A note on the γ\gamma-coefficients of the "tree Eulerian polynomial"

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. B. Drake proved that this polynomial factors completely over the integers. From his product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these positive coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of the author and Wachs related to the poset of weighted partitions and the free multibracketed Lie algebra.Comment: 13 pages, 6 figures, Interpretations derived from results in arXiv:1309.5527 and arXiv:1408.541

A Lie algebra related to the universal Askey-Wilson algebra

Let $\mathbb{F}$ denote an algebraically closed field. Denote the three-element set by $\mathcal{X}=\{A,B,C\}$, and let \mathbb{F}\left denote the free unital associative $\mathbb{F}$-algebra on $\mathcal{X}$. Fix a nonzero $q\in\mathbb{F}$ such that $q^4\neq 1$. The universal Askey-Wilson algebra $\Delta$ is the quotient space \mathbb{F}\left/\mathbb{I}, where $\mathbb{I}$ is the two-sided ideal of \mathbb{F}\left generated by the nine elements $UV-VU$, where $U$ is one of $A,B,C$, and $V$ is one of \begin{equation} (q+q^{-1}) A+\frac{qBC-q^{-1}CB}{q-q^{-1}},\nonumber \end{equation} \begin{equation} (q+q^{-1}) B+\frac{qCA-q^{-1}AC}{q-q^{-1}},\nonumber \end{equation} \begin{equation} (q+q^{-1}) C+\frac{qAB-q^{-1}BA}{q-q^{-1}}.\nonumber \end{equation} Turn \mathbb{F}\left into a Lie algebra with Lie bracket $\left[ X,Y\right] = XY-YX$ for all X,Y\in\mathbb{F}\left. Let $\mathcal{L}$ denote the Lie subalgebra of \mathbb{F}\left generated by $\mathcal{X}$, which is also the free Lie algebra on $\mathcal{X}$. Let $L$ denote the Lie subalgebra of $\Delta$ generated by $A,B,C$. Since the given set of defining relations of $\Delta$ are not in $\mathcal{L}$, it is natural to conjecture that $L$ is freely generated by $A,B,C$. We give an answer in the negative by showing that the kernel of the canonical map \mathbb{F}\left\rightarrow\Delta has a nonzero intersection with $\mathcal{L}$. Denote the span of all Hall basis elements of $\mathcal{L}$ of length $n$ by $\mathcal{L}_n$, and denote the image of $\sum_{i=1}^n\mathcal{L}_i$ under the canonical map $\mathcal{L}\rightarrow L$ by $L_n$. We study some properties of $L_4$ and $L_5$

Lie polynomials in an algebra defined by a linearly twisted commutation relation

We present an elementary approach in characterizing Lie polynomials in the generators $A,B$ of an algebra with a defining relation that is in the form of a deformed or twisted commutation relation $AB=\sigma(BA)$ where the deformation or twisting map $\sigma$ is a linear polynomial with a slope parameter that is not a root of unity. The class of algebras defined as such encompasses $q$-deformed Heisenberg algebras, rotation algebras, and some types of $q$-oscillator algebras whose deformation parameters are not roots of unity, and so we have a general solution for the Lie polynomial characterization problem for these algebras

Pop III GRBs: an estimative of the event rate for future surveys

We discuss the theoretical event rate of gamma-ray bursts (GRBs) from the collapse of massive primordial stars. We construct a theoretical model to calculate the rate and detectability of these GRBs taking into account all important feedback and recent results from numerical simulations of pristine gas. We expect to observe a maximum of N $\lesssim$ 0.2 GRBs per year integrated over at z > 6 with \textit{Swift} and N $\lesssim$ 10 GRBs per year integrated over at z > 6 with EXIST.Comment: 6 pages, 2 figures, published in Proceedings of the Gamma-Ray Bursts 2012 Conference (GRB 2012

Explosive synchronization with partial degree-frequency correlation

Networks of Kuramoto oscillators with a positive correlation between the oscillators frequencies and the degree of the their corresponding vertices exhibits the so-called explosive synchronization behavior, which is now under intensive investigation. Here, we study and report explosive synchronization in a situation that has not yet been considered, namely when only a part, typically small, of the vertices is subjected to a degree frequency correlation. Our results show that in order to have explosive synchronization, it suffices to have degree-frequency correlations only for the hubs, the vertices with the highest degrees. Moreover, we show that a partial degree-frequency correlation does not only promotes but also allows explosive synchronization to happen in networks for which a full degree-frequency correlation would not allow it. We perform exhaustive numerical experiments for synthetic networks and also for the undirected and unweighted version of the neural network of the worm Caenorhabditis elegans. The latter is an explicit example where partial degree-frequency correlation leads to explosive synchronization with hysteresis, in contrast with the fully correlated case, for which no explosive synchronization is observed.Comment: 10 pages, 6 figures, final version to appear in PR

Optimal synchronization of Kuramoto oscillators: a dimensional reduction approach

A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronization of a Kuramoto network demands the maximization of the quadratic function $\omega^T L \omega$, where $\omega$ stands for the vector of the natural frequencies of the oscillators, and $L$ for the network Laplacian matrix. Many recently obtained numerical results can be re-obtained analytically and in a simpler way from our maximization condition. A computationally efficient {hill climb} rewiring algorithm is proposed to generate networks with optimal synchronization properties. Our approach can be easily adapted to the case of the Kuramoto models with both attractive and repulsive interactions, and again many recent numerical results can be rederived in a simpler and clearer analytical manner.Comment: 6 pages, 6 figures, final version to appear in PR