Quantizing the line element field

A metric with signature (-+++) can be constructed from a metric with signature (++++) and a double-sided vector field called the line element field. Some of the classical and quantum properties of this vector field are studied.Comment: 9 page

Reduced and declining physical function in prevalent dialysis patients – identifying the vulnerable

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Growth kinetics of colloidal chains and labyrinths

Particles interacting by a combination of isotropic short-range attraction and long-range repulsion have been shown to form complex phases despite the apparent simplicity of the interparticle potential. Using computer simulations we study the behavior of two-dimensional systems of colloids with such an interaction, focusing on how area fraction and repulsion range at fixed repulsion gradient may be used to tune the resulting kinetics and nonequilibrium structure. While the short-range attraction leads to aggregation, the long-range repulsion encourages growth of chains of particles due to repulsive intercluster interactions. Depending on area fraction/ repulsion range we observe chain labyrinths, chain-compact aggregate coexistence, and connected networks of chains. The kinetics of cluster growth displays a sequence of connected networks and disconnected cluster or chain systems with increasing repulsion range, indicating the competing roles of connectivity of growing chains and repulsion-driven breakup of chains into compact aggregates. Chain-dominated systems show approximately logarithmic coarsening at late time that we interpret as the result of chains performing random walks in the randomly fluctuating potential landscape created by their neighbors, a situation reminiscent of glassy systems

Polar Polytopes and Recovery of Sparse Representations

Suppose we have a signal y which we wish to represent using a linear combination of a number of basis atoms a_i, y=sum_i x_i a_i = Ax. The problem of finding the minimum L0 norm representation for y is a hard problem. The Basis Pursuit (BP) approach proposes to find the minimum L1 norm representation instead, which corresponds to a linear program (LP) that can be solved using modern LP techniques, and several recent authors have given conditions for the BP (minimum L1 norm) and sparse (minimum L0 solutions) representations to be identical. In this paper, we explore this sparse representation problem} using the geometry of convex polytopes, as recently introduced into the field by Donoho. By considering the dual LP we find that the so-called polar polytope P of the centrally-symmetric polytope P whose vertices are the atom pairs +-a_i is particularly helpful in providing us with geometrical insight into optimality conditions given by Fuchs and Tropp for non-unit-norm atom sets. In exploring this geometry we are able to tighten some of these earlier results, showing for example that the Fuchs condition is both necessary and sufficient for L1-unique-optimality, and that there are situations where Orthogonal Matching Pursuit (OMP) can eventually find all L1-unique-optimal solutions with m nonzeros even if ERC fails for m, if allowed to run for more than m steps