Continued Fractions and $\operatorname{SL}(2,{\mathbb{Z}})$ Conjugacy Classes. Elements of Gauss’s Reduction Theory. Markov Spectrum

2013 ◽  
pp. 67-85
Author(s):  
Oleg Karpenkov
Keyword(s):  
Continued Fractions ◽  
Conjugacy Classes ◽  
Reduction Theory ◽  
Markov Spectrum

scholarly journals ON BINARY QUADRATIC FORMS AND THE HECKE GROUPS

2009 ◽  
Vol 05 (08) ◽  
pp. 1401-1418 ◽  
Author(s):  
WENDELL RESSLER
Keyword(s):  
Fixed Points ◽  
Continued Fractions ◽  
Quadratic Forms ◽  
Modular Group ◽  
Reduction Theory ◽  
Binary Quadratic Forms ◽  
Hecke Groups ◽  
Hecke Group ◽  
Hyperbolic Fixed Points

We present a reduction theory for certain binary quadratic forms with coefficients in ℤ[λ], where λ is the minimal translation in a Hecke group. We generalize from the modular group Γ(1) = PSL(2,ℤ) to the Hecke groups and make extensive use of modified negative continued fractions. We also define and characterize "reduced" and "simple" hyperbolic fixed points of the Hecke groups.

The Practice and Evolution of Torque and Drag Reduction: Theory and Field Results

2011 ◽  
Author(s):  
John Edward McCormick ◽  
Chad D. Evans ◽  
Joe Le ◽  
TzuFang Chiu
Keyword(s):  
Drag Reduction ◽  
Reduction Theory

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

scholarly journals Groups with boundedly finite conjugacy classes of commutators

2018 ◽  
Vol 69 (3) ◽  
pp. 1047-1051 ◽  
Author(s):  
Gláucia Dierings ◽  
Pavel Shumyatsky
Keyword(s):  
Conjugacy Classes ◽  
Finite Conjugacy
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