A heat trace anomaly on polygons

Let $\Omega_0$ be a polygon in \RR^2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that \Omega_\e is a family of surfaces with \calC^\infty boundary which converges to $\Omega_0$ smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov \cite{Fe}, Kac \cite{K} and McKean-Singer \cite{MS} recognized that certain heat trace coefficients, in particular the coefficient of $t^0$, are not continuous as \e \searrow 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain $Z$ which models the corner formation. The result applies both for Dirichlet and Neumann conditions. We also include a discussion of what one might expect in higher dimensions.Comment: Revision includes treatment of the Neumann problem and a discussion of the higher dimensional case; some new reference

Securities Class Actions, CAFA and a Countrywide Crisis: A Call for Clarity and Consistency

The unfolding of the credit crisis raises novel issues in securities litigation. This Note explores the conflict between the nonremoval provision of the Securities Act of 1933 (’33 Act) and the removal provisions of the Class Action Fairness Act of 2005 (CAFA), and their interplay in the context of class actions involving mortgage-backed securities. Circuits are currently split over whether or not such class actions are removable under CAFA. The Seventh Circuit and the Southern District of New York have held that class actions asserting only ’33 Act claims are removable under CAFA unless they fall within one of CAFA’s exceptions, while the Ninth Circuit has found that the ’33 Act’s nonremoval provision trumps CAFA. This Note unpacks the historical purposes and legislative histories of CAFA and the federal securities laws, and examines them under the lens of the current financial crisis. The Note argues that the Seventh Circuit interpretation is superior because it gives effect to all of CAFA’s provisions, as well as the historical purposes of the ’33 Act. CAFA applies to all class actions, including securities class actions, but not to individual securities actions. Individual securities actions are not removable under the ’33 Act, thus giving effect to the nonremoval provision and its historical purposes of providing investor protection. While looking to the past is instructive, courts should consider the present situation. Though they are not traded on a national exchange, mortgage-backed securities are currently at the heart of the countrywide credit crisis. To that end, this Note proposes that the approach with the most clarity and consistency is to allow for removal to federal courts of securities cases that are of real national importance

Ricci flow of conformally compact metrics

In this paper we prove that given a smoothly conformally compact metric there is a short-time solution to the Ricci flow that remains smoothly conformally compact. We adapt recent results of Schn\"urer, Schulze and Simon to prove a stability result for conformally compact Einstein metrics sufficiently close to the hyperbolic metric.Comment: 26 pages, 2 figures. Version 2 includes stronger stability result and fixes several typo

Impact of the High Speed Trains on the European Cities Hierarchy

The European space is marked by the recent beginning of the dualism nation-region. In it the cities take on a fundamental role because their success becomes the success of the territories around. Manuel Castell has maintained that the city is the social structure in which any territorial phenomena (from the economical development processes to the relations between classes or ethnic groups, from the public intervention to the financial accumulation) takes on its bigger strength because in it are concentrated the focusing in the territorial transformations. Obvious the cities are not the same, for physical or functional dimension; besides every innovation adds and modifies the relational system previously created. Aim of the paper is to analyse the factors generating the urban hierarchies to the European level and the impact on it of the new high velocity nets. In the first section it is carried out a reading/analysis of the hierarchies in the urban European system, as outlined in a series of studies. The second section analyses the role of the communication infrastructures in the building of the hierarchies and, in the third, is deepen the impact of the building of European high speed network on the fluctuations in the cities hierarchy. The paper asserts that the hierarchy is influenced by the growing of this infrastructure only for the second level positions, while the head positions are not influenced by it. One of the possible conclusion is that in a mature situation as the European territorial system, the urban structure seems to be well organized around poles with a strong persistence. This does not mean that a city could not climb the hierarchies, although this is possible only if a number of preconditions and of support policies are verified and with the remarks that this does not seem to affect the head positions, characterized by large stability.

Fredholm theory for elliptic operators on quasi-asymptotically conical spaces

We consider the mapping properties of generalized Laplace-type operators ${\mathcal L} = \nabla^* \nabla + {\mathcal R}$ on the class of quasi-asymptotically conical (QAC) spaces, which provide a Riemannian generalization of the QALE manifolds considered by Joyce. Our main result gives conditions under which such operators are Fredholm when between certain weighted Sobolev or weighted H\"older spaces. These are generalizations of well-known theorems in the asymptotically conical (or asymptotically Euclidean) setting, and also sharpen and extend corresponding theorems by Joyce. The methods here are based on heat kernel estimates originating from old ideas of Moser and Nash, as developed further by Grigor'yan and Saloff-Coste. As demonstrated by Joyce's work, the QAC spaces here contain many examples of gravitational instantons, and this work is motivated by various applications to manifolds with special holonomy

Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space

Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $\Delta u = f(u)$ on hyperbolic space \HH^n must reduce to functions of one variable provided they admit asymptotic boundary values on the infinite boundary of \HH^n which are invariant under a cohomogeneity one subgroup of the group of isometries of \HH^n. We also prove existence of these one-dimensional solutions.Comment: 24 page