8,272 research outputs found
Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie--Weiss model
Let be an exchangeable pair. Assume that
where is a dominated term and is negligible. Let
and define , where is a
properly chosen constant and .
Let be a random variable with the probability density function . It is
proved that converges to in distribution when the conditional second
moment of given satisfies a law of large numbers. A Berry-Esseen
type bound is also given. We use this technique to obtain a Berry-Esseen error
bound of order in the noncentral limit theorem for the
magnetization in the Curie-Weiss ferromagnet at the critical temperature.
Exponential approximation with application to the spectrum of the
Bernoulli-Laplace Markov chain is also discussed.Comment: Published in at http://dx.doi.org/10.1214/10-AAP712 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The extremal spectral radii of -uniform supertrees
In this paper, we study some extremal problems of three kinds of spectral
radii of -uniform hypergraphs (the adjacency spectral radius, the signless
Laplacian spectral radius and the incidence -spectral radius).
We call a connected and acyclic -uniform hypergraph a supertree. We
introduce the operation of "moving edges" for hypergraphs, together with the
two special cases of this operation: the edge-releasing operation and the total
grafting operation. By studying the perturbation of these kinds of spectral
radii of hypergraphs under these operations, we prove that for all these three
kinds of spectral radii, the hyperstar attains uniquely the
maximum spectral radius among all -uniform supertrees on vertices. We
also determine the unique -uniform supertree on vertices with the second
largest spectral radius (for these three kinds of spectral radii). We also
prove that for all these three kinds of spectral radii, the loose path
attains uniquely the minimum spectral radius among all
-th power hypertrees of vertices. Some bounds on the incidence
-spectral radius are given. The relation between the incidence -spectral
radius and the spectral radius of the matrix product of the incidence matrix
and its transpose is discussed
The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs
Let and be the adjacency tensor, Laplacian tensor and signless Laplacian
tensor of uniform hypergraph , respectively. Denote by the largest H-eigenvalue of tensor . Let be a
uniform hypergraph, and be obtained from by inserting a new
vertex with degree one in each edge. We prove that Denote by
the th power hypergraph of an ordinary graph with maximum degree
. We will prove that
is a strictly decreasing sequence, which imply Conjectrue 4.1 of Hu, Qi and
Shao in \cite{HuQiShao2013}. We also prove that
converges to when goes to
infinity. The definiton of th power hypergraph has been generalized
as We also prove some eigenvalues properties about which generalize some known results. Some related results
about are also mentioned
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