scholarly journals On the critical exponent of generalized Thue-Morse words

2007 ◽  
Vol Vol. 9 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Alexandre B. Massé ◽  
Srečko Brlek ◽  
Amy Glen ◽  
Sébastien Labbé
Keyword(s):  
Critical Exponent ◽  
Rational Numbers ◽  
International Audience

Automata, Logic and Semantics International audience For certain generalized Thue-Morse words t, we compute the critical exponent, i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.

scholarly journals For each $\alpha > 2$ there is an Infinite Binary Word with Critical Exponent $\alpha$

10.37236/909 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
James D. Currie ◽  
Narad Rampersad
Keyword(s):  
Critical Exponent ◽  
Rational Numbers ◽  
Alpha Power ◽  
Binary Word ◽  
Infinite Word ◽  
Open Question

The critical exponent of an infinite word ${\bf w}$ is the supremum of all rational numbers $\alpha$ such that ${\bf w}$ contains an $\alpha$-power. We resolve an open question of Krieger and Shallit by showing that for each $\alpha > 2$ there is an infinite binary word with critical exponent $\alpha$.

scholarly journals Structure of spanning trees on the two-dimensional Sierpinski gasket

2011 ◽  
Vol Vol. 12 no. 3 (Combinatorics) ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Probability Distribution ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Sierpinski Gasket ◽  
Rational Numbers ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Sierpiński Gasket ◽  
Average Probability ◽  
International Audience

Combinatorics International audience Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.

scholarly journals 3x+1 Minus the +

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Kenneth G. Monks
Keyword(s):  
Positive Integer ◽  
Rational Numbers ◽  
General Mechanism ◽  
Arbitrary Positive Integer ◽  
Arbitrary Integer ◽  
Constant Multiple ◽  
International Audience ◽  
Positive Integers

International audience We use Conway's \emphFractran language to derive a function R:\textbfZ^+ → \textbfZ^+ of the form R(n) = r_in if n ≡ i \bmod d where d is a positive integer, 0 ≤ i < d and r_0,r_1, ... r_d-1 are rational numbers, such that the famous 3x+1 conjecture holds if and only if the R-orbit of 2^n contains 2 for all positive integers n. We then show that the R-orbit of an arbitrary positive integer is a constant multiple of an orbit that contains a power of 2. Finally we apply our main result to show that any cycle \ x_0, ... ,x_m-1 \ of positive integers for the 3x+1 function must satisfy \par ∑ _i∈ \textbfE \lfloor x_i/2 \rfloor = ∑ _i∈ \textbfO \lfloor x_i/2 \rfloor +k. \par where \textbfO=\ i : x_i is odd \ , \textbfE=\ i : x_i is even \ , and k=|\textbfO|. \par The method used illustrates a general mechanism for deriving mathematical results about the iterative dynamics of arbitrary integer functions from \emphFractran algorithms.

scholarly journals Further Variations on the Six Exponentials Theorem.

2005 ◽  
Vol Volume 28 ◽  
Author(s):  
Michel Waldschmidt
Keyword(s):  
Present Note ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Linearly Independent ◽  
International Audience ◽  
Rank 2

International audience According to the Six Exponentials Theorem, a $2\times 3$ matrix whose entries $\lambda_{ij}$ ($i=1,2$, $j=1,2,3$) are logarithms of algebraic numbers has rank $2$, as soon as the two rows as well as the three columns are linearly independent over the field $\BbbQ$ of rational numbers. The main result of the present note is that one at least of the three $2\times 2$ determinants, viz. $$ \lambda_{21}\lambda_{12}-\lambda_{11}\lambda_{22}, \quad \lambda_{22}\lambda_{13}-\lambda_{12}\lambda_{23}, \quad \lambda_{23}\lambda_{11}-\lambda_{13}\lambda_{21} $$ is transcendental.

scholarly journals Simple Equational Specifications of Rational Arithmetic

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Lawrence S. Moss
Keyword(s):  
Rational Numbers ◽  
Absolute Value ◽  
Error Constant ◽  
Greatest Integer Function ◽  
International Audience ◽  
Rational Arithmetic ◽  
Equational Specifications

International audience We exhibit an initial specification of the rational numbers equipped with addition, subtraction, multiplication, greatest integer function, and absolute value. Our specification uses only the sort of rational numbers. It uses one hidden function; that function is unary. But it does not use an error constant, or extra (hidden) sorts, or conditional equations. All of our work is elementary and self-contained.

scholarly journals Extended Rate, more GFUN

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Martin Rubey
Keyword(s):  
Software Package ◽  
Rational Functions ◽  
Rational Numbers ◽  
International Audience

International audience We present a software package that guesses formulas for sequences of, for example, rational numbers or rational functions, given the first few terms. Thereby we extend and complement Christian Krattenthaler’s program $\mathtt{Rate}$ and the relevant parts of Bruno Salvy and Paul Zimmermann’s $\mathtt{GFUN}$.

scholarly journals Arithmetics in β-numeration

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Julien Bernat
Keyword(s):  
Analysis Of Algorithms ◽  
Fractional Part ◽  
Rational Numbers ◽  
Pisot Number ◽  
Real Numbers ◽  
International Audience ◽  
Perron Number ◽  
Decimal Numbers

Analysis of Algorithms International audience The β-numeration, born with the works of Rényi and Parry, provides a generalization of the notions of integers, decimal numbers and rational numbers by expanding real numbers in base β, where β>1 is not an integer. One of the main differences with the case of numeration in integral base is that the sets which play the role of integers, decimal numbers and rational numbers in base β are not stable under addition or multiplication. In particular, a fractional part may appear when one adds or multiplies two integers in base β. When β is a Pisot number, which corresponds to the most studied case, the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β are bounded by constants which only depend on β. We prove that, for any Perron number β, the set of finite or ultimately periodic fractional parts of the sum, or the product, of two integers in base β is finite. Additionally, we prove that it is possible to compute this set for the case of addition when β is a Parry number. As a consequence, we deduce that, when β is a Perron number, there exist bounds, which only depend on β, for the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β. Moreover, when β is a Parry number, the bound associated with the case of addition can be explicitly computed.

scholarly journals On the computability of the topological entropy of subshifts

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Jakob Grue Simonsen
Keyword(s):  
Topological Entropy ◽  
Error Bound ◽  
Decision Procedure ◽  
Rational Numbers ◽  
Sofic Shifts ◽  
International Audience

International audience We prove that the topological entropy of subshifts having decidable language is uncomputable in the following sense: For no error bound less than 1/4 does there exists a program that, given a decision procedure for the language of a subshift as input, will approximate the entropy of the subshift within the error bound. In addition, we prove that not only is the topological entropy of sofic shifts computable to arbitary precision (a well-known fact), but all standard comparisons of the topological entropy with rational numbers are decidable.

scholarly journals Effective irrationality measures for quotients of logarithms of rational numbers

2015 ◽  
Vol Volume 38 ◽  
Author(s):  
Yann Bugeaud
Keyword(s):  
Rational Numbers ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
International Audience

International audience We establish uniform irrationality measures for the quotients of the logarithms of two rational numbers which are very close to 1. Our proof is based on a refinement in the theory of linear forms in logarithms which goes back to a paper of Shorey.

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