Duality And Definability In First Order Logic
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| Author | : Michael Makkai |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 122 |
| Release | : 1993 |
| Genre | : Mathematics |
| ISBN | : 0821825658 |
We develop a duality theory for small Boolean pretoposes in which the dual of the [italic capital]T is the groupoid of models of a Boolean pretopos [italic capital]T equipped with additional structure derived from ultraproducts. The duality theorem states that any small Boolean pretopos is canonically equivalent to its double dual. We use a strong version of the duality theorem to prove the so-called descent theorem for Boolean pretoposes which says that category of descent data derived from a conservative pretopos morphism between Boolean pretoposes is canonically equivalent to the domain-pretopos. The descent theorem contains the Beth definability theorem for classical first order logic. Moreover, it gives, via the standard translation from the language of categories to symbolic logic, a new definability theorem for classical first order logic concerning set-valued functors on models, expressible in purely syntactical (arithmetical) terms.
| Author | : John Patrick Campbell Greenlees |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 193 |
| Release | : 1995 |
| Genre | : Mathematics |
| ISBN | : 0821826034 |
Let [italic capital]G be a compact Lie group, [italic capitals]EG a contractible free [italic capital]G-space and let [italic capitals]E~G be the unreduced suspension of [italic capitals]EG with one of the cone points as basepoint. Let [italic]k*[over][subscript italic capital]G be a [italic capital]G-spectrum. Let [italic capital]X+ denote the disjoint union of [italic capital]X and a [italic capital]G-fixed basepoint. Define the [italic capital]G-spectra [italic]f([italic]k*[over][subscript italic capital]G) = [italic]k*[over][subscript italic capital]G [up arrowhead symbol] [italic capitals]EG+, [italic]c([italic]k*[over][subscript italic capital]G) = [italic capital]F([italic capitals]EG+,[italic]k*[over][subscript italic capital]G), and [italic]t([italic]k[subscript italic capital]G)* = [italic capital]F([italic capitals]EG+,[italic]k*[over][subscript italic capital]G) [up arrowhead symbol] [italic capitals]E~G. The last of these is the [italic capital]G-spectrum representing the generalized Tate homology and cohomology theories associated to [italic]k[subscript italic capital]G. Here [italic capital]F([italic capitals]EG+,[italic]k*[over][subscript italic capital]G) is the function space spectrum. The authors develop the properties of these theories, illustrating the manner in which they generalize the classical Tate-Swan theories.
| Author | : Tom Lindstrøm |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 140 |
| Release | : 1990 |
| Genre | : Mathematics |
| ISBN | : 0821824848 |
Lindstrom (U. of Oslo) constructs Brownian motion on a reasonably general class of self-similar fractals. He deals with diffusions, self-similar fractals, fractal Laplacians, asymptotic distribution of eigenvalues, nonstandard analysis. Annotation copyright Book News, Inc. Portland, Or.
| Author | : Vladimir I︠A︡kovlevich Lin |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 90 |
| Release | : 1994 |
| Genre | : Mathematics |
| ISBN | : 0821826042 |
This work studies equivariant linear second order elliptic operators [italic capital]P on a connected noncompact manifold [italic capital]X with a given action of a group [italic capital]G. The action is assumed to be cocompact, meaning that [italic capitals]GV = [italic capital]X for some compact subset of [italic capital]V of [italic capital]X. The aim is to study the structure of the convex cone of all positive solutions of [italic capital]P[italic]u = 0.
| Author | : Roger D. Nussbaum |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 148 |
| Release | : 1988 |
| Genre | : Mathematics |
| ISBN | : 0821824546 |
| Author | : Kevin W. J. Kadell |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 93 |
| Release | : 1994 |
| Genre | : Mathematics |
| ISBN | : 0821825526 |
Macdonald and Morris gave a series of constant term [italic]q-conjectures associated with root systems. Selberg evaluated a multivariable beta-type integral which plays an important role in the theory of constant term identities associated with root systems. K. Aomoto recently gave a simple and elegant proof of a generalization of Selberg's integral. Kadell extended this proof to treat Askey's conjectured [italic]q-Selberg integral, which was proved independently by Habsieger. We use a constant term formulation of Aomoto's argument to treat the [italic]q-Macdonald-Morris conjecture for the root system [italic capitals]BC[subscript italic]n. We show how to obtain the required functional equations using only the q-transportation theory for [italic capitals]BC[subscript italic]n.
| Author | : A. L. Levin |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 166 |
| Release | : 1994 |
| Genre | : Mathematics |
| ISBN | : 0821825992 |
Bounds for orthogonal polynomials which hold on the 'whole' interval of orthogonality are crucial to investigating mean convergence of orthogonal expansions, weighted approximation theory, and the structure of weighted spaces. This book focuses on a method of obtaining such bounds for orthogonal polynomials (and their Christoffel functions) associated with weights on [-1,1]. Also presented are uniform estimates of spacing of zeros of orthogonal polynomials and applications to weighted approximation theory.
| Author | : Anthony Valiant Phillips |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 90 |
| Release | : 1993 |
| Genre | : Mathematics |
| ISBN | : 0821825666 |
We examine the general problem of computing characteristic invariants of principal bundles whose structural group [italic capital]G is a topological group. Under the hypothesis that [italic capital]G has real cohomology finitely generated as an [bold]R-module, we are able to give a completely topological, local method for computing representative cocycles for real characteristic classes; our method applies, for example, to the (homologically) 10-dimensional non-Lie group of Hilton-Roitberg-Stasheff.
| Author | : I. V. Evstigneev |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 113 |
| Release | : 1994 |
| Genre | : Mathematics |
| ISBN | : 0821825976 |
Various notions of the Markov property relative to a partial ordering have been proposed by both physicists and mathematicians. This work develops techniques for stying Markov fields on partially ordered sets. We introduce random transformations of the index set which preserves the Markov property of the field. These transformations yield new classes of Markov fields starting from relatively simple ones. Examples include a model for crack formation and a model for the distribution of fibres in a composite material.
| Author | : Anthony Ayers Iarrobino |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 128 |
| Release | : 1994 |
| Genre | : Mathematics |
| ISBN | : 0821825763 |
In 1904, Macaulay described the Hilbert function of the intersection of two plane curve branches: It is the sum of a sequence of functions of simple form. This monograph describes the structure of the tangent cone of the intersection underlying this symmetry. Iarrobino generalizes Macaulay's result beyond complete intersections in two variables to Gorenstein Artin algebras in an arbitrary number of variables. He shows that the tangent cone of a Gorenstein singularity contains a sequence of ideals whose successive quotients are reflexive modules. Applications are given to determining the multiplicity and orders of generators of Gorenstein ideals and to problems of deforming singular mapping germs. Also included are a survey of results concerning the Hilbert function of Gorenstein Artin algebras and an extensive bibliography.
