On Some Aspects of Oscillation Theory and Geometry
On Some Aspects of Oscillation Theory and Geometry

Author: Bruno Bianchini

Publisher: American Mathematical Soc.

Published: 2013-08-23

Total Pages: 208

ISBN-13: 0821887998

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The aim of this paper is to analyze some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. With this motivation the authors prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE's that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep their investigation basically self-contained, the authors also collect some, more or less known, material which often appears in the literature in various forms and for which they give, in some instances, new proofs according to their specific point of view.

Recent Trends in Nonlinear Partial Differential Equations I
Recent Trends in Nonlinear Partial Differential Equations I

Author: James B. Serrin

Publisher: American Mathematical Soc.

Published: 2013-07-22

Total Pages: 323

ISBN-13: 082188736X

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This book is the first of two volumes which contain the proceedings of the Workshop on Nonlinear Partial Differential Equations, held from May 28-June 1, 2012, at the University of Perugia in honor of Patrizia Pucci's 60th birthday. The workshop brought t

Combinatorial Floer Homology
Combinatorial Floer Homology

Author: Vin de Silva

Publisher: American Mathematical Soc.

Published: 2014-06-05

Total Pages: 126

ISBN-13: 0821898868

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The authors define combinatorial Floer homology of a transverse pair of noncontractible nonisotopic embedded loops in an oriented -manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original Lagrangian Floer homology. Their proof uses a formula for the Viterbo-Maslov index for a smooth lune in a -manifold.

Effective Hamiltonians for Constrained Quantum Systems
Effective Hamiltonians for Constrained Quantum Systems

Author: Jakob Wachsmuth

Publisher: American Mathematical Soc.

Published: 2014-06-05

Total Pages: 96

ISBN-13: 0821894897

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The authors consider the time-dependent Schrödinger equation on a Riemannian manifold with a potential that localizes a certain subspace of states close to a fixed submanifold . When the authors scale the potential in the directions normal to by a parameter , the solutions concentrate in an -neighborhood of . This situation occurs for example in quantum wave guides and for the motion of nuclei in electronic potential surfaces in quantum molecular dynamics. The authors derive an effective Schrödinger equation on the submanifold and show that its solutions, suitably lifted to , approximate the solutions of the original equation on up to errors of order at time . Furthermore, the authors prove that the eigenvalues of the corresponding effective Hamiltonian below a certain energy coincide up to errors of order with those of the full Hamiltonian under reasonable conditions.

Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions
Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions

Author: Ioan Bejenaru

Publisher: American Mathematical Soc.

Published: 2014-03-05

Total Pages: 120

ISBN-13: 0821892150

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The authors consider the Schrödinger Map equation in 2+1 dimensions, with values into \mathbb{S}^2. This admits a lowest energy steady state Q, namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. The authors prove that Q is unstable in the energy space \dot H^1. However, in the process of proving this they also show that within the equivariant class Q is stable in a stronger topology X \subset \dot H^1.

On the Spectra of Quantum Groups
On the Spectra of Quantum Groups

Author: Milen Yakimov

Publisher: American Mathematical Soc.

Published: 2014-04-07

Total Pages: 104

ISBN-13: 082189174X

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Joseph and Hodges-Levasseur (in the A case) described the spectra of all quantum function algebras on simple algebraic groups in terms of the centers of certain localizations of quotients of by torus invariant prime ideals, or equivalently in terms of orbits of finite groups. These centers were only known up to finite extensions. The author determines the centers explicitly under the general conditions that the deformation parameter is not a root of unity and without any restriction on the characteristic of the ground field. From it he deduces a more explicit description of all prime ideals of than the previously known ones and an explicit parametrization of .

Global and Local Regularity of Fourier Integral Operators on Weighted and Unweighted Spaces
Global and Local Regularity of Fourier Integral Operators on Weighted and Unweighted Spaces

Author: David Dos Santos Ferreira

Publisher: American Mathematical Soc.

Published: 2014-04-07

Total Pages: 86

ISBN-13: 0821891197

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The authors investigate the global continuity on spaces with of Fourier integral operators with smooth and rough amplitudes and/or phase functions subject to certain necessary non-degeneracy conditions. In this context they prove the optimal global boundedness result for Fourier integral operators with non-degenerate phase functions and the most general smooth Hörmander class amplitudes i.e. those in with . They also prove the very first results concerning the continuity of smooth and rough Fourier integral operators on weighted spaces, with and (i.e. the Muckenhoupt weights) for operators with rough and smooth amplitudes and phase functions satisfying a suitable rank condition.

Weighted Bergman Spaces Induced by Rapidly Increasing Weights
Weighted Bergman Spaces Induced by Rapidly Increasing Weights

Author: Jose Angel Pelaez

Publisher: American Mathematical Soc.

Published: 2014-01-08

Total Pages: 136

ISBN-13: 0821888021

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This monograph is devoted to the study of the weighted Bergman space $A^p_\omega$ of the unit disc $\mathbb{D}$ that is induced by a radial continuous weight $\omega$ satisfying $\lim_{r\to 1^-}\frac{\int_r^1\omega(s)\,ds}{\omega(r)(1-r)}=\infty.$ Every such $A^p_\omega$ lies between the Hardy space $H^p$ and every classical weighted Bergman space $A^p_\alpha$. Even if it is well known that $H^p$ is the limit of $A^p_\alpha$, as $\alpha\to-1$, in many respects, it is shown that $A^p_\omega$ lies ``closer'' to $H^p$ than any $A^p_\alpha$, and that several finer function-theoretic properties of $A^p_\alpha$ do not carry over to $A^p_\omega$.

Nonlinear Stability of Ekman Boundary Layers in Rotating Stratified Fluids
Nonlinear Stability of Ekman Boundary Layers in Rotating Stratified Fluids

Author: Hajime Koba

Publisher: American Mathematical Soc.

Published: 2014-03-05

Total Pages: 142

ISBN-13: 0821891332

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A stationary solution of the rotating Navier-Stokes equations with a boundary condition is called an Ekman boundary layer. This book constructs stationary solutions of the rotating Navier-Stokes-Boussinesq equations with stratification effects in the case when the rotating axis is not necessarily perpendicular to the horizon. The author calls such stationary solutions Ekman layers. This book shows the existence of a weak solution to an Ekman perturbed system, which satisfies the strong energy inequality. Moreover, the author discusses the uniqueness of weak solutions and computes the decay rate of weak solutions with respect to time under some assumptions on the Ekman layers and the physical parameters. The author also shows that there exists a unique global-in-time strong solution of the perturbed system when the initial datum is sufficiently small. Comparing a weak solution satisfying the strong energy inequality with the strong solution implies that the weak solution is smooth with respect to time when time is sufficiently large.