scholarly journals A Square Root Map on Sturmian Words

10.37236/6074 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Jarkko Peltomäki ◽  
Markus A. Whiteland
Keyword(s):  
Fixed Points ◽  
General Setting ◽  
Square Root ◽  
Sturmian Word ◽  
Word Equation ◽  
Sturmian Words ◽  
Infinite Set ◽  
Curious Property

We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope $\alpha$, there exists exactly six minimal squares in its language (a minimal square does not have a square as a proper prefix). A Sturmian word $s$ of slope $\alpha$ can be written as a product of these six minimal squares: $s = X_1^2 X_2^2 X_3^2 \cdots$. The square root of $s$ is defined to be the word $\sqrt{s} = X_1 X_2 X_3 \cdots$. The main result of this paper is that $\sqrt{s}$ is also a Sturmian word of slope $\alpha$. Further, we characterize the Sturmian fixed points of the square root map, and we describe how to find the intercept of $\sqrt{s}$ and an occurrence of any prefix of $\sqrt{s}$ in $s$. Related to the square root map, we characterize the solutions of the word equation $X_1^2 X_2^2 \cdots X_n^2 = (X_1 X_2 \cdots X_n)^2$ in the language of Sturmian words of slope $\alpha$ where the words $X_i^2$ are minimal squares of slope $\alpha$.We also study the square root map in a more general setting. We explicitly construct an infinite set of non-Sturmian fixed points of the square root map. We show that the subshifts $\Omega$ generated by these words have a curious property: for all $w \in \Omega$ either $\sqrt{w} \in \Omega$ or $\sqrt{w}$ is periodic. In particular, the square root map can map an aperiodic word to a periodic word.

scholarly journals CONSTRAINTS IN QUANTUM GEOMETRODYNAMICS

2004 ◽  
Vol 19 (10) ◽  
pp. 1609-1638 ◽  
Author(s):  
ADRIAN P. GENTLE ◽  
NATHAN D. GEORGE ◽  
ARKADY KHEYFETS ◽  
WARNER A. MILLER
Keyword(s):  
Quantum Gravity ◽  
General Setting ◽  
Square Root ◽  
Dynamical Equations ◽  
Dynamic Content ◽  
Quantization Formalism ◽  
Canonical Quantum Gravity ◽  
First Time ◽  
Quantum Geometrodynamics ◽  
Canonical Quantum

We compare different treatments of the constraints in canonical quantum gravity. The standard approach on the superspace of 3-geometries treats the constraints as the sole carriers of the dynamic content of the theory, thus rendering the traditional dynamical equations obsolete. Quantization of the constraints in both the Dirac and ADM square root Hamiltonian approaches leads to the well known problems of time evolution. These problems of time are of both an interpretational and technical nature. In contrast, the geometrodynamic quantization procedure on the superspace of the true dynamical variables separates the issues of quantization from the enforcement of the constraints. The resulting theory takes into account states that are off-shell with respect to the constraints, and thus avoids the problems of time. We develop, for the first time, the geometrodynamic quantization formalism in a general setting and show that it retains all essential features previously illustrated in the context of homogeneous cosmologies.

scholarly journals The Maximal Number of Runs in Standard Sturmian Words

10.37236/2473 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Paweł Baturo ◽  
Marcin Piątkowski ◽  
Wojciech Rytter
Keyword(s):  
Explicit Formula ◽  
Special Class ◽  
Linear Time ◽  
Infinite Sequence ◽  
Integer Sequence ◽  
Sturmian Word ◽  
Compact Representations ◽  
Sturmian Words ◽  
Standard Word

We investigate some repetition problems for a very special class $\mathcal{S}$ of strings called the standard Sturmian words, which  have very compact representations in terms of sequences of integers. Usually the size of this word is exponential with respect to the size of its integer sequence, hence we are dealing with repetition problems in compressed strings. An explicit formula is given for the number $\rho(w)$ of runs in a standard word $w$. We show that $\rho(w)/|w|\le 4/5$ for each $w\in S$, and  there is an infinite sequence of strictly growing words $w_k\in {\mathcal{S}}$ such that $\lim_{k\rightarrow \infty} \frac{\rho(w_k)}{|w_k|} = \frac{4}{5}$. Moreover, we show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation.

STABILITY AND BIFURCATION ANALYSIS OF A DISCRETE PREY–PREDATOR MODEL WITH SQUARE-ROOT FUNCTIONAL RESPONSE AND OPTIMAL HARVESTING

2020 ◽  
Vol 28 (01) ◽  
pp. 91-110
Author(s):  
PRABIR CHAKRABORTY ◽  
UTTAM GHOSH ◽  
SUSMITA SARKAR
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Functional Response ◽  
Population Model ◽  
Optimal Harvesting ◽  
Herd Behavior ◽  
Square Root ◽  
Optimal Harvesting Policy ◽  
Optimal Value ◽  
Predator Model

In this paper, we have considered a discrete prey–predator model with square-root functional response and optimal harvesting policy. This type of functional response is used to study the dynamics of the prey–predator model where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The considered population model has three fixed points; one is trivial, the second one is axial and the last one is an interior fixed point. The first two fixed points are always feasible but the last one depends on the parameter value. The interior fixed point experiences the flip and Neimark–Sacker bifurcations depending on the predator harvesting coefficient. Finally, an optimal harvesting policy has been introduced and the optimal value of the harvesting coefficient is determined.

Lines and Hyperplanes associated with Families of Closed and Bounded Sets in Conjugate Banach Spaces

1970 ◽  
Vol 22 (5) ◽  
pp. 933-938
Author(s):  
M. Edelstein
Keyword(s):  
General Setting ◽  
Locally Convex Space ◽  
Finite Family ◽  
Topological Spaces ◽  
Compact Sets ◽  
Straight Line ◽  
Straight Lines ◽  
Bounded Sets ◽  
Infinite Set ◽  
Locally Convex

Let be a family of sets in a linear space X. A hyperplane π is called a k-secant of if π intersects exactly k members of . The existence of k-secants for families of compact sets in linear topological spaces has been discussed in a number of recent papers (cf. [3–7]). For X normed (and a finite family of two or more disjoint non-empty compact sets) it was proved [5] that if the union of all members of is an infinite set which is not contained in any straight line of X, then has a 2-secant. This result and related ones concerning intersections of members of by straight lines have since been extended in [4] to the more general setting of a Hausdorff locally convex space.

scholarly journals Sumsets of Finite Beatty Sequences

10.37236/1614 ◽  
2000 ◽  
Vol 8 (2) ◽  
Author(s):  
Jane Pitman
Keyword(s):  
Arithmetic Progression ◽  
Continued Fractions ◽  
Sturmian Word ◽  
Beatty Sequences ◽  
Sturmian Words ◽  
Beatty Sequence

An investigation of the size of $S+S$ for a finite Beatty sequence $S=(s_i)=(\lfloor i\alpha+\gamma \rfloor)$, where $\lfloor \hphantom{x} \rfloor$ denotes "floor", $\alpha$, $\gamma$ are real with $\alpha\ge 1$, and $0\le i \le k-1$ and $k\ge 3$. For $\alpha>2$, it is shown that $|S+S|$ depends on the number of "centres" of the Sturmian word $\Delta S=(s_i-s_{i-1})$, and hence that $3(k-1)\le |S+S|\le 4k-6$ if $S$ is not an arithmetic progression. A formula is obtained for the number of centres of certain finite periodic Sturmian words, and this leads to further information about $|S+S|$ in terms of finite nearest integer continued fractions.

scholarly journals Entropy and variational principle for one-dimensional lattice systems with a generala prioriprobability: positive and zero temperature

2014 ◽  
Vol 35 (6) ◽  
pp. 1925-1961 ◽  
Author(s):  
A. O. LOPES ◽  
J. K. MENGUE ◽  
J. MOHR ◽  
R. R. SOUZA
Keyword(s):  
Infinite Product ◽  
A Priori ◽  
Transfer Operator ◽  
Thermodynamic Formalism ◽  
General Setting ◽  
Lattice Systems ◽  
Ruelle Operator ◽  
Finite Set ◽  
Zero Entropy ◽  
Infinite Set

We generalize several results of the classical theory of thermodynamic formalism by considering a compact metric space$M$as the state space. We analyze the shift acting on$M^{\mathbb{N}}$and consider a generala prioriprobability for defining the transfer (Ruelle) operator. We study potentials$A$which can depend on the infinite set of coordinates in$M^{\mathbb{N}}$. We define entropy and by its very nature it is always a non-positive number. The concepts of entropy and transfer operator are linked. If$M$is not a finite set there exist Gibbs states with arbitrary negative value of entropy. Invariant probabilities with support in a fixed point will have entropy equal to minus infinity. In the case$M=S^{1}$, and thea priorimeasure is Lebesgue$dx$, the infinite product of$dx$on$(S^{1})^{\mathbb{N}}$will have zero entropy. We analyze the Pressure problem for a Hölder potential$A$and its relation with eigenfunctions and eigenprobabilities of the Ruelle operator. Among other things we analyze the case where temperature goes to zero and we show some selection results. Our general setting can be adapted in order to analyze the thermodynamic formalism for the Bernoulli space with countable infinite symbols. Moreover, the so-called$XY$model also fits under our setting. In this last case$M$is the unitary circle$S^{1}$. We explore the differentiable structure of$(S^{1})^{\mathbb{N}}$by considering a certain class of smooth potentials and we show some properties of the corresponding main eigenfunctions.

scholarly journals Words with many Palindrome Pair Factors

10.37236/5583 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Adam Borchert ◽  
Narad Rampersad
Keyword(s):  
Infinite Words ◽  
Sturmian Word ◽  
Proper Factor ◽  
Sturmian Words

Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many $n$, every length-$n$ factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue—Morse word does not have this property. We investigate finite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair.

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