The Experts below are selected from a list of 309 Experts worldwide ranked by ideXlab platform
Arnaud Chéritat - One of the best experts on this subject based on the ideXlab platform.
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Upper bound for the size of quadratic Siegel disks
Inventiones mathematicae, 2004Co-Authors: Xavier Buff, Arnaud ChéritatAbstract:If α is an Irrational Number, we let { p _ n / q _ n }_ n ≥0, be the approximants given by its continued fraction expansion. The Bruno series B (α) is defined as $$B(\alpha)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial P _α: z ↦ e ^2 i πα z + z ^2 has an indifferent fixed point at the origin. If P _α is linearizable, we let r (α) be the conformal radius of the Siegel disk and we set r (α)=0 otherwise. Yoccoz proved that if B (α)=∞, then r (α)=0 and P _α is not linearizable. In this article, we present a different proof and we show that there exists a constant C such that for all Irrational Number α with B (α)
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On the Size of Quadratic Siegel Disks: Part I
arXiv: Dynamical Systems, 2003Co-Authors: Xavier Buff, Arnaud ChéritatAbstract:If $\a$ is an Irrational Number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial $P_\a:z\mapsto e^{2i\pi \a}z+z^2$ has an indifferent fixed point at the origin. If $P_\a$ is linearizable, we let $r(\a)$ be the conformal radius of the Siegel disk and we set $r(\a)=0$ otherwise. Yoccoz proved that if $B(\a)=\infty$, then $r(\a)=0$ and $P_\a$ is not linearizable. In this article, we present a different proof and we show that there exists a constant $C$ such that for all Irrational Number $\a$ with $B(\a)
Xavier Buff - One of the best experts on this subject based on the ideXlab platform.
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Upper bound for the size of quadratic Siegel disks
Inventiones mathematicae, 2004Co-Authors: Xavier Buff, Arnaud ChéritatAbstract:If α is an Irrational Number, we let { p _ n / q _ n }_ n ≥0, be the approximants given by its continued fraction expansion. The Bruno series B (α) is defined as $$B(\alpha)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial P _α: z ↦ e ^2 i πα z + z ^2 has an indifferent fixed point at the origin. If P _α is linearizable, we let r (α) be the conformal radius of the Siegel disk and we set r (α)=0 otherwise. Yoccoz proved that if B (α)=∞, then r (α)=0 and P _α is not linearizable. In this article, we present a different proof and we show that there exists a constant C such that for all Irrational Number α with B (α)
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On the Size of Quadratic Siegel Disks: Part I
arXiv: Dynamical Systems, 2003Co-Authors: Xavier Buff, Arnaud ChéritatAbstract:If $\a$ is an Irrational Number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial $P_\a:z\mapsto e^{2i\pi \a}z+z^2$ has an indifferent fixed point at the origin. If $P_\a$ is linearizable, we let $r(\a)$ be the conformal radius of the Siegel disk and we set $r(\a)=0$ otherwise. Yoccoz proved that if $B(\a)=\infty$, then $r(\a)=0$ and $P_\a$ is not linearizable. In this article, we present a different proof and we show that there exists a constant $C$ such that for all Irrational Number $\a$ with $B(\a)
Andrew Haas - One of the best experts on this subject based on the ideXlab platform.
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The relative growth rate for partial quotients
2008Co-Authors: Andrew HaasAbstract:The rate of growth of the partial quotients of an Irrational Number is studied relative to the rate of approximation of the Number by its convergents. The focus is on the Hausdorff dimension of exceptional sets on which different growth rates are achieved.
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The relative growth rate for partial quotients
arXiv: Number Theory, 2007Co-Authors: Andrew HaasAbstract:We look at the rate of growth of the partial quotients of the infinite continued fraction expansion of an Irrational Number relative to the rate of approximation of the Number by its convergents. In non-generic cases the Hausdorff dimension of some exceptional sets is computed.
Esra Aksoy - One of the best experts on this subject based on the ideXlab platform.
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learning difficulties about the relationship between Irrational Number set with rational or real Number sets
Turkish Journal of Computer and Mathematics Education (TURCOMAT), 2016Co-Authors: Yusuf Emre Ercire, Serkan Narli, Esra AksoyAbstract:The aim of this study is to investigate students’ difficulties about the relation between Irrational Number set, rational Number set and real Number set. For this purpose, ‘Irrational Number Concept Test’ which was composed of open-ended questions has been developed. The Data collection instrument was applied to 58 students in grade 8 and 50 students in grade 9. Semi-structured interviews with ten students who were selected from different levels with the maximum diversity sampling were conducted. In each grade, it was found that students had difficulties in understanding the relationship between real Number set and other Number sets. There have been some thoughts such as ‘all of the Irrational Numbers are not real Numbers’ and ‘a Number can be both rational and Irrational’. It is found that the rate of students that have these wrong thoughts in 8th grades is more than those in 9th grades.
Jun Wu - One of the best experts on this subject based on the ideXlab platform.
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beta expansion and continued fraction expansion
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Bing Li, Jun WuAbstract:For any real Number β>1, let e(1,β)=(e1(1),e2(1),…,en(1),…) be the infinite β-expansion of 1. Define ln=sup{k⩾0:en+j(1)=0for all1⩽j⩽k}. Let x∈[0,1) be an Irrational Number. We denote by kn(x) the exact Number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x. If {ln,n⩾1} is bounded, we obtain that for all x∈[0,1)∖Q, lim infn→+∞kn(x)n=logβ2β*(x),lim supn→+∞kn(x)n=logβ2β*(x), where β*(x), β*(x) are the upper and lower Levy constants, which generalize the result in [J. Wu, Continued fraction and decimal expansions of an Irrational Number, Adv. Math. 206 (2) (2006) 684–694]. Moreover, if lim supn→+∞lnn=0, we also get the similar result except a small set.
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continued fraction and decimal expansions of an Irrational Number
Advances in Mathematics, 2006Co-Authors: Jun WuAbstract:Abstract For an Irrational Number x and n ⩾ 1 , we denote by k n ( x ) the exact Number of partial quotients in the continued fraction expansion of x given by the first n decimals of x and p n ( x ) / q n ( x ) the nth convergent of x. Let β ∗ ( x ) = lim inf n → ∞ log q n ( x ) n , β ∗ ( x ) = lim sup n → ∞ log q n ( x ) n . We prove that lim sup n → ∞ k n ( x ) n = log 10 2 β ∗ ( x ) , lim inf n → ∞ k n ( x ) n = log 10 2 β ∗ ( x ) . This result significantly strengthens the results of G. Lochs and C. Faivre.
