12,233 research outputs found
Near-optimal mean value estimates for multidimensional Weyl sums
We obtain sharp estimates for multidimensional generalisations of
Vinogradov's mean value theorem for arbitrary translation-dilation invariant
systems, achieving constraints on the number of variables approaching those
conjectured to be the best possible. Several applications of our bounds are
discussed
Bounding Stochastic Dependence, Complete Mixability of Matrices, and Multidimensional Bottleneck Assignment Problems
We call a matrix completely mixable if the entries in its columns can be
permuted so that all row sums are equal. If it is not completely mixable, we
want to determine the smallest maximal and largest minimal row sum attainable.
These values provide a discrete approximation of of minimum variance problems
for discrete distributions, a problem motivated by the question how to estimate
the -quantile of an aggregate random variable with unknown dependence
structure given the marginals of the constituent random variables. We relate
this problem to the multidimensional bottleneck assignment problem and show
that there exists a polynomial -approximation algorithm if the matrix has
only columns. In general, deciding complete mixability is
-complete. In particular the swapping algorithm of Puccetti et
al. is not an exact method unless . For a
fixed number of columns it remains -complete, but there exists a
PTAS. The problem can be solved in pseudopolynomial time for a fixed number of
rows, and even in polynomial time if all columns furthermore contain entries
from the same multiset
Weighted Sum of Correlated Lognormals: Convolution Integral Solution
Probability density function (pdf) for sum of n
correlated lognormal variables is deducted as a special
convolution integral. Pdf for weighted sums (where weights can
be any real numbers) is also presented. The result for four
dimensions was checked by Monte Carlo simulation
Laws of the single logarithm for delayed sums of random fields
We extend a law of the single logarithm for delayed sums by Lai to delayed
sums of random fields. A law for subsequences, which also includes the
one-dimensional case, is obtained in passing.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ103 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Translation invariance, exponential sums, and Waring's problem
We describe mean value estimates for exponential sums of degree exceeding 2
that approach those conjectured to be best possible. The vehicle for this
recent progress is the efficient congruencing method, which iteratively
exploits the translation invariance of associated systems of Diophantine
equations to derive powerful congruence constraints on the underlying
variables. There are applications to Weyl sums, the distribution of polynomials
modulo 1, and other Diophantine problems such as Waring's problem.Comment: Submitted to Proceedings of the ICM 201
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