Related papers: Standard Lyndon loop words: weighted orders
We give a necessary and sufficient PBW basis criterion for Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great…
A new representation of Quantum Gravity is developed. This formulation is based on an extension of the group of loops. The enlarged group, that we call the Extended Loop Group, behaves locally as an infinite dimensional Lie group. Quantum…
A codeword is associated to a linearized polynomial. The weight distribution of the codewords is determined as the linearized polynomial varies in a family of fixed degree. There is a corresponding result on Wenger graphs from linearized…
Using the notion of generalized weight we improve estimates on the parameters of quantum codes obtained by Steane's construction from binary codes. This yields several new families of quantum codes.
Motivated by applications to string processing, we introduce variants of the Lyndon factorization called inverse Lyndon factorizations. Their factors, named inverse Lyndon words, are in a class that strictly contains anti-Lyndon words, that…
The notion of a $p$-Riordan graph generalizes that of a Riordan graph, which, in turn, generalizes the notions of a Pascal graph and a Toeplitz graph. In this paper we introduce the notion of a $p$-Riordan word, and show how to encode…
Originally motivated by questions of P. Etingof related to growth rates of tensor powers in symmetric tensor categories, we obtain general bounds on the order of finite subgroups of ${\rm GL}(n,\mathbb{C})$ with restricted composition…
In this paper, we extend the notion of Lyndon word to transfinite words. We prove two main results. We first show that, given a transfinite word, there exists a unique factorization in Lyndon words that are densely non-increasing, a…
The coinvariant algebra is a quotient of the polynomial ring $\mathbb{Q}[x_1,\ldots,x_n]$ whose algebraic properties are governed by the combinatorics of permutations of length $n$. A word $w = w_1 \dots w_n$ over the positive integers is…
Let $X= \{x_1, x_2, \cdots, x_n\}$ be a finite alphabet, and let $K$ be a field. We study classes $\mathfrak{C}(X, W)$ of graded $K$-algebras $A = K\langle X\rangle / I$, generated by $X$ and with a fixed set of obstructions $W$. Initially…
We study the asymptotics and fine-scale behavior of quantitative combinatorial measures of infinite words and related dynamical and algebraic structures. We construct infinite recurrent words $w$ whose complexity functions $p_w(n)$ are…
Recently, a new characterization of Lyndon words that are also perfectly clustering was proposed by Lapointe and Reutenauer (2024). A word over a ternary alphabet {a,b,c} is called perfectly clustering Lyndon if and only if it is the…
This paper describes experiments showing that some tasks in natural language processing (NLP) can already be performed using quantum computers, though so far only with small datasets. We demonstrate various approaches to topic…
In this paper we consider the normalized lengths of the factors of some factorizations of random words. First, for the \emph{Lyndon factorization} of finite random words with $n$ independent letters drawn from a finite or infinite totally…
The class of input-output systems representable as Chen-Fliess series arises often in control theory. One well known drawback of this representation, however, is that the iterated integrals which appear in these series are algebraically…
Universal enveloping algebras of braided m-Lie algebras and PBW theorem are obtained by means of combinatorics on words.
Firstly, we give a formula on the generalized Hamming weight of linear codes constructed generically by defining sets. Secondly, by choosing properly the defining set we obtain a class of cyclotomic linear codes and then present two…
We study combinatorial Dyson Schwinger equations, expressed in the Hopf algebra of words with a quasi shuffle product. We map them into an algebra of polynomials in one indeterminate L and show that the leading log expansion one obtains…
A simple model is constructed which allows to compute modified dispersion relations with effects from loop quantum gravity. Different quantization choices can be realized and their effects on the order of corrections studied explicitly. A…
A combinatorial formula of G.-C. Rota and J.A. Stein is taken to perform Wick re-ordering in quantum field theory. Wick's theorem becomes a Hopf algebraic identity called Cliffordization. The combinatorial method relying on Hopf algebras is…