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538
The density of prime divisors in the arithmetic dynamics of quadratic polynomials
 J. Lond. Math. Soc
"... Abstract. Let f ∈ Z[x], and consider the recurrence given by an = f(an−1), with a0 ∈ Z. Denote by P(f, a0) the set of prime divisors of this recurrence, i.e., the set of primes p dividing some nonzero an, and denote the natural density of this set by D(P(f, a0)). The problem of determining D(P(f, a ..."
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Cited by 32 (10 self)
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Abstract. Let f ∈ Z[x], and consider the recurrence given by an = f(an−1), with a0 ∈ Z. Denote by P(f, a0) the set of prime divisors of this recurrence, i.e., the set of primes p dividing some nonzero an, and denote the natural density of this set by D(P(f, a0)). The problem of determining D(P(f, a0)) when f is linear has attracted significant study, although it remains unresolved in full generality. In this paper we consider the case of f quadratic, where previously D(P(f, a0)) was known only in a few cases. We show D(P(f, a0)) = 0 regardless of a0 for four infinite families of f, including f = x 2 + k, k ∈ Z\{−1}. The proof relies on tools from group theory and probability theory to formulate a sufficient condition for D(P(f, a0)) = 0 in terms of arithmetic properties of the forward orbit of the critical point of f. This provides an analogy to results in real and complex dynamics, where analytic properties of the forward orbit of the critical point have been shown to determine many global dynamical properties of a quadratic polynomial. The article also includes apparently new work on the irreducibility of iterates of quadratic polynomials. 1.
The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, Enseign
 Math
, 1994
"... Abstract. We give a modern exposition and an elementary proof of the topological equivalence between periodic homeomorphisms of the disc and the sphere and euclidean isometries. 1. ..."
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Cited by 31 (3 self)
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Abstract. We give a modern exposition and an elementary proof of the topological equivalence between periodic homeomorphisms of the disc and the sphere and euclidean isometries. 1.
Finite State Machines and Recurrent Neural Networks  Automata and Dynamical Systems Approaches
 Neural Networks and Pattern Recognition
, 1998
"... We present two approaches to the analysis of the relationship between a recurrent neural network (RNN) and the finite state machine M the network is able to exactly mimic. First, the network is treated as a state machine and the relationship between the RNN and M is established in the context of alg ..."
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Cited by 29 (11 self)
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We present two approaches to the analysis of the relationship between a recurrent neural network (RNN) and the finite state machine M the network is able to exactly mimic. First, the network is treated as a state machine and the relationship between the RNN and M is established in the context of algebraic theory of automata. In the second approach, the RNN is viewed as a set of discretetime dynamical systems associated with input symbols of M. In particular, issues concerning network representation of loops and cycles in the state transition diagram of M are shown to provide a basis for the interpretation of learning process from the point of view of bifurcation analysis. The circumstances under which a loop corresponding to an input symbol x is represented by an attractive fixed point of the underlying dynamical system associated with x are investigated. For the case of two recurrent neurons, under some assumptions on weight values, bifurcations can be understood in the geometrical c...
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 28 (14 self)
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this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where
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 27 (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
Regionbased image watermarking
 Image Processing, IEEE Transactions on
, 2001
"... Abstract—We introduce a novel method for embedding and detecting a chaotic watermark in the digital spatial image domain, based on segmenting the image and locating regions that are robust to several image manipulations. The robustness of the method is confirmed by experimental results that display ..."
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Cited by 26 (1 self)
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Abstract—We introduce a novel method for embedding and detecting a chaotic watermark in the digital spatial image domain, based on segmenting the image and locating regions that are robust to several image manipulations. The robustness of the method is confirmed by experimental results that display the immunity of the embedded watermark to several kinds of attacks, such as compression, filtering, scaling, cropping, and rotation. Index Terms—Chaos, copyright protection, correlators, feature extraction, image segmentation, signal detection.
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.