1,754 research outputs found

    Explicit bounds on integrals of eigenfunctions over curves in surfaces of nonpositive curvature

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    Let (M,g)(M,g) be a compact Riemannian surface with nonpositive sectional curvature and let γ\gamma be a closed geodesic in MM. And let eλe_\lambda be an L2L^2-normalized eigenfunction of the Laplace-Beltrami operator Δg\Delta_g with Δgeλ=λ2eλ-\Delta_g e_\lambda = \lambda^2 e_\lambda. Sogge, Xi, and Zhang showed using the Gauss-Bonnet theorem that γeλds=O((logλ)1/2), \int_\gamma e_\lambda \, ds = O((\log\lambda)^{-1/2}), an improvement over the general O(1)O(1) bound. We show this integral enjoys the same decay for a wide variety of curves, where MM has nonpositive sectional curvature. These are the curves γ\gamma whose geodesic curvature avoids, pointwise, the geodesic curvature of circles of infinite radius tangent to γ\gamma.Comment: 23 pages, 3 figures, arXiv admin note: text overlap with arXiv:1702.0355

    Direct measurement of xenon flashtube opacity

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    Opacity measurement of xenon flash tube - optical mase

    Application of DOT-MORSE coupling to the analysis of three-dimensional SNAP shielding problems

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    The use of discrete ordinates and Monte Carlo techniques to solve radiation transport problems is discussed. A general discussion of two possible coupling schemes is given for the two methods. The calculation of the reactor radiation scattered from a docked service and command module is used as an example of coupling discrete ordinates (DOT) and Monte Carlo (MORSE) calculations

    Fourier coefficients of restrictions of eigenfunctions

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    Let {ej}\{e_j\} be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold (M,g)(M,g). Let HMH \subset M be a submanifold and let {ψk}\{\psi_k\} be an orthonormal basis of Laplace eigenfunctions of HH with the induced metric. We obtain joint asymptotics for the Fourier coefficients γHej,ψkL2(H)=HejψkdVH, \langle \gamma_H e_j, \psi_k \rangle_{L^2(H)} = \int_H e_j \overline \psi_k \, dV_H, of restrictions γHej\gamma_H e_j of eje_j to HH. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum {(μk,λj)}j,k0\{(\mu_k, \lambda_j)\}_{j,k - 0}^{\infty} of the (square roots of the) Laplacian ΔM\Delta_M on MM and the Laplacian ΔH\Delta_H on HH in a family of suitably `thick' regions in R2\mathbb R^2. Thick regions include (1) the truncated cone μk/λj[a,b](0,1)\mu_k/\lambda_j \in [a,b] \subset (0,1) and λjλ\lambda_j \leq \lambda, and (2) the slowly thickening strip μkcλjw(λ)|\mu_k - c\lambda_j| \leq w(\lambda) and λjλ\lambda_j \leq \lambda, where w(λ)w(\lambda) is monotonic and 1w(λ)λ11/n1 \ll w(\lambda) \lesssim \lambda^{1 - 1/n}. Key tools for obtaining these asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.Comment: 36 pages. Referees' suggestions incorporated. To appear in Sci. China Mat

    The VC-dimension of quadratic residues in finite fields

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    We study the Vapnik-Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, Fq\mathbb F_q, when considered as a subset of the additive group. We conjecture that as qq \to \infty, the squares have the maximum possible VC-dimension, viz. (1+o(1))log2q(1+o(1))\log_2 q. We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is (12+o(1))log2q\geq (\frac{1}{2} + o(1))\log_2 q. We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups ΓFq×\Gamma \subseteq \mathbb F_q^\times of bounded index.Comment: 21 pages, 3 figure
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