Consider the recovery of an unknown signal x from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that x is sparse, and that the measurements are of the form sign( a i , x ) ∈ {±1}. Since such measurements give no information on the norm of x, recovery methods typically assume that x 2 = 1. We show that if one allows more generally for quantized affine measurements of the form sign( a i , x + b i ), and if the vectors a i are random, an appropriate choice of the affine shifts b i allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed x in the annulus r ≤ x 2 ≤ R, one may estimate the norm x 2 up to additive error δ from m R 4 r −2 δ −2 such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors x in a Euclidean ball of known radius.

Wearable sensor data is relatively easily collected and provides direct measurements of movement that can be used to develop useful behavioral biomarkers. Sensitive and specific behavioral biomarkers for neurodegenerative diseases are critical to supporting early detection, drug development efforts, and targeted treatments. In this paper, we use autoregressive hidden Markov models and a time-frequency approach to create meaningful quantitative descriptions of behavioral characteristics of cerebellar ataxias from wearable inertial sensor data gathered during movement. We create a flexible and descriptive set of features derived from accelerometer and gyroscope data collected from wearable sensors worn while participants perform clinical assessment tasks, and use these data to estimate disease status and severity. A short period of data collection (<5 min) yields enough information to effectively separate patients with ataxia from healthy controls with very high accuracy, to separate ataxia from other neurodegenerative diseases such as Parkinson’s disease, and to provide estimates of disease severity.

An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic α in a hyperbolic 3-manifold M, we find sufficient conditions for it to be an unknotting tunnel. In particular, if α corresponds to a 4-bracelet, 5-bracelet or 6-bracelet in the universal cover and has short enough length, it must be an unknotting tunnel. Furthermore, we consider a vertical geodesic α that satisfies the elder sibling property, which means that in the universal cover, every horoball except the one centered at ∞ is connected to a larger horoball by a lift of α. Such an α with length less than ln(2) is then shown to be an unknotting tunnel.2010 Mathematics subject classification: primary 57M50.

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An important question in some voting rights and redistricting litigation in the U.S. is whether and to what degree voting is racially polarized. In the setting of voting rights cases, there is a family of methods called "ecological inference" (see especially King, 1997) that uses observed data, pairing voting outcomes with demographic information for each precinct in a given polity, to infer voting patterns for each demographic group.