Thomas Decomposition of Algebraic and Differential Systems

In this paper we consider disjoint decomposition of algebraic and non-linear partial differential systems of equations and inequations into so-called simple subsystems. We exploit Thomas decomposition ideas and develop them into a new algorithm. For algebraic systems simplicity means triangularity, squarefreeness and non-vanishing initials. For differential systems the algorithm provides not only algebraic simplicity but also involutivity. The algorithm has been implemented in Maple

Modeling active electrolocation in weakly electric fish

In this paper, we provide a mathematical model for the electrolocation in weakly electric fishes. We first investigate the forward complex conductivity problem and derive the approximate boundary conditions on the skin of the fish. Then we provide a dipole approximation for small targets away from the fish. Based on this approximation, we obtain a non-iterative location search algorithm using multi-frequency measurements. We present numerical experiments to illustrate the performance and the stability of the proposed multi-frequency location search algorithm. Finally, in the case of disk- and ellipse-shaped targets, we provide a method to reconstruct separately the conductivity, the permittivity, and the size of the targets from multi-frequency measurements.Comment: 37 pages, 11 figure

RNA-DNA strand exchange by the Drosophila Polycomb complex PRC2.

Polycomb Group (PcG) proteins form memory of transient transcriptional repression that is necessary for development. In Drosophila, DNA elements termed Polycomb Response Elements (PREs) recruit PcG proteins. How PcG activities are targeted to PREs to maintain repressed states only in appropriate developmental contexts has been difficult to elucidate. PcG complexes modify chromatin, but also interact with both RNA and DNA, and RNA is implicated in PcG targeting and function. Here we show that R-loops form at many PREs in Drosophila embryos, and correlate with repressive states. In vitro, both PRC1 and PRC2 can recognize R-loops and open DNA bubbles. Unexpectedly, we find that PRC2 drives formation of RNA-DNA hybrids, the key component of R-loops, from RNA and dsDNA. Our results identify R-loop formation as a feature of Drosophila PREs that can be recognized by PcG complexes, and RNA-DNA strand exchange as a PRC2 activity that could contribute to R-loop formation

Injection of Orbital Angular Momentum and Storage of Quantized Vortices in Polariton Superfluids

We report the experimental investigation and theoretical modeling of a rotating polariton superfluid relying on an innovative method for the injection of angular momentum. This novel, multipump injection method uses four coherent lasers arranged in a square, resonantly creating four polariton populations propagating inwards. The control available over the direction of propagation of the superflows allows injecting a controllable nonquantized amount of optical angular momentum. When the density at the center is low enough to neglect polariton-polariton interactions, optical singularities, associated with an interference pattern, are visible in the phase. In the superfluid regime resulting from the strong nonlinear polariton-polariton interaction, the interference pattern disappears and only vortices with the same sign are persisting in the system. Remarkably, the number of vortices inside the superfluid region can be controlled by controlling the angular momentum injected by the pumps