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RAPIDLY CONVERGING EXPANSIONS WITH FIBONACCI COEFFICIENTS
, 1984
"... Properties of Fibonacci numbers have been known for a very long time. Their origin dates back to the year 1202 with the publication of the Liber Abaci by the Italian mathematician Leonardo of Pisa, better known to us by the nickname "Fibonacci, " a short form of Filius Bonacci, meaning &qu ..."
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Properties of Fibonacci numbers have been known for a very long time. Their origin dates back to the year 1202 with the publication of the Liber Abaci by the Italian mathematician Leonardo of Pisa, better known to us by the nickname "Fibonacci, " a short form of Filius Bonacci, meaning "Son of Bonacci." Fibonacci seems to have had a sense of humor apart from his mathematical talents: Liber was a Latin God, son of Ceres and brother of Proserpina. The Romans assimilated this God to Bacchus or Dyonisus, the Greek god of wine. Festivals, known as "Liberalia, " were celebrated every year honoring Liber Bacus. Since Liber Abaci means Book of the Abacus * Fibonacci may have amused himself by naming his book, at a time of strong domination by the Roman Catholic Church, in a way reminiscent of a pagan god of wine and fertility. We know Fibonacci was fond of play on words. For instance, he signed some of his work "Leonardo Bigollo. " Bigollo is a work meaning both "traveler, " which Fibonacci certainly was, and "blockhead. " It has been said that Fibonacci had in mind the latter meaning to tease his contemporaries who had ridiculed him for his interest in HinduArabic numerals and methods. Fibonacci had become a very successful mathematician whith these methods. Fibonacci did not discover any of the properties of the sequence which bears his name. He simply proposed, and solved, in the Liber Abaci * the problem of how many rabbits would be born in one year starting from a given pair. With some natural assumptions about the breeding habits of rabbits, the population of rabbit pairs per month correspond to the elements of the Fibonacci sequence— 1, 1, 2, 3, 4, 8, 13, etc.—where, beginning with zero and one, each term of the sequence is the sum of the two preceding ones. With the passage of time, this sequence would appear in so many areas with no possible connection to the breeding of rabbits that, in 1877, Edward Lucas proposed naming it Fibonacci Sequence and its terms Fibonacci Numbers. The fertility of this sequence seems to be inexhaustible, and every year new and curious properties of it are discovered. Fn has become the standard symbol for Fibonacci numbers, and their defining relation is F = F n + F o9 Fn = 0, F1 = 1. n n \ n2 s
Rapid Convergence to Feature Layer Correspondences
"... We describe a neural network able to rapidly establish correspondence between neural feature layers. Each of the network’s two layers consists of interconnected cortical columns and each column consists of inhibitorily coupled subpopulations of excitatory neurons. The dynamics of the system builds ..."
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We describe a neural network able to rapidly establish correspondence between neural feature layers. Each of the network’s two layers consists of interconnected cortical columns and each column consists of inhibitorily coupled subpopulations of excitatory neurons. The dynamics of the system
RAPIDLY CONVERGING APPROXIMATIONS AND REGULARITY THEORY
, 906
"... Abstract. We consider distributions on a closed compact manifold M as maps on smoothing operators. Thus spaces of maps between Ψ − ∞ (M) → C ∞ (M) are considered as generalized functions. For any collection of regularizing processes we produce an algebra of generalized functions and a diffeomorphis ..."
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Abstract. We consider distributions on a closed compact manifold M as maps on smoothing operators. Thus spaces of maps between Ψ − ∞ (M) → C ∞ (M) are considered as generalized functions. For any collection of regularizing processes we produce an algebra of generalized functions and a diffeomorphism equivariant embedding of distributions into this algebra. We provide examples invariant under certain group actions. The regularity for such generalized functions is provided in terms of a certain tameness of maps between graded Frechét spaces. This notion of regularity implies the regularity in Colombeau algebras in the G ∞ sense. 1.
Rapid object detection using a boosted cascade of simple features
 ACCEPTED CONFERENCE ON COMPUTER VISION AND PATTERN RECOGNITION 2001
, 2001
"... This paper describes a machine learning approach for visual object detection which is capable of processing images extremely rapidly and achieving high detection rates. This work is distinguished by three key contributions. The first is the introduction of a new image representation called the " ..."
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Cited by 3222 (9 self)
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This paper describes a machine learning approach for visual object detection which is capable of processing images extremely rapidly and achieving high detection rates. This work is distinguished by three key contributions. The first is the introduction of a new image representation called
Rapidly convergent spectral representations for periodic Green functions
"... Spectral representations for the Green function for 2D and 3D configurations with 1D periodicities are introduced. The representations rely on the expansion of the Green function in terms of the continuous longitudinal spectrum. These representations allow obtaining the Green functions in terms of a ..."
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of a small number of direct source contributions, a small number of pole (Floquet mode) contributions, and a rapidly convergent integral that can be evaluated using quadrature rules with a small number of nodes. The presented schemes work seamlessly even for small/vanishing sourceobserver separations
Delayed internet routing convergence
 ACM SIGCOMM Computer Communication Review
, 2000
"... Abstract—This paper examines the latency in Internet path failure, failover, and repair due to the convergence properties of interdomain routing. Unlike circuitswitched paths which exhibit failover on the order of milliseconds, our experimental measurements show that interdomain routers in the pac ..."
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Cited by 414 (5 self)
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Abstract—This paper examines the latency in Internet path failure, failover, and repair due to the convergence properties of interdomain routing. Unlike circuitswitched paths which exhibit failover on the order of milliseconds, our experimental measurements show that interdomain routers
On A Rapidly Converging Series For The Riemann Zeta Function
"... To evaluate Riemann’s zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series for the Rie ..."
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To evaluate Riemann’s zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series
Evaluating the Accuracy of SamplingBased Approaches to the Calculation of Posterior Moments
 IN BAYESIAN STATISTICS
, 1992
"... Data augmentation and Gibbs sampling are two closely related, samplingbased approaches to the calculation of posterior moments. The fact that each produces a sample whose constituents are neither independent nor identically distributed complicates the assessment of convergence and numerical accurac ..."
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Cited by 583 (14 self)
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Data augmentation and Gibbs sampling are two closely related, samplingbased approaches to the calculation of posterior moments. The fact that each produces a sample whose constituents are neither independent nor identically distributed complicates the assessment of convergence and numerical
The Ant System: Optimization by a colony of cooperating agents
 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART B
, 1996
"... An analogy with the way ant colonies function has suggested the definition of a new computational paradigm, which we call Ant System. We propose it as a viable new approach to stochastic combinatorial optimization. The main characteristics of this model are positive feedback, distributed computation ..."
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Cited by 1241 (45 self)
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computation, and the use of a constructive greedy heuristic. Positive feedback accounts for rapid discovery of good solutions, distributed computation avoids premature convergence, and the greedy heuristic helps find acceptable solutions in the early stages of the search process. We apply the proposed
RAPIDLYCONVERGING METHODS FOR SOLVING MULTILEVEL TRANSFER PROBLEMS
"... It is well known that lambda iterations can be used to solve multilevel nonLTE transfer equations in a reasonable number of iterations when the lambda operator is preconditioned, e.g., when the diagonal part of the operator is combined with other terms analytically. This approach is currently used ..."
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in this way. Here we show that 1) a hybrid approach involving such a direct solution for a few of the strongest transitions, and lambda iterations for the rest, gives rapid convergence, often with oscillations that need to be damped, and 2) this approach should include preconditioning of the lambda operator
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