3,175 research outputs found
Fast simulation of new coins from old
Let S\subset (0,1). Given a known function f:S\to (0,1), we consider the
problem of using independent tosses of a coin with probability of heads p
(where p\in S is unknown) to simulate a coin with probability of heads f(p). We
prove that if S is a closed interval and f is real analytic on S, then f has a
fast simulation on S (the number of p-coin tosses needed has exponential
tails). Conversely, if a function f has a fast simulation on an open set, then
it is real analytic on that set.Comment: Published at http://dx.doi.org/10.1214/105051604000000549 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Jacobi Polynomials, Bernstein-type Inequalities and Dispersion Estimates for the Discrete Laguerre Operator
The present paper is about Bernstein-type estimates for Jacobi polynomials
and their applications to various branches in mathematics. This is an old topic
but we want to add a new wrinkle by establishing some intriguing connections
with dispersive estimates for a certain class of Schr\"odinger equations whose
Hamiltonian is given by the generalized Laguerre operator. More precisely, we
show that dispersive estimates for the Schr\"odinger equation associated with
the generalized Laguerre operator are connected with Bernstein-type
inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi
polynomials to establish some new dispersive estimates. In turn, the optimal
dispersive decay estimates lead to new Bernstein-type inequalities.Comment: 25 page
Inequalities for Lorentz polynomials
We prove a few interesting inequalities for Lorentz polynomials including
Nikolskii-type inequalities. A highlight of the paper is a sharp Markov-type
inequality for polynomials of degree at most n with real coefficients and with
derivative not vanishing in the open unit disk. The result may be compared with
Erdos's classical Markov-type inequality (1940) for polynomials of degree at
most n having only real zeros outside the interval (-1,1)
Conformally Mapped Polynomial Chaos Expansions for Maxwell's Source Problem with Random Input Data
Generalized Polynomial Chaos (gPC) expansions are well established for
forward uncertainty propagation in many application areas. Although the
associated computational effort may be reduced in comparison to Monte Carlo
techniques, for instance, further convergence acceleration may be important to
tackle problems with high parametric sensitivities. In this work, we propose
the use of conformal maps to construct a transformed gPC basis, in order to
enhance the convergence order. The proposed basis still features orthogonality
properties and hence, facilitates the computation of many statistical
properties such as sensitivities and moments. The corresponding surrogate
models are computed by pseudo-spectral projection using mapped quadrature
rules, which leads to an improved cost accuracy ratio. We apply the methodology
to Maxwell's source problem with random input data. In particular, numerical
results for a parametric finite element model of an optical grating coupler are
given
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