Decision Problems for Equational Theories of Relation Algebras
Decision Problems for Equational Theories of Relation Algebras

Author: H. Andréka

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 146

ISBN-13: 0821805959

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"We prove that any variety of relation algebras which contains an algebra with infinitely many elements below the identity, or which contains the full group relation algebra on some infinite group (or on arbitrarily large finite groups), must have an undecidable equational theory. Then we construct an embedding of the lattice of all subsets of the natural numbers into the lattice of varieties of relation algebras such that the variety correlated with a set [italic capital]X of natural numbers has a decidable equational theory if and only if [italic capital]X is a decidable (i.e., recursive) set. Finally, we construct an example of an infinite, finitely generated, simple, representable relation algebra that has a decidable equational theory.'' -- Abstract.

Decision Problems for Equational Theories of Relation Algebras
Decision Problems for Equational Theories of Relation Algebras

Author: H. Andréka

Publisher: American Mathematical Soc.

Published: 1997-01-01

Total Pages: 148

ISBN-13: 9780821863275

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This work presents a systematic study of decision problems for equational theories of algebras of binary relations (relation algebras). For example, an easily applicable but deep method, based on von Neumann's coordinatization theorem, is developed for establishing undecidability results. The method is used to solve several outstanding problems posed by Tarski. In addition, the complexity of intervals of equational theories of relation algebras with respect to questions of decidability is investigated. Using ideas that go back to Jonsson and Lyndon, the authors show that such intervals can have the same complexity as the lattice of subsets of the set of the natural numbers. Finally, some new and quite interesting examples of decidable equational theories are given. The methods developed in the monograph show promise of broad applicability. The provide researchers in algebra and logc with a new arsenal of techniques for resolving decision questions in various domains of algebraic logic.

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.

Cutting Brownian Paths
Cutting Brownian Paths

Author: Richard F. Bass

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 113

ISBN-13: 0821809687

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A long open problem in probability theory has been the following: Can the graph of planar Brownian motion be split by a straight line? In this volume, the authors provide a solution, discuss related works, and present a number of open problems.

Study of the Critical Points at Infinity Arising from the Failure of the Palais-Smale Condition for n-Body Type Problems
Study of the Critical Points at Infinity Arising from the Failure of the Palais-Smale Condition for n-Body Type Problems

Author: Hasna Riahi

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 127

ISBN-13: 0821808737

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In this work, the author examines the following: When the Hamiltonian system $m i \ddot{q} i + (\partial V/\partial q i) (t,q) =0$ with periodicity condition $q(t+T) = q(t),\; \forall t \in \germ R$ (where $q {i} \in \germ R{\ell}$, $\ell \ge 3$, $1 \le i \le n$, $q = (q {1},...,q {n})$ and $V = \sum V {ij}(t,q {i}-q {j})$ with $V {ij}(t,\xi)$ $T$-periodic in $t$ and singular in $\xi$ at $\xi = 0$) is posed as a variational problem, the corresponding functional does not satisfy the Palais-Smale condition and this leads to the notion of critical points at infinity. This volume is a study of these critical points at infinity and of the topology of their stable and unstable manifolds. The potential considered here satisfies the strong force hypothesis which eliminates collision orbits. The details are given for 4-body type problems then generalized to n-body type problems.

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.

Matching of Orbital Integrals on $GL(4)$ and $GSp(2)$
Matching of Orbital Integrals on $GL(4)$ and $GSp(2)$

Author: Yuval Zvi Flicker

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 127

ISBN-13: 0821809598

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The trace formula is the most powerful tool currently available to establish liftings of automorphic forms, as predicted by Langlands principle of functionality. The geometric part of the trace formula consists of orbital integrals, and the lifting is based on the fundamental lemma. The latter is an identity of the relevant orbital integrals for the unit elements of the Hecke algebras. This volume concerns a proof of the fundamental lemma in the classically most interesting case of Siegel modular forms, namely the symplectic group Sp(2). These orbital integrals are compared with those on GL(4), twisted by the transpose inverse involution. The technique of proof is elementary. Compact elements are decomposed into their absolutely semi-simple and topologically unipotent parts also in the twisted case; a double coset decomposition of the form H\ G/K--where H is a subgroup containing the centralizer--plays a key role.

Hajnal Andréka and István Németi on Unity of Science
Hajnal Andréka and István Németi on Unity of Science

Author: Judit Madarász

Publisher: Springer Nature

Published: 2021-05-31

Total Pages: 517

ISBN-13: 3030641872

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This book features more than 20 papers that celebrate the work of Hajnal Andréka and István Németi. It illustrates an interaction between developing and applying mathematical logic. The papers offer new results as well as surveys in areas influenced by these two outstanding researchers. They also provide details on the after-life of some of their initiatives. Computer science connects the papers in the first part of the book. The second part concentrates on algebraic logic. It features a range of papers that hint at the intricate many-way connections between logic, algebra, and geometry. The third part explores novel applications of logic in relativity theory, philosophy of logic, philosophy of physics and spacetime, and methodology of science. They include such exciting subjects as time travelling in emergent spacetime. The short autobiographies of Hajnal Andréka and István Németi at the end of the book describe an adventurous journey from electric engineering and Maxwell’s equations to a complex system of computer programs for designing Hungary’s electric power system, to exploring and contributing deep results to Tarskian algebraic logic as the deepest core theory of such questions, then on to applications of the results in such exciting new areas as relativity theory in order to rejuvenate logic itself.

Annihilating Fields of Standard Modules of $\mathfrak {sl}(2, \mathbb {C})^\sim $ and Combinatorial Identities
Annihilating Fields of Standard Modules of $\mathfrak {sl}(2, \mathbb {C})^\sim $ and Combinatorial Identities

Author: Arne Meurman

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 105

ISBN-13: 0821809237

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In this volume, the authors show that a set of local admissible fields generates a vertex algebra. For an affine Lie algebra $\tilde{\frak g}$, they construct the corresponding level $k$ vertex operator algebra and show that level $k$ highest weight $\tilde{\frak g}$-modules are modules for this vertex operator algebra. They determine the set of annihilating fields of level $k$ standard modules and study the corresponding loop $\tilde{\frak g}$-module--the set of relations that defines standard modules. In the case when $\tilde{\frak g}$ is of type $A{(1)} 1$, they construct bases of standard modules parameterized by colored partitions, and as a consequence, obtain a series of Rogers-Ramanujan type combinatorial identities.