http://hdl.handle.net/1957/13818 2013-05-25T21:20:53Z 2013-05-25T21:20:53Z Petsche, Clayton http://hdl.handle.net/1957/37907 2013-03-29T21:50:07Z 2012-11-01T00:00:00Z

Critically Separable Rational Maps in Families Petsche, Clayton Given a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps in these families, we prove a finiteness theorem which is analogous to Shafarevich’s theorem for elliptic curves. We also define the minimal critical discriminant, a global object which can be viewed as a measure of arithmetic complexity of a rational map. We formulate a conjectural bound on the minimal critical discriminant, which is analogous to Szpiro’s conjecture for elliptic curves, and we prove that a special case of our conjecture implies Szpiro’s conjecture in the semistable case. The version of record is embargoed until 10-12-2013. The final peer reviewed, accepted manuscript is available without an embargo. The published article is copyrighted by Foundation Compositio Mathematica and published by Cambridge University Press. It can be found at: http://journals.cambridge.org/action/displayJournal?jid=com.

2012-11-01T00:00:00Z Chueshov, Igor Lasiecka, Irena Webster, Justin T. http://hdl.handle.net/1957/37397 2013-03-09T01:24:09Z 2013-02-15T00:00:00Z

Evolution Semigroups in Supersonic Flow-Plate Interactions Chueshov, Igor; Lasiecka, Irena; Webster, Justin T. We consider the well-posedness of a model for a flow-structure interaction. This model describes the dynamics of an elastic flexible plate with clamped boundary conditions immersed in a supersonic flow. A perturbed wave equation describes the flow potential. The plateʼs out-of-plane displacement can be modeled by various nonlinear plate equations (including von Karman and Berger). Supersonic regimes corresponding to the flow provide for new mathematical challenge that is related to the loss of ellipticity in a stationary dynamics. This difficulty is present also in the linear model. We show that the linearized model is well-posed on the state space (as given by finite energy considerations) and generates a strongly continuous semigroup. We make use of these results along with sharp regularity of Airyʼs stress function (obtained by compensated compactness method) to conclude global-in-time well-posedness for the fully nonlinear model. The proof of generation has two novel features, namely: (1) we introduce a new flow potential velocity-type variable which makes it possible to cover both subsonic and supersonic cases, and to split the dynamics generating operator into a skew-adjoint component and a perturbation acting outside of the state space. Performing semigroup analysis also requires a nontrivial approximation of the domain of the generator. The latter is due to the loss of ellipticity. And (2) we make critical use of hidden trace regularity for the flow component of the model (in the abstract setup for the semigroup problem) which allows us to develop a fixed point argument and eventually conclude well-posedness. This well-posedness result for supersonic flows (in the absence of regularizing rotational inertia) has been hereto open. The use of semigroup methods to obtain well-posedness opens this model to long-time behavior considerations. This is the author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier and can be found at: http://www.journals.elsevier.com/journal-of-differential-equations/.

2013-02-15T00:00:00Z Geredeli, Pelin G. Lasiecka, Irena Webster, Justin T. http://hdl.handle.net/1957/37345 2013-03-05T21:56:19Z 2013-02-01T00:00:00Z

Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer Geredeli, Pelin G.; Lasiecka, Irena; Webster, Justin T. In this paper dynamic von Karman equations with localized interior damping supported in a boundary collar are considered. Hadamard well-posedness for von Karman plates with various types of nonlinear damping are well known, and the long-time behavior of nonlinear plates has been a topic of recent interest. Since the von Karman plate system is of “hyperbolic type” with critical nonlinearity (noncompact with respect to the phase space), this latter topic is particularly challenging in the case of geometrically constrained, nonlinear damping. In this paper we first show the existence of a compact global attractor for finite energy solutions, and we then prove that the attractor is both smooth and finite dimensional. Thus, the hyperbolic-like flow is stabilized asymptotically to a smooth and finite dimensional set. This is the author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier and can be found at: http://www.journals.elsevier.com/journal-of-differential-equations/.

2013-02-01T00:00:00Z Zeng, Wei Guo, Ren Luo, Feng Gu, Xianfeng http://hdl.handle.net/1957/37107 2013-02-22T19:35:42Z 2012-07-01T00:00:00Z

Discrete Heat Kernel Determines Discrete Riemannian Metric Zeng, Wei; Guo, Ren; Luo, Feng; Gu, Xianfeng The Laplace-Beltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the Laplace-Beltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given a Euclidean polyhedral surface, its Riemannian metric is represented as edge lengths, satisfying triangle inequalities on all faces. The Laplace-Beltrami operator is formulated using the cotangent formula, where the edge weight is defined as the sum of the cotangent of angles against the edge. We prove that the edge lengths can be determined by the edge weights unique up to a scaling using the variational approach. The constructive proof leads to a computational algorithm that finds the unique metric on a triangle mesh from a discrete Laplace-Beltrami operator matrix. This is the author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier and can be found at: http://www.journals.elsevier.com/graphical-models/.

2012-07-01T00:00:00Z