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Computing Order Statistics in the Farey Sequence
"... Abstract. We study the problem of computing the kth term of the Farey sequence of order n, for given n and k. Several methods for generating the entire Farey sequence are known. However, these algorithms require at least quadratic time, since the Farey sequence has Θ(n 2) elements. For the problem ..."
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Abstract. We study the problem of computing the kth term of the Farey sequence of order n, for given n and k. Several methods for generating the entire Farey sequence are known. However, these algorithms require at least quadratic time, since the Farey sequence has Θ(n 2) elements. For the problem of finding the kth element, we obtain an algorithm that runs in time O(n lg n) and uses space O ( √ n). The same bounds hold for the problem of determining the rank in the Farey sequence of a given fraction. A more complicated solution can reduce the space to O(n 1/3 (lg lg n) 2/3), and, for the problem of determining the rank of a fraction, reduce the time to O(n). We also argue that an algorithm with running time O(poly(lg n)) is unlikely to exist, since that would give a polynomialtime algorithm for integer factorization. 1
1 A Recursion for the Farey Sequence
, 2009
"... Represent a rational number f as an ordered pair (n, d) where n is the numerator and d is the denominator. The Farey sequence of order m, Fm, is given by the recursion di−1 + m fi+1 = di fi − fi−1 where f1 = (0, 1) and f2 = (1, m) ([Har02], [Far16], [Hal70]). The Farey sequence Fm is the sequence of ..."
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Represent a rational number f as an ordered pair (n, d) where n is the numerator and d is the denominator. The Farey sequence of order m, Fm, is given by the recursion di−1 + m fi+1 = di fi − fi−1 where f1 = (0, 1) and f2 = (1, m) ([Har02], [Far16], [Hal70]). The Farey sequence Fm is the sequence of irreducible rational numbers { n d} with 0 ≤ n ≤ d ≤ m arranged in increasing order. It is occasionally useful to have available a expression for Fm+1 in terms of Fm. 2 A Recursion for the Farey Sequence Sequence Represent an element f of the Farey sequence Fm as an ordered triple (n, d, s) where n is the numerator, d is the denominator and s is the number of Farey sequences following Fm until a new fraction appears immediately after f. In 1 this notation F2 is given by
Approximating Rational Numbers by Fractions
"... Abstract. In this paper we show a polynomialtime algorithm to find the best rational approximation of a given rational number within a given interval. As a special case, we show how to find the best rational number that after evaluating and rounding exactly matches the input number. In both results ..."
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Abstract. In this paper we show a polynomialtime algorithm to find the best rational approximation of a given rational number within a given interval. As a special case, we show how to find the best rational number that after evaluating and rounding exactly matches the input number. In both results, “best ” means “having the smallest possible denominator”. 1
Bifurcation in Complex Quadratic Polynomial and Some Folk Theorems Involving the Geometry of Bulbs of the Mandelbrot Set
, 2001
"... senrices bibliographiques ..."
The Mandelbrot Set and The Farey Tree
 Am. Math. Monthly
, 1999
"... this paper is to explain and to make precise several "folk theorems" involving the Mandelbrot set and the Farey tree [D]. Recall that the Mandelbrot set is the parameter plane for iteration of the complex quadratic function Q c (z) = z ..."
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this paper is to explain and to make precise several "folk theorems" involving the Mandelbrot set and the Farey tree [D]. Recall that the Mandelbrot set is the parameter plane for iteration of the complex quadratic function Q c (z) = z
Hard Unknots and Collapsing Tangles
"... Classical knot theory is about the classification, up to isotopy, of embedded closed curves in threedimensional space. Two closed curves embedded in threedimensional space are said to be isotopic if there is a continuous family of embeddings starting with one curve and ending with the other curve. ..."
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Classical knot theory is about the classification, up to isotopy, of embedded closed curves in threedimensional space. Two closed curves embedded in threedimensional space are said to be isotopic if there is a continuous family of embeddings starting with one curve and ending with the other curve. This
Geometry of the Antennas in the Mandelbrot Set
, 2000
"... In the Mandelbrot set, the bulbs attached directly to the main cardioid are called the p=qbulbs. The reason for this is that the largest component of the interior of these bulbs consists of cvalues for which the quadratic function Q c (z) = z 2 + c admits an attracting cycle with rotation numbe ..."
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In the Mandelbrot set, the bulbs attached directly to the main cardioid are called the p=qbulbs. The reason for this is that the largest component of the interior of these bulbs consists of cvalues for which the quadratic function Q c (z) = z 2 + c admits an attracting cycle with rotation number p=q. In this paper we give a geometric method to read off p=q from the geometry of the antenna attached to the bulb. 1 1 Introduction Our goal in this paper is to describe a geometric method of "reading off" dynamical information about the orbits of Q c (z) = z 2 + c from geometric information about the parameter plane for this family, the well known Mandelbrot set. Recall that the Mandelbrot set M is given by the set of complex cvalues for which the orbit of 0 under the quadratic function Q c (z) = z 2 + c does not tend to infinity. It is known that many of the components of the interior of M consist of cvalues for which the orbit of 0 is attracted to an Boston University, Bo...
ARNOL ′ D TONGUES ARISING FROM A GRAZINGSLIDING BIFURCATION OF A PIECEWISESMOOTH SYSTEM RÓBERT SZALAI ⋆ AND
"... Abstract. The NeĭmarkSacker bifurcation, or Hopf bifurcation for maps, is a wellknown bifurcation for smooth dynamical systems. At a NeĭmarkSacker bifurcation a periodic orbit loses stability and, except for certain socalled strong resonances, an invariant torus is born; the dynamics on the toru ..."
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Abstract. The NeĭmarkSacker bifurcation, or Hopf bifurcation for maps, is a wellknown bifurcation for smooth dynamical systems. At a NeĭmarkSacker bifurcation a periodic orbit loses stability and, except for certain socalled strong resonances, an invariant torus is born; the dynamics on the torus can be either quasiperiodic or phase locked, which is organized by Arnol ′ d tongues in parameter space. Inside the Arnol ′ d tongues phaselocked periodic orbits exist that disappear in saddlenode bifurcations on the tongue boundaries. In this paper we investigate whether a piecewisesmooth system with sliding regions may exhibit an equivalent of the NeĭmarkSacker bifurcation. The vector field defining such a system changes from one region in phase space to the next and the dividing socalled switching surface contains a sliding region if the vector fields on both sides point towards the switching surface. The existence of a sliding region has a superstabilizing effect on periodic orbits interacting with it. In particular, the associated Poincaré map is noninvertible. We consider the grazingsliding bifurcation at which a periodic orbit becomes tangent to the sliding region. We provide conditions under which the grazingsliding bifurcation can be thought of as a NeĭmarkSacker bifurcation. We give a normal form of the Poincaré map derived at the grazingsliding bifurcation and show that the resonances are again organized in Arnol ′ d tongues. The associated periodic orbits