Singularities in Speckled Speckle

Speckle patterns produced by random optical fields with two (or more) widely different correlation lengths exhibit speckle spots that are themselves highly speckled. Using computer simulations and analytic theory we present results for the point singularities of speckled speckle fields: optical vortices in scalar (one polarization component) fields; C points in vector (two polarization component) fields. In single correlation length fields both types of singularities tend to be more{}-or{}-less uniformly distributed. In contrast, the singularity structure of speckled speckle is anomalous: for some sets of source parameters vortices and C points tend to form widely separated giant clusters, for other parameter sets these singularities tend to form chains that surround large empty regions. The critical point statistics of speckled speckle is also anomalous. In scalar (vector) single correlation length fields phase (azimuthal) extrema are always outnumbered by vortices (C points). In contrast, in speckled speckle fields, phase extrema can outnumber vortices, and azimuthal extrema can outnumber C points, by factors that can easily exceed $10^{4}$ for experimentally realistic source parameters

Linear systems solvers - recent developments and implications for lattice computations

We review the numerical analysis' understanding of Krylov subspace methods for solving (non-hermitian) systems of equations and discuss its implications for lattice gauge theory computations using the example of the Wilson fermion matrix. Our thesis is that mature methods like QMR, BiCGStab or restarted GMRES are close to optimal for the Wilson fermion matrix. Consequently, preconditioning appears to be the crucial issue for further improvements.Comment: 7 pages, LaTeX using espcrc2.sty, 2 figures, 9 eps-files, Talk presented at LATTICE96(algorithms), submitted to Nucl. Phys. B, Proc. Supp

A Categorical Construction of Bachmann-Howard Fixed Points

Peter Aczel has given a categorical construction for fixed points of normal functors, i.e. dilators which preserve initial segments. For a general dilator $X\mapsto T_X$ we cannot expect to obtain a well-founded fixed point, as the order type of $T_X$ may always exceed the order type of $X$. In the present paper we show how to construct a Bachmann-Howard fixed point of $T$, i.e. an order $\operatorname{BH}(T)$ with an "almost" order preserving collapse $\vartheta:T_{\operatorname{BH}(T)}\rightarrow\operatorname{BH}(T)$. Building on previous work, we show that $\Pi^1_1$-comprehension is equivalent to the assertion that $\operatorname{BH}(T)$ is well-founded for any dilator $T$.Comment: This version has been accepted for publication in the Bulletin of the London Mathematical Societ

The trade response to global downturns : historical evidence

The author examines the impact of historical global downturns on trade flows. The results provide insight into why trade has dropped so dramatically in the current crisis, what is likely to happen in the coming years, how global imbalances are affected, and which regions and industries suffer most heavily. The author finds that the elasticity of global trade volumes to real world GDP has increased gradually from around 2 in the 1960s to above 3 now. The author also finds that trade is more responsive to GDP during global downturns than in tranquil times. The results suggest that the overall drop in real trade this year is likely to exceed 15 percent. There is significant variation across industries, with food and beverages the least affected and crude materials and fuels the most affected. On the positive side, trade tends to rebound very rapidly when the outlook brightens. The author also finds evidence that global downturns often lead to persistent improvements in the ratio of the trade balance to GDP in borrower countries.Economic Theory&Research,Emerging Markets,Free Trade,Trade Policy,Currencies and Exchange Rates

Optical M0bius Strips in Three Dimensional Ellipse Fields: Lines of Linear Polarization

The minor axes of, and the normals to, the polarization ellipses that surround singular lines of linear polarization in three dimensional optical ellipse fields are shown to be organized into Mobius strips and into structures we call rippled rings (r-rings). The Mobius strips have two full twists, and can be either right- or left-handed. The major axes of the surrounding ellipses generate cone-like structures. Three orthogonal projections that give rise to 15 indices are used to characterize the different structures. These indices, if independent, could generate 839,808 geometrically and topologically distinct lines; selection rules are presented that reduce the number of lines to 8,248, some 5,562 of which have been observed in a computer simulation. Statistical probabilities are presented for the most important index combinations in random fields. It is argued that it is presently feasible to perform experimental measurements of the Mobius strips, r-rings, and cones described here theoretically

Pade-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems

A standard approach to reduced-order modeling of higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for reduced-order modeling of first-order systems. While this approach results in reduced-order models that are characterized as Pade-type or even true Pade approximants of the system's transfer function, in general, these models do not preserve the form of the original higher-order system. In this paper, we present a new approach to reduced-order modeling of higher-order systems based on projections onto suitably partitioned Krylov basis matrices that are obtained by applying Krylov-subspace techniques to an equivalent first-order system. We show that the resulting reduced-order models preserve the form of the original higher-order system. While the resulting reduced-order models are no longer optimal in the Pade sense, we show that they still satisfy a Pade-type approximation property. We also introduce the notion of Hermitian higher-order linear dynamical systems, and we establish an enhanced Pade-type approximation property in the Hermitian case