We prove the endpoint case of the multilinear Kakeya conjecture of Bennett, Carbery, and Tao. The proof uses the polynomial method introduced by Dvir. In [1], Bennett, Carbery, and Tao formulated a multilinear Kakeya conjecture, and they proved the conjecture except for the endpoint case. In this paper, we slightly sharpen their result by proving the endpoint case of the conjecture. Our method ...

This paper describes the use of a microsphere sensor technology that allows simple fabrication of vapor sensor arrays with reproducible response patterns. Microsphere sensor fabrication protocols are uncomplicated and yield billions of highly reproducible sensors. Microsphere sensor arrays combined with a generalized Whitney-Mann-Wilcoxen (GWMW) classifier were used to discriminate between the ...

In this paper, we propose two regularization approaches for the Boltz-mann collision operator. The constructed operators preserve the mass, momentum and energy; their equilibrium states are Maxwellians and they satisfy the H theorem. In the rst approach, the regularization consists in allowing microscopic collisions which do not exactly preserve energy and momentum. However, the limit of the mo...

Increased emphasis on aircraft inspection to ensure aviation safety has resulted in the need for a better trained workforce of aircraft maintenance personnel. As almost 90% of inspection is visual in nature, it has led to the development of computer simulators to train human inspectors performing the inspection task. However, the lack of immersion and interaction has led to limited effectivenes...