Spacetime Slices and Surfaces of Revolution

Under certain conditions, a $(1+1)$-dimensional slice $\hat{g}$ of a spherically symmetric black hole spacetime can be equivariantly embedded in $(2+1)$-dimensional Minkowski space. The embedding depends on a real parameter that corresponds physically to the surface gravity $\kappa$ of the black hole horizon. Under conditions that turn out to be closely related, a real surface that possesses rotational symmetry can be equivariantly embedded in 3-dimensional Euclidean space. The embedding does not obviously depend on a parameter. However, the Gaussian curvature is given by a simple formula: If the metric is written $g = \phi(r)^{-1} dr^2 + \phi(r) d\theta^2$, then \K_g=-{1/2}\phi''(r). This note shows that metrics $g$ and $\hat{g}$ occur in dual pairs, and that the embeddings described above are orthogonal facets of a single phenomenon. In particular, the metrics and their respective embeddings differ by a Wick rotation that preserves the ambient symmetry. Consequently, the embedding of $g$ depends on a real parameter. The ambient space is not smooth, and $\kappa$ is inversely proportional to the cone angle at the axis of rotation. Further, the Gaussian curvature of $\hat{g}$ is given by a simple formula that seems not to be widely known.Comment: 15 pages, added reference

The Wigner caustic on shell and singularities of odd functions

We study the Wigner caustic on shell of a Lagrangian submanifold L of affine symplectic space. We present the physical motivation for studying singularities of the Wigner caustic on shell and present its mathematical definition in terms of a generating family. Because such a generating family is an odd deformation of an odd function, we study simple singularities in the category of odd functions and their odd versal deformations, applying these results to classify the singularities of the Wigner caustic on shell, interpreting these singularities in terms of the local geometry of L.Comment: 24 page

Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

We define the Global Centre Symmetry set (GCS) of a smooth closed m-dimensional submanifold M of R^n, $n \leq 2m$, which is an affinely invariant generalization of the centre of a k-sphere in R^{k+1}. The GCS includes both the centre symmetry set defined by Janeczko and the Wigner caustic defined by Berry. We develop a new method for studying generic singularities of the GCS which is suited to the case when M is lagrangian in R^{2m} with canonical symplectic form. The definition of the GCS, which slightly generalizes one by Giblin and Zakalyukin, is based on the notion of affine equidistants, so, we first study singularities of affine equidistants of Lagrangian submanifolds, classifying all the stable ones. Then, we classify the affine-Lagrangian stable singularities of the GCS of Lagrangian submanifolds and show that, already for smooth closed convex curves in R^2, many singularities of the GCS which are affine stable are not affine-Lagrangian stable.Comment: 26 pages, 2 figure

Population Dynamics and Community Composition of Ammonia Oxidizers in Salt Marshes after the Deepwater Horizon Oil Spill

The recent oil spill in the Gulf of Mexico had significant effects on microbial communities in the Gulf, but impacts on nitrifying communities in adjacent salt marshes have not been investigated. We studied persistent effects of oil on ammonia-oxidizing archaeal (AOA) and bacterial (AOB) communities and their relationship to nitrification rates and soil properties in Louisiana marshes impacted by the Deepwater Horizon oil spill. Soils were collected at oiled and unoiled sites from Louisiana coastal marshes in July 2012, 2 years after the spill, and analyzed for community differences based on ammonia monooxygenase genes (amoA). Terminal Restriction Fragment Polymorphism and DNA sequence analyses revealed significantly different AOA and AOB communities between the three regions, but few differences were found between oiled and unoiled sites. Community composition of nitrifiers was best explained by differences in soil moisture and nitrogen content. Despite the lack of significant oil effects on overall community composition, we identified differences in correlations of individual populations with potential nitrification rates between oiled and unoiled sites that help explain previously published correlation patterns. Our results suggest that exposure to oil, even 2 years post-spill, led to subtle changes in population dynamics. How, or if, these changes may impact ecosystem function in the marshes, however, remains uncertain

Judicial Development of Standards of Liability in Government Enforcement Actions under the Comprehensive Environmental Response, Compensation and Liability Act

The Comprehensive Environmental Response, Compensation and Liability Act of 1980 (CERCLA) represents the first major attempt on a national level to address the problem of abandoned hazardous waste sites. CERCLA permits an action to be brought in federal court for recovery of amounts disbursed from the Superfund against, inter alia, any person who arranges for treatment or disposal of wastes at a site, typically the generator of the hazardous wastes. CERCLA\u27s enforcement provisions contain numerous ambiguities and apparent inconsistencies on issues that directly affect the potential liability of CERCLA\u27s defendants. Many of the inadequacies probably can be traced to the last-minute compromise which led to the passage of the statute. The net effect of inadequate drafting and consideration of Superfund legislation is that the task of forging meaningful standards for liability under the final product is left largely to the courts. This Article discusses the significant legal issues which arise in government enforcement actions regarding the imposition of liability under CERCLA, and examines the role that courts have played in developing applicable standards of liability. This Article also examines the state of the law in the context of two common scenarios: (a) actions under section 107 of CERCLA for recovery of costs of response at a hazardous waste site; and (b) actions under section 106 of CERCLA for injunctive relief. Finally, this Article addresses: (1) who may be held liable; (2) under what conditions liability may arise; (3) what standard for liability exists; and (4) the extent of potential liability