Related papers: Christoffel words and Markoff triples
Diophantine quadruples are sets of four distinct positive integers such that the product of any two is one less than a square. All known examples belong to an infinite set which can be constructed recursively. Some observations on these…
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…
To give a parametrization of the Diophantine equation $A^{3}+B^{3}=C^{3}+D^{3}$ in terms of integral binary quadratic forms in a constructive way.
Among a triangle's exparabolas (parabolas escribed to the triangle), three are distinguished by having locally maximal parameter. They are determined by a simple cubic equation and characterized by having axes that contain the triangle's…
Prime factorization is an outstanding problem in arithmetic, with important consequences in a variety of fields, most notably cryptography. Here we employ the intriguing analogy between prime factorization and optical interferometry in…
Certain upper triangular matrices, termed as Parikh matrices, are often used in the combinatorial study of words. Given a word, the Parikh matrix of that word elegantly computes the number of occurrences of certain predefined subwords in…
We give a parametrization of test configurations in the sense of Donaldson via spherical buildings, and show the existence of "optimal" destabilizing test configurations for unstable varieties, in the wake of Mumford and Kempf. We also give…
The interplay between bifurcations and random switching processes of vector fields is studied. More precisely, we provide a classification of piecewise deterministic Markov processes arising from stochastic switching dynamics near fold,…
The described tagger is based on a hidden Markov model and uses tags composed of features such as part-of-speech, gender, etc. The contextual probability of a tag (state transition probability) is deduced from the contextual probabilities…
The data on 13 typologically different languages have been processed using a two-parameter word length model, based on 1-displaced uniform Poisson distribution. Statistical dependencies of the 2nd parameter on the 1st one are revealed for…
We present an explicit description of the Hopf 2-cocycles involved in the classification of finite dimensional pointed and copointed Hopf algebras over the symmetric group on three letters. We determine which cocycles are exponentials of…
We consider the previously defined notion of finite-state independence and we focus specifically on normal words. We characterize finite-state independence of normal words in three different ways, using three different kinds of asynchronous…
We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning…
We investigate the extent to which Linear Temporal Logic (LTL) formulas can be uniquely characterized by a finite set of labeled examples. We consider different types of examples, ranging from finite words to transfinite words, as well as…
We consider the Markoff-Rosenberger equation $$ax^2+by^2+cz^2=dxyz$$ with $(x,y,z)=(U_i,U_j,U_k),$ where $U_i$ denotes the $i$-th generalized Lucas number of first/second kind. We provide upper bound for the minimum of the indices and we…
Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its…
In this work we survey on connections of Markov chains and the theory of multiple orthogonality. Here we mainly concentrate on give a procedure to generate stochastic tetra diagonal Hessenberg matrices, coming from some specific families of…
We classify non-factorial nodal Fano threefolds with $1$ node and class group of rank $2$.
Four integer parametrizations for the bi-orthogonal monoclinic Diophantine parallelepiped are given.
Multivariate orthogonal polynomials can be introduced by using a moment functional defined on the linear space of polynomials in several variables with real coefficients. We study the so-called Uvarov and Christoffel modifications obtained…