A Geometric Setting for Hamiltonian Perturbation Theory

A Geometric Setting for Hamiltonian Perturbation Theory PDF

Author: Anthony D. Blaom

Publisher: American Mathematical Soc.

Published: 2001

Total Pages: 137

ISBN-13: 0821827200

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In this text, the perturbation theory of non-commutatively integrable systems is revisited from the point of view of non-Abelian symmetry groups. Using a co-ordinate system intrinsic to the geometry of the symmetry, the book generalizes and geometrizes well-known estimates of Nekhoroshev (1977), in a class of systems having almost $G$-invariant Hamiltonians. These estimates are shown to have a natural interpretation in terms of momentum maps and co-adjoint orbits. The geometric framework adopted is described explicitly in examples, including the Euler-Poinsot rigid body.

Geometric Perturbation Theory In Physics

Geometric Perturbation Theory In Physics PDF

Author: S M Omohundro

Publisher: World Scientific

Published: 1986-10-31

Total Pages: 588

ISBN-13: 9814603430

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This book which focusses on mechanics, waves and statistics, describes recent developments in the application of differential geometry, particularly symplectic geometry, to the foundations of broad areas of physics. Throughout the book, intuitive descriptions and diagrams are used to elucidate the mathematical theory. It develops a coordinate-free framework for perturbation theory and uses this to show how underlying symplectic structures arise from physical asymptotes. It describes a remarkable parity between classical mechanics which arises asymptotically from quantum mechanics and classical thermodynamics which arises asymptotically from statistical mechanics. Included here is a section with one hundred unanswered questions for further research.

Some Generalized Kac-Moody Algebras with Known Root Multiplicities

Some Generalized Kac-Moody Algebras with Known Root Multiplicities PDF

Author: Peter Niemann

Publisher: American Mathematical Soc.

Published: 2002

Total Pages: 137

ISBN-13: 0821828886

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Starting from Borcherds' fake monster Lie algebra, this text construct a sequence of six generalized Kac-Moody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly. The root systems decompose space into convex holes, of finite and affine type, similar to the situation in the case of the Leech lattice. As a corollary, we obtain strong upper bounds for the root multiplicities of a number of hyperbolic Lie algebras, including $AE_3$.

Triangulations of Oriented Matroids

Triangulations of Oriented Matroids PDF

Author: Francisco Santos

Publisher: American Mathematical Soc.

Published: 2002

Total Pages: 95

ISBN-13: 0821827693

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We consider the concept of triangulation of an oriented matroid. We provide a definition which generalizes the previous ones by Billera-Munson and by Anderson and which specializes to the usual notion of triangulation (or simplicial fan) in the realizable case. Then we study the relation existing between triangulations of an oriented matroid $\mathcal{M}$ and extensions of its dual $\mathcal{M}^*$, via the so-called lifting triangulations. We show that this duality behaves particularly well in the class of Lawrence matroid polytopes. In particular, that the extension space conjecture for realizable oriented matroids is equivalent to the restriction to Lawrence polytopes of the Generalized Baues problem for subdivisions of polytopes. We finish by showing examples and a characterization of lifting triangulations.

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