445 research outputs found
A Note on Rate of Convergence in Probability to Semicircular Law
In the present paper, we prove that under the assumption of the finite sixth
moment for elements of a Wigner matrix, the convergence rate of its empirical
spectral distribution to the Wigner semicircular law in probability is
when the dimension tends to infinity.Comment: 13 page
Convergence of the empirical spectral distribution function of Beta matrices
Let ,
where and are two independent sample covariance
matrices with dimension and sample sizes and , respectively. This is
the so-called Beta matrix. In this paper, we focus on the limiting spectral
distribution function and the central limit theorem of linear spectral
statistics of . Especially, we do not require or
to be invertible. Namely, we can deal with the case where
and . Therefore, our results cover many important
applications which cannot be simply deduced from the corresponding results for
multivariate matrices.Comment: Published at http://dx.doi.org/10.3150/14-BEJ613 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Circle packings and total geodesic curvatures in hyperbolic background geometry
In this paper, we study a new type of circle packings in hyperbolic
background geometry. Horocycles and hypercycles are also considered in this
packing. We give the existence and rigidity of this type of circle packing with
conical singularities in terms of the total geodesic curvature. Moreover, we
introduce the combinatorial curvature flow on surfaces to find the desired
circle packing with the prescribed total geodesic curvature
Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary
This paper investigates a generalized hyperbolic circle packing (including
circles, horocycles or hypercycles) with respect to the total geodesic
curvatures on the surface with boundary. We mainly focus on the existence and
rigidity of circle packing whose contact graph is the -skeleton of a finite
polygonal cellular decomposition, which is analogous to the construction of
Bobenko and Springborn [4]. Motivated by Colin de Verdi\`ere's method [6], we
introduce the variational principle for generalized hyperbolic circle packings
on polygons. By analyzing limit behaviours of generalized circle packings on
polygons, we give an existence and rigidity for the generalized hyperbolic
circle packing with conical singularities regarding the total geodesic
curvature on each vertex of the contact graph. As a consequence, we introduce
the combinatoral Ricci flow to find a desired circle packing with a prescribed
total geodesic curvature on each vertex of the contact graph.Comment: 26 pages, 7 figure
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