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Several optical microscopy methods are now available for characterizing scientific and industrial processes at sub-micron resolution. However, they are often ill-suited for imaging rapid events. Limited by the trade-off between camera frame-rate and sensitivity, or the need for mechanical scanning, current microscopes are optimized for imaging at hundreds of frames-per-second (fps), well-below what is needed in processes such as neuronal signaling or moving parts in manufacturing lines. Here, we present a scan-less technology that allows sub-micrometric imaging at thousands of fps. It is based on combining a single-pixel camera with parallelized encoded illumination. We use two acousto-optic deflectors (AODs) placed in a Mach–Zehnder interferometer and drive them simultaneously with multiple and unique acoustic frequencies. As a result, orthogonal light stripes are obtained that interfere with the sample plane, forming a two-dimensional array of flickering spots – each with its modulation frequency. The light from the sample is collected with a single photodiode that, after spectrum analysis, allows for image reconstruction at speeds only limited by the AOD’s bandwidth and laser power. We describe the working principle of our approach, characterize its imaging performance as a function of the number of pixels – up to 400 × 400 – and characterize dynamic events at 5000 fps.

In this paper, we prove that every equivalence class in the quotient group of integral 1-currents modulo p in Euclidean space contains an integral current, with quantitative estimates on its mass and the mass of its boundary. Moreover, we show that the validity of this statement for m-dimensional integral currents modulo p implies that the family of ( m - 1 ) {(m-1)} -dimensional flat chains of the form pT, with T a flat chain, is closed with respect to the flat norm. In particular, we deduce that such closedness property holds for 0-dimensional flat chains, and, using a proposition from The structure of minimizing hypersurfaces mod 4 by Brian White, also for flat chains of codimension 1. [ABSTRACT FROM AUTHOR]

The Steiner tree problem can be stated in terms of finding a connected set of minimal length containing a given set of finitely many points. We show how to formulate it as a mass-minimization problem for 1-dimensional currents with coefficients in a suitable normed group. The representation used for these currents allows to state a calibration principle for this problem. We also exhibit calibrations in some examples. [ABSTRACT FROM AUTHOR]