69,470 research outputs found
Rich, Sturmian, and trapezoidal words
In this paper we explore various interconnections between rich words,
Sturmian words, and trapezoidal words. Rich words, first introduced in
arXiv:0801.1656 by the second and third authors together with J. Justin and S.
Widmer, constitute a new class of finite and infinite words characterized by
having the maximal number of palindromic factors. Every finite Sturmian word is
rich, but not conversely. Trapezoidal words were first introduced by the first
author in studying the behavior of the subword complexity of finite Sturmian
words. Unfortunately this property does not characterize finite Sturmian words.
In this note we show that the only trapezoidal palindromes are Sturmian. More
generally we show that Sturmian palindromes can be characterized either in
terms of their subword complexity (the trapezoidal property) or in terms of
their palindromic complexity. We also obtain a similar characterization of rich
palindromes in terms of a relation between palindromic complexity and subword
complexity.Comment: 7 page
Enumeration and Structure of Trapezoidal Words
Trapezoidal words are words having at most distinct factors of length
for every . They therefore encompass finite Sturmian words. We give
combinatorial characterizations of trapezoidal words and exhibit a formula for
their enumeration. We then separate trapezoidal words into two disjoint
classes: open and closed. A trapezoidal word is closed if it has a factor that
occurs only as a prefix and as a suffix; otherwise it is open. We investigate
open and closed trapezoidal words, in relation with their special factors. We
prove that Sturmian palindromes are closed trapezoidal words and that a closed
trapezoidal word is a Sturmian palindrome if and only if its longest repeated
prefix is a palindrome. We also define a new class of words, \emph{semicentral
words}, and show that they are characterized by the property that they can be
written as , for a central word and two different letters .
Finally, we investigate the prefixes of the Fibonacci word with respect to the
property of being open or closed trapezoidal words, and show that the sequence
of open and closed prefixes of the Fibonacci word follows the Fibonacci
sequence.Comment: Accepted for publication in Theoretical Computer Scienc
On prefixal factorizations of words
We consider the class of all infinite words over
a finite alphabet admitting a prefixal factorization, i.e., a factorization
where each is a non-empty prefix of With
each one naturally associates a "derived" infinite word
which may or may not admit a prefixal factorization. We are
interested in the class of all words of
such that for all . Our primary
motivation for studying the class stems from its connection
to a coloring problem on infinite words independently posed by T. Brown in
\cite{BTC} and by the second author in \cite{LQZ}. More precisely, let be the class of all words such that for every finite
coloring there exist and a factorization
with for each In \cite{DPZ}
we conjectured that a word if and only if is purely
periodic. In this paper we show that so
in other words, potential candidates to a counter-example to our conjecture are
amongst the non-periodic elements of We establish several
results on the class . In particular, we show that a
Sturmian word belongs to if and only if is
nonsingular, i.e., no proper suffix of is a standard Sturmian word
On Christoffel and standard words and their derivatives
We introduce and study natural derivatives for Christoffel and finite
standard words, as well as for characteristic Sturmian words. These
derivatives, which are realized as inverse images under suitable morphisms,
preserve the aforementioned classes of words. In the case of Christoffel words,
the morphisms involved map to (resp.,~) and to
(resp.,~) for a suitable . As long as derivatives are
longer than one letter, higher-order derivatives are naturally obtained. We
define the depth of a Christoffel or standard word as the smallest order for
which the derivative is a single letter. We give several combinatorial and
arithmetic descriptions of the depth, and (tight) lower and upper bounds for
it.Comment: 28 pages. Final version, to appear in TC
The sequence of open and closed prefixes of a Sturmian word
A finite word is closed if it contains a factor that occurs both as a prefix
and as a suffix but does not have internal occurrences, otherwise it is open.
We are interested in the {\it oc-sequence} of a word, which is the binary
sequence whose -th element is if the prefix of length of the word is
open, or if it is closed. We exhibit results showing that this sequence is
deeply related to the combinatorial and periodic structure of a word. In the
case of Sturmian words, we show that these are uniquely determined (up to
renaming letters) by their oc-sequence. Moreover, we prove that the class of
finite Sturmian words is a maximal element with this property in the class of
binary factorial languages. We then discuss several aspects of Sturmian words
that can be expressed through this sequence. Finally, we provide a linear-time
algorithm that computes the oc-sequence of a finite word, and a linear-time
algorithm that reconstructs a finite Sturmian word from its oc-sequence.Comment: Published in Advances in Applied Mathematics. Journal version of
arXiv:1306.225
Some characterizations of Sturmian words in terms of the lexicographic order
In this paper we present three new characterizations of Sturmian words based
on the lexicographic ordering of their factors
Hund's metals, explained
A possible practical definition for a Hund's metal is given, as a metallic
phase - arising consistently in realistic simulations and experiments in
Fe-based superconductors and other materials - with three features: large
electron masses, high-spin local configurations dominating the paramagnetic
fluctuations and orbital-selective correlations. These features are triggered
by, and increase with the proximity to, a Hund's coupling-favored Mott
insulator that is realized for half-filled conduction bands. A clear crossover
line is found where these three features get enhanced, departing from the Mott
transition at half filling and extending in the interaction/doping plane,
between a normal (at weak interaction and large doping) and a Hund's metal (at
strong interaction and small doping). This phenomenology is found identically
in models with featureless bands, highlighting the generality of this physics
and its robustness by respect to the details of the material band structures.
Some analytical arguments are also given to gain insight into these defining
features. Finally the attention is brought on the recent theoretical finding of
enhanced/diverging electronic compressibility near the Hund's metal crossover,
pointing to enhanced quasiparticle interactions that can cause or boost
superconductivity or other instabilities.Comment: Lecture prepared for the Autumn School on Correlated Electrons, 25-29
September 2017, Juelich. To appear on: E. Pavarini, E. Koch, R. Scalettar,
and R. Martin (eds.) The Physics of Correlated Insulators, Metals, and
Superconductors Modeling and Simulation Vol. 7 Forschungszentrum Juelich,
2017, ISBN 978-3-95806-224-5 http://www.cond- mat.de/events/correl1
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