In this paper, we develop and analyze a trilinear immersed finite element method for solving three-dimensional elliptic interface problems. The proposed can be utilized on interface-unfitted meshes such as Cartesian grids consisting of cuboids. We establish the trace inverse inequalities IFE functions elements with arbitrary interface-cutting configuration. Optimal priori error estimates are ri...

Robbins’s visionary 1951 paper can be seen as an exercise in binary classification, but also as a precursor to the outpouring of recent work on high-dimensional data analysis and multiple testing. It can also be seen as the birth of empirical Bayes methods. Our objective in the present note is to use this problem and several variants of it to provide a glimpse into the evolution of empirical Ba...

Sufficient number of linear and noisy measurements for exact and approximate sparsity pattern/support set recovery in the high dimensional setting is derived. Although this problem as been addressed in the recent literature, there is still considerable gaps between those results and the exact limits of the perfect support set recovery. To reduce this gap, in this paper, the sufficient con...

The ability to uniquely identify a quantum state is integral to quantum science, but for nonorthogonal states, quantum mechanics precludes deterministic, error-free discrimination. However, using the nondeterministic protocol of unambiguous state discrimination enables the error-free differentiation of states, at the cost of a lower frequency of success. We discriminate experimentally between n...

A rigorous proof of an error estimate for a numerical method for two-dimensional scalar conservation laws is presented. The numerical method under consideration is based on the use of dimensional splitting and front tracking to solve the one-dimensional equations. It is shown that the error is bounded by C(((t) 1=2 + ((x) 1=2 +), where x is the space step, t is the time step, is the parameter m...