597,808 research outputs found
On new sum-product-type estimates
New lower bounds involving sum, difference, product, and ratio sets for a set
A\subset \C are given. The estimates involving the sum set match, up to
constants, the one obtained by Solymosi for the reals and are obtained by
generalising his approach to the complex plane. The bounds involving the
difference set are slightly weaker. They improve on the best known ones,
including the case , which also due to Solymosi, by means of
combining the use of the Szemer\'edi-Trotter theorem with an arithmetic
combinatorics technique.Comment: 19pp. This is a new extended version, accepted for publication to
SIAM J. Disc. Math. Note: the earlier homonymous preprint arXiv_math:
1111.4977 of the Second Author contained weaker estimates involving the
sum-set. The present estimate for the sum-set was erroneously claimed in
arXiv:0812.145
A bound on the multiplicative energy of a sum set and extremal sum-product problems
In recent years some near-optimal estimates have been established for certain
sum-product type estimates. This paper gives some first extremal results which
provide information about when these bounds may or may not be tight. The main
tool is a new result which provides a nontrivial upper bound on the
multiplicative energy of a sum set or difference set.Comment: 13 page
Bilinear wavelet representation of Calderón-Zygmund forms
We represent a bilinear Calderón–Zygmund operator at a given smoothness level as a finite sum of cancellative, complexity-zero operators, involving smooth wavelet forms, and continuous paraproduct forms. This representation results in a sparse
T
(
1
)
-type bound, which in turn yields directly new sharp weighted bilinear estimates on Lebesgue and Sobolev spaces. Moreover, we apply the representation theorem to study fractional differentiation of bilinear operators, establishing Leibniz-type rules in weighted Sobolev spaces which are new even in the simplest case of the pointwise product
Point-plane incidences and some applications in positive characteristic
The point-plane incidence theorem states that the number of incidences
between points and planes in the projective three-space over a
field , is where is the maximum number
of collinear points, with the extra condition if has
characteristic . This theorem also underlies a state-of-the-art
Szemer\'edi-Trotter type bound for point-line incidences in , due to
Stevens and de Zeeuw.
This review focuses on some recent, as well as new, applications of these
bounds that lead to progress in several open geometric questions in , for
. These are the problem of the minimum number of distinct nonzero
values of a non-degenerate bilinear form on a point set in , the analogue
of the Erd\H os distinct distance problem in and additive energy
estimates for sets, supported on a paraboloid and sphere in . It avoids
discussing sum-product type problems (corresponding to the special case of
incidences with Cartesian products), which have lately received more attention.Comment: A survey, with some new results, for the forthcoming Workshop on
Pseudorandomness and Finite Fields in at RICAM in Linz 15-19 October, 2018;
24p
A Filon-Clenshaw-Curtis-Smolyak rule for multi-dimensional oscillatory integrals with application to a UQ problem for the Helmholtz equation
In this paper, we combine the Smolyak technique for multi-dimensional
interpolation with the Filon-Clenshaw-Curtis (FCC) rule for one-dimensional
oscillatory integration, to obtain a new Filon-Clenshaw-Curtis-Smolyak (FCCS)
rule for oscillatory integrals with linear phase over the dimensional cube
. By combining stability and convergence estimates for the FCC rule
with error estimates for the Smolyak interpolation operator, we obtain an error
estimate for the FCCS rule, consisting of the product of a Smolyak-type error
estimate multiplied by a term that decreases with
, where is the wavenumber and is
the number of oscillatory dimensions. If all dimensions are oscillatory, a
higher negative power of appears in the estimate. As an application, we
consider the forward problem of uncertainty quantification (UQ) for a
one-space-dimensional Helmholtz problem with wavenumber and a random
heterogeneous refractive index, depending in an affine way on i.i.d.
uniform random variables. After applying a classical hybrid
numerical-asymptotic approximation, expectations of functionals of the solution
of this problem can be formulated as a sum of oscillatory integrals over
, which we compute using the FCCS rule. We give numerical results for
the FCCS rule and the UQ algorithm showing that accuracy improves when both
and the order of the rule increase. We also give results for dimension-adaptive
sparse grid FCCS quadrature showing its efficiency as dimension increases
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