Analytic Deformations of the Spectrum of a Family of Dirac Operators on an Odd-Dimensional Manifold with Boundary
Analytic Deformations of the Spectrum of a Family of Dirac Operators on an Odd-Dimensional Manifold with Boundary

Author: Paul Kirk

Publisher: American Mathematical Soc.

Published: 1996

Total Pages: 73

ISBN-13: 082180538X

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The analytic perturbation theory for eigenvalues of Dirac operators on odd dimensional manifolds with boundary is described in terms of [italic]extended L2 eigenvectors [end italics] on manifolds with cylindrical ends. These are generalizations of the Atiyah-Patodi-Singer extended [italic capital]L2 kernel of a Dirac operator. We prove that they form a discrete set near zero and deform analytically, in contrast to [italic capital]L2 eigenvectors, which can be absorbed into the continuous spectrum under deformations when the tangential operator is not invertible. We show that the analytic deformation theory for extended [italic capital]L2 eigenvectors and Atiyah-Patodi-Singer eigenvectors coincides.

Bosonic Construction of Vertex Operator Para-Algebras from Symplectic Affine Kac-Moody Algebras
Bosonic Construction of Vertex Operator Para-Algebras from Symplectic Affine Kac-Moody Algebras

Author: Michael David Weiner

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 121

ISBN-13: 0821808664

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Begins with the bosonic construction of four level -1/2 irreducible representations of the symplectic affine Kac-Moody Lie algebra Cl. The direct sum of two of these is given the structure of a vertex operator algebra (VOA), and the direct sum of the other two is given the structure of a twisted VOA-module. The dissertation includes the bosonic analog of the fermionic construction of a vertex operator superalgebra from the four level 1 irreducible modules of type Dl. No index. Annotation copyrighted by Book News, Inc., Portland, OR

Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions
Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions

Author: Christina Q. He

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 114

ISBN-13: 0821805975

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This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of "fractal drums" (and especially of "fractal strings"). In this work, the authors extend previous results in this area by using the notionof generalized Minkowski content which is defined through some suitable "gauge functions" other than power functions. (This content is used to measure the irregularity (or "fractality") of the boundary of an open set in R]n by evaluating the volume of its small tubular neighborhoods). In the situation when the power function is not the natural "gauge function", this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators. This text will also be of interest to those working in mathematical physics.

Nonlinear Eigenvalues and Analytic-Hypoellipticity
Nonlinear Eigenvalues and Analytic-Hypoellipticity

Author: Ching-Chau Yu

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 106

ISBN-13: 0821807846

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Explores the failure of analytic-hypoellipticity of two partial differential operators. The operators are sums of squares of real analytic vector fields and satisfy Hormander's condition. By reducing to an ordinary differential operator, the author shows the existence of non-linear eigenvalues, which is used to disprove analytic- hypoellipticity of the original operators. No index. Annotation copyrighted by Book News, Inc., Portland, OR

Lie Groups and Subsemigroups with Surjective Exponential Function
Lie Groups and Subsemigroups with Surjective Exponential Function

Author: Karl Heinrich Hofmann

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 189

ISBN-13: 0821806416

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In the structure theory of real Lie groups, there is still information lacking about the exponential function. Most notably, there are no general necessary and sufficient conditions for the exponential function to be surjective. It is surprising that for subsemigroups of Lie groups, the question of the surjectivity of the exponential function can be answered. Under nature reductions setting aside the "group part" of the problem, subsemigroups of Lie groups with surjective exponential function are completely classified and explicitly constructed in this memoir. There are fewer than one would think and the proofs are harder than one would expect, requiring some innovative twists. The main protagonists on the scene are SL(2, R) and its universal covering group, almost abelian solvable Lie groups (ie. vector groups extended by homotheties), and compact Lie groups. This text will also be of interest to those working in algebra and algebraic geometry.

The Finite Irreducible Linear 2-Groups of Degree 4
The Finite Irreducible Linear 2-Groups of Degree 4

Author: Dane Laurence Flannery

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 93

ISBN-13: 0821806254

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This memoir contains a complete classification of the finite irreducible 2-subgroups of GL(4, C). Specifically, the author provides a parametrized list of representatives for the conjugacy classes of such groups, where each representative is defined by generating a set of monomial matrices. The problem is treated by a variety of techniques, including: elementary character theory; a method for describing Hasse diagrams of submodule lattices; and calculation of 2-cohomology by means of the Lyndon-Hochschild-Serre spectral sequence. Related questions concerning isomorphism between the listed groups and Schur indices of their defining characters are also considered

Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies
Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies

Author: Wolfgang Bulla

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 97

ISBN-13: 0821808087

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In this work, the authors provide a self-contained discussion of all real-valued quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies of completely integrable evolution equations. The approach utilizes algebro-geometric methods, factorization techniques for finite difference expressions, as well as Miura-type transformations. Detailed spectral theoretic properties of Lax pairs and theta function representations of the solutions are derived. Features: Simple and unified treatment of the topic. Self-contained development. Novel results for the Kac-van Moerbeke hierarchy and its algebro-geometric solutions.

Some Connections between Isoperimetric and Sobolev-type Inequalities
Some Connections between Isoperimetric and Sobolev-type Inequalities

Author: Serguei Germanovich Bobkov

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 127

ISBN-13: 0821806424

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For Borel probability measures on metric spaces, this text studies the interplay between isoperimetric and Sobolev-type inequalities. In particular the question of finding optimal constants via isoperimetric quantities is explored. Also given are necessary and sufficient conditions for the equivalence between the extremality of some sets in the isoperimetric problem and the validity of some analytic inequalities. The book devotes much attention to: the probability distributions on the real line; the normalized Lebesgue measure on the Euclidean sheres; and the canonical Gaussian measure on the Euclidean space.

The Structure of $k$-$CS$- Transitive Cycle-Free Partial Orders
The Structure of $k$-$CS$- Transitive Cycle-Free Partial Orders

Author: Richard Warren

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 183

ISBN-13: 082180622X

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The class of cycle-free partial orders (CFPOs) is defined, and the CFPOs fulfilling a natural transitivity assumption, called k-connected set transitivity (k-CS-transitivity), are analysed in some detail. Classification in many of the interesting cases is given. This work generlizes Droste's classification of the countable k-transitive trees (k>1). In a CFPO, the structure can be branch downwards as well as upwards, and can do so repeatedely (though it neverr returns to the starting point by a cycle). Mostly it is assumed that k>2 and that all maximal chains are finite. The main classification splits into the sporadic and skeletal cases. The former is complete in all cardinalities. The latter is performed only in the countable case. The classification is considerably more complicated than for trees, and skeletal CFPOs exhibit rich, elaborate and rather surprising behaviour.