12,233 research outputs found

    Near-optimal mean value estimates for multidimensional Weyl sums

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    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

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    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 α\alpha-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 22-approximation algorithm if the matrix has only 33 columns. In general, deciding complete mixability is NP\mathcal{NP}-complete. In particular the swapping algorithm of Puccetti et al. is not an exact method unless NP⊆ZPP\mathcal{NP}\subseteq\mathcal{ZPP}. For a fixed number of columns it remains NP\mathcal{NP}-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

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    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

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    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

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    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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