Related papers: Continued fractions, modular symbols, and non-comm…
In this work we generalize the concept of injective module and develop a theory of divisibility for modules over a general ring, which provides a general and unified framework to study Kummer-like field extensions arising from commutative…
We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit…
In this article we continue a previous work in which we have generalized the Rogers Ramanujan continued fraction (RR) introducing what we call, the Ramanujan-Quantities (RQ). We use the Mathematica package to give several modular equations…
We investigate two Stieltjes continued fractions given by the paperfolding sequence and the Rudin-Shapiro sequence. By explicitly describing certain subsequences of the convergents $P_n(x)/Q_n(x)$ modulo $4$, we give the formal power series…
We develop a noncommutative invariant theory for ordinary linear differential operators on Riemann surfaces. For a monic binomially normalized operator $L=\sum_{k=0}^n {n\choose k}a_kD^{\,n-k}$, $a_0=1$, with coefficients in an associative…
Multidimensional Continued Fraction Algorithms are generalizations of the Euclid algorithm and find iteratively the gcd of two or more numbers. They are defined as linear applications on some subcone of $\mathbb{R}^d$. We consider…
Let $F$ be a totally real number field. We prove that a character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely…
Consider the sequence of continued fraction convergents $p_n/q_n$ to a random irrational number. We study the distribution of the sequences $p_n \pmod{m}$ and $q_n \pmod{m}$ with a fixed modulus $m$, and more generally, the distribution of…
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization…
Bayesian nonparametric mixture models are common for modeling complex data. While these models are well-suited for density estimation, recent results proved posterior inconsistency of the number of clusters when the true number of…
We study the involution of the real line induced by the outer automorphism of the extended modular group PGL(2,Z). This `modular' involution is discontinuous at rationals but satisfies a surprising collection of functional equations. It…
The quantum modularity conjecture, first introduced by Don Zagier, is a general statement about a relation between $\mathfrak{sl}_2$ quantum invariants of links and 3-manifolds at roots of unity related by a modular transformation. In this…
In this paper we develop a functorial language of probabilistic morphisms and apply it to some basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic morphisms proposed by Lawvere and Giry…
For a complex polynomial $D(t)$ of even degree, one may define the continued fraction of $\sqrt{D(t)}$. This was found relevant already by Abel in 1826, and later by Chebyshev, concerning integration of (hyperelliptic) differentials; they…
Let G be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible…
We extend the close interplay between continued fractions, orthogonal polynomials, and Gaussian quadrature rules to several variables in a special but natural setting which we characterize in terms of moment sequences. The crucial condition…
We interpret certain Seiberg-like dualities of two-dimensional N=(2,2) quiver gauge theories with unitary groups as cluster mutations in cluster algebras, originally formulated by Fomin and Zelevinsky. In particular, we show how the…
We consider an interval map which is a generalization of the R\'enyi transformation. For the continued fraction expansion arising from this transformation, we prove a result concerning the asymptotic behavior of the distribution functions…
In his influential paper on quantum modular forms, Zagier developed a conjectural framework describing the behavior of certain quantum knot invariants under the action of the modular group on their arguments. More precisely, when $J_{K,0}$…
We attempt to investigate a two-dimensional Gauss-Kuzmin theorem for R\'enyi-type continued fraction expansions. More precisely speaking, our focus is to obtain specific lower and upper bounds for the error term considered which imply the…