18,657 research outputs found
A note on the -coefficients of the "tree Eulerian polynomial"
We consider the generating polynomial of the number of rooted trees on the
set 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 -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 -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 denote an algebraically closed field. Denote the
three-element set by , and let
\mathbb{F}\left denote the free unital associative
-algebra on . Fix a nonzero such that
. The universal Askey-Wilson algebra is the quotient space
\mathbb{F}\left/\mathbb{I}, where is the
two-sided ideal of \mathbb{F}\left generated by the nine
elements , where is one of , and 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
for all X,Y\in\mathbb{F}\left.
Let denote the Lie subalgebra of
\mathbb{F}\left generated by , which is also
the free Lie algebra on . Let denote the Lie subalgebra of
generated by . Since the given set of defining relations of
are not in , it is natural to conjecture that is
freely generated by . 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 . Denote the span of all Hall basis elements of
of length by , and denote the image of
under the canonical map
by . We study some properties of and
Lie polynomials in an algebra defined by a linearly twisted commutation relation
We present an elementary approach in characterizing Lie polynomials in the
generators of an algebra with a defining relation that is in the form of
a deformed or twisted commutation relation where the
deformation or twisting map is a linear polynomial with a slope
parameter that is not a root of unity. The class of algebras defined as such
encompasses -deformed Heisenberg algebras, rotation algebras, and some types
of -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 0.2 GRBs per year
integrated over at z > 6 with \textit{Swift} and N 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 , where stands for the vector of
the natural frequencies of the oscillators, and 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
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