Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer

Let $G$ be an $n$-node planar graph. In a visibility representation of $G$, each node of $G$ is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of $G$ are vertically visible to each other. In the present paper we give the best known compact visibility representation of $G$. Given a canonical ordering of the triangulated $G$, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained from Schnyder's realizer for the triangulated $G$ yields a visibility representation of $G$ no wider than $\frac{22n-40}{15}$. Our easy-to-implement O(n)-time algorithm bypasses the complicated subroutines for four-connected components and four-block trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant's open question about whether $\frac{3n-6}{2}$ is a worst-case lower bound on the required width. Also, if $G$ has no degree-three (respectively, degree-five) internal node, then our visibility representation for $G$ is no wider than $\frac{4n-9}{3}$ (respectively, $\frac{4n-7}{3}$). Moreover, if $G$ is four-connected, then our visibility representation for $G$ is no wider than $n-1$, matching the best known result of Kant and He. As a by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem on realizers, due to Bonichon, Sa\"{e}c, and Mosbah.Comment: 11 pages, 6 figures, the preliminary version of this paper is to appear in Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Berlin, Germany, 200

Microstructure control during twin roll casting of an AZ31 magnesium alloy

The existing twin roll casting technique for magnesium alloys suffers heterogeneity in both microstructure and chemistry and downstream processing is required to improve the strip quality, resulting in cost rise. In the present work, twin roll casting was carried out using an AZ31 magnesium alloy, with the application of intensive shearing melt conditioning prior to casting. The effect of process parameters such as pouring temperature and casting speed on microstructure control during casting and subsequent downstream processing was studied. Experimental results showed that the melt conditioning treatment allowed the production of AZ31 strips with uniform and refined microstructure free of centreline segregations. It was also shown that an optimized combination of pouring temperature and casting speed, in conjunction with a strip thickness control operation, resulted in uniformly distributed stored energies due to enhanced plastic deformation, which promoted recrystallization during casting and subsequent heat treatment. Strips prepared by twin roll casting and homogenization developed similar microstructural features to those prepared by twin roll casting followed by lengthy downstream processing by homogenization, hot rolling and annealing and displayed a weaker basal texture, exhibiting a potentially better formability.The EPSRC (UK

Pollution Taxes and Location Decision under Free Entry Oligopoly

This paper examines the impact of a pollution tax as a pollution control device on the output and location decisions of undifferentiated oligopolistic firms with free entry. It shows that the optimum output and location of an oligopolistic firm is independent of a change in the pollution tax if the demand function is linear. Furthermore, an increase in the pollution tax will increase (decrease) output and move the plant location toward (away from) the CBD if the demand function is concave (convex). It also shows that a higher pollution tax will increase the pollution damage if the demand function is linear and the location effect dominates the demand effect. These results are significantly different from the conventional results based on the monopolistic location model. It indicates that the demand condition plays an important role in the determination of the impact of a pollution tax on the location decision of an oligopolistic firm and the pollution damage to the CBD residents.

Twin roll casting and melt conditioned twin-roll casting of magnesium alloys

Recently, BCAST at Brunel University has developed a MCAST (melt conditioning by advanced shear technology) process for conditioning liquid metal at temperature either above or bellow the alloy liquidus using a high shear twin-screw mechanism. The MCAST process has now been combined with the twin roll casting (TRC) process to form an innovative technology, namely, the melt conditioned twin roll casting (MC-TRC) process for casting Al-alloy and Mg-alloy strips. During the MC-TRC process, liquid alloy with a specified temperature is continuously fed into the MCAST machine. By intensive shearing under the high shear rate and high intensity of turbulence, the liquid is transformed into conditioned melt with uniform temperature and composition throughout the whole volume. The conditioned melt is then fed continuously into the twin-roll caster for strip production. The experimental results show that the AZ91D MC-TRC strips with different thicknesses have fine and uniform microstructure. The strip consists of equiaxed grains with a mean size of 60-70μm. The strip displays extremely uniform grain size and composition throughout the whole cross-section. Investigation also shows that both TRC and MC-TRC processes with reduced deformation are effective to reduce the formation of defects, particularly the formation of the central line segregations

Inversion formula and Parsval theorem for complex continuous wavelet transforms studied by entangled state representation

In a preceding Letter (Opt. Lett. 32, 554 (2007)) we have proposed complex continuous wavelet transforms (CCWTs) and found Laguerre--Gaussian mother wavelets family. In this work we present the inversion formula and Parsval theorem for CCWT by virtue of the entangled state representation, which makes the CCWT theory complete. A new orthogonal property of mother wavelet in parameter space is revealed.Comment: 4 pages no figur

A V-Diagram for the Design of Integrated Health Management for Unmanned Aerial Systems

Designing Integrated Vehicle Health Management (IVHM) for Unmanned Aerial Systems (UAS) is inherently complex. UAS are a system of systems (SoS) and IVHM is a product-service, thus the designer has to take into account many factors, such as: the design of the other systems of the UAS (e.g. engines, structure, communications), the split of functions between elements of the UAS, the intended operation/mission of the UAS, the cost verses benefit of monitoring a system/component/part, different techniques for monitoring the health of the UAS, optimizing the health of the fleet and not just the individual UAS, amongst others. The design of IVHM cannot sit alongside, or after, the design of UAS, but itself be integrated into the overall design to maximize IVHM’s potential. Many different methods exist to help design complex products and manage the process. One method used is the V-diagram which is based on three concepts: decomposition & definition; integration & testing; and verification & validation. This paper adapts the V-diagram so that it can be used for designing IVHM for UAS. The adapted v-diagram splits into different tracks for the different system elements of the UAS and responses to health states (decomposition and definition). These tracks are then combined into an overall IVHM provision for the UAS (integration and testing), which can be verified and validated. The stages of the adapted V-diagram can easily be aligned with the stages of the V-diagram being used to design the UAS bringing the design of the IVHM in step with the overall design process. The adapted V-diagram also allows the design IVHM for a UAS to be broken down in to smaller tasks which can be assigned to people/teams with the relevant competencies. The adapted V-diagram could also be used to design IVHM for other SoS and other vehicles or products