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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.

Book Synopsis On Some Aspects of Oscillation Theory and Geometry by : Bruno Bianchini

Download or read book On Some Aspects of Oscillation Theory and Geometry written by Bruno Bianchini and published by American Mathematical Soc.. This book was released on 2013-08-23 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Geometric Analysis of Quasilinear Inequalities on Complete Manifolds

Author: Bruno Bianchini

Publisher: Springer Nature

Published: 2021-01-18

Total Pages: 291

ISBN-13: 3030627047

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This book demonstrates the influence of geometry on the qualitative behaviour of solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among others, from the theory of submanifolds, the authors study classes of coercive elliptic differential inequalities on domains of a manifold M with very general nonlinearities depending on the variable x, on the solution u and on its gradient. The book highlights the mean curvature operator and its variants, and investigates the validity of strong maximum principles, compact support principles and Liouville type theorems. In particular, it identifies sharp thresholds involving curvatures or volume growth of geodesic balls in M to guarantee the above properties under appropriate Keller-Osserman type conditions, which are investigated in detail throughout the book, and discusses the geometric reasons behind the existence of such thresholds. Further, the book also provides a unified review of recent results in the literature, and creates a bridge with geometry by studying the validity of weak and strong maximum principles at infinity, in the spirit of Omori-Yau’s Hessian and Laplacian principles and subsequent improvements.

Book Synopsis Geometric Analysis of Quasilinear Inequalities on Complete Manifolds by : Bruno Bianchini

Download or read book Geometric Analysis of Quasilinear Inequalities on Complete Manifolds written by Bruno Bianchini and published by Springer Nature. This book was released on 2021-01-18 with total page 291 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book demonstrates the influence of geometry on the qualitative behaviour of solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among others, from the theory of submanifolds, the authors study classes of coercive elliptic differential inequalities on domains of a manifold M with very general nonlinearities depending on the variable x, on the solution u and on its gradient. The book highlights the mean curvature operator and its variants, and investigates the validity of strong maximum principles, compact support principles and Liouville type theorems. In particular, it identifies sharp thresholds involving curvatures or volume growth of geodesic balls in M to guarantee the above properties under appropriate Keller-Osserman type conditions, which are investigated in detail throughout the book, and discusses the geometric reasons behind the existence of such thresholds. Further, the book also provides a unified review of recent results in the literature, and creates a bridge with geometry by studying the validity of weak and strong maximum principles at infinity, in the spirit of Omori-Yau’s Hessian and Laplacian principles and subsequent improvements.

Cohomology for Quantum Groups via the Geometry of the Nullcone

Author: Christopher P. Bendel

Publisher: American Mathematical Soc.

Published: 2014-04-07

Total Pages: 110

ISBN-13: 0821891758

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In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p ) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible G -modules stipulates that p=h. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H (u ? ,C) of the small quantum group.

Book Synopsis Cohomology for Quantum Groups via the Geometry of the Nullcone by : Christopher P. Bendel

Download or read book Cohomology for Quantum Groups via the Geometry of the Nullcone written by Christopher P. Bendel and published by American Mathematical Soc.. This book was released on 2014-04-07 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p ) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible G -modules stipulates that p=h. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H (u ? ,C) of the small quantum group.

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.

Book Synopsis Combinatorial Floer Homology by : Vin de Silva

Download or read book Combinatorial Floer Homology written by Vin de Silva and published by American Mathematical Soc.. This book was released on 2014-06-05 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

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.

Book Synopsis Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions by : Ioan Bejenaru

Download or read book Near Soliton Evolution for Equivariant Schrodinger Maps in Two Spatial Dimensions written by Ioan Bejenaru and published by American Mathematical Soc.. This book was released on 2014-03-05 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

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 .

Book Synopsis On the Spectra of Quantum Groups by : Milen Yakimov

Download or read book On the Spectra of Quantum Groups written by Milen Yakimov and published by American Mathematical Soc.. This book was released on 2014-04-07 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 .

Topics in Modern Differential Geometry

Author: Stefan Haesen

Publisher: Springer

Published: 2016-12-21

Total Pages: 284

ISBN-13: 9462392404

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A variety of introductory articles is provided on a wide range of topics, including variational problems on curves and surfaces with anisotropic curvature. Experts in the fields of Riemannian, Lorentzian and contact geometry present state-of-the-art reviews of their topics. The contributions are written on a graduate level and contain extended bibliographies. The ten chapters are the result of various doctoral courses which were held in 2009 and 2010 at universities in Leuven, Serbia, Romania and Spain.

Book Synopsis Topics in Modern Differential Geometry by : Stefan Haesen

Download or read book Topics in Modern Differential Geometry written by Stefan Haesen and published by Springer. This book was released on 2016-12-21 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: A variety of introductory articles is provided on a wide range of topics, including variational problems on curves and surfaces with anisotropic curvature. Experts in the fields of Riemannian, Lorentzian and contact geometry present state-of-the-art reviews of their topics. The contributions are written on a graduate level and contain extended bibliographies. The ten chapters are the result of various doctoral courses which were held in 2009 and 2010 at universities in Leuven, Serbia, Romania and Spain.

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.

Book Synopsis Nonlinear Stability of Ekman Boundary Layers in Rotating Stratified Fluids by : Hajime Koba

Download or read book Nonlinear Stability of Ekman Boundary Layers in Rotating Stratified Fluids written by Hajime Koba and published by American Mathematical Soc.. This book was released on 2014-03-05 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials

Author: Florica C. Cîrstea

Publisher: American Mathematical Soc.

Published: 2014-01-08

Total Pages: 97

ISBN-13: 0821890220

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In particular, for b = 1 and λ = 0, we find a sharp condition on h such that the origin is a removable singularity for all non-negative solutions of [[eqref]]one, thus addressing an open question of Vázquez and Véron.

Book Synopsis A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials by : Florica C. Cîrstea

Download or read book A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials written by Florica C. Cîrstea and published by American Mathematical Soc.. This book was released on 2014-01-08 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: In particular, for b = 1 and λ = 0, we find a sharp condition on h such that the origin is a removable singularity for all non-negative solutions of [[eqref]]one, thus addressing an open question of Vázquez and Véron.

Oscillation Theory

Author: K. Kreith

Publisher: Springer

Published: 2006-11-15

Total Pages: 115

ISBN-13: 3540400052

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Book Synopsis Oscillation Theory by : K. Kreith

Download or read book Oscillation Theory written by K. Kreith and published by Springer. This book was released on 2006-11-15 with total page 115 pages. Available in PDF, EPUB and Kindle. Book excerpt: