Elliptic functions, continued fractions and doubled permutations

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Combinatorial families of multilabelled increasing trees and hook-length formulas

In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in the tree can get multiple labels. For all tree classes we show characterizations of suitable generating functions for the tree enumeration sequence via differential equations. Furthermore, for several combinatorial classes of multilabelled increasing tree families we present explicit enumeration results. We also present multilabelled increasing tree families of an elliptic nature, where the exponential generating function can be expressed in terms of the Weierstrass-p function or the lemniscate sine function. Furthermore, we show how to translate enumeration formulas for multilabelled increasing trees into hook-length formulas for trees and present a general "reverse engineering" method to discover hook-length formulas associated to such tree families.Comment: 37 page

Spectral analysis of non-self-adjoint Jacobi operator associated with Jacobian elliptic functions

We perform the spectral analysis of a family of Jacobi operators $J(\alpha)$ depending on a complex parameter $\alpha$. If $|\alpha|\neq1$ the spectrum of $J(\alpha)$ is discrete and formulas for eigenvalues and eigenvectors are established in terms of elliptic integrals and Jacobian elliptic functions. If $|\alpha|=1$, $\alpha \neq \pm 1$, the essential spectrum of $J(\alpha)$ covers the entire complex plane. In addition, a formula for the Weyl $m$-function as well as the asymptotic expansions of solutions of the difference equation corresponding to $J(\alpha)$ are obtained. Finally, the completeness of eigenvectors and Rodriguez-like formulas for orthogonal polynomials, studied previously by Carlitz, are proved.Comment: published version, 2 figures added; 21 pages, 3 figure

A simple algorithm for expanding a power series as a continued fraction

I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980)

Rates of mixing in models of fluid devices with discontinuities

In the simplest sense, mixing acts on an initially heterogeneous system, transforming it to a homogeneous state through the actions of stirring and diffusion. The theory of dynamical systems has been successful in improving understanding of underlying features in fluid mixing, and how smooth stirring fields, coherent structures and boundaries affect mixing rates. The main stirring mechanism in fluids at low Reynolds number is the stretching and folding of fluid elements, although this is not the only mechanism to achieve complicated dynamics. Mixing by cutting and shuffling occurs in many situations, for example in micro–fluidic split and recombine flows, through the closing and re-orientation of values in sink–source flows, and within the bulk flow of granular material. The dynamics of this mixing mechanism are subtle and not well understood. Here, mixing rates arising from fundamental models capturing the essence of discontinuous, chaotic stirring with diffusion are investigated. In purely cutting and shuffling flows it is found that the number of cuts introduced iteratively is the most important mechanism driving the approach to uniformity. A balance between cutting, shuffling and diffusion achieves a long-time exponential mixing rate, but similar mechanisms dominate the finite time mixing observed through the interaction of many slowly decaying eigenfunctions. The time to achieve a mixed condition varies polynomially with diffusivity rate κ, obeying t ∝ κ^{−η} . For the transformations meeting good stirring criteria, η < 1. Considering the time to achieve a mixed condition to be governed by a balance between cutting, shuffling, and diffusion derives η ∼ 1/2, which shows good agreement with numerical results. In stirring fields which are predominantly chaotic and exponentially mixing, it is observed that the addition of discontinuous transformations contaminates mixing when the stretching rates are uniform, or close to uniform. The contamination comes from an increase in scales of the concentration field by the reassembly of striations when cut and shuffled. Mixing stemming from this process is unpredictable, and the discontinuities destroy the possibility to approximate early mixing rates from stretching histories. A speed up in mixing rate can be achieved if the discontinuity aids particle transport into islands of the original transformation, or chops and rearranges large striations generated from highly non-uniform stretching. The long-time mixing rates and time to achieve a mixed condition are shown to behave counter-intuitively when varying the diffusivity rate. A deceleration of mixing with increasing diffusion coefficient is observed, sometimes overshooting analytically derived bounds