Showing papers on "Measure (mathematics) published in 1991"

Multidimensional orientation estimation with applications to texture analysis and optical flow

Abstract: The problem of detection of orientation in finite dimensional Euclidean spaces is solved in the least squares sense. The theory is developed for the case when such orientation computations are necessary at all local neighborhoods of the n-dimensional Euclidean space. Detection of orientation is shown to correspond to fitting an axis or a plane to the Fourier transform of an n-dimensional structure. The solution of this problem is related to the solution of a well-known matrix eigenvalue problem. The computations can be performed in the spatial domain without actually doing a Fourier transformation. Along with the orientation estimate, a certainty measure, based on the error of the fit, is proposed. Two applications in image analysis are considered: texture segmentation and optical flow. The theory is verified by experiments which confirm accurate orientation estimates and reliable certainty measures in the presence of noise. The comparative results indicate that the theory produces algorithms computing robust texture features as well as optical flow. >

A Distance-Based Attribute Selection Measure for Decision Tree Induction

Abstract: This note introduces a new attribute selection measure for ID3-like inductive algorithms. This measure is based on a distance between partitions such that the selected attribute in a node induces the partition which is closest to the correct partition of the subset of training examples corresponding to this node. The relationship of this measure with Quinlan's information gain is also established. It is also formally proved that our distance is not biased towards attributes with large numbers of values. Experimental studies with this distance confirm previously reported results showing that the predictive accuracy of induced decision trees is not sensitive to the goodness of the attribute selection measure. However, this distance produces smaller trees than the gain ratio measure of Quinlan, especially in the case of data whose attributes have significantly different numbers of values.

From Timed to Hybrid Systems

Abstract: We propose a framework for the formal specification and verification of timed and hybrid systems. For timed systems we propose a specification language that refers to time only through age functions which measure the length of the most recent time interval in which a given formula has been continuously true.

Discrete all-pole modeling

Abstract: A method for parametric modeling and spectral envelopes when only a discrete set of spectral points is given is introduced. This method, called discrete all-pole (DAP) modeling, uses a discrete version of the Itakura-Saito distortion measure as its error criterion. One result is an autocorrelation matching condition that overcomes the limitations of linear prediction and produces better fitting spectral envelopes for spectra that are representable by a relatively small discrete set of values, such as in voiced speech. An iterative algorithm for DAP modeling that is shown to converge to a unique global minimum is presented. Results of applying DAP modeling to real and synthetic speech are also presented. DAP modeling is extended to allow frequency-dependent weighting of the error measure, so that spectral accuracy can be enhanced in certain frequency regions. >

Measure of the nonclassicality of nonclassical states

Abstract: A continuous parameter introduced into the convolution transformation between P and Q functions leads to a measure of how nonclassical quantum states are with values ranging from 0 to 1: For photon-number states, the value is 1, the maximum possible. For squeezed vacuum states, it is a monotonically increasing function of the squeeze parameter with values varying from 0 to 1/2. This measure is identical to the minimum number of thermal photon necessary to destroy whatever nonclassical effects existing in the quantum states.

H ∞ control with transients

Abstract: In $H_\infty $ (or uniformly optimal) control problems, it is usually assumed that the system initial conditions are zero. In this paper, an $H_\infty $-like control problem that incorporates uncertainty in initial conditions is formulated. This is done by defining a worst-case performance measure. Both finite and infinite horizon problems are considered. Necessary and sufficient conditions are derived for the existence of controllers that yield a closed-loop system for which the above-mentioned performance measure is less than a prespecified value. State-space formulae for the controllers are also presented.

Quantitative stability of variational systems. I. The epigraphical distance

Abstract: This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems).

Ergodic theory of equilibrium states for rational maps

Abstract: Let T be a rational map of degree d>or=2 of the Riemann sphere C=C union ( infinity ). The authors develop the theory of equilibrium states for the class of Holder continuous functions f for which the pressure is larger than sup f. They show that there exist a unique conformal measure (reference measure) and a unique equilibrium state, which is equivalent to the conformal measure with a positive continuous density. The associated Perron-Frobenius operator acting on the space of continuous functions is almost periodic and they show that the system is exact with respect to the equilibrium measure.

Uncertainty in the dempster-shafer Theory - A Critical Re-examination

Abstract: Measures of two types of uncertainty that coexist in the Dempster-Shafer theory are overivewed. A measure of one type of uncertainty, which expresses nonspecificity of evidential claims, is well justified on both intuitive and mathermatical grounds. Proposed measures of the other types of uncertainty, which attempt to capture conflicts among evidential claims, are shown to have some deficiencies. To alleviate these deficiencies, a new measure is proposed. This measure, which is called a measure of discord, is not only satisfactory on intuitive grounds, but has alos desirable mathematical properties. A measure of total uncertainty, which is defined as the sum of nonspecificity and discord, is also discussed. The paper focuses on conceptual issues. Mathematical properties of the measure of idscord are only stated ; their proofs are given in a companion paper

Quantum nondemolition measurements of photon number by atomic beam deflection

Abstract: We present a new quantum nondemolition measurement scheme in which a measure of the photon statistics of a cavity field can be achieved by monitoring the deflection of atoms interacting with the field. Repeated measurements result in the collapse of the photon distribution in the field to a number state. Quantum jumps in the photon number are observed when the cavity field is coupled to an external reservoir

Optimal morphological pattern restoration from noisy binary images

Abstract: A theoretical analysis of morphological filters for the optimal restoration of noisy binary images is presented. The problem is formulated in a general form, and an optimal solution is obtained by using fundamental tools from mathematical morphology and decision theory. Consideration is given to the set-difference distance function as a measure of comparison between images. This function is used to introduce the mean-difference function as a quantitative measure of the degree of geometrical and topological distortion introduced by morphological filtering. It is proved that the class of alternating sequential filters is a set of parametric, smoothing morphological filters that best preserve the crucial structure of input images in the least-mean-difference sense. >

Measurable dynamics of $S$-unimodal maps of the interval

Abstract: — In this paper we sum up our results on one-dimensional measurable dynamics reducing them to the S-unimodal case (compare Appendix 2). Let / be an S-unimodal map of the interval having no limit cycles. Then / is ergodic with respect to the Lebesgue measure, and has a unique attractor A in the sense of Milnor. This attractor coincides with the conservative kernel of/. There are no strongly wandering sets of positive measure. If / has a finite a. c. i. (absolutely continuous invariant) measure a, then it has positive entropy: h^(f)>0. This result is closely related to the following: the measure of Feigenbaum-like attractors is equal to zero. Some extra topological properties of Cantor attractors are studied.

Quasi-random set systems

Abstract: There are many properties of mathematical objects that satisfy what is sometimes called a 0-1 law, in the following sense Under some natural probability measure on the set of objects, the measure of the subset of objects having the given property is either 0 or 1 In the latter case we can say that almost all the objects have the property Familiar examples of this phenomenon are the following: almost all real numbers are transcendental (or normal to every base), almost all integers are composite, almost all continuous real functions are nondifferentiable, etc It is often the case that the objects under consideration can be partitioned into a countable number of finite classes Cn, with the probability assigned to an object in Cn being just 1/ICn I In this case, we say that a property Pn satisfies a 0-1 law if the fraction of the number of objects in Cn that satisfy Pn either tends to 0 or tends to 1 as n -x 0o For example, almost all graphs on n vertices have maximum cliques and maximum independent sets of size at most 2 log n, almost all Boolean functions with n variables have circuit complexity (1 + o( 1 ))2n and almost all binary codes of length n with at most 2nR codewords (with R less than the binary symmetric channel capacity C) have arbitrarily small error probability (a special case of Shannon's coding theorem; see [S48]) One of the first general results of this type was the theorem of Fagin [F76] and Glebskii et al [GKLT69], which asserts that every property of graphs that can be expressed in first-order logic satisfies a 0-1 law (see [SS88] for recent striking developments in this topic) One obvious method for finding explicit objects having some property Pn shared by almost all objects in Cn is simply to select one at random With overwhelming probability (tending to 1 as n -x oc), the selected object will have property Pn Unfortunately, it may be (and often is) extremely difficult to prove that any particular object does indeed satisfy Pn It is our purpose in this paper to describe a method that can to a certain extent circumvent this difficulty We will show that, for a variety of families, it is possible to identify a natural hierarchy of equivalence classes of properties, all of which are shared by almost all objects in the family Any object satisfying

Quantum Yang-Mills on a Riemann surface

Abstract: We obtain the quantum expectations of gauge-invariant functions of the connection on aG=SU(N) product bundle over a Riemann surface of genusg. We show that the spaceA/Gm of connections modulo those gauge transformations which are the identity at one point is itself a principal bundle with affine linear fiber. The base space Path2gG consists of 2g-tuples of paths inG subject to a relation on their endpoint values. Quantum expectations are iterated path integrals over first the fiber then over Path2gG, each with respect to the push-forward toA/Gm of the measuree−S(A)DA. Here,S(A) denotes the Yang-Mills action onA. We exhibit a global section ofA/Gm to define a choice of origin in each fiber, relative to which the measure on the fiber is Gaussian. The induced measure on Path2gG is the product of Wiener measures on the component paths, conditioned to preserve the endopoint relation. Conformal transformations of the metric onM act by reparametrizing these paths. We explicitly compute the partition function in the general case and the expectations of functions of certain products of Wilson loops in the caseg=1.

Kolmogorov's contributions to the physical and geometrical understanding of small-scale turbulence and recent developments

Abstract: This paper reviews how Kolmogorov postulated for the first time the existence of a steady statistical state for small-scale turbulence, and its defining parameters of dissipation rate and kinematic viscosity. Thence he made quantitative predictions of the statistics by extending previous methods of dimensional scaling to multiscale random processes. We present theoretical arguments and experimental evidence to indicate when the small-scale motions might tend to a universal form (paradoxically not necessarily in uniform flows when the large scales are gaussian and isotropic), and discuss the implications for the kinematics and dynamics of the fact that there must be singularities in the velocity field associated with the - inertial range spectrum. These may be particular forms of eddy or 'eigenstructure' such as spiral vortices, which may not be unique to turbulent flows. Also, they tend to lead to the notable spiral contours of scalars in turbulence, whose self-similar structure enables the 'boxcounting' technique to be used to measure the 'capacity' DK of the contours themselves or of their intersections with lines, DK. Although the capacity, a term invented by Kolmogorov (and studied thoroughly by Kolmogorov & Tikhomirov), is like the exponent 2p of a spectrum in being a measure of the distribution of length scales (D' being related to 2p in the limit of very high Reynolds numbers), the capacity is also different in that experimentally it can be evaluated at local regions within a flow and at lower values of the Reynolds number. Thus Kolmogorov & Tikhomirov provide the basis for a more widely applicable measure of the self-similar structure of turbulence. Finally, we also review how Kolmogorov's concept of the universal spatial structure of the small scales, together with appropriate additional physical hypotheses, enables other aspects of turbulence to be understood at these scales; in particular the general forms of the temporal statistics such as the highfrequency (inertial range) spectra in eulerian and lagrangian frames of reference, and the perturbations to the small scales caused by non-isotropic, non-gaussian and inhomogeneous large-scale motions. 1. Kolmogorov's papers: review and comments (a) Introduction In this review we join with the other contributors to this special publication in celebrating some of Kolmogorov's great contributions to fluid mechanics and mathematics, and showing in some small way how his genius has inspired further

On Weyuker's axioms for software complexity measures

Abstract: Properties for software complexity measures are discussed. It is shown that a collection of nine properties suggested by E.J. Weyuker is inadequate for determining the quality of a software complexity measure. (see ibid., vol.14, p.1357-65, 1988). A complexity measure which satisfies all nine of the properties, but which has absolutely no practical utility in measuring the complexity of a program is presented. It is concluded that satisfying all of the nine properties is a necessary, but not sufficient, condition for a good complexity measure. >

What is Perfect Competition

Abstract: We provide a mathematical formulation of the idea of perfect competition for an economy with infinitely many agents and commodities. We conclude that in the presence of infinitely many commodities the Aumann (1964, 1966) measure space of agents, i.e., the interval [0,1] endowed with Lebesgue measure, is not appropriate to model the idea of perfect competition and we provide a characterization of the “appropriate” measure space of agents in an infinite dimensional commodity space setting. The latter is achieved by modeling precisely the idea of an economy with “many more” agents than commodities. For such an economy the existence of a competitive equilibrium is proved. The convexity assumption on preferences is not needed in the existence proof. We wish to thank Tom Armstrong for useful comments. As always we are responsible for any remaining errors.

Poisson Boundaries of Random Walks on Discrete Solvable Groups

Abstract: Let G be a topological group, and μ — a probability measure on G. A function f on G is called harmonic if it satisfies the mean value property $$ f(g) = \int {f(gx)d\mu (x)} $$ for all g ∈ G. It is well known that under natural assumptions on the measure μ there exists a measure G-space Γ with a quasi-invariant measure v such that the Poisson formula $$ f(g) = $$ states an isometric isomorphism between the Banach space H ∞(G, μ) of bounded harmonic functions with sup-norm and the space X∞(Γ, μ). The space (Γ, v) is called the Poisson boundary of the pair (G, μ). Thus triviality of the Poisson boundary is equivalent to absence of non-constant bounded harmonic functions for the pair (G, μ) (the Liouville property).

Iterated function systems: theory, applications and the inverse problem

Abstract: The basic theory and properties of Iterated Function Systems are given, comparing the original approaches of Hutchinson [47] and Barnsley and Demko [8]. Some examples and applications are discussed, along with computational aspects. A generalized recurrent IFS is introduced, with suitably constructed measure space, from which the existence of an invariant measure follows. This yields a Collage Theorem for Measures on generalized RIFS. Finally, the inverse problem of fractal/measure construction is discussed. Some recent applications of Genetic Algorithms as a stochastic optimization method for (i) moment matching and (ii) Collage Theorem for measures are reported.

Markov finite approximation of Frobenius-Perron operator

Abstract: THE PROBLEM of existence and computation of absolutely continuous invariant measures for nonsingular transformations on measure spaces is one of the main concerns in the modern ergodic theory [3]. Lasota and Yorke have established the existence of the invariant measures for a class of nonsingular measurable transformations S from [0, l] to itself [4]. Later in [6], Li and Yorke gave a sufficient condition for the uniqueness of this invariant density and thus for the ergodicity of the mapping. The classical Birkhoff’s Individual Ergodic Theorem says that, if ,U is an ergodic invariant probability measure under S, then for any measurable set A C [0, 11, the time average lim .+,(1/n) C~:AX~(S~(X)), which measures the “average time” spent in A under iteration of S, exists to be &I) for p-almost all x, where xA is the characteristic function of A (= 1 on A and = 0 off A). This seems to suggest that we may use the Cesaro sum of the above form to calculate the invariant measure. A simple and important example in [5], however, shows that the computer round-off error can completely dominate the calculation and make the implementation difficult. In order to overcome this difficulty, Li [5] proposed a rigorous numerical procedure which can be implemented on a computer with negligible round-off error problem. A mapping S: [0, l] + [0, l] is called piecewise C2, if there is a partition 0 = a,, < a, < ... < a, = 1 of [0, l] such that for k = 1, . . . , r, the restriction Sk of S on (uk_, , a,) is a C2-function which can be extended to the closed interval [ak_i, ak] as a C2-function. S need not be continuous at the point ak. For A c [0, 11, we write S-‘(A) for (x: S(x) E A). Throughout the paper we assume S is a nonsingular measurable transformation, i.e. S is measurable and for any measurable subset A c [0, l] with m(A) = 0, m(S-‘(A)) = 0. Here, m denotes the Lebesgue measure on (0, 11. The operator Ps: L'(0, 1) + L’(0, 1) defined by