4,510 research outputs found
Sharp norm estimates for composition operators and Hilbert-type inequalities
Let denote the Hardy space of Dirichlet series with square summable coefficients and suppose that
is a symbol generating a composition operator on by
. Let denote the Riemann zeta
function and the unique positive solution of the equation
. We obtain sharp upper bounds for the norm of
on when
, by relating such sharp
upper bounds to the best constant in a family of discrete Hilbert-type
inequalities.Comment: This paper has been accepted for publication in Bulletin of the LM
Ages of records in random walks
We consider random walks with continuous and symmetric step distributions. We
prove universal asymptotics for the average proportion of the age of the kth
longest lasting record for k=1,2,... and for the probability that the record of
the kth longest age is broken at step n. Furthermore, we show that the ranked
sequence of proportions of ages converges to the Poisson-Dirichlet
distribution.Comment: 15 pages, 1 figur
Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics
We explicitly construct the series expansion for a certain class of solutions
to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric
solutions. We express the Taylor coefficients through some universal
combinatorial constants and find recurrence relations for them. These results
are used to obtain new formulas for the genus 0 double Hurwitz numbers. They
can also serve as a starting point for a constructive approach to the Riemann
mapping problem and the inverse potential problem in 2D.Comment: 26 page
Generalized Species Sampling Priors with Latent Beta reinforcements
Many popular Bayesian nonparametric priors can be characterized in terms of
exchangeable species sampling sequences. However, in some applications,
exchangeability may not be appropriate. We introduce a {novel and
probabilistically coherent family of non-exchangeable species sampling
sequences characterized by a tractable predictive probability function with
weights driven by a sequence of independent Beta random variables. We compare
their theoretical clustering properties with those of the Dirichlet Process and
the two parameters Poisson-Dirichlet process. The proposed construction
provides a complete characterization of the joint process, differently from
existing work. We then propose the use of such process as prior distribution in
a hierarchical Bayes modeling framework, and we describe a Markov Chain Monte
Carlo sampler for posterior inference. We evaluate the performance of the prior
and the robustness of the resulting inference in a simulation study, providing
a comparison with popular Dirichlet Processes mixtures and Hidden Markov
Models. Finally, we develop an application to the detection of chromosomal
aberrations in breast cancer by leveraging array CGH data.Comment: For correspondence purposes, Edoardo M. Airoldi's email is
[email protected]; Federico Bassetti's email is
[email protected]; Michele Guindani's email is
[email protected] ; Fabrizo Leisen's email is
[email protected]. To appear in the Journal of the American
Statistical Associatio
Selberg's orthonormality conjecture and joint universality of L-functions
In the paper we introduce the new approach how to use an orthonormality
relation of coefficients of Dirichlet series defining given L-functions from
the Selberg class to prove joint universality
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