Pairwise Well-Formed Modes and Transformations

One of the most significant attitudinal shifts in the history of music occurred in the Renaissance, when an emerging triadic consciousness moved musicians towards a new scalar formation that placed major thirds on a par with perfect fifths. In this paper we revisit the confrontation between the two idealized scalar and modal conceptions, that of the ancient and medieval world and that of the early modern world, associated especially with Zarlino. We do this at an abstract level, in the language of algebraic combinatorics on words. In scale theory the juxtaposition is between well-formed and pairwise well-formed scales and modes, expressed in terms of Christoffel words or standard words and their conjugates, and the special Sturmian morphisms that generate them. Pairwise well-formed scales are encoded by words over a three-letter alphabet, and in our generalization we introduce special positive automorphisms of $F3$, the free group over three letters.Comment: 12 pages, 3 figures, paper presented at the MCM2017 at UNAM in Mexico City on June 27, 2017, keywords: pairwise well-formed scales and modes, well-formed scales and modes, well-formed words, Christoffel words, standard words, central words, algebraic combinatorics on words, special Sturmian morphism

An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications

We provide a new algorithm for generating the Baker--Campbell--Hausdorff (BCH) series Z = \log(\e^X \e^Y) in an arbitrary generalized Hall basis of the free Lie algebra $\mathcal{L}(X,Y)$ generated by $X$ and $Y$. It is based on the close relationship of $\mathcal{L}(X,Y)$ with a Lie algebraic structure of labeled rooted trees. With this algorithm, the computation of the BCH series up to degree 20 (111013 independent elements in $\mathcal{L}(X,Y)$) takes less than 15 minutes on a personal computer and requires 1.5 GBytes of memory. We also address the issue of the convergence of the series, providing an optimal convergence domain when $X$ and $Y$ are real or complex matrices.Comment: 30 page

Subexponential estimations in Shirshov's height theorem (in English)

In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: "Suppose that F_{2, m} is a 2-generated associative ring with the identity x^m=0. Is it true, that the nilpotency degree of F_{2, m} has exponential growth?" We show that the nilpotency degree of l-generated associative algebra with the identity x^d=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2^{18} l (nd)^{3 log_3 (nd)+13}d^2. We give the definitive answer to E. I. Zelmanov by this result. It is the consequence of one fact, which is based on combinatorics of words. Let l, n and d>n be positive integers. Then all the words over alphabet of cardinality l which length is greater than Psi(n,d,l) are either n-divided or contain d-th power of subword, where a word W is n-divided, if it can be represented in the following form W=W_0 W_1...W_n such that W_1 >' W_2>'...>'W_n. The symbol >' means lexicographical order here. A. I. Shirshov proved that the set of non n-divided words over alphabet of cardinality l has bounded height h over the set Y consisting of all the words of degree <n. Original Shirshov's estimation was just recursive, in 1982 double exponent was obtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation. We show, that h<Phi(n,l), where Phi(n,l) = 2^{87} n^{12 log_3 n + 48} l. Our proof uses Latyshev idea of Dilworth theorem application.Comment: 21 pages, Russian version of the article is located at the link arXiv:1101.4909; Sbornik: Mathematics, 203:4 (2012), 534 -- 55

The complexity of tangent words

In a previous paper, we described the set of words that appear in the coding of smooth (resp. analytic) curves at arbitrary small scale. The aim of this paper is to compute the complexity of those languages.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Computer aided synthesis: a game theoretic approach

In this invited contribution, we propose a comprehensive introduction to game theory applied in computer aided synthesis. In this context, we give some classical results on two-player zero-sum games and then on multi-player non zero-sum games. The simple case of one-player games is strongly related to automata theory on infinite words. All along the article, we focus on general approaches to solve the studied problems, and we provide several illustrative examples as well as intuitions on the proofs.Comment: Invitation contribution for conference "Developments in Language Theory" (DLT 2017

Certain Distance-Based Topological Indices of Parikh Word Representable Graphs

Relating graph structures with words which are finite sequences of symbols, Parikh word representable graphs ($PWRGs$) were introduced. On the other hand in chemical graph theory, graphs have been associated with molecular structures. Also several topological indices have been defined in terms of graph parameters and studied for different classes of graphs. In this paper, we derive expressions for computing certain topological indices of $PWRGs$ of binary core words, thereby enriching the study of $PWRGs.

Computing the kk-binomial complexity of the Thue--Morse word

Two words are $k$-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most $k$ with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The $k$-binomial complexity of an infinite word $\mathbf{x}$ maps the integer $n$ to the number of classes in the quotient, by this $k$-binomial equivalence relation, of the set of factors of length $n$ occurring in $\mathbf{x}$. This complexity measure has not been investigated very much. In this paper, we characterize the $k$-binomial complexity of the Thue--Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue--Morse word is aperiodic, its $k$-binomial complexity eventually takes only two values. In this paper, we first obtain general results about the number of occurrences of subwords appearing in iterates of the form $\Psi^\ell(w)$ for an arbitrary morphism $\Psi$. We also thoroughly describe the factors of the Thue--Morse word by introducing a relevant new equivalence relation

On the Language of Standard Discrete Planes and Surfaces

International audienceA standard discrete plane is a subset of Z^3 verifying the double Diophantine inequality mu =< ax+by+cz < mu + omega, with (a,b,c) != (0,0,0). In the present paper we introduce a generalization of this notion, namely the (1,1,1)-discrete surfaces. We first study a combinatorial representation of discrete surfaces as two-dimensional sequences over a three-letter alphabet and show how to use this combinatorial point of view for the recognition problem for these discrete surfaces. We then apply this combinatorial representation to the standard discrete planes and give a first attempt of to generalize the study of the dual space of parameters for the latter [VC00]

The role of RAS oncogene in survival of patients with lung cancer: a systematic review of the literature with meta-analysis

The proto-oncogene RAS, coding for a 21 kDa protein (p21), is mutated in 20% of lung cancer. However, the literature remains controversial on its prognostic significance for survival in lung cancer. We performed a systematic review of the literature with meta-analysis to assess its possible prognostic value on survival. Published studies on lung cancer assessing prognostic value of RAS mutation or p21 overexpression on survival were identified by an electronic search. After a methodological assessment, we estimated individual hazard ratios (HR) estimating RAS protein alteration or RAS mutation effect on survival and combined them using meta-analytic methods. In total, 53 studies were found eligible, with 10 concerning the same cohorts of patients. Among the 43 remaining studies, the revelation method was immunohistochemistry (IHC) in nine and polymerase chain reaction (PCR) in 34. Results in terms of survival were significantly pejorative, significantly favourable, not significant and not conclusive in 9, 1, 31, 2, respectively. In total, 29 studies were evaluable for meta-analysis but we aggregated only the 28 dealing with non-small-cell lung cancer (NSCLC) and not the only one dealing with small-cell-lung cancer (SCLC). The quality scores were not statistically significantly different between studies with or without significant results in terms of survival, allowing us to perform a quantitative aggregation. The combined HR was 1.35 (95% CI: 1.16–1.56), showing a worse survival for NSCLC with KRAS2 mutations or p21 overexpression and, particularly, in adenocarcinomas (ADC) (HR 1.59; 95% CI 1.26–2.02) and in studies using PCR (HR 1.40; 95% CI 1.18–1.65) but not in studies using IHC (HR 1.08; 95% CI 0.86–1.34). RAS appears to be a pejorative prognostic factor in terms of survival in NSCLC globally, in ADC and when it is studied by PCR

Ki-67 expression and patients survival in lung cancer: systematic review of the literature with meta-analysis

Among new biological markers that could become useful prognostic factors for lung carcinoma, Ki-67 is a nuclear protein involved in cell proliferation regulation. Some studies have suggested an association between Ki-67 and poor survival in lung cancer patients. In order to clarify this point, we have performed a systematic review of the literature, using the methodology already described by our Group, the European Lung Cancer Working Party. In total, 37 studies, including 3983 patients, were found to be eligible. In total, 49% of the patients were considered as having a tumour positive for the expression of Ki-67 according to the authors cutoff. In all, 29 of the studies dealt with non-small-cell lung carcinoma (NSCLC), one with small-cell carcinoma (SCLC), two with carcinoid tumours and five with any histology. In terms of survival results, Ki-67 was a bad prognosis factor for survival in 15 studies while it was not in 22. As there was no statistical difference in quality scores between the significant and nonsignificant studies evaluable for the meta-analysis, we were allowed to aggregate the survival results. The combined hazard ratio for NSCLC, calculated using a random-effects model was 1.56 (95% CI: 1.30-1.87), showing a worse survival when Ki-67 expression is increased. In conclusion, our meta-analysis shows that the expression of Ki-67 is a factor of poor prognosis for survival in NSCLC.Journal ArticleMeta-AnalysisResearch Support, Non-U.S. Gov'tSCOPUS: re.jinfo:eu-repo/semantics/publishe