Analysis of Spatially-Coupled Counter Braids

A counter braid (CB) is a novel counter architecture introduced by Lu et al. in 2007 for per-flow measurements on high-speed links. CBs achieve an asymptotic compression rate (under optimal decoding) that matches the entropy lower bound of the flow size distribution. Spatially-coupled CBs (SC-CBs) have recently been proposed. In this work, we further analyze single-layer CBs and SC-CBs using an equivalent bipartite graph representation of CBs. On this equivalent representation, we show that the potential and area thresholds are equal. We also show that the area under the extended belief propagation (BP) extrinsic information transfer curve (defined for the equivalent graph), computed for the expected residual CB graph when a peeling decoder equivalent to the BP decoder stops, is equal to zero precisely at the area threshold. This, combined with simulations and an asymptotic analysis of the Maxwell decoder, leads to the conjecture that the area threshold is in fact equal to the Maxwell decoding threshold and hence a lower bound on the maximum a posteriori (MAP) decoding threshold. Finally, we present some numerical results and give some insight into the apparent gap of the BP decoding threshold of SC-CBs to the conjectured lower bound on the MAP decoding threshold.Comment: To appear in the IEEE Information Theory Workshop, Jeju Island, Korea, October 201

Conley: Computing connection matrices in Maple

In this work we announce the Maple package conley to compute connection and C-connection matrices. conley is based on our abstract homological algebra package homalg. We emphasize that the notion of braids is irrelevant for the definition and for the computation of such matrices. We introduce the notion of triangles that suffices to state the definition of (C)-connection matrices. The notion of octahedra, which is equivalent to that of braids is also introduced.Comment: conley is based on the package homalg: math.AC/0701146, corrected the false "counter example

Evolution of a barotropic shear layer into elliptical vortices

When a barotropic shear layer becomes unstable, it produces the well known Kelvin-Helmholtz instability (KH). The non-linear manifestation of KH is usually in the form of spiral billows. However, a piecewise linear shear layer produces a different type of KH characterized by elliptical vortices of constant vorticity connected via thin braids. Using direct numerical simulation and contour dynamics, we show that the interaction between two counter-propagating vorticity waves is solely responsible for this KH formation. We investigate the oscillation of the vorticity wave amplitude, the rotation and nutation of the elliptical vortex, and straining of the braids. Our analysis also provides possible explanation behind the formation and evolution of elliptical vortices appearing in geophysical and astrophysical flows, e.g. meddies, Stratospheric polar vortices, Jovian vortices, Neptune's Great Dark Spot and coherent vortices in the wind belts of Uranus.Comment: 7 pages, 4 figures, Accepted in Physical Review

Estimating topological entropy from the motion of stirring rods

Stirring a two-dimensional viscous fluid with rods is often an effective way to mix. The topological features of periodic rod motions give a lower bound on the topological entropy of the induced flow map, since material lines must `catch' on the rods. But how good is this lower bound? We present examples from numerical simulations and speculate on what affects the 'gap' between the lower bound and the measured topological entropy. The key is the sign of the rod motion's action on first homology of the orientation double cover of the punctured disk.Comment: 10 pages, 20 figures. IUTAM Procedia style (included). Submitted to volume "Topological Fluid Dynamics II.

Evolution method and "differential hierarchy" of colored knot polynomials

We consider braids with repeating patterns inside arbitrary knots which provides a multi-parametric family of knots, depending on the "evolution" parameter, which controls the number of repetitions. The dependence of knot (super)polynomials on such evolution parameters is very easy to find. We apply this evolution method to study of the families of knots and links which include the cases with just two parallel and anti-parallel strands in the braid, like the ordinary twist and 2-strand torus knots/links and counter-oriented 2-strand links. When the answers were available before, they are immediately reproduced, and an essentially new example is added of the "double braid", which is a combination of parallel and anti-parallel 2-strand braids. This study helps us to reveal with the full clarity and partly investigate a mysterious hierarchical structure of the colored HOMFLY polynomials, at least, in (anti)symmetric representations, which extends the original observation for the figure-eight knot to many (presumably all) knots. We demonstrate that this structure is typically respected by the t-deformation to the superpolynomials.Comment: 31 page