerate quadratic form is defined almost everywhere. The author proves the ergodicity of the billiard map in the cardioid. The methods used in this proof allow one to check which if the billiards with Pesin region of measure one are ergodic.

Locally monotonic regression

Abstract: The concept of local monotonicity appears in the study of the set of root signals of the median filter and provides a measure of the smoothness of the signal. The median filter is a suboptimal smoother under this measure of smoothness, since a filter pass does necessarily yield a locally monotonic output; even if a locally monotonic output does result, there is no guarantee that it will possess other desirable properties such as optimal similarity to the original signal. Locally monotonic regression is a technique for the optimal smoothing of finite-length discrete real signals under such a criterion. A theoretical framework in which the existence of locally monotonic regression is proved and algorithms for their computation are given. Regression is considered as an approximation problem in R/sub n/, the criterion of approximation is derived from a semimetric, and the approximating set is the collection of signals sharing the property of being locally monotonic. >

Local influence: a new approach

Abstract: The concept of local influence was introduced by Cook(1986). Closer study of the idea of perturbations suggests that it is important to distinguish between those of the data and those of the model, and that in the latter case Cook's definition has a theoretical difficulty. Here a new measure is proposed, which has the incidental benefit of being simpler to compute.

Quadratic and quasi-quadratic functionals

Abstract: In this note we show how Jordan *-derivations arise as a "measure" of the representability of quasi-quadratic functionals by sesquilinear ones. Our main result can be considered as an extension of the Jordan-von Neumann characterization of pre-Hilbert space.

A system-theoretic property of serial production lines: improvability

Abstract: A production system is called improvable if the limited resources involved in its operation can be redistributed so that a performance measure is improved. In this paper, the property of improvability is analyzed for the case of a particular system-the serial production line. Improvability of the production rate with respect to the workforce and work-in-process distribution is analyzed, appropriate indicators of improvability are derived, and their utilization in the process of continuous improvement is discussed. It is shown, in particular, that in a well designed system each buffer is, on the average, half full, and each intermediate machine has equal frequencies of blockages and starvations. >

Algorithmic complexity of points in dynamical systems

Abstract: This work is based on the author's dissertation. We examine the algorithmic complexity (in the sense of Kolmogorov and Chaitin) of the orbits of points in dynamical systems. Extending a theorem of A. A. Brudno, we note that for any ergodic invariant probability measure on a compact dynamical system, almost every trajectory has a limiting complexity equal to the entropy of the system. We use these results to show that for minimal dynamical systems, and for systems with the tracking property (a weaker version of specification), the set of points whose trajectories have upper complexity equal to the topological entropy is residual. We give an example of a topologically transitive system with positive entropy for which an uncountable open set of points has upper complexity equal to zero. We use techniques from universal data compression to prove a recurrence theorem: if a compact dynamical system has a unique measure of maximal entropy, then any point whose lower complexity is equal to the topological entropy is generic for that unique measure. Finally, we discuss algorithmic versions of the theorem of Kamae on preservation of the class of normal sequences under selection by sequences of zero Kamae-entropy.

The Measure Representation: A Correction

Abstract: Wakker (1991) and Puppe (1990) point out a mistake in theorem 1 in Segal (1989). This theorem deals with representing preference relations over lotteries by the measure of their epigraphs. An error in the theorem is that it gives wrong conditions concerning the continuity of the measure. This article corrects the error. Another problem is that the axioms do not imply that the measure is bounded; therefore, the measure representation applies only to subsets of the space of lotteries, although these subsets can become arbitrarily close to the whole space of lotteries. Some additional axioms (Segal, 1989, 1990) implying that the measure is a product measure (and hence anticipated utility) also guarantee that the measure is bounded.