Glaucoma is a leading cause of blindness and real-time monitoring of intraocular pressure is of great demand. We present a stretchable sensor inside a contact lens capable of monitoring change in the curvature of cornea caused by IOP fluctuations.

Let M be a smooth d-dimensional submanifold of R N with boundary that's equipped with the Euclidean (chordal) metric, and choose m ≤ N . In this paper we consider the probability that a random matrix A ∈ R m×N will serve as a bi-Lipschitz function A : M → R m with bi-Lipschitz constants close to one for three different types of distributions on the m × N matrices A, including two whose realizations are guaranteed to have fast matrix-vector multiplies.In doing so we generalize prior randomized metric space embedding results of this type for submanifolds of R N by allowing for the presence of boundary while also retaining, and in some cases improving, prior lower bounds on the achievable embedding dimensions m for which one can expect small distortion with high probability. In particular, motivated by recent modewise embedding constructions for tensor data, herein we present a new class of highly structured distributions on matrices which outperform prior structured matrix distributions for embedding sufficiently low-dimensional submanifolds of R N (with d √ N ) with respect to both achievable embedding dimension, and computationally efficient realizations. As a consequence we are able to present, for example, a general new class of Johnson-Lindenstrauss embedding matrices for O(log c N )-dimensional submanifolds of R N which enjoy O(N log(log N ))-time matrix vector multiplications.

Let M be a smooth, connected, compact submanifold of R n without boundary and of dimension k ≥ 2. Let S k ⊂ R k+1 ⊂ R n denote the k-dimesnional unit sphere. We show if M has reach equal to one, then its volume satisfies vol(M ) ≥ vol(S k ) with equality holding only if M is congruent to S k .

No abstract

No abstract