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

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

Direct measurement of xenon flashtube opacity

Opacity measurement of xenon flash tube - optical mase

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

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

Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{\psi_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients $$\langle \gamma_H e_j, \psi_k \rangle_{L^2(H)} = \int_H e_j \overline \psi_k \, dV_H,$$ of restrictions $\gamma_H e_j$ of $e_j$ to $H$. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\{(\mu_k, \lambda_j)\}_{j,k - 0}^{\infty}$ of the (square roots of the) Laplacian $\Delta_M$ on $M$ and the Laplacian $\Delta_H$ on $H$ in a family of suitably `thick' regions in $\mathbb R^2$. Thick regions include (1) the truncated cone $\mu_k/\lambda_j \in [a,b] \subset (0,1)$ and $\lambda_j \leq \lambda$, and (2) the slowly thickening strip $|\mu_k - c\lambda_j| \leq w(\lambda)$ and $\lambda_j \leq \lambda$, where $w(\lambda)$ is monotonic and $1 \ll w(\lambda) \lesssim \lambda^{1 - 1/n}$. 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

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