Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs

It is shown that the edges of any n-point vertex expander can be replaced by new edges so that the resulting graph is an edge expander, and such that any two vertices that are joined by a new edge are at distance in the original graph. This result is optimal, and is shown to have various geometric consequences. In particular, it is used to obtain an alternative perspective on the recent algorithm of Arora et al. [Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, 2004, pp. 222-231.] for approximating the edge expansion of a graph, and to give a nearly optimal lower bound on the ratio between the observable diameter and the diameter of doubling metric measure spaces which are quasisymmetrically embeddable in Hilbert space.     

Dislocation glide through non-randomly distributed point obstacles

Abstract

Classical meso-scale models for dislocation–obstacle interactions have, by and large, assumed a random distribution of obstacles on the glide plane. While a good approximation in many situations, this does not represent materials where obstacles are clustered on the glide plane. In this work, we have investigated the statistical problem of a dislocation sampling a set of clustered point obstacles in the glide plane using a modified areal-glide model. The results of these simulations show two clear regimes. For weak obstacles, the spatial distribution does not matter and the critically resolved shear stress is found to be independent of the degree of clustering. In contrast, above a critical obstacle strength determined by the degree of clustering, the critical resolved shear strength becomes constant. It is shown that this behaviour can be explained semi-analytically by considering the probability of interaction between the dislocation line and obstacles at a given level of stress. The consequences for alloys exhibiting solute clustering are discussed.     

Mobile geometric graphs: detection, coverage and percolation

We consider the following dynamic Boolean model introduced by van den Berg et al. (Stoch. Process. Appl. 69:247–257, 1997). At time 0, let the nodes of the graph be a Poisson point process in ${\mathbb{R}^d}$ with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes whose distance is at most r. We study three fundamental problems in this model: detection (the time until a target point—fixed or moving—is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis.     

Synthetic infrared spectra

We have designed, microfabricated, and characterized a diffractive optical element that reproduces the infrared spectrum of HF from 3600 to 4300 cm(-1) . The reflection-mode diffractive optic consists of 4096 lines, each 4.5mum wide, at 16 discrete depths relative to the substrate from 0 to 1.2 mum and was fabricated upon a silicon wafer by anisotropic reactive ion-beam etching in a four-mask-level process. We envisage the use of diffractive optical elements of this type as the basis for a new class of miniaturized, remote chemical sensor systems based on correlation spectroscopy.     

Microchip Nd:vanadate lasers at 1342 and 671nm

As much as 105 mW of single-frequency output at 1342 nm was obtained from a diode-laser-pumped Nd:YVO(4) microchip laser. An intracavity frequency-doubled device generated 10 mW of single-frequency red radiation.     

Directed neural stem cell differentiation on polyaniline-coated high strength hydrogels

Stem-cell-based neural regeneration has received significant attention, as it has potential to restore functionality to diseased or damaged neural tissues that have a limited ability to self-repair or regenerate. Culturing neural stem cells (NSCs) on hydrogel substrates has been shown to facilitate differentiation to neural progenitors, but this has only been achieved on very soft hydrogels, greatly increasing the difficulty of manufacture and limiting their wide applications. Here, we realized the differentiation of NSCs to neural and glial progenitors on high-strength hydrogels. Hydrogen-bonding-strengthened conductive hydrogels (PVV-PANI) were synthesized through one-pot copolymerization of 2-vinyl-4,6-diamino-1,3,5-triazine, 1-vinylimidazole and polyethylene glycol diacrylate, followed by post-coating with polyaniline (PANI). Diaminotriazine-diaminotriazine hydrogen bonding dramatically increases their mechanical strength, while copolymerization with VI pronouncedly promotes the adsorption of PANI particles, endowing the hydrogels with electrical conductivity. These hydrogels exhibit tensile strengths up to 1.16 MPa, a 559% breaking strain, a 9.9 MPa compressive strength and up to 16.7 mS/cm conductivity. Importantly, PVV-PANI hydrogels support the attachment, proliferation, and differentiation of NSCs, and allow the efficient induction of neural and glial differentiation via electrical stimulation. This work demonstrates high-strength conductive hydrogels can serve as an electroactive soft-wet platform for modulating the specific differentiation of NSCs, a significant step towards cell-based therapies for neurological diseases.     

Interpreting weak value amplification with a toy realist model

Constructing an ontology for quantum theory is challenging, in part due to measurement back-action. The Aharonov-Albert-Vaidman weak measurement formalism provides a method to predict measurement results (weak values) when back-action is negligible. The weak value appears analogous to a classical conditional mean, yet can be complex and unbounded. We study weak values in the context of a recent quantum optical experiment involving two-photon interactions. The results of the experiment are reinterpreted within a realist ‘stochastic optics’ model of light. We show that the conditional means of the intensities in the model correspond to the experimentally observed weak values and study the breakdown of the model outside the experimentally probed regime in the limit where the weak value predicts ‘anomalous’ results.     

The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2 × 2 matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.     

On the generation of exact solutions for evaluating numerical schemes and estimating discretization error

In this paper we further develop the Method of Nearby Problems (MNP) for generating exact solutions to realistic partial differential equations by extending it to two dimensions. We provide an extensive discussion of the 2D spline fitting approach which provides C^k continuity (continuity of the solution value and its first k derivatives) along spline boundaries and is readily extendable to higher dimensions. A detailed one-dimensional example is given to outline the general concepts, then the two-dimensional spline fitting approach is applied to two problems: heat conduction with a distributed source term and the viscous, incompressible flow in a lid-driven cavity with both a constant lid velocity and a regularized lid velocity (which removes the strong corner singularities). The spline fitting approach results in very small spline fitting errors for the heat conduction problem and the regularized driven cavity, whereas the fitting errors in the standard lid-driven cavity case are somewhat larger due to the singular behaviour of the pressure near the driven lid. The MNP approach is used successfully as a discretization error estimator for the driven cavity cases, outperforming Richardson extrapolation which requires two grid levels. However, MNP has difficulties with the simpler heat conduction case due to the discretization errors having the same magnitude as the spline fitting errors.