scholarly journals Entropy ratio for infinite sequences with positive entropy

2018 ◽  
Vol 40 (3) ◽  
pp. 751-762 ◽  
Author(s):  
CHRISTIAN MAUDUIT ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Exponential Growth ◽  
Fractal Dimensions ◽  
Combinatorial Structure ◽  
Finite Alphabet ◽  
Exponential Growth Rate ◽  
Positive Entropy ◽  
Infinite Words ◽  
Complexity Function ◽  
Infinite Word ◽  
Infinite Sequences

The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. For any given function $f$ with exponential growth, we introduced in [Complexity and fractal dimensions for infinite sequences with positive entropy. Commun. Contemp. Math. to appear] the notion of word entropy$E_{W}(f)$ associated to $f$ and we described the combinatorial structure of sets of infinite words with a complexity function bounded by $f$. The goal of this work is to give estimates on the word entropy $E_{W}(f)$ in terms of the limiting lower exponential growth rate of $f$.

scholarly journals Complexity and fractal dimensions for infinite sequences with positive entropy

2019 ◽  
Vol 21 (06) ◽  
pp. 1850068
Author(s):  
Christian Mauduit ◽  
Carlos Gustavo Moreira
Keyword(s):  
Exponential Growth ◽  
Fractal Dimensions ◽  
Combinatorial Structure ◽  
Finite Alphabet ◽  
Combinatorial Proof ◽  
Infinite Words ◽  
Entropy Formula ◽  
Complexity Function ◽  
Infinite Word ◽  
Infinite Sequences

The complexity function of an infinite word [Formula: see text] on a finite alphabet [Formula: see text] is the sequence counting, for each non-negative [Formula: see text], the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of the infinite word [Formula: see text]. The goal of this work is to estimate the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of an infinite word [Formula: see text] with a complexity function bounded by a given function [Formula: see text] with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the word entropy [Formula: see text] associated to a given function [Formula: see text] and we determine the fractal dimensions of sets of infinite sequences with complexity function bounded by [Formula: see text] in terms of its word entropy. We present a combinatorial proof of the fact that [Formula: see text] is equal to the topological entropy of the subshift of infinite words whose complexity is bounded by [Formula: see text] and we give several examples showing that even under strong conditions on [Formula: see text], the word entropy [Formula: see text] can be strictly smaller than the limiting lower exponential growth rate of [Formula: see text].

scholarly journals Defect Effect of Bi-infinite Words in the Two-element Case

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Ján Maňuch
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Alphabet ◽  
Infinite Words ◽  
Infinite Word ◽  
The Family ◽  
International Audience ◽  
Necessary And Sufficient ◽  
Primitive Roots ◽  
Element Set

International audience Let X be a two-element set of words over a finite alphabet. If a bi-infinite word possesses two X-factorizations which are not shiftequivalent, then the primitive roots of the words in X are conjugates. Note, that this is a strict sharpening of a defect theorem for bi-infinite words stated in \emphKMP. Moreover, we prove that there is at most one bi-infinite word possessing two different X-factorizations and give a necessary and sufficient conditions on X for the existence of such a word. Finally, we prove that the family of sets X for which such a word exists is parameterizable.

Generalized Hausdorff dimensions of sets of real numbers with zero entropy expansion

2011 ◽  
Vol 32 (3) ◽  
pp. 1073-1089 ◽  
Author(s):  
CHRISTIAN MAUDUIT ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Continued Fraction ◽  
Negative Integer ◽  
Finite Alphabet ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Hausdorff Dimensions ◽  
Subexponential Growth ◽  
Complexity Function ◽  
Infinite Word ◽  
Zero Entropy

AbstractThe complexity function of an infinite wordwon a finite alphabetAis the sequence counting, for each non-negative integern, the number of words of lengthnon the alphabetAthat are factors of the infinite wordw. Letfbe a given function with subexponential growth. The goal of this work is to estimate the generalized Hausdorff dimensions of the set of real numbers whoseq-adic expansion has a complexity function bounded byfand the set of real numbers whose continued fraction expansion is bounded byqand has a complexity function bounded byf.

A NEW COMPLEXITY FUNCTION FOR WORDS BASED ON PERIODICITY

2013 ◽  
Vol 23 (04) ◽  
pp. 963-987 ◽  
Author(s):  
FILIPPO MIGNOSI ◽  
ANTONIO RESTIVO
Keyword(s):  
The Other ◽  
Factorization Theorem ◽  
Complexity Measures ◽  
Infinite Words ◽  
Average Value ◽  
Complexity Function ◽  
Binary Word ◽  
Infinite Word

Motivated by the extension of the critical factorization theorem to infinite words, we study the (local) periodicity function, i.e. the function that, for any position in a word, gives the size of the shortest square centered in that position. We prove that this function characterizes any binary word up to exchange of letters. We then introduce a new complexity function for words (the periodicity complexity) that, for any position in the word, gives the average value of the periodicity function up to that position. The new complexity function is independent from the other commonly used complexity measures as, for instance, the factor complexity. Indeed, whereas any infinite word with bounded factor complexity is periodic, we will show a recurrent non-periodic word with bounded periodicity complexity. Further, we will prove that the periodicity complexity function grows as Θ( log n) in the case of the Fibonacci infinite word and that it grows as Θ(n) in the case of the Thue–Morse word. Finally, we will show examples of infinite recurrent words with arbitrary high periodicity complexity.

scholarly journals Optimal Strategies for Prudent Investors

1998 ◽  
Vol 01 (04) ◽  
pp. 473-486 ◽  
Author(s):  
Roberto Baviera ◽  
Michele Pasquini ◽  
Maurizio Serva ◽  
Angelo Vulpiani
Keyword(s):  
Growth Rate ◽  
Random Walk ◽  
Exponential Growth ◽  
Optimal Strategies ◽  
Maximum Amount ◽  
Exponential Growth Rate ◽  
One Dimensional ◽  
Economic Level ◽  
Random Walk With Drift ◽  
Analytical Expressions

We consider a stochastic model of investment on an asset in a stock market for a prudent investor. she decides to buy permanent goods with a fraction α of the maximum amount of money owned in her life in order that her economic level never decreases. The optimal strategy is obtained by maximizing the exponential growth rate for a fixed α. We derive analytical expressions for the typical exponential growth rate of the capital and its fluctuations by solving an one-dimensional random walk with drift.

LANGUAGES WITH A FINITE ANTIDICTIONARY: SOME GROWTH QUESTIONS

2014 ◽  
Vol 25 (08) ◽  
pp. 937-953
Author(s):  
ARSENY M. SHUR
Keyword(s):  
Growth Rate ◽  
Structural Properties ◽  
Exponential Growth ◽  
Regular Languages ◽  
The Other ◽  
Exponential Growth Rate ◽  
Strong Component ◽  
Order Of Growth ◽  
Finite Sets ◽  
Complex Order

We study FAD-languages, which are regular languages defined by finite sets of forbidden factors, together with their “canonical” recognizing automata. We are mainly interested in the possible asymptotic orders of growth for such languages. We analyze certain simplifications of sets of forbidden factors and show that they “almost” preserve the canonical automata. Using this result and structural properties of canonical automata, we describe an algorithm that effectively lists all canonical automata having a sink strong component isomorphic to a given digraph, or reports that no such automata exist. This algorithm can be used, in particular, to prove the existence of a FAD-language over a given alphabet with a given exponential growth rate. On the other hand, we give an example showing that the algorithm cannot prove non-existence of a FAD-language having a given growth rate. Finally, we provide some examples of canonical automata with a nontrivial condensation graph and of FAD-languages with a “complex” order of growth.

Validation and Analysis of Modeled Predictions of Growth of Bacillus cereus Spores in Boiled Rice

2000 ◽  
Vol 63 (2) ◽  
pp. 268-272 ◽  
Author(s):  
DANA M. McELROY ◽  
LEE-ANN JAYKUS ◽  
PEGGY M. FOEGEDING
Keyword(s):  
Growth Rate ◽  
Bacillus Cereus ◽  
Exponential Growth ◽  
Generation Time ◽  
Time Lag ◽  
Lag Phase ◽  
Exponential Growth Rate ◽  
Phase Duration ◽  
Margin Of Safety ◽  
Fail Safe

The growth of psychrotrophic Bacillus cereus 404 from spores in boiled rice was examined experimentally at 15, 20, and 30°C. Using the Gompertz function, observed growth was modeled, and these kinetic values were compared with kinetic values for the growth of mesophilic vegetative cells as predicted by the U.S. Department of Agriculture's Pathogen Modeling Program, version 5.1. An analysis of variance indicated no statistically significant difference between observed and predicted values. A graphical comparison of kinetic values demonstrated that modeled predictions were “fail safe” for generation time and exponential growth rate at all temperatures. The model also was fail safe for lag-phase duration at 20 and 30°C but not at l5°C. Bias factors of 0.55, 0.82, and 1.82 for generation time, lag-phase duration, and exponential growth rate, respectively, indicated that the model generally was fail safe and hence provided a margin of safety in its growth predictions. Accuracy factors of 1.82, 1.60, and 1.82 for generation time, lag-phase duration, and exponential growth rate, respectively, quantitatively demonstrated the degree of difference between predicted and observed values. Although the Pathogen Modeling Program produced reasonably accurate predictions of the growth of psychrotrophic B. cereus from spores in boiled rice, the margin of safety provided by the model may be more conservative than desired for some applications. It is recommended that if microbial growth modeling is to be applied to any food safety or processing situation, it is best to validate the model before use. Once experimental data are gathered, graphical and quantitative methods of analysis can be useful tools for evaluating specific trends in model prediction and identifying important deviations between predicted and observed data.

TOPOLOGICAL ENTROPY VERSUS GEODESIC ENTROPY

1994 ◽  
Vol 05 (02) ◽  
pp. 213-218 ◽  
Author(s):  
GABRIEL P. PATERNAIN ◽  
MIGUEL PATERNAIN
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Exponential Growth ◽  
Geodesic Flow ◽  
Compact Riemannian Manifold ◽  
Betti Numbers ◽  
Exponential Growth Rate ◽  
Sphere Bundle ◽  
Space Of Paths ◽  
Simply Connected

Using Yomdin's Theorem [8], we show that for a compact Riemannian manifold M, the geodesic entropy — defined as the exponential growth rate of the average number of geodesic segments between two points — is ≤ the topological entropy of the geodesic flow of M. We also show that if M is simply connected and N ⊂ M is a compact simply connected submanifold, then the exponential growth rate of the sequence given by the Betti numbers of the space of paths starting in N and ending in a fixed point of M, is bounded above by the topological entropy of the geodesic flow on the normal sphere bundle of N.

scholarly journals THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY

2015 ◽  
Vol 3 ◽  
Author(s):  
VAN CYR ◽  
BRYNA KRA
Keyword(s):  
Automorphism Group ◽  
Finite Groups ◽  
Linear Growth ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Complexity Function ◽  
Algebraic Properties

For a finite alphabet ${\mathcal{A}}$ and shift $X\subseteq {\mathcal{A}}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group $\text{Aut}(X)$. For such systems, we show that every finitely generated subgroup of $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if $X$ is transitive, then $\text{Aut}(X)$ is virtually $\mathbb{Z}$; if $X$ has dense aperiodic points, then $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$. We also classify all finite groups that arise as the automorphism group of a shift.

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