Optimal three-ball inequalities and quantitative uniqueness for the Lam\'e system with Lipschitz coefficients

In this paper we study the local behavior of a solution to the Lam\'e system with \emph{Lipschitz} coefficients in dimension $n\ge 2$. Our main result is the bound on the vanishing order of a nontrivial solution, which immediately implies the strong unique continuation property. This paper solves the open problem of the strong uniqueness continuation property for the Lam\'e system with Lipschitz coefficients in any dimension

Distributed Training Large-Scale Deep Architectures

Scale of data and scale of computation infrastructures together enable the current deep learning renaissance. However, training large-scale deep architectures demands both algorithmic improvement and careful system configuration. In this paper, we focus on employing the system approach to speed up large-scale training. Via lessons learned from our routine benchmarking effort, we first identify bottlenecks and overheads that hinter data parallelism. We then devise guidelines that help practitioners to configure an effective system and fine-tune parameters to achieve desired speedup. Specifically, we develop a procedure for setting minibatch size and choosing computation algorithms. We also derive lemmas for determining the quantity of key components such as the number of GPUs and parameter servers. Experiments and examples show that these guidelines help effectively speed up large-scale deep learning training

Experimental and Numerical Analysis of High-Resolution Injection Technique for Capillary Electrophoresis Microchip

This study presents an experimental and numerical investigation on the use of high-resolution injection techniques to deliver sample plugs within a capillary electrophoresis (CE) microchip. The CE microfluidic device was integrated into a U-shaped injection system and an expansion chamber located at the inlet of the separation channel, which can miniize the sample leakage effect and deliver a high-quality sample plug into the separation channel so that the detection performance of the device is enhanced. The proposed 45° U-shaped injection system was investigated using a sample of Rhodamine B dye. Meanwhile, the analysis of the current CE microfluidic chip was studied by considering the separation of Hae III digested ϕx-174 DNA samples. The experimental and numerical results indicate that the included 45° U-shaped injector completely eliminates the sample leakage and an expansion separation channel with an expansion ratio of 2.5 delivers a sample plug with a perfect detection shape and highest concentration intensity, hence enabling an optimal injection and separation performance