Léo Mathis · arXiv (Cornell University) · 2022
We study the Vitale zonoid (a convex body associated to a probability distribution) associated to a non--centered Gaussian vector. This defines a family of convex bodies, that contains and generalizes ellipsoids, which we call Gaussian zonoids. We show that each Gaussian zonoid can be approximated by an ellipsoid that we compute explicitely. We use this result to give new estimates for the expectation of the absolute value of the determinant of a non--centered Gaussian matrix in terms of mixed volume of ellipsoids. Finally, exploiting a recent link between random fields and zonoids uncovered by Stecconi and the author, we apply our results to the study of the zero set of non--centered Gaussian random fields. We show how these can be approximated by a suitable centered Gaussian random field and give a quantitative asymptotic in the limit where the variance goes to zero.
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Francesco Pampaloni, Jörg Enderlein · ArXiv.org · 2004
The paper aims at presenting a didactic and self-contained overview of Gauss-Hermite and Gauss-Laguerre laser beam modes. The usual textbook approach for deriving these modes is to solve the Helmoltz electromagnetic wave equation within the paraxial approximation. Here, a different technique is presented: Using the plane wave representation of the fundamental Gaussian mode as seed function, all higher-order beam modes are derived by acting with differential operators on this fundamental solution. Even special beam modes as the recently introduced Bessel beams are easily described within that framework.
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Guillaume Aubrun, StanisÅaw Szarek · Mathematical surveys and monographs · 2017
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Julien Barral, Xiong Jin, Rémi Rhodes et al. · 2013
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Shashi Shekhar, Hui Xiong, Xun Zhou · Encyclopedia of GIS · 2017
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Kirk T. Mcdonald · 2012
(Compare [1] and chap. 3 [2].) We first consider a single grid plane, as shown in Fig. 1, that consists of wires along the z axis, separated by distance a in the plane y =0. Figure 1: A grid of wires with spacing a in the plane y =0. The simplest analysis of this grid is when each wire carries charge λ per unit length (and no other charges are present). The form of the potential V (x, y) can be obtained via separation of variables as in Appendix 1, or via conjugate functions [3]. The result is (in
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Ekkehard Krätzel, E. Krätzel · 2014
Abstract. We investigate the number of lattice points in special three-dimensional convex bodies. They are called convex bodies of pseudo revolution, because we have in one special case a body of revolution and in another case even a super sphere. These bodies have lines at the boundary, where all points have Gaussian curvature zero. We consider the influence of these points to the lattice rest in the asymptotic representation of the number of lattice points.
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C. A. L. Bailer-jones · 2008
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Greg Fasshauer · 2011
Φ(x) = e −ε2 ‖x ‖ 2, ε> 0, (1) is strictly positive definite (and radial) on R s for any s (see Fourier transform). Moreover, Gaussians are infinitely differentiable.
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Anton Polishko, Nadia Ponts, Karine G. Le Roch et al. · 2013
Vol. 28 ISMB 2012, pages i242–i249 doi:10.1093/bioinformatics/bts206 NORMAL: accurate nucleosome positioning using a modified
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SpringerReference · 2011
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S. Kawano, S. Konishi, Shuichi Kawano et al. · 2009
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Y. Gordon · 2013
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Suganda Jutamulia · 2016
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Reiner Schulz, Multi-layer Perceptron · 2011
And many others whose copyright I shamelessly infringed by including their illustrations in this presentation.
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O. Treu Filhoa, J. C. Pinheirob, R. T. Kondod et al. · 2003
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Francesco Russo Frederi, Hal Id Hal, Francesco Russo et al. · 2014
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Marbac-Lourdelle, M. (Matthieu), Biernacki, C. (Christophe), Vandewalle, V. (Vincent) et al. · LillOA (Université de Lille (University Of Lille)) · 2020
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Christophe Biernacki, Matthieu Marbac, Vincent Vandewalle · Journal of Classification · 2020
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Christophe Biernacki, Matthieu Marbac-Lourdelle, Vincent Vandewalle · HAL (Le Centre pour la Communication Scientifique Directe) · 2019
International audience
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