84 research outputs found
New coins from old, smoothly
Given a (known) function , we consider the problem of
simulating a coin with probability of heads by tossing a coin with
unknown heads probability , as well as a fair coin, times each, where
may be random. The work of Keane and O'Brien (1994) implies that such a
simulation scheme with the probability equal to 1 exists iff
is continuous. Nacu and Peres (2005) proved that is real analytic in an
open set iff such a simulation scheme exists with the
probability decaying exponentially in for every . We
prove that for non-integer, is in the space if
and only if a simulation scheme as above exists with , where \Delta_n(x)\eqbd \max \{\sqrt{x(1-x)/n},1/n \}.
The key to the proof is a new result in approximation theory:
Let \B_n be the cone of univariate polynomials with nonnegative Bernstein
coefficients of degree . We show that a function is in
if and only if has a series representation
with F_n \in \B_n and for all and . We also provide a
counterexample to a theorem stated without proof by Lorentz (1963), who claimed
that if some \phi_n \in \B_n satisfy for all and , then .Comment: 29 pages; final version; to appear in Constructive Approximatio
Bernstein-type polynomials on several intervals
We construct the analogues of Bernstein polynomials on the set Js of s finitely many intervals. Two cases are considered: first when there are no restrictions on Js, and then when Js has a so-called T-polynomial. On such sets we define approximating operators resembling the classic Bernstein polynomials. Reproducing and interpolation properties as well as estimates for the rate of convergence are given. © Springer International Publishing AG 2017
Rearranging Edgeworth-Cornish-Fisher Expansions
This paper applies a regularization procedure called increasing rearrangement
to monotonize Edgeworth and Cornish-Fisher expansions and any other related
approximations of distribution and quantile functions of sample statistics.
Besides satisfying the logical monotonicity, required of distribution and
quantile functions, the procedure often delivers strikingly better
approximations to the distribution and quantile functions of the sample mean
than the original Edgeworth-Cornish-Fisher expansions.Comment: 17 pages, 3 figure
Physically Similar Systems - A History of the Concept
PreprintThe concept of similar systems arose in physics, and appears to have originated with Newton in the
seventeenth century. This chapter provides a critical history of the concept of physically similar
systems, the twentieth century concept into which it developed. The concept was used in the
nineteenth century in various fields of engineering (Froude, Bertrand, Reech), theoretical physics (van
der Waals, Onnes, Lorentz, Maxwell, Boltzmann) and theoretical and experimental hydrodynamics
(Stokes, Helmholtz, Reynolds, Prandtl, Rayleigh). In 1914, it was articulated in terms of ideas
developed in the eighteenth century and used in nineteenth century mathematics and mechanics:
equations, functions and dimensional analysis. The terminology physically similar systems was
proposed for this new characterization of similar systems by the physicist Edgar Buckingham.
Related work by Vaschy, Bertrand, and Riabouchinsky had appeared by then. The concept is very
powerful in studying physical phenomena both theoretically and experimentally. As it is not currently
part of the core curricula of STEM disciplines or philosophy of science, it is not as well known as it
ought to be
- …