scholarly journals Renormalisation of correlations in a barrier billiard: Quadratic irrational trajectories

2014 ◽  
Vol 270 ◽  
pp. 30-45 ◽  
Author(s):  
L.N.C. Adamson ◽  
A.H. Osbaldestin
Keyword(s):  
Quadratic Irrational

scholarly journals CONTINUED FRACTIONS WITH BOUNDED PARTIAL QUOTIENTS

2002 ◽  
Vol 45 (3) ◽  
pp. 653-671 ◽  
Author(s):  
J. L. Davison
Keyword(s):  
Rate Of Convergence ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Subject Classification ◽  
Specific Sequence ◽  
Secondary 11B37 ◽  
Quadratic Irrational ◽  
Infinite Word ◽  
Partial Quotients

AbstractPrecise bounds are given for the quantity$$ L(\alpha)=\frac{\limsup_{m\rightarrow\infty}(1/m)\ln q_m}{\liminf_{m\rightarrow\infty}(1/m)\ln q_m}, $$where $(q_m)$ is the classical sequence of denominators of convergents to the continued fraction $\alpha=[0,u_1,u_2,\dots]$ and $(u_m)$ is assumed bounded, with a distribution.If the infinite word $\bm{u}=u_1u_2\dots$ has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number $\gamma$, called the segment-repetition factor.If $\alpha$ is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to $\alpha$, the rate of convergence given in terms of $L$ and $\gamma$. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form $[0,u_1,u_2,\dots]$ with $u_m=1+\lfloor m\theta\rfloor\Mod n$, $n\geq2$, and $\theta$ an irrational number which satisfies any of a given set of conditions.AMS 2000 Mathematics subject classification: Primary 11A55. Secondary 11B37

Properties of real quadratic irrational numbers under the action of the group H = áx, y: x2 = y4 = 1ñ

2005 ◽  
Vol 42 (4) ◽  
pp. 371-386
Author(s):  
M. Aslam Malik ◽  
S. M. Husnine ◽  
Abdul Majeed
Keyword(s):  
Quadratic Field ◽  
Infinite Group ◽  
Algebraic Structures ◽  
Real Quadratic Field ◽  
Irrational Numbers ◽  
The Real ◽  
Basic Properties ◽  
Quadratic Irrational ◽  
Real Quadratic

Studying groups through their actions on different sets and algebraic structures has become a useful technique to know about the structure of the groups. The main object of this work is to examine the action of the infinite group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H = \langle x,y : x^{2} = y^{4} = 1\rangle$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $x (z) = \frac{-1}{2z}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $y (z) = \frac{-1}{2(z+1)}$ \end{document} on the real quadratic field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} and find invariant subsets of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H$ \end{document}. We also discuss some basic properties of elements of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group H.

scholarly journals Short proofs of theorems of Lekkerkerker and Ballieu

1982 ◽  
Vol 5 (3) ◽  
pp. 609-612 ◽  
Author(s):  
Max Riederle
Keyword(s):  
Irrational Number ◽  
Quadratic Irrational ◽  
Accumulation Points

For any irrational numberξletA(ξ)denote the set of all accumulation points of{z:z=q(qξ−p),   p∈ℤ,   q∈ℤ−{0},   p   and   q   relatively prime}. In this paper the following theorem of Lekkerkerker is proved in a short and elementary way: The setA(ξ)is discrete and does not contain zero if and only ifξis a quadratic irrational. The procedure used for this proof simultaneously takes care of a theorem of Ballieu.

Coset Diagram for the Action of Picard Group on $\mathbb{Q}(i,\sqrt{3})$

2016 ◽  
Vol 23 (01) ◽  
pp. 33-44 ◽  
Author(s):  
Qaiser Mushtaq ◽  
Saima Anis
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Picard Group ◽  
Closed Path ◽  
Irrational Numbers ◽  
Unique Pattern ◽  
Graphical Interpretation ◽  
Quadratic Irrational ◽  
Coset Diagrams ◽  
Real Quadratic

In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group Γ=PSL(2,ℤ[i]) on [Formula: see text]. Graphical interpretation of amalgamation of the components of Γ is also given. Some elements [Formula: see text] of [Formula: see text] and their conjugates [Formula: see text] over ℚ(i) have different signs in the orbits of the biquadratic field [Formula: see text] when acted upon by Γ. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Γα, and they form a closed path which is the only closed path in the orbit Γα. We also devise a procedure to obtain ambiguous numbers of the form [Formula: see text], where b is a positive integer.

Modular group acting on real quadratic fields

1988 ◽  
Vol 37 (2) ◽  
pp. 303-309 ◽  
Author(s):  
Q. Mushtaq
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Modular Group ◽  
Quadratic Fields ◽  
Closed Path ◽  
Irrational Numbers ◽  
Real Quadratic Fields ◽  
Quadratic Irrational ◽  
Coset Diagrams ◽  
Real Quadratic

Coset diagrams for the orbit of the modular group G = 〈x, y: x2 = y3 = 1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that for a fixed value of n, a non-square positive integer, there are only a finite number of real quadratic irrational numbers of the form , where θ and its algebraic conjugate have different signs, and that part of the coset diagram containing such numbers forms a single circuit (closed path) and it is the only circuit in the orbit of θ.

Length of the period of a quadratic irrational

1982 ◽  
Vol 18 (6) ◽  
pp. 919-923 ◽  
Author(s):  
E. V. Podsypanin
Keyword(s):  
Quadratic Irrational
Sign in / Sign up

Export Citation Format

Share Document