scholarly journals A Sturmian sequence related to the uniqueness conjecture for Markoff numbers

2009 ◽  
Vol 410 (30-32) ◽  
pp. 2864-2869 ◽  
Author(s):  
Yann Bugeaud ◽  
Christophe Reutenauer ◽  
Samir Siksek
Keyword(s):  
Sturmian Sequence ◽  
Markoff Numbers

scholarly journals The uniqueness of the Markoff numbers

1976 ◽  
Vol 30 (134) ◽  
pp. 361-361 ◽  
Author(s):  
Gerhard Rosenberger
Keyword(s):  
Markoff Numbers

Numerology

2018 ◽  
pp. 69-74
Author(s):  
Christophe Reutenauer
Keyword(s):  
Quadratic Forms ◽  
Triple P ◽  
Mathematical Objects ◽  
Markoff Numbers

This chapter gives several examples, which may help the reader to work in concrete terms with Markoff numbers, Christoffel words, Markoff constants, and quadratic forms. In particular the thirteen Markoff numbers <1000 are given, together with the associated mathematical objects considered before in the book:Markoff constants, Christoffel words, the associated matrices by the representation of Chapter 3, theMarkoff quadratic numbers whose expansion is given by the Christoffel word, the Markoff quadratic forms. Some results of Frobenius, Aigner, andClemens are given. In particular thematrix associated with a Christoffel word may be computed directly from its Markoff triple.

scholarly journals On the Frobenius conjecture for Markoff numbers

2013 ◽  
Vol 133 (7) ◽  
pp. 2363-2373 ◽  
Author(s):  
Feng-Juan Chen ◽  
Yong-Gao Chen
Keyword(s):  
Markoff Numbers

scholarly journals Congruence and uniqueness of certain Markoff numbers

2007 ◽  
Vol 128 (3) ◽  
pp. 295-301 ◽  
Author(s):  
Ying Zhang
Keyword(s):  
Markoff Numbers

scholarly journals Sturmian Sequences and Invertible Substitutions

2011 ◽  
Vol Vol. 13 no. 2 (Combinatorics) ◽  
Author(s):  
Li Peng ◽  
Bo Tan
Keyword(s):  
Fixed Point ◽  
The Other ◽  
Sturmian Sequence ◽  
Other Hand ◽  
Angle Alpha ◽  
International Audience ◽  
Sturmian Sequences ◽  
Irrational Angle

Combinatorics International audience It is known that a Sturmian sequence S can be defined as a coding of the orbit of rho (called the intercept of S) under a rotation of irrational angle alpha (called the slope). On the other hand, a fixed point of an invertible substitution is Sturmian. Naturally, there are two interrelated questions: (1) Given an invertible substitution, we know that its fixed point is Sturmian. What is the slope and intercept? (2) Which kind of Sturmian sequences can be fixed by certain non-trivial invertible substitutions? In this paper we give a unified treatment to the two questions. We remark that though the results are known, our proof is very elementary and concise.

scholarly journals Cobham’s Theorem and Automaticity

2019 ◽  
Vol 30 (08) ◽  
pp. 1363-1379
Author(s):  
Lucas Mol ◽  
Narad Rampersad ◽  
Jeffrey Shallit ◽  
Manon Stipulanti
Keyword(s):  
Upper Bounds ◽  
Sturmian Sequence ◽  
Automatic Sequence ◽  
Common Prefix

We make certain bounds in Krebs’ proof of Cobham’s theorem explicit and obtain corresponding upper bounds on the length of a common prefix of an aperiodic [Formula: see text]-automatic sequence and an aperiodic [Formula: see text]-automatic sequence, where [Formula: see text] and [Formula: see text] are multiplicatively independent. We also show that an automatic sequence cannot have arbitrarily large factors in common with a Sturmian sequence.

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