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277
Random Generators and Normal Numbers
 EXPERIMENTAL MATHEMATICS
, 2000
"... Pursuant to the authors' previous chaoticdynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental ..."
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Cited by 25 (11 self)
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Pursuant to the authors' previous chaoticdynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental constant log 2 = P n2Z + 1=(n2 n ) is not yet known to be 2normal (i.e. normal to base 2), we are able to establish bnormality (and transcendency) for constants of the form P 1=(nb n ) but with the index n constrained to run over certain subsets of Z + . In this way we demonstrate, for example, that the constant 2;3 = P n=3;3 2 ;3 3 ;::: 1=(n2 n ) is 2normal. The constants share with ; log 2 and others the property that isolated digits can be directly calculated, but for the new class such computation is extraordinarily rapid. For example, we find that the googolth (i.e. 10 100  th) binary bit of 2;3 is 0. We also present a collection of other results  such as density results and irrationality proofs based on PRNG ideas  for various special numbers.
Fool's Gold: Extracting Finite State Machines From Recurrent Network Dynamics
"... Several recurrent networks have been proposed as representations for the task of formal language learning. After training a recurrent network recognize a formal language or predict the next symbol of a sequence, the next logical step is to understand the information processing carried out by the net ..."
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Cited by 25 (0 self)
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Several recurrent networks have been proposed as representations for the task of formal language learning. After training a recurrent network recognize a formal language or predict the next symbol of a sequence, the next logical step is to understand the information processing carried out by the network. Some researchers have begun to extracting finite state machines from the internal state trajectories of their recurrent networks. This paper describes how sensitivity to initial conditions and discrete measurements can trick these extraction methods to return illusory finite state descriptions.
Numerical Analysis of Dynamical Systems
, 1995
"... This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long time intervals. Standard error estimates comparing individual trajectories are of no direct use in this context since the error ..."
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Cited by 24 (3 self)
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This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long time intervals. Standard error estimates comparing individual trajectories are of no direct use in this context since the error constant typically grows like the exponential of the time interval under consideration. Instead of comparing trajectories, the effect of discretization on various sets which are invariant under the evolution of the underlying differential equation is studied. Such invariant sets are crucial in determining long time dynamics. The particular invariant sets which are studied are equilibrium points, together with their unstable manifolds and local phase portraits, periodic solutions, quasiperiodic solutions and strange attractors. Particular attention is paid to the development of a unified theory and to the development of an existence theory for invariant sets of the underlying differential equation which may be used directly to construct an analogous existence theory (and hence a simple approximation theory) for the numerical method. To appear in Acta Numerica 1994, Cambridge University Press CONTENTS
Rates of Convergence for Data Augmentation on Finite Sample Spaces
 Ann. Appl. Prob
, 1993
"... this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where ..."
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Cited by 24 (13 self)
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this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where
Continuoustime Relaxation Labeling Processes
, 1998
"... We study the properties of two new relaxation labeling schemes described in terms of differential equations, and hence evolving in countinuous time. This contrasts with the customary approach to defining relaxation labeling algorithms which prefers discrete time. Continuoustime dynamical systems ar ..."
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Cited by 19 (4 self)
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We study the properties of two new relaxation labeling schemes described in terms of differential equations, and hence evolving in countinuous time. This contrasts with the customary approach to defining relaxation labeling algorithms which prefers discrete time. Continuoustime dynamical systems are particularly attractive because they can be implemented directly in hardware circuitry, and the study of their dynamical properties is simpler and more elegant. They are also more plausible as models of biological visual computation. We prove that the proposed models enjoy exactly the same dynamical properties as the classical relaxation labeling schemes, and show how they are intimately related to Hummel and Zucker's now classical theory of constraint satisfaction. In particular, we prove that, when a certain symmetry condition is met, the dynamical systems' behavior is governed by a Liapunov function which turns out to be (the negative of) a wellknown consistency measure. Moreover, we p...
Limit measures for affine cellular automata II
, 2008
"... If M is a monoid, and A is an abelian group, then AM is a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F: AM − → AM that commutes with all shift maps. If F is diffusive, and µ is a harmonically mixing (HM) probability measure on AM, then the sequence {FNµ} ∞ ..."
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Cited by 17 (8 self)
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If M is a monoid, and A is an abelian group, then AM is a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F: AM − → AM that commutes with all shift maps. If F is diffusive, and µ is a harmonically mixing (HM) probability measure on AM, then the sequence {FNµ} ∞ N=1 weak*converges to the Haar measure on AM, in density. Fully supported Markov measures on AZ are HM, and nontrivial LCA on A (ZD) are diffusive when A = Z/p is a prime cyclic group. In the present work, we provide sufficient conditions for diffusion of LCA on A (ZD) when A = Z/n is any cyclic group or when A = ( ) J Z /pr (p prime). We show that any fully supported Markov random field A (ZD) is HM (where A is any abelian group). We also provide examples of HM Markov measures not having full support, and measures on A Z which have the Kolmogorov property but which are not HM.
Dual billiards, twist maps and impact oscillators
 Nonlinearity
, 1996
"... Abstract. In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve Γ in the plane. It is shown that there do not exist invariant circles near Γ when there is a point on Γ where the radius of curvature vanishes or is di ..."
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Cited by 17 (0 self)
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Abstract. In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve Γ in the plane. It is shown that there do not exist invariant circles near Γ when there is a point on Γ where the radius of curvature vanishes or is discontinuous. In addition, when the radius of curvature is not C 1 there are examples with orbits that converge to a point of Γ. If the derivative of the radius of curvature is bounded, such orbits cannot exist. The final section of the paper concerns an impact oscillator whose dynamics are the same as a dual billiards map. The appendix is a remark on the connection of the inverse problems for invariant circles in billiards and dual billiards. Dual billiards is a dynamical system defined on the exterior of an oriented convex closed curve Γ in the plane. If z is a point in the unbounded component of R 2 − Γ, then its image under the dual billiards map Φ is the reflection about the point of tangency in the oriented supporting line to Γ (see Figure 0.1). It is clear that Φ is an areapreserving, and if Γ is
A fast algorithm for Julia sets of hyperbolic rational functions
 Proc. of CCA 2004, in ENTCS, vol 120
, 2005
"... Although numerous computer programs have been written to compute sets of points which claim to approximate Julia sets, no reliable high precision pictures of nontrivial Julia sets are currently known. Usually, no error estimates are added and even those algorithms which work reliable in theory, beco ..."
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Cited by 16 (0 self)
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Although numerous computer programs have been written to compute sets of points which claim to approximate Julia sets, no reliable high precision pictures of nontrivial Julia sets are currently known. Usually, no error estimates are added and even those algorithms which work reliable in theory, become unreliable in practice due to rounding errors and the use of fixed length floating point numbers. In this paper we prove the existence of polynomial time algorithms to approximate the Julia sets of given hyperbolic rational functions. We will give a strict computable error estimation w.r.t. the Hausdorff metric on the complex sphere. This extends a result on polynomials z ↦ → z 2 + c, where c  < 1/4, in [RW03] and an earlier result in [Zho98] on the recursiveness of the Julia sets of hyperbolic polynomials. The algorithm given in this paper computes Julia sets locally in time O(k · M(k)) (where M(k) denotes the time needed to multiply two kbit numbers). Roughly speaking, the local time complexity is the number of Turing machine steps to decide a set of disks of spherical diameter 2 −k so that the union of these disks has Hausdorff distance at most 2 −k+2. This allows to give reliable pictures of Julia sets to arbitrary precision. Key words: Julia Sets, Computational Complexity. 1
Mathematical Complexity Of Simple Economics
 Notices of the American Mathematical Society
"... . Even simple, standard price adjustment models from economics  used to model the "invisible hand" story of Adam Smith  admit highly chaotic behavior. After relating these dynamical conclusions to complexity problems from numerical analysis and showing the mathematical reason why these result ..."
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Cited by 15 (2 self)
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. Even simple, standard price adjustment models from economics  used to model the "invisible hand" story of Adam Smith  admit highly chaotic behavior. After relating these dynamical conclusions to complexity problems from numerical analysis and showing the mathematical reason why these results arise, it is suggested why similar counterintuitive conclusions permeate the social sciences. A lesson learned from modern dynamics is that natural systems can be surprisingly complex. No longer are we astonished to discover that systems from, say, biology (e.g., [GOI, Ma1, Ma2]) or the Newtonian nbody problem (e.g., [MM, Mo, Mk, SX, X]) admit all sorts of previously unexpected dynamical behavior. This seeming randomness, however, sharply contrasts with what we have been conditioned to expect from economics. On the evening news and talk shows, in the newspapers, and during political debate we hear about the powerful moderating force of the market which, if just left alone, would steadi...