Curve diffusion and straightening flows on parallel lines

In this paper, we study families of immersed curves $\gamma:(-1,1)\times[0,T)\rightarrow\mathbb{R}^2$ with free boundary supported on parallel lines $\{\eta_1, \eta_2\}:\mathbb{R}\rightarrow\mathbb{R}^2$ evolving by the curve diffusion flow and the curve straightening flow. The evolving curves are orthogonal to the boundary and satisfy a no-flux condition. We give estimates and monotonicity on the normalised oscillation of curvature, yielding global results for the flows.Comment: 35 pages, 3 figure

A checklist of the ants of Wyoming (Hymenoptera: Formicidae)

The history of myrmrecology in Wyoming began when Fore1 described a new species, obscuripes in the genus Formica from Green River in 1886, four years before Wyoming Territory became a state. There was a long hiatus until 1932 when Cole cited ten records for Pogonomyrmex occidentalis. Since then a dozen authors have published one to many records each. During the summers of 1957, 1958, 1961, 1963, 1964,1965 and 1966, while we were still at the University of North Dakota (Grand Forks), we made a dozen field trips into Wyoming to observe and collect ants in 12 of the 23 counties and in Yellowstone National Park. These expeditions yielded a total of 168 records for 45 species (a record is a species in a locality)

Mean curvature flow with free boundary outside a hypersphere

The purpose of this paper is twofold: firstly, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface, and secondly, to illustrate the application of Killing vector fields in the preservation of graphicality for the mean curvature flow with free boundary. To this end we focus on the mean curvature flow of a topological annulus with inner boundary meeting a standard n-sphere in \R^{n+1} perpendicularly and outer boundary fixed to an (n-1)-sphere with radius R>1 at a fixed height h. We call this the \emph{sphere problem}. Our work is set in the context of graphical mean curvature flow with either symmetry or mean concavity/convexity restrictions. For rotationally symmetric initial data we obtain, depending on the exact configuration of the initial graph, either long time existence and convergence to a minimal hypersurface with boundary or the development of a finite-time curvature singularity. With reflectively symmetric initial data we are able to use Killing vector fields to preserve graphicality of the flow and uniformly bound the mean curvature pointwise along the flow. Finally we prove that the mean curvature flow of an initially mean concave/convex graphical surface exists globally in time and converges to a piece of a minimal surface.Comment: 23 page

Concentration-compactness and finite-time singularities for Chen's flow

Chen's flow is a fourth-order curvature flow motivated by the spectral decomposition of immersions, a program classically pushed by B.-Y. Chen since the 1970s. In curvature flow terms the flow sits at the critical level of scaling together with the most popular extrinsic fourth-order curvature flow, the Willmore and surface diffusion flows. Unlike them however the famous Chen conjecture indicates that there should be no stationary nonminimal data, and so in particular the flow should drive all closed submanifolds to singularities. We investigate this idea, proving that (1) closed data becomes extinct in finite time in all dimensions and for any codimension; (2) singularities are characterised by concentration of curvature in $L^n$ for intrinsic dimension $n \in \{2,4\}$ and any codimension (a Lifespan Theorem); and (3) for $n = 2$ and in one codimension only, there exists an explicit small constant $\varepsilon_2$ such that if the $L^2$ norm of the tracefree curvature is initially smaller than $\varepsilon_2$, the flow remains smooth until it shrinks to a point, and that the blowup of that point is an embedded smooth round sphere.Comment: 48 page

On a curvature flow model for embryonic epidermal wound healing

The paper studies a curvature flow linked to the physical phenomenon of wound closure. Under the flow we show that a closed, initially convex or close-to-convex curve shrinks to a round point in finite time. We also study the singularity, showing that the singularity profile after continuous rescaling is that of a circle. We additionally give a maximal time estimate, with an application to the classification of blowups.Comment: 43 page

Effect of Change in Apportionment Formula on Georgia Corporate Tax Liability - Brief

The purpose of this brief is to identify the number and characteristics of corporations that are expected to benefit from the switch to a single-factor apportionment formula and the number and characteristics of those that are not. FRC Brief 20

European Union state aid, public subsidies and analogue switch-off/digital switchover

This article considers the European Union’s (EU) role in the media and communications field concerning analogue switch-off/digital switchover. It focuses on the EU Competition Directorate’s approach concerning the application of the State Aid mechanism with regard to those Member States who have used public subsidies for digital switchover. Therefore, this analysis considers how the Directorate has sought to balance its market-driven set of rules with the need to be adaptable to the Member States’ specific requirements. In turn, this account will discuss how the demand to achieve the 2012 completion deadline, alongside the requirement to release analogue spectrum to realize a digital dividend, impacted upon the EU’s principles of competition. Finally, this analysis will reflect on how the employment of State Aid in relation to digital switchover relates to a wider debate concerning the EU’s neo-liberal agendas against Member States’ interests to promote their national cultures

Analysis and Recommendations for the Property Tax on Motor Vehicles in Georgia

This report discusses the economic effects, including revenue effects, of eliminating or reducing the state property tax in motor vehicles. FRC Report 14

Mass Shootings and the Media: How Race and Ethnicity Influence Media Coverage

Topics related to crime and the criminal justice system comprise a majority of topics discussed in the media. This study explores how media coverage of mass shootings varies based on the racial or ethnic identity of the shooter. Topics examined include popular beliefs and misconceptions about the race of mass murderers, the role mass media plays in influencing public perceptions about race and crime related topics, and the alarming rate at which the public recognizes and accepts the media as a reliable source of information when official data sources provide a much different picture. Further analysis of the Columbine High School massacre and Virginia Tech shooting provides a more targeted and detailed examination of how the media dealt with the issue of race when reporting on these two shootings. Additional research focusing on a larger selection of mass shootings is needed to provide a more comprehensive understanding of the problem in its entirety. Regardless, the mass media’s representation of a shooter’s race and ethnicity has serious policy implications for the criminal justice system and public intergroup relations, indicating that extensive changes in how the media reports information surrounding these events are required