A Search for Gamma-Ray Burst Optical Emission with the Automated Patrol Telescope

The Automated Patrol Telescope (APT) is a wide-field (5 X 5 deg.s), modified Schmidt capable of covering large gamma-ray burst (GRB) localization regions to produce a high rate of GRB optical emission measurements. Accounting for factors such as bad weather and incomplete overlap of our field and large GRB localization regions, we estimate our search will image the actual location of 20-41 BATSE GRB sources each year. Long exposures will be made for these images, repeated for several nights, to detect delayed optical transients (OTs) with light curves similar to those already discovered. The APT can also respond within about 20 sec. to GRB alerts from BATSE to search for prompt emission from GRBs. We expect to image more than 2.4 GRBs/yr. during gamma-ray emission. More than 5.1 will be imaged/yr. within about 20 sec. of emission. The APT's 50 cm aperture is much larger than other currently operating experiments used to search for prompt emission, and the APT is the only GRB dedicated telescope in the Southern Hemisphere. Given the current rate of about 25% OTs per X/gamma localization, we expect to produce a sample of about 10 OTs for detailed follow-up observations in 1-2 years of operation.Comment: 4 pages latex + 3 ps figures. Download a single tar file of ps at http://panisse.lbl.gov/public/bruce/optgrbsearch.tar.g

Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric scattering transform on manifolds. Similar to the Euclidean scattering transform, the geometric scattering transform is based on a cascade of wavelet filters and pointwise nonlinearities. It is invariant to local isometries and stable to certain types of diffeomorphisms. Empirical results demonstrate its utility on several geometric learning tasks. Our results generalize the deformation stability and local translation invariance of Euclidean scattering, and demonstrate the importance of linking the used filter structures to the underlying geometry of the data.Comment: 35 pages; 3 figures; 2 tables; v3: Revisions based on reviewer comment

Martingales, Singular Integrals, and Fourier Multipliers

Many probabilistic constructions have been created to study the Lp-boundedness, 1 \u3c p \u3c ∞, of singular integrals and Fourier multipliers. We will use a combination of analytic and probabilistic methods to study analytic properties of these constructions and obtain results which cannot be obtained using probability alone. In particular, we will show that a large class of operators, including many that are obtained as the projection of martingale transforms with respect to the background radiation process of Gundy and Varapolous or with respect to space-time Brownian motion, satisfy the assumptions of Calderón-Zygmund theory and therefore boundedly map L1 to weak- L1. We will also use a method of rotations to study the L p boundedness, 1 \u3c p \u3c ∞, of Fourier multipliers which are obtained as the projections of martingale transforms with respect to symmetric α-stable processes, 0 \u3c α \u3c 2. Our proof does not use the fact that 0 \u3c α \u3c 2 and therefore allows us to obtain a larger class of multipliers, indexed by a parameter, 0 \u3c r \u3c ∞, which are bounded on L p. As in the case of the multipliers which arise as the projection of martingale transforms, these new multipliers also have potential applications to the study of the Beurling-Ahlfors transform and are related to the celebrated conjecture of T. Iwaniec concerning its exact Lp norm