Enumeration and Structure of Trapezoidal Words

Trapezoidal words are words having at most $n+1$ distinct factors of length $n$ for every $n\ge 0$. 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 $uxyu$, for a central word $u$ and two different letters $x,y$. 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

Energy bounds for vertex operator algebra extensions

Let V be a simple unitary vertex operator algebra and U be a (polynomially) energy-bounded unitary subalgebra containing the conformal vector of V. We give two sufficient conditions implying that V is energy-bounded. The first condition is that U is a compact orbifold for some compact group G of unitary automorphisms of V. The second condition is that V is exponentially energy-bounded and it is a finite direct sum of simple U-modules. As consequence of the second condition, we prove that if U is a regular energy-bounded unitary subalgebra of a simple unitary vertex operator V, then $V$ is energy-bounded. In particular, every simple unitary extension (with the same conformal vector) of a simple unitary affine vertex operator algebra associated with a semisimple Lie algebra is energy-bounded.Comment: 19 page

Harmonic and gold Sturmian words

AbstractIn the combinatorics of Sturmian words an essential role is played by the set PER of all finite words w on the alphabet A={a,b} having two periods p and q which are coprime and such that |w|=p+q−2. As is well known, the set St of all finite factors of all Sturmian words equals the set of factors of PER. Moreover, the elements of PER have many remarkable structural properties. In particular, the relation Stand=A∪PER{ab,ba} holds, where Stand is the set of all finite standard Sturmian words. In this paper we introduce two proper subclasses of PER that we denote by Harm and Gold. We call an element of Harm a harmonic word and an element of Gold a gold word. A harmonic word w beginning with the letter x is such that the ratio of two periods p/q, with p<q, is equal to its slope, i.e., (|w|y+1)/(|w|x+1), where {x,y}={a,b}. A gold word is an element of PER such that p and q are primes. Some characterizations of harmonic words are given and the number of harmonic words of each length is computed. Moreover, we prove that St is equal to the set of factors of Harm and to the set of factors of Gold. We introduce also the classes Harm and Gold of all infinite standard Sturmian words having infinitely many prefixes in Harm and Gold, respectively. We prove that Gold∩Harm contain continuously many elements. Finally, some conjectures are formulated

Codes of central Sturmian words

AbstractA central Sturmian word, or simply central word, is a word having two coprime periods p and q and length equal to p+q-2. We consider sets of central words which are codes. Some general properties of central codes are shown. In particular, we prove that a non-trivial maximal central code is infinite. Moreover, it is not maximal as a code. A central code is called prefix central code if it is a prefix code. We prove that a central code is a prefix (resp., maximal prefix) central code if and only if the set of its ‘generating words’ is a prefix (resp., maximal prefix) code. A suitable arithmetization of the theory is obtained by considering the bijection θ, called ratio of periods, from the set of all central words to the set of all positive irreducible fractions defined as: θ(ε)=1/1 and θ(w)=p/q (resp., θ(w)=q/p) if w begins with the letter a (resp., letter b), p is the minimal period of w, and q=|w|-p+2. We prove that a central code X is prefix (resp., maximal prefix) if and only if θ(X) is an independent (resp., independent and full) set of fractions. Finally, two interesting classes of prefix central codes are considered. One is the class of Farey codes which are naturally associated with the Farey series; we prove that Farey codes are maximal prefix central codes. The other is given by uniform central codes. A noteworthy property related to the number of occurrences of the letter a in the words of a maximal uniform central code is proved

A Characterization of Bispecial Sturmian Words

A finite Sturmian word w over the alphabet {a,b} is left special (resp. right special) if aw and bw (resp. wa and wb) are both Sturmian words. A bispecial Sturmian word is a Sturmian word that is both left and right special. We show as a main result that bispecial Sturmian words are exactly the maximal internal factors of Christoffel words, that are words coding the digital approximations of segments in the Euclidean plane. This result is an extension of the known relation between central words and primitive Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the set of Sturmian words.Comment: Accepted to MFCS 201

Correction to: Rip current hazard assessment on a sandy beach in Liguria, NW Mediterranean

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Sea-level change and the supralittoral environment: Potential impact on a splashpool habitat on the Ligurian coast (NW Mediterranean)

Climate change represents one of the major drivers of habitat modification that is affecting a wide variety of environments. In coastal environments, great effort is being put in trying to understand and forecast the possible effects of such processes, and the Sea-Level Rise (SLR) is one of the most investigated phenomena. This paper describes the possible effects of different 2100 sea-level scenarios related to greenhouse gas mitigation policies (Representative Concentration Pathways - RCPs). This work was conducted on a supralittoral habitat situated in Genova (Ligurian Sea), and has covered an eventual change of environmental conditions driven by SLR, which might impact the Culicid Acartomyiamariae, a resident species. The wave run-up stemming from the different RCPs was simulated using the XBeach model, and to infer SLR effects on A. mariae life cycle; the results were coupled with data obtained from field surveys. The model outputs highlighted a variation in the wave run-up oscillations under common wave conditions, which might affect the supralittoral area in terms of water input and hydric balance, and the A. mariae life cycle, which is highly dependent on temperature and salinity

On the Number of Closed Factors in a Word

A closed word (a.k.a. periodic-like word or complete first return) is a word whose longest border does not have internal occurrences, or, equivalently, whose longest repeated prefix is not right special. We investigate the structure of closed factors of words. We show that a word of length $n$ contains at least $n+1$ distinct closed factors, and characterize those words having exactly $n+1$ closed factors. Furthermore, we show that a word of length $n$ can contain $\Theta(n^{2})$ many distinct closed factors.Comment: Accepted to LATA 201

A linear algorithm for string reconstruction in the reverse complement equivalence model

In the reverse complement equivalence model, it is not possible to distinguish a string from its reverse complement. We show that one can still reconstruct a string of length n, up to reverse complement, using a linear number of subsequence queries of bounded length. We first give the proof for strings over a binary alphabet, and then extend it to arbitrary finite alphabets. A simple information theoretic lower bound proves the number of queries to be asymptotically tight. Furthermore, our result is optimal w.r.t. the bound on the query length given in Erdos et al. (2006) [6]

The flavonoid compound apigenin prevents colonic inflammation and motor dysfunctions associated with high fat diet-induced obesity

When compared to SD mice, HFD animals displayed increased body weight, epididymal fat weight and metabolic indexes. HFD mice showed increments in colonic MDA, IL-1 beta and IL-6 levels, as well as a decrease in let-7f expression in both colonic and epididymal tissues. HFD mice displayed an increase in colonic eosinophil infiltration. Immunohistochemistry revealed an increase in SP and iNOS expression in myenteric ganglia of HFD mice. In preparations from HFD mice, electrically evoked contractions upon NOS blockade or mediated by tachykininergic stimulation were enhanced. In HFD mice, Apigenin counteracted the increase in body and epididymal fat weight, as well as the alterations of metabolic indexes. Apigenin reduced also MDA, IL-1 beta and IL-6 colonic levels as well as eosinophil infiltration, SP and iNOS expression, along with a normalization of electrically evoked tachykininergic and nitrergic contractions. In addition, apigenin normalized let-7f expression in epididymal fat tissues, but not in colonic specimens