A PDE viewpoint on basic properties of coordination algorithms with symmetries

Several recent control applications consider the coordination of subsystems through local interaction. Often the interaction has a symmetry in state space, e.g. invariance with respect to a uniform translation of all subsystem values. The present paper shows that in presence of such symmetry, fundamental properties can be highlighted by viewing the distributed system as the discrete approximation of a partial differential equation. An important fact is that the symmetry on the state space differs from the popular spatial invariance property, which is not necessary for the present results. The relevance of the viewpoint is illustrated on two examples: (i) ill-conditioning of interaction matrices in coordination/consensus problems and (ii) the string instability issue. ©2009 IEEE.

reducing symmetries to generate easier sat instances

Finding countermodels is an effective way of disproving false conjectures. In first-order predicate logic, model finding is an undecidable problem. But if a finite model exists, it can be found by exhaustive search. The finite model generation problem in the first-order logic can also be translated to the satisfiability problem in the propositional logic. But a direct translation may not be very efficient. This paper discusses how to take the symmetries into account so as to make the resulting problem easier. A static method for adding constraints is presented, which can be thought of as an approximation of the least number heuristic (LNH). Also described is a dynamic method, which asks a model searcher like SEM to generate a set of partial models, and then gives each partial model to a propositional prover. The two methods are analyzed, and compared with each other.

Non-relativistic conformal symmetries in fluid mechanics

The symmetries of a free incompressible fluid span the Galilei group, augmented with independent dilations of space and time. When the fluid is compressible, the symmetry is enlarged to the expanded Schrodinger group, which also involves, in addition, Schrodinger expansions. While incompressible fluid dynamics can be derived as an appropriate non-relativistic limit of a conformally invariant relativistic theory, the recently discussed conformal Galilei group, obtained by contraction from the relativistic conformal group, is not a symmetry. This is explained by the subtleties of the non-relativistic limit.

Explanation for Micromorphologies Around Broken Fibers in Fiber-Reinforced Composites

Experimental observations on micromorphologies around broken fibers in glass-fiber-reinforced epoxy matrix composites reveal different kinds of highly oriented patches at the circumambience of broken fibers, whereas the bulk of the matrix has been observed to be largely isotropic. These patches are interpreted to correlated areas where the stress gradients of the matrix are formed after fiber breaking, but the underlying cause for the orientation is still unknown. The authors have modified an embedded cell model to explain the experimental phenomena. The finite element simulation indicates that the surfaces around broken fibers display a change from an extension micromorphology to a mixed tension and shear micromorphology with the increase of applied strain.

Local Rotational Symmetries

This thesis describes a new representation for two-dimensional round regions called Local Rotational Symmetries. Local Rotational Symmetries are intended as a companion to Brady's Smoothed Local Symmetry Representation for elongated shapes. An algorithm for computing Local Rotational Symmetry representations at multiple scales of resolution has been implemented and results of this implementation are presented. These results suggest that Local Rotational Symmetries provide a more robustly computable and perceptually accurate description of round regions than previous proposed representations. In the course of developing this representation, it has been necessary to modify the way both Smoothed Local Symmetries and Local Rotational Symmetries are computed. First, grey-scale image smoothing proves to be better than boundary smoothing for creating representations at multiple scales of resolution, because it is more robust and it allows qualitative changes in representations between scales. Secondly, it is proposed that shape representations at different scales of resolution be explicitly related, so that information can be passed between scales and computation at each scale can be kept local. Such a model for multi-scale computation is desirable both to allow efficient computation and to accurately model human perceptions.

Public sector pay in the UK: has the historic pattern of public/private sector pay movements been broken?

The historic pattern of public sector pay movements in the UK has been counter-cyclical with private sector pay growth. Periods of relative decline in public sector pay against private sector movements have been followed by periods of ‘catch-up’ as Government controls are eased to remedy skill shortages or deal with industrial unrest among public servants. Public sector ‘catch up’ increases have therefore come at awkward times for Government, often coinciding with economic downturn in the private sector (Trinder 1994, White 1996, Bach 2002). Several such epochs of public sector pay policy can be identified since the 1970s. The question is whether the current limits on public sector pay being imposed by the UK Government fit this historic pattern or whether the pattern has been broken and, if so, how and why? This paper takes a historical approach in considering the context to public sector pay determination in the UK. In particular the paper seeks to review the period since Labour came into office (White and Hatchett 2003) and the various pay ‘modernisation’ exercises that have been in process over the last decade (White 2004). The paper draws on national statistics on public sector employment and pay levels to chart changes in public sector pay policy and draws on secondary literature to consider both Government policy intentions and the impact of these policies for public servants.

Broken symmetry and pseudogaps in ropes of carbon nanotubes

Since the discovery of carbon nanotubes, it has been speculated that these materials should behave like nanoscale wires with unusual electronic properties and exceptional strength. Recently, 'ropes' of close-packed single-wall nanotubes have been synthesized in high yield. The tubes in these ropes are mainly of the (10,10) type3, which is predicted to be metallic. Experiments on individual nanotubes and ropes indicate that these systems indeed have transport properties that qualify them to be viewed as nanoscale quantum wires at low temperature. It has been expected that the close-packing of individual nanotubes into ropes does not change their electronic properties significantly. Here, however, we present first-principles calculations which show that a broken symmetry of the (10,10) tube caused by interactions between tubes in a rope induces a pseudogap of about 0.1 eV at the Fermi level. This pseudogap strongly modifies many of the fundamental electronic properties: we predict a semimetal-like temperature dependence of the electrical conductivity and a finite gap in the infrared absorption spectrum. The existence of both electron and hole charge carriers will lead to qualitatively different thermopower and Hall-effect behaviours from those expected for a normal metal.

Broken symmetry and pseudogaps in ropes of carbon nanotubes

We investigate the influence of tube-tube interactions in ropes of (10,10) carbon nanotubes, and find that these effects induce a pseudogap in the density of state (DOS) of the rope of width 0.1 eV at the Fermi level. In an isolated (n,n) carbon nanotube there are two bands that cross in a linear fashion at the Fermi level, making the nanotube metallic with a DOS that is constant in a 1.5 eV wide window around the Fermi energy. The presence of the neighbouring tubes causes these two bands to repel, opening up a band gap that can be as large as 0.3 eV. The small dispersion in the plane perpendicular to the rope smears out this gap for a rope with a large cross-sectional area, and we see a pseudogap at the Fermi energy in the DOS where the DOS falls to one third of its value for the isolated tube. This phenomenon should affect many properties of the behavior of ropes of (n,n) nanotubes, which should display a more semimetallic character than expected in transport and doping experiments, with the existence of both hole and electron carriers leading to qualitatively different thermopower and Hall-effect behaviors from those expected for a normal metal. Band repulsion like this can be expected to occur for any tube perturbed by a sufficiently strong interaction, for example, from contact with a surface or with other tubes.