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41
PseudoRandom Generation from OneWay Functions
 PROC. 20TH STOC
, 1988
"... Pseudorandom generators are fundamental to many theoretical and applied aspects of computing. We show howto construct a pseudorandom generator from any oneway function. Since it is easy to construct a oneway function from a pseudorandom generator, this result shows that there is a pseudorandom gene ..."
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Cited by 859 (21 self)
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Pseudorandom generators are fundamental to many theoretical and applied aspects of computing. We show howto construct a pseudorandom generator from any oneway function. Since it is easy to construct a oneway function from a pseudorandom generator, this result shows that there is a pseudorandom generator iff there is a oneway function.
Estimating fractal dimension
 Journal of the Optical Society of America A
, 1990
"... Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools ..."
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Cited by 96 (3 self)
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Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor. Confusion is a word we have invented for an order which is not understood.Henry Miller, "Interlude," Tropic of Capricorn Numerical coincidence is a common path to intellectual perdition in our quest for meaning. We delight in catalogs of disparate items united by the same number, and often feel in our gut that some unity must underlie it all.
Quantum Algorithm For Hilberts Tenth Problem
 Int.J.Theor.Phys
, 2003
"... We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert’s tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomp ..."
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Cited by 60 (10 self)
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We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert’s tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle—that is, if certain hamiltonian and its ground state can be physically constructed according to the proposal—quantum computability would surpass classical computability as delimited by the ChurchTuring thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles. 1
Anomalous scaling laws in multifractal objects
 Phys. Rep
, 1987
"... Contents: 0. Introduction 149 3.6. A thermodynamical formalism for unidimensional1. Chaotic attractors as inhomogeneous fractals 152 maps 192 1.1. Why study attractors ’ dimensions? 152 3.7. Partial dimensions and entropies 195 ..."
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Cited by 44 (4 self)
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Contents: 0. Introduction 149 3.6. A thermodynamical formalism for unidimensional1. Chaotic attractors as inhomogeneous fractals 152 maps 192 1.1. Why study attractors ’ dimensions? 152 3.7. Partial dimensions and entropies 195
Families of Alpha Beta and GammaDivergences: Flexible and Robust Measures of Similarities
, 2010
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Stability of Tsallis entropy and instabilities of Rényi and normalized Tsallis entropies: A basis for qexponential distributions
 Nonextensive Statistical Mechanics and Its Applications
, 2001
"... A basis for qexponential distributions ..."
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Mutual information of population codes and distance measures in probability space
 Phys Rev Lett
, 2001
"... We studied the mutual information between a stimulus and a large system consisting of stochastic, statistically independent elements that respond to a stimulus. The Mutual Information (MI) of the system saturates exponentially with system size. A theory of the rate of saturation of the MI is develop ..."
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Cited by 15 (0 self)
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We studied the mutual information between a stimulus and a large system consisting of stochastic, statistically independent elements that respond to a stimulus. The Mutual Information (MI) of the system saturates exponentially with system size. A theory of the rate of saturation of the MI is developed. We show that this rate is controlled by a distance function between the response probabilities induced by different stimuli. This function, which we term the Confusion Distance between two probabilities, is related to the Renyi αInformation.
Random Heuristic Search
 Theoretical Computer Science
, 1999
"... There is a developing theory of growing power which, at its current stage of development (indeed, for a number of years now), speaks to qualitative and quantitative aspects of search strategies. Although it has been specialized and applied to genetic algorithms, it's implications and applicabil ..."
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Cited by 13 (1 self)
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There is a developing theory of growing power which, at its current stage of development (indeed, for a number of years now), speaks to qualitative and quantitative aspects of search strategies. Although it has been specialized and applied to genetic algorithms, it's implications and applicability are far more general. This paper deals with the broad outlines of the theory, introducing basic principles and results rather than analyzing or specializing to particular algorithms. A few specific examples are included for illustrative purposes, but the theory's basic structure, as opposed to applications, remains the focus. Key words: Random Heuristic Search, Modeling Evolutionary Algorithms, Degenerate Royal Road Functions. 1 Introduction Vose [20] introduced a rigorous dynamical system model for the binary representation genetic algorithm with proportional selection, mutation determined by a rate, and onepoint crossover, using the simplifying assumption of an infinite population. 1 ...
Universal geometric approach to uncertainty, entropy and infromation, Phys. Rev. A 59
, 1999
"... It is shown that a unique measure of volume is associated with any statistical ensemble, which directly quantifies the inherent spread or localisation of the ensemble. It is applicable whether the ensemble is classical or quantum, continuous or discrete, and may be derived from a small number of the ..."
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Cited by 11 (0 self)
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It is shown that a unique measure of volume is associated with any statistical ensemble, which directly quantifies the inherent spread or localisation of the ensemble. It is applicable whether the ensemble is classical or quantum, continuous or discrete, and may be derived from a small number of theoryindependent geometric postulates. Remarkably, this unique ensemble volume is proportional to the exponential of the ensemble entropy, and hence provides a novel geometric characterisation of the latter quantity. Applications include unified volumebased derivations of the Holevo and Shannon bounds in quantum and classical information theory, a precise geometric interpretation of thermodynamic entropy for equilibrium ensembles, a geometric derivation of semiclassical uncertainty relations, a new means for defining classical and quantum localization for arbitrary evolution processes, a geometric interpretation of relative entropy, and a new proposed definition for the spotsize of an optical beam. Advantages of ensemble volume over other measures of localization (rootmeansquare deviation, Renyi entropies, and inverse participation ratio) are discussed. PACS Numbers: 03.65.Bz, 03.67.a, 05.45+b, 42.60.Jf I