Related papers: Christoffel words and Markoff triples
We find the two-variables generating function for the statistic which counts the number of variations in a word bounded by $1$. Thus, we refine and extend previous results concerning staircase words, which are words in which the variation…
Global Markov properties in mixed graphs are usually formulated in terms of the path-oriented m-separation or by use of augmented graphs (similar to moral graphs in the case of directed acyclic graphs). We provide an alternative…
An alternative expression for the Christoffel--Darboux formula for multiple orthogonal polynomials of mixed type is derived from the $LU$ factorization of the moment matrix of a given measure and two sets of weights. We use the action of…
A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating…
Mixed orthogonal Laurent polynomials on the unit circle of CMV type are constructed utilizing a matrix of moments and its Gauss--Borel factorization and employing a multiple extension of the CMV ordering. A systematic analysis of the…
We propose an equivalent condition for the Markov triples, which was mentioned by H. Rademacher essentially. As an application, we study the solvability of the Diophantine equation $a^2+b^2+c^2=abcf(a,b,c)$.
An analysis of Wolfenstein parametrization for the Kobayashi-Maskawa matrix shows that it has a serious flaw: it depends on {\em three} independent parameters instead of {\em four} as it should be. Because this approximation is currently…
Darboux transformations for polynomial perturbations of a real multivariate measure are found. The 1D Christoffel formula is extended to the multidimensional realm: multivariate orthogonal polynomials are expressed in terms of last…
We pursue the investigation of generalizations of the Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. The finite…
A parametrization of the Kasner indices in terms of a continuous parameter is constructed by exploiting their representation as trilinear coordinates. This provides a clear picture of their variation through their entire range vis a vis…
We study the properties of the uncountable set of Stewart words. These are Toeplitz words specified by infinite sequences of Toeplitz patterns of the form $\alpha\beta\gamma$, where $\alpha,\beta,\gamma$ is any permutation of the symbols…
Monolingual word alignment is important for studying fine-grained editing operations (i.e., deletion, addition, and substitution) in text-to-text generation tasks, such as paraphrase generation, text simplification, neutralizing biased…
In this paper, we consider the Dirichlet problem associated to an elliptic Kirchhoff-type equation depending on two parameters. Under rather general and natural assumptions, we prove that, for certain values of the parameters, the problem…
To enumerate 3-manifold triangulations with a given property, one typically begins with a set of potential face pairing graphs (also known as dual 1-skeletons), and then attempts to flesh each graph out into full triangulations using an…
Conditional independence and Markov properties are powerful tools allowing expression of multidimensional probability distributions by means of low-dimensional ones. As multidimensional possibilistic models have been studied for several…
We describe estimators $\chi_n(X_0,X_1,...,X_n)$, which when applied to an unknown stationary process taking values from a countable alphabet ${\cal X}$, converge almost surely to $k$ in case the process is a $k$-th order Markov chain and…
We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of "the CF…
We find generating functions for the number of words avoiding certain patterns or sets of patterns on at most 2 distinct letters and determine which of them are equally avoided. We also find the exact number of words avoiding certain…
We introduce partial Markov categories as a synthetic framework for synthetic probabilistic inference, blending the work of Cho and Jacobs, Fritz, and Golubtsov on Markov categories with the work of Cockett and Lack on cartesian restriction…
Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation. We investigate asymptotic properties of geometric progressions modulo one and observe significantly analogous results on the set \[ {\mathcal…