The Loewner driving function of trajectory arcs of quadratic differentials

We obtain a first order differential equation for the driving function of the chordal Loewner differential equation in the case where the domain is slit by a curve which is a trajectory arc of certain quadratic differentials. In particular this includes the case when the curve is a path on the square, triangle or hexagonal lattice in the upper halfplane or, indeed, in any domain with boundary on the lattice. We also demonstrate how we use this to calculate the driving function numerically. Equivalent results for other variants of the Loewner differential equation are also obtained: Multiple slits in the chordal Loewner differential equation and the radial Loewner differential equation. The method also works for other versions of the Loewner differential equation. The proof of our formula uses a generalization of Schwarz-Christoffel mapping to domains bounded by trajectory arcs of rotations of a given quadratic differential that is of interest in its own right.Comment: 22 pages, 4 figures Changes in v2: Changed some definitions and exchanged ordering of theorems for clarity purposes. Typos corrected. Changes in v3: Mistakes corrected. Added new Lemma 2.2. Overall clarity improve

Flex-Gears

Flex-Gears are being developed as an alternative to brushes and slip rings to conduct electricity across a rotating joint. Flex-Gears roll in the annulus of sun and ring gears for electrical contact while maintaining their position by using a novel application of involute gears. A single Flex-Gear is predicted to transfer up to 2.8 amps, thereby allowing a six inch diameter device, holding 30 Flex-Gears, to transfer over 80 amps. Semi-rigid Flex-Gears are proposed to decrease Flex-Gear stress and insure proper gear meshing

Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice

In this paper we present the proof of the convergence of the critical bond percolation exploration process on the square lattice to the trace of SLE$_{6}$. This is an important conjecture in mathematical physics and probability. The case of critical site percolation on the hexagonal lattice was established in the seminal work of Smirnov via proving Cardy's formula. Our proof uses a series of transformations and conditioning to construct a pair of paths: the $+\partial$CBP and the $-\partial$CBP. The convergence in the site percolation case on the hexagonal lattice allows us to obtain certain estimates on the scaling limit of the $+\partial$CBP and the $-\partial$CBP. By considering a path which is the concatenation of $+\partial$CBPs and $-\partial$CBPs in an alternating manner, we can prove the convergence in the case of bond percolation on the square lattice.Comment: This is a preliminary version. The first two authors attended the Planar Statistical Models workshop in Sanya from Jan 5 to Jan 8, 2013. We received many critical comments from the participants. We will revise the paper and provide a clean proo

Exploring the origins of the power-law properties of energy landscapes: An egg-box model

Multidimensional potential energy landscapes (PELs) have a Gaussian distribution for the energies of the minima, but at the same time the distribution of the hyperareas for the basins of attraction surrounding the minima follows a power-law. To explore how both these features can simultaneously be true, we introduce an ``egg-box'' model. In these model landscapes, the Gaussian energy distribution is used as a starting point and we examine whether a power-law basin area distribution can arise as a natural consequence through the swallowing up of higher-energy minima by larger low-energy basins when the variance of this Gaussian is increased sufficiently. Although the basin area distribution is substantially broadened by this process,it is insufficient to generate power-laws, highlighting the role played by the inhomogeneous distribution of basins in configuration space for actual PELs.Comment: 7 pages, 8 figure

Use of Mutated Self-Cleaving 2A Peptides as a Molecular Rheostat to Direct Simultaneous Formation of Membrane and Secreted Anti-HIV Immunoglobulins

In nature, B cells produce surface immunoglobulin and secreted antibody from the same immunoglobulin gene via alternative splicing of the pre-messenger RNA. Here we present a novel system for genetically programming B cells to direct the simultaneous formation of membrane-bound and secreted immunoglobulins that we term a “Molecular Rheostat”, based on the use of mutated “self-cleaving” 2A peptides. The Molecular Rheostat is designed so that the ratio of secreted to membrane-bound immunoglobulins can be controlled by selecting appropriate mutations in the 2A peptide. Lentiviral transgenesis of Molecular Rheostat constructs into B cell lines enables the simultaneous expression of functional b12-based IgM-like BCRs that signal to the cells and mediate the secretion of b12 IgG broadly neutralizing antibodies that can bind and neutralize HIV-1 pseudovirus. We show that these b12-based Molecular Rheostat constructs promote the maturation of EU12 B cells in an in vitro model of B lymphopoiesis. The Molecular Rheostat offers a novel tool for genetically manipulating B cell specificity for B-cell based gene therapy