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Bruno O.P.,Computing and Mathematical science | Delourme B.,University of Paris 13
Journal of Computational Physics | Year: 2014

We introduce a new methodology, based on new quasi-periodic Green functions which converge rapidly even at and around Wood-anomaly configurations, for the numerical solution of problems of scattering by periodic rough surfaces in two-dimensional space. As is well known the classical quasi-periodic Green function ceases to exist at Wood anomalies. The approach introduced in this text produces fast Green function convergence throughout the spectrum on the basis of a certain "finite-differencing" approach and smooth windowing of the classical Green function lattice sum. The resulting Green-function convergence is super-algebraically fast away from Wood anomalies, and it reduces to an arbitrarily-high (user-prescribed) algebraic order of convergence at Wood anomalies. © 2013 Elsevier Inc. Source

Ligett K.,Computing and Mathematical science | Roth A.,University of Pennsylvania
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2012

In this paper, we consider the problem of estimating a potentially sensitive (individually stigmatizing) statistic on a population. In our model, individuals are concerned about their privacy, and experience some cost as a function of their privacy loss. Nevertheless, they would be willing to participate in the survey if they were compensated for their privacy cost. These cost functions are not publicly known, however, nor do we make Bayesian assumptions about their form or distribution. Individuals are rational and will misreport their costs for privacy if doing so is in their best interest. Ghosh and Roth recently showed in this setting, when costs for privacy loss may be correlated with private types, if individuals value differential privacy, no individually rational direct revelation mechanism can compute any non-trivial estimate of the population statistic. In this paper, we circumvent this impossibility result by proposing a modified notion of how individuals experience cost as a function of their privacy loss, and by giving a mechanism which does not operate by direct revelation. Instead, our mechanism has the ability to randomly approach individuals from a population and offer them a take-it-or-leave-it offer. This is intended to model the abilities of a surveyor who may stand on a street corner and approach passers-by. © 2012 Springer-Verlag. Source

Cohen G.,Computing and Mathematical science | Shinkar I.,Courant Institute of Mathematical Sciences
ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science | Year: 2016

We study depth 3 circuits of the form OR AND XOR, or equivalently-DNF of parities. This model was first explicitly studied by Jukna (CPC'06) who obtained a 2(n) lower bound for explicit functions in this model. Several related models have gained attention in the last few years, such as parity decision trees, the parity kill number and AC0 XOR circuits. For a function f : f0; 1gn ! f0; 1g, we denote by DNFδ(f) the least integer s for which there exists an OR AND XOR circuit, with OR gate of fan-in s, that computes f. We summarize some of our results: For any affine disperser f : f0; 1gn ! f0; 1g for dimension k, it holds that DNF (f) ≥ 2n-2k. By plugging Shaltiel's affine disperser (FOCS'11) we obtain an explicit 2n-no(1) lower bound. We give a non-Trivial general upper bound by showing that DNF (f) ≥ O(2n=n) for any function f on n bits. This bound is shown to be tight up to an O(log n) factor. We show that for any symmetric function f it holds that DNF(f) ≤ 1:5n poly(n). Furthermore, there exists a symmetric function f for which this bound is tight up to a polynomial factor. We show tighter bounds for symmetric threshold functions. For example, we show that the majority function has DNF complexity of ω2n=2 poly(n). This is also tight up to a polynomial factor. For the inner product function IP on n inputs we show that DNFδ(IP) = 2n=2-1. Previously, Jukna gave a lower bound of (2n=4) for the DNF complexity of this function. We further give bounds for any low degree polynomial. Finally, we obtain a 2n-o(n) average case lower bound for the parity decision tree model using affine extractors. Source

Crane K.,Computing and Mathematical science | Desbrun M.,Computing and Mathematical science | Schroder P.,Computing and Mathematical science | Schroder P.,TU Munich
Computer Graphics Forum | Year: 2010

This paper presents a straightforward algorithm for constructing connections on discrete surfaces that are as smooth as possible everywhere but on a set of isolated singularities with given index. We compute these connections by solving a single linear system built from standard operators. The solution can be used to design rotationally symmetric direction fields with user-specified singularities and directional constraints. Journal compilation © 2010 The Eurographics Association and Blackwell Publishing Ltd. Source

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