A Proof of the $q$-Macdonald-Morris Conjecture for $BC_n$
A Proof of the $q$-Macdonald-Morris Conjecture for $BC_n$

Author: Kevin W. J. Kadell

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 93

ISBN-13: 0821825526

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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.

Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces
Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces

Author: Yongsheng Han

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 138

ISBN-13: 0821825925

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In this work, Han and Sawyer extend Littlewood-Paley theory, Besov spaces, and Triebel-Lizorkin spaces to the general setting of a space of homogeneous type. For this purpose, they establish a suitable analogue of the Calder 'on reproducing formula and use it to extend classical results on atomic decomposition, interpolation, and T1 and Tb theorems. Some new results in the classical setting are also obtained: atomic decompositions with vanishing b-moment, and Littlewood-Paley characterizations of Besov and Triebel-Lizorkin spaces with only half the usual smoothness and cancellation conditions on the approximate identity.

Some Special Properties of the Adjunction Theory for $3$-Folds in $\mathbb P^5$
Some Special Properties of the Adjunction Theory for $3$-Folds in $\mathbb P^5$

Author: Mauro Beltrametti

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 79

ISBN-13: 0821802348

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This work studies the adjunction theory of smooth 3-folds in P]5. Because of the many special restrictions on such 3-folds, the structure of the adjunction theoretic reductions are especially simple, e.g. the 3-fold equals its first reduction, the second reduction is smooth except possibly for a few explicit low degrees, and the formulae relating the projective invariants of the given 3-fold with the invariants of its second reduction are very explicit. Tables summarizing the classification of such 3-folds up to degree 12 are included. Many of the general results are shown to hold for smooth projective n-folds embedded in P]N with N 2n -1.

On Finite Groups and Homotopy Theory
On Finite Groups and Homotopy Theory

Author: Ran Levi

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 121

ISBN-13: 0821804014

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In part 1 we study the homology, homotopy, and stable homotopy of [capital Greek]Omega[italic capital]B[lowercase Greek]Pi[up arrowhead][over][subscript italic]p, where [italic capital]G is a finite [italic]p-perfect group. In part 2 we define the concept of resolutions by fibrations over an arbitrary family of spaces.

The Cohen-Macaulay and Gorenstein Rees Algebras Associated to Filtrations
The Cohen-Macaulay and Gorenstein Rees Algebras Associated to Filtrations

Author: Shirō Gotō

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 149

ISBN-13: 0821825844

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At first, this volume was intended to be an investigation of symbolic blow-up rings for prime ideals defining curve singularities. The motivation for that has come from the recent 3-dimensional counterexamples to Cowsik's question, given by the authors and Watanabe: it has to be helpful, for further researches on Cowsik's question and a related problem of Kronecker, to generalize their methods to those of a higher dimension. However, while the study was progressing, it proved apparent that the framework of Part I still works, not only for the rather special symbolic blow-up rings but also in the study of Rees algebras R(F) associated to general filtrations F = {F[subscript]n} [subscript]n [subscript][set membership symbol][subscript bold]Z of ideals. This observation is closely explained in Part II of this volume, as a general ring-theory of Rees algebras R(F). We are glad if this volume will be a new starting point for the further researchers on Rees algebras R(F) and their associated graded rings G(F).

Christoffel Functions and Orthogonal Polynomials for Exponential Weights on $[-1, 1]$
Christoffel Functions and Orthogonal Polynomials for Exponential Weights on $[-1, 1]$

Author: A. L. Levin

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 166

ISBN-13: 0821825992

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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.

The Major Counting of Nonintersecting Lattice Paths and Generating Functions for Tableaux
The Major Counting of Nonintersecting Lattice Paths and Generating Functions for Tableaux

Author: Christian Krattenthaler

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 122

ISBN-13: 0821826131

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A theory of counting nonintersecting lattice paths by the major index and its generalizations is developed. We obtain determinantal expressions for the corresponding generating functions for families of nonintersecting lattice paths with given starting points and given final points, where the starting points lie on a line parallel to [italic]x + [italic]y = 0. In some cases these determinants can be evaluated to result in simple products. As applications we compute the generating function for tableaux with [italic]p odd rows, with at most [italic]c columns, and with parts between 1 and [italic]n. Moreover, we compute the generating function for the same kind of tableaux which in addition have only odd parts. We thus also obtain a closed form for the generating function for symmetric plane partitions with at most [italic]n rows, with parts between 1 and [italic]c, and with [italic]p odd entries on the main diagonal. In each case the result is a simple product. By summing with respect to [italic]p we provide new proofs of the Bender-Knuth and MacMahon (ex-)conjectures, which were first proved by Andrews, Gordon, and Macdonald. The link between nonintersecting lattice paths and tableaux is given by variations of the Knuth correspondence.

Diagram Cohomology and Isovariant Homotopy Theory
Diagram Cohomology and Isovariant Homotopy Theory

Author: Giora Dula

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 97

ISBN-13: 0821825895

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Obstruction theoretic methods are introduced into isovariant homotopy theory for a class of spaces with group actions; the latter includes all smooth actions of cyclic groups of prime power order. The central technical result is an equivalence between isovariant homotopy and specific equivariant homotopy theories for diagrams under suitable conditions. This leads to isovariant Whitehead theorems, an obstruction-theoretic approach to isovariant homotopy theory with obstructions in cohomology groups of ordinary and equivalent diagrams, and qualitative computations for rational homotopy groups of certain spaces of isovariant self maps of linear spheres. The computations show that these homotopy groups are often far more complicated than the rational homotopy groups for the corresponding spaces of equivariant self maps. Subsequent work will use these computations to construct new families of smooth actions on spheres that are topologically linear but differentiably nonlinear.

Inverse Nodal Problems: Finding the Potential from Nodal Lines
Inverse Nodal Problems: Finding the Potential from Nodal Lines

Author: Ole H. Hald

Publisher: American Mathematical Soc.

Published: 1996

Total Pages: 162

ISBN-13: 0821804863

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In this paper we consider an eigenvalue problem which arises in the study of rectangular membranes. The mathematical model is an elliptic equation, in potential form, with Dirichlet boundary conditions. We show that the potential is uniquely determined, up to an additive constant, by a subset of the nodal lines of the eigenfunctions. A formula is shown which, when the additive constant is given, yields an approximation to the potential at a dense set of points. We present an estimate for the error made by the formula. A substantial part of this work is the derivation of the asymptotic forms for a rich set of eigenvalues and eigenfunctions for a large set of rectangles.