Showing papers on "Bounded function published in 1993"

One-dimensional dynamics

Abstract: 0. Introduction.- I. Circle Diffeomorphisms.- 1. The Combinatorial Theory of Poincare.- 2. The Topological Theory of Denjoy.- 2.a The Denjoy Inequality.- 2.b Ergodicity.- 3. Smooth Conjugacy Results.- 4. Families of Circle Diffeomorphisms Arnol'd tongues.- 5. Counter-Examples to Smooth Linearizability.- 6. Frequency of Smooth Linearizability in Families.- 7. Some Historical Comments and Further Remarks.- II. The Combinatorics of One-Dimensional Endomorphisms.- 1. The Theorem of Sarkovskii.- 2. Covering Maps of the Circle as Dynamical Systems.- 3. The Kneading Theory and Combinatorial Equivalence.- 3.a Examples.- 3.b Hofbauer's Tower Construction.- 4. Full Families and Realization of Maps.- 5. Families of Maps and Renormalization.- 6. Piecewise Monotone Maps can be Modelled by Polynomial Maps.- 7. The Topological Entropy.- 8. The Piecewise Linear Model.- 9. Continuity of the Topological Entropy.- 10. Monotonicity of the Kneading Invariant for the Quadratic Family.- 11. Some Historical Comments and Further Remarks.- III. Structural Stability and Hyperbolicity.- 1. The Dynamics of Rational Mappings.- 2. Structural Stability and Hyperbolicity.- 3. Hyperbolicity in Maps with Negative Schwarzian Derivative.- 4. The Structure of the Non-Wandering Set.- 5. Hyperbolicity in Smooth Maps.- 6. Misiurewicz Maps are Almost Hyperbolic.- 7. Some Further Remarks and Open Questions.- IV. The Structure of Smooth Maps.- 1. The Cross-Ratio: the Minimum and Koebe Principle.- l.a Some Facts about the Schwarzian Derivative.- 2. Distortion of Cross-Ratios.- 2.a The Zygmund Conditions.- 3. Koebe Principles on Iterates.- 4. Some Simplifications and the Induction Assumption.- 5. The Pullback of Space: the Koebe/Contraction Principle.- 6. Disjointness of Orbits of Intervals.- 7. Wandering Intervals Accumulate on Turning Points.- 8. Topological Properties of a Unimodal Pullback.- 9. The Non-Existence of Wandering Intervals.- 10. Finiteness of Attractors.- 11. Some Further Remarks and Open Questions.- V. Ergodic Properties and Invariant Measures.- 1. Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures.- 2. Invariant Measures for Markov Maps.- 3. Constructing Invariant Measures by Inducing.- 4. Constructing Invariant Measures by Pulling Back.- 5. Transitive Maps Without Finite Continuous Measures.- 6. Frequency of Maps with Positive Liapounov Exponents in Families and Jakobson's Theorem.- 7. Some Further Remarks and Open Questions.- VI. Renormalization.- 1. The Renormalization Operator.- 2. The Real Bounds.- 3. Bounded Geometry.- 4. The PullBack Argument.- 5. The Complex Bounds.- 6. Riemann Surface Laminations.- 7. The Almost Geodesic Principle.- 8. Renormalization is Contracting.- 9. Universality of the Attracting Cantor Set.- 10. Some Further Remarks and Open Questions.- VII. Appendix.- 1. Some Terminology in Dynamical Systems.- 2. Some Background in Topology.- 3. Some Results from Analysis and Measure Theory.- 4. Some Results from Ergodic Theory.- 5. Some Background in Complex Analysis.- 6. Some Results from Functional Analysis.

Global adaptive output-feedback control of nonlinear systems. II. Nonlinear parameterization

Abstract: For pt.I, see ibid., p.17-32 (1993). The problem of designing global output-feedback robust stabilizing controls for a class of single-input single-output minimum-phase uncertain nonlinear systems with known and constant relative degree is addressed. They are assumed to be linear with respect to the input and nonlinear with respect to an unknown constant parameter vector. The nonlinearities depend on the output only. The nonlinearities may be uncertain and are only required to be bounded by known smooth functions. The order of the robust compensator is equal to the relative degree minus one and is static when the relative degree is one. A self-tuning version of the robust control capable of achieving set point regulation is developed in which the control gains are tuned by an output-feedback adaptive algorithm. When the parameter vector enters linearly, the self-tuning regulator does not require the knowledge of parameter bounds and guarantees set point regulation for the same class of systems considered in Part I. >

Scheduling dynamic dataflow graphs with bounded memory using the token flow model

Abstract: The authors build upon research by E. A. Lee (1991) concerning the token flow model, an analytical model for the behavior of dataflow graphs with data-dependent control flow, by analyzing the properties of cycles of the schedule: sequences of actor executions that return the graph to its initial state. Necessary and sufficient conditions are given for the existence of a bounded cyclic schedule as well as sufficient conditions for execution of the graph in bounded memory. The techniques presented apply to a more general class of dataflow graphs than previous methods. >

Deformation Quantization for Actions of R ]D

Abstract: Oscillatory integrals The deformed product Function algebras The algebra of bounded operators Functoriality for the operator norm Norms of deformed deformations Smooth vectors, and exactness Continuous fields Strict deformation quantization Old examples The quantum Euclidean closed disk and quantum quadrant The algebraists quantum plane, and quantum groups References.

Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory

Abstract: We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Our main results apply to local (in position space) RG maps acting on systems of bounded spins (compact single-spin space). Regarding regularity, we show that the RG map, defined on a suitable space of interactions (=formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce, and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension d⩾3, these pathologies occur in a full neighborhood {β>β0, ¦h¦

Local error estimates for radial basis function interpolation of scattered data

Abstract: Introducing a suitable variational formulation for the local error of scattered data intepolation by radial basis functions φ(r), the error can be bounded by a term depending on the Fourier transform of the interpolated function f and a certain «Kriging function», which allows a formulation as an integral involving the Fourier transform of φ. The explicit construction of locally well-behaving admissible coefficient vectors makes the Kriging function bounded by some power of the local density h of data points

Wavelet-like bases for the fast solutions of second-kind integral equations

Abstract: A class of vector-space bases is introduced for the sparse representation of discretizations of integral operators An operator with a smooth, nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix, to high precision A method is presented that employs these bases for the numerical solution of second-kind integral equations in time bounded by $O(n\log ^2 n)$, where n is the number of points in the discretization Numerical results are given which demonstrate the effectiveness of the approach, and several generalizations and applications of the method are discussed

Performance and stability of communication networks via robust exponential bounds

Abstract: A method for evaluating the performance of packet switching communication networks under a fixed, session-based, routing strategy is proposed. The approach is based on properly bounding the probability distribution functions of the system input processes. The suggested bounds which are decaying exponentials, possess three convenient properties. When the inputs to an isolated network element are all bounded, they result in bounded outputs and assure that the delays and queues in this element have exponentially decaying distributions. In some network settings, bounded inputs result in bounded outputs. Natural traffic processes can be shown to satisfy such bounds. Consequently, this method enables the analysis of various previously intractable setups. Sufficient conditions are provided for the stability of such networks, and derive upper bounds for the parameters of network performance are derived. >

Explicit construction of quadratic lyapunov functions for the small gain, positivity, circle, and popov theorems and their application to robust stability. part II: Discrete-time theory

Abstract: The purpose of this paper is to construct Lyapunov functions to prove the key fundamental results of linear system theory, namely, the small gain (bounded real), positivity (positive real), circle, and Popov theorems. For each result a suitable Riccati-like matrix equation is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time-invariant system and a memoryless nonlinearity. Lyapunov functions for the small gain and positivity results are also constructed for the interconnection of two transfer functions. A multivariable version of the circle criterion, which yields the bounded real and positive real results as limiting cases, is also derived. For a multivariable extension of the Popov criterion, a Lure-Postnikov Lyapunov function involving both a quadratic term and an integral of the nonlinearity, is constructed. Each result is specialized to the case of linear uncertainty for the problem of robust stability. In the case of the Popov criterion, the Lyapunov function is a parameter-dependent quadratic Lyapunov function.

Complexity of Bezout’s Theorem II Volumes and Probabilities

Abstract: In this paper we study volume estimates in the space of systems of n homegeneous polynomial equations of fixed degrees d i with respect to a natural Hermitian structure on the space of such systems invariant under the action of the unitary group We show that the average number of real roots of real systems is D 1/2 where D = Π d i is the Be zout number We estimate the volume of the subspace of badly conditioned problems and show that volume is bounded by a small degree polynomial in n, N and D times the reciprocal of the condition number to the fourth power Here N is the dimension of the space of systems

Statistical limit points

Abstract: Following the concept of a statistically convergent sequence x, we define a statistical limit point of x as a number λ that is the limit of a subsequence {x k(j) } of x such that the set {k(j): j ∈ N} does not have density zero. Similarly, a statistical cluster point of x is a number γ such that for every e > 0 the set {k ∈ N: |x k − γ| < e} does not have density zero. These concepts, which are not equivalent, are compared to the usual concept of limit point of a sequence. Statistical analogues of limit point results are obtained. For example, if x is a bounded sequence then x has a statistical cluster point but not necessarily a statistical limit point. Also, if the set M:= {k ∈ N: x k+1 } has density one and x is bounded on M, then x is statistically convergent

Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal formalism, effective conductivities, and weak approximation

Abstract: We consider the effect of measuring randomly varying local hydraulic conductivities K(x) on one's ability to predict steady state flow within a bounded domain, driven by random source and boundary functions. More precisely, we consider the prediction of local hydraulic head h(x) and Darcy flux q(x) by means of their unbiased ensemble moments 〈h(x)〉κ and 〈q(x)〉κ conditioned on measurements of K(x). These predictors satisfy a deterministic flow equation in which 〈q(x)〉κ = −κ(x)∇〈h(x)〉κ + rκ(x), where κ(x) is a relatively smooth unbiased estimate of K(x) and rκ(x) is a “residual flux.” We derive a compact integral expression for rκ(x) which is rigorously valid for a broad class of K(x) fields, including fractals. It demonstrates that 〈q(x)〉κ is nonlocal and non-Darcian so that an effective hydraulic conductivity does not generally exist. We show analytically that under uniform mean flow the effective conductivity may be a scalar, a symmetric or a nonsymmetric tensor, or a set of directional scalars which do not form a tensor. We demonstrate numerically that in two-dimensional mean radial flow it may increase from the harmonic mean of K(x) near interior and boundary sources to the geometric mean far from such sources. For cases where rκ(x) can neither be expressed nor approximated by a local expression, we propose a weak (integral) approximation (closure) which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of K(x) measurements increase. The nonlocal deterministic flow equation can be solved numerically by standard methods; our theory shows clearly how the scale of grid discretization should relate to the scale, quantity, and quality of available data. After providing explicit approximations for the second moments of head and flux prediction errors, we conclude by discussing practical methods to compute κ(x) from noisy measurements of K(x) and to calculate required second moments of the associated estimation errors when K(x) is lognormal.

EFFECTIVE BANDWIDTH OF GENERAL MARKOVIAN TRAFFIC SOURCES AND ADMISSION CONTROL OF HIGH SPEED NETWORKS (Extended Abstract)

Abstract: The emerging high speed networks, notably the ATMbased Broadband-ISDN, are expected to integrate through statistical multiplexing large numbers of traflc sources having a broad range of burstiness characteristics. A prime instrument for controlling congestion in the network is admission control which limits calls and guarantees a grade of service determined by delay and loss probability in the multiplexel: We show, for general Markovian trafJic sources, that it is possible to assign a notional effective bandwidth to each source which is an explicitly identijied, simply computed quantity with provably correct properties in the natural asymptotic regime of small loss probabilities. It is the maximal real eigenvalue of a matrix which is directly obtained from the source characteristics and the admission criterion, and for several sources it is simply additive. We consider both fluid and point process models and obtain parallel results. Numerical results show that the acceptance set for heterogeneous classes of sources is closely approximated and conservatively bounded by the set obtained from the effective bandwidth approximation.

Beurling type density theorems in the unit disk.

Abstract: We consider two equivalent density concepts for the unit disk that provide a complete description of sampling and interpolation inA −n (the Banach space of functionsf analytic in the unit disk with (1-|z|2) n |f(z)| bounded). This study reveals a ‘Nyquist density’: A sequence of points is (roughly speaking) a set of sampling if and only if its density in every part of the disk is strictly larger thann, and it is a set of interpolation if and only if its density in every part of the disk is strictly smaller thann. Similar density theorems are also obtained for weighted Bergman spaces.

Greed is good : approximating independent sets in sparse and bounded-degree graphs

Abstract: Theminimum-degree greedy algorithm, or Greedy for short, is a simple and well-studied method for finding independent sets in graphs. We show that it achieves a performance ratio of (Δ+2)/3 for approximating independent sets in graphs with degree bounded by Δ. The analysis yields a precise characterization of the size of the independent sets found by the algorithm as a function of the independence number, as well as a generalization of Turan's bound. We also analyze the algorithm when run in combination with a known preprocessing technique, and obtain an improved $$(2\bar d + 3)/5$$ performance ratio on graphs with average degree $$\bar d$$ , improving on the previous best $$(\bar d + 1)/2$$ of Hochbaum. Finally, we present an efficient parallel and distributed algorithm attaining the performance guarantees of Greedy.

Neural net robot controller with guaranteed tracking performance

Abstract: A neural net (NN) controller for a general serial-link robot arm is developed. The NN has two layers so that linearity in the parameters holds, but the "net functional reconstruction error" is taken as nonzero. The structure of the NN controller is derived using a filtered error/passivity approach. It is shown that standard backpropagation, when used for real time closed-loop control, can yield unbounded NN weights if (1) the net cannot exactly reconstruct a certain required control function, or (2) there are bounded unknown disturbances in the robot dynamics. An online weight tuning algorithm including a correction term to backpropagation guarantees tracking as well as bounded weights. The notions of a passive NN and a robust NN are introduced. >

Cosmological theory without singularities

Abstract: A theory of gravitation is constructed in which all homogeneous and isotropic solutions are nonsingular, and in which all curvature invariants are bounded All solutions for which curvature invariants approach their limiting values approach de Sitter space The action for this theory is obtained by a higher-derivative modification of Einstein's theory We expect that our model can easily be generalized to solve the singularity problem also for anisotropic cosmologies

An existence result for a class of shape optimization problems

Abstract: Given a bounded open subset Ω of R n, we prove the existence of a minimum point for a functional F defined on the family A(Ω) of all “quasiopen” subsets of Ω, under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A(Ω) with respect to a suitable topology, related to the resolvents of the Laplace operator with Dirichlet boundary condition. Applications are given to the existence of sets of prescribed volume with minimal k th eigenvalue (or with minimal capacity) with respect to a given elliptic operator.

Strong localized perturbations of eigenvalue problems

Abstract: This paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded o...

On the regularity of spherically symmetric wave maps

Abstract: Wave maps are critical points U: M → N of the Lagrangian ℒ[U] = ∞M ‖dU‖2, where M is an Einsteinian manifold and N a Riemannian one. For the case M = ℝ2,1 and U a spherically symmetric map, it is shown that the solution to the Cauchy problem for U with smooth initial data of arbitrary size is smooth for all time, provided the target manifold N satisfies the two conditions that: (1) it is either compact or there exists an orthonormal frame of smooth vectorfields on N whose structure functions are bounded; and (2) there are two constants c and C such that the smallest eigenvalue λ and the largest eigenvalue λ of the second fundamental form kAB of any geodesic sphere Σ(p, s) of radius s centered at p ϵ N satisfy sλ ≧ c and s A ≦ C(1 + s). This is proved by first analyzing the energy-momentum tensor and using the second condition to show that near the first possible singularity, the energy of the solution cannot concentrate, and hence is small. One then proves that for targets satisfying the first condition, initial data of small energy imply global regularity of the solution. © 1993 John Wiley & Sons, Inc.

Hit-and-run algorithms for generating multivariate distributions

Abstract: We introduce a general class of Hit-and-Run algorithms for generating essentially arbitrary absolutely continuous distributions on Rd. They include the Hypersphere Directions algorithm and the Coordinate Directions algorithm that have been proposed for identifying nonredundant linear constraints and for generating uniform distributions over subsets of Rd. Given a bounded open set S in Rd, an absolutely continuous probability distribution π on S the target distribution and an arbitrary probability distribution I? on the boundary of the d-dimensional unit sphere centered at the origin the direction distribution, the I?, π-Hit-and-Run algorithm produces a sequence of iteration points as follows. Given the nth iteration point x, choose a direction I? according to the distribution I? and then choose the n + 1st iteration point according to the conditionalization of the distribution π along the line defined by x and x + I?. Under mild conditions, we show that this sequence of points is a Harris recurrent reversible Markov chain converging in total variation to the target distribution π.

Bounding the Vapnik-Chervonenkis dimension of concept classes parameterized by real numbers

Abstract: The Vapnik-Chervonenkis (V-C) dimension is an important combinatorial tool in the analysis of learning problems in the PAC framework. For polynomial learnability, we seek upper bounds on the V-C dimension that are polynomial in the syntactic complexity of concepts. Such upper bounds are automatic for discrete concept classes, but hitherto little has been known about what general conditions guarantee polynomial bounds on V-C dimension for classes in which concepts and examples are represented by tuples of real numbers. In this paper, we show that for two general kinds of concept class the V-C dimension is polynomially bounded in the number of real numbers used to define a problem instance. One is classes where the criterion for membership of an instance in a concept can be expressed as a formula (in the first-order theory of the reals) with fixed quantification depth and exponentially-bounded length, whose atomic predicates are polynomial inequalities of exponentially-bounded degree, The other is classes where containment of an instance in a concept is testable in polynomial time, assuming we may compute standard arithmetic operations on reals exactly in constant time. Our results show that in the continuous case, as in the discrete, the real barrier to efficient learning in the Occam sense is complexity-theoretic and not information-theoretic. We present examples to show how these results apply to concept classes defined by geometrical figures and neural nets, and derive polynomial bounds on the V-C dimension for these classes.

Geometry and integrability of topological-antitopological fusion

Abstract: Integrability of equations of topological-antitopological fusion (being proposed by Cecotti and Vafa) describing the ground state metric on a given 2D topological field theory (TFT) model, is proved. For massive TFT models these equations are reduced to a universal form (being independent on the given TFT model) by gauge transformations. For massive perturbations of topological conformal field theory models the separatrix solutions of the equations bounded at infinity are found by the isomonodromy deformations method. Also it is shown that the ground state metric together with some part of the underlined TFT structure can be parametrized by pluriharmonic maps of the coupling space to the symmetric space of real positive definite quadratic forms.

Rank one property for derivatives of functions with bounded variation

Abstract: In this paper we introduce a new tool in geometric measure theory and then we apply it to study the rank properties of the derivatives of vector functions with bounded variation.