2,701 research outputs found
Multivariate Fuss-Catalan numbers
Catalan numbers enumerate binary trees and
Dyck paths. The distribution of paths with respect to their number of
factors is given by ballot numbers .
These integers are known to satisfy simple recurrence, which may be visualised
in a ``Catalan triangle'', a lower-triangular two-dimensional array. It is
surprising that the extension of this construction to 3 dimensions generates
integers that give a 2-parameter distribution of , which may be called order-3 Fuss-Catalan numbers, and
enumerate ternary trees. The aim of this paper is a study of these integers
. We obtain an explicit formula and a description in terms of trees
and paths. Finally, we extend our construction to -dimensional arrays, and
in this case we obtain a -parameter distribution of , the number of -ary trees
Matricially free random variables
We show that the operatorial framework developed by Voiculescu for free
random variables can be extended to arrays of random variables whose
multiplication imitates matricial multiplication. The associated notion of
independence, called matricial freeness, can be viewed as a generalization of
both freeness and monotone independence. At the same time, the sums of
matricially free random variables, called random pseudomatrices, are closely
related to Gaussian random matrices. The main results presented in this paper
concern the standard and tracial central limit theorems for random
pseudomatrices and the corresponding limit distributions which can be viewed as
matricial generalizations of semicirle laws.Comment: 38 pages, 4 figure
Sub-Wavelength Plasmonic Crystals: Dispersion Relations and Effective Properties
We obtain a convergent power series expansion for the first branch of the
dispersion relation for subwavelength plasmonic crystals consisting of
plasmonic rods with frequency-dependent dielectric permittivity embedded in a
host medium with unit permittivity. The expansion parameter is , where is the norm of a fixed wavevector, is the period of
the crystal and is the wavelength, and the plasma frequency scales
inversely to , making the dielectric permittivity in the rods large and
negative. The expressions for the series coefficients (a.k.a., dynamic
correctors) and the radius of convergence in are explicitly related to
the solutions of higher-order cell problems and the geometry of the rods.
Within the radius of convergence, we are able to compute the dispersion
relation and the fields and define dynamic effective properties in a
mathematically rigorous manner. Explicit error estimates show that a good
approximation to the true dispersion relation is obtained using only a few
terms of the expansion. The convergence proof requires the use of properties of
the Catalan numbers to show that the series coefficients are exponentially
bounded in the Sobolev norm
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