Completely Positive Hypergroup Actions
Completely Positive Hypergroup Actions

Author: Ajit Iqbal Singh

Publisher: American Mathematical Soc.

Published: 1996

Total Pages: 87

ISBN-13: 0821805398

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It is now well know that the measure algebra [script capital]M([italic capital]G) of a locally compact group can be regarded as a subalgebra of the operator algebra [italic capital]B([italic capital]B([italic capital]L2([italic capital]G))) of the operator algebra [italic capital]B([italic capital]L2([italic capital]G)) of the Hilbert space [italic capital]L2([italic capital]G). We study the situation in hypergroups and find that, in general, the analogous map for them is neither an isometry nor a homomorphism. However, it is completely positive and completely bounded in certain ways. This work presents the related general theory and special examples.

Classification of Simple $C$*-algebras: Inductive Limits of Matrix Algebras over Trees
Classification of Simple $C$*-algebras: Inductive Limits of Matrix Algebras over Trees

Author: Liangqing Li

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 138

ISBN-13: 0821805967

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In this paper, it is shown that the simple unital C*-algebras arising as inductive limits of sequences of finite direct sums of matrix algebras over [italic capital]C([italic capital]X[subscript italic]i), where [italic capital]X[subscript italic]i are arbitrary variable trees, are classified by K-theoretical and tracial data. This result generalizes the result of George Elliott of the case of [italic capital]X[subscript italic]i = [0, 1]. The added generality is useful in the classification of more general inductive limit C*-algebras.

Wandering Vectors for Unitary Systems and Orthogonal Wavelets
Wandering Vectors for Unitary Systems and Orthogonal Wavelets

Author: Xingde Dai

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 82

ISBN-13: 0821808001

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Investigates topological and structural properties of the set W(U) of all complete wandering vectors for a system U of unitary operators acting on a Hilbert space. The authors parameterize W(U) in terms of a fixed vector y and the set of all unitary operators which locally commute with U at y. No index. Annotation copyrighted by Book News, Inc., Portland, OR

Generalized Symplectic Geometries and the Index of Families of Elliptic Problems
Generalized Symplectic Geometries and the Index of Families of Elliptic Problems

Author: Liviu I. Nicolaescu

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 98

ISBN-13: 0821806211

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In this book, an index theorem is proved for arbitrary families of elliptic boundary value problems for Dirac operators and a surgery formula for the index of a family of Dirac operators on a closed manifold. Also obtained is a very general result on the cobordism invariance of the index of a family. All results are established by first symplectically rephrasing the problems and then using a generalized symplectic reduction technique. This provides a unified approach to all possible parameter spaces and all possible symmetries of a Dirac operator (eigh symmetries in the real case and two in the complex case). This text will also be of interest to those working in geometry and topology.

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.

Gauge Theory on Compact Surfaces
Gauge Theory on Compact Surfaces

Author: Ambar Sengupta

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 98

ISBN-13: 0821804847

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In this paper we develop a concrete description of connections on principal bundles, possibly non-trivial, over compact surfaces and use this description to construct the Yang-Mills measure which underlies the Euclidean quantum theory of gauge fields, involving compact gauge groups, on compact connected two-dimensional Riemannian manifolds (possibly with boundary). Using this measure we compute expectation values of important random variables, the Wilson loops variables, corresponding to a broad class of configurations of loops on the surface.

Locally Finite, Planar, Edge-Transitive Graphs
Locally Finite, Planar, Edge-Transitive Graphs

Author: Jack E. Graver

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 89

ISBN-13: 0821805568

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The nine finite, planar, 3-connected, edge-transitive graphs have been known and studied for many centuries. The infinite, locally finite, planar, 3-connected, edge-transitive graphs can be classified according to the number of their end. The 1-ended graphs in this class were identified by Grünbaum and Shephard; Watkins characterized the 2-ended members. Any remaining graphs in this class must have uncountably may ends. In this work, infinite-ended members of this class are shown to exist. A more detailed classification scheme in terms of the types of Petrie walks in the graphs in this class and the local structure of their automorphism groups is presented.

The Integral Manifolds of the Three Body Problem
The Integral Manifolds of the Three Body Problem

Author: Christopher Keil McCord

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 106

ISBN-13: 0821806920

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The phase space of the spatial three-body problem is an open subset in R18. Holding the ten classical integrals of energu, center of mass, linear and angular momentum fixed defines an eight dimensional manifold. For fixed nonzero angular momentum, the topology of this manifold depends only on the energy. This volume computes the homology of this manifold for all energy values. This table of homology shows that for negative energy, the integral manifolds undergo seven bifurcations. Four of these are the well-known bifurcations due to central configurations, and three are due to "critical points at infinity". This disproves Birkhoffs conjecture that the bifurcations occur only at central configurations.

Short-Time Geometry of Random Heat Kernels
Short-Time Geometry of Random Heat Kernels

Author: Richard Bucher Sowers

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 145

ISBN-13: 0821806491

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This volume studies the behaviour of a random heat kernel associated with a stochastic partial differential equation, and gives short-time expansion of this heat kernel. The author finds that the dominant exponential term is classical and depends only on the Riemannian distance function. The second exponential term is a work term and also has classical meaning. There is also a third non-negligible exponential term which blows up. The author finds an expression for this third exponential term which involves a random translation of the index form and the equations of Jacobi fields. In the process, he develops a method to approximate the heat kernel to any arbitrary degree of precision.