**Auteur :** Gunnar Fløystad

**la langue :** en

**Éditeur:** American Mathematical Soc.

**Date de sortie :** 1998

Given a homogeneous ideal $I$ and a monomial order, one may form the initial ideal $\textnormal{in}(I)$. The initial ideal gives information about $I$, for instance $I$ and $\textnormal{in}(I)$ have the same Hilbert function. However, if $\mathcal I$ is the sheafification of $I$ one cannot read the higher cohomological dimensions $h^i({\mathbf P}^n, \mathcal I(\nu)$ from $\textnormal{in}(I)$. This work remedies this by defining a series of higher initial ideals $\textnormal{in}_s(I)$ for $s\geq0$. Each cohomological dimension $h^i({\mathbf P}^n, \mathcal I(\nu))$ may be read from the $\textnormal{in}_s(I)$. The $\textnormal{in}_s(I)$ are however more refined invariants and contain considerably more information about the ideal $I$. This work considers in particular the case where $I$ is the homogeneous ideal of a curve in ${\mathbf P}^3$ and the monomial order is reverse lexicographic.Then the ordinary initial ideal $\textnormal{in}_0(I)$ and the higher initial ideal $\textnormal{in}_1(I)$ have very simple representations in the form of plane diagrams. It enables one to visualize cohomology of projective schemes in ${\mathbf P}^n$. It provides an algebraic approach to studying projective schemes. It gives structures which are generalizations of initial ideals.

**Auteur :** Edward Norman Dancer

**la langue :** en

**Éditeur:** American Mathematical Soc.

**Date de sortie :** 1999

This book is intended for graduate students and research mathematicians working in partial differential equations.

**Auteur :** Harvey I. Blau

**la langue :** en

**Éditeur:** American Mathematical Soc.

**Date de sortie :** 2000

This book features homogeneous integral table algebras of degree three with a faithful real element. The algebras of the title are classified to exact isomorphism; that is, the sets of structure constants which arise from the given basis are completely determined. Other results describe all possible extensions (pre-images), with a faithful element which is not necessarily real, of certain simple homogeneous integral table algebras of degree three. On antisymmetric homogeneous integral table algebras of degree three. This paper determines the homogeneous integral table algebras of degree three in which the given basis has a faithful element and has no nontrivial elements that are either real (symmetric) or linear, and where an additional hypothesis is satisfied.It is shown that all such bases must occur as the set of orbit sums in the complex group algebra of a finite abelian group under the action of a fixed-point-free automorphism of order three. Homogeneous integral table algebras of degree three with no nontrivial linear elements. The algebras of the title which also have a faithful element are determined to exact isomorphism. All of the simple homogeneous integral table algebras of degree three are displayed, and the commutative association schemes in which all the nondiagonal relations have valency three and where some relation defines a connected graph on the underlying set are classified up to algebraic isomorphism.

**Auteur :** Xingde Dai

**la langue :** en

**Éditeur:** American Mathematical Soc.

**Date de sortie :** 1998

This volume concerns some general methods for the analysis of those orthonormal bases for a separable complex infinite dimensional Hilbert space which are generated by the action of a system of unitary transformations on a single vector, which is called a complete wandering vector for the system. The main examples are the orthonormal wavelet bases. Topological and structural properties of the set of all orthonormal dyadic wavelets are investigated in this way by viewing them as complete wandering vectors for an affiliated unitary system and then applying techniques of operator algebra and operator theory.It describes an operator-theoretic perspective on wavelet theory that is accessible to functional analysts. It describes some natural generalizations of standard wavelet systems. It contains numerous examples of computationally elementary wavelets. It poses many open questions and directions for further research. This book is particularly accessible to operator theorists and operator algebraists who are interested in a functional analytic approach to some of the pure mathematics underlying wavelet theory.

**Auteur :** Wolfgang Bulla

**la langue :** en

**Éditeur:** American Mathematical Soc.

**Date de sortie :** 1998

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. It features a simple and unified treatment of the topic. It has self-contained development. There are novel results for the Kac-van Moerbeke hierarchy and its algebro-geometric solutions.

**Auteur :** Yael Karshon

**la langue :** en

**Éditeur:** American Mathematical Soc.

**Date de sortie :** 1999

Abstract - we classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian $S^1$-spaces. Additionally, we show that all these spaces are Kahler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if the fixed points are isolated then the space is a toric variety.

**Auteur :** Michael David Weiner

**la langue :** en

**Éditeur:** American Mathematical Soc.

**Date de sortie :** 1998

Inspired by mathematical structures found by theoretical physicists and by the desire to understand the "monstrous moonshine" of the Monster group, Borcherds, Frenkel, Lepowsky, and Meurman introduced the definition of vertex operator algebra (VOA). An important part of the theory of VOAs concerns their modules and intertwining operators between modules. Feingold, Frenkel, and Ries defined a structure, called a vertex operator para-algebra (VOPA), where a VOA, its modules and their intertwining operators are unified. In this work, for each $n \geq 1$, the author uses the bosonic construction (from a Weyl algebra) of four level $- 1/2$ irreducible representations of the symplectic affine Kac-Moody Lie algebra $C_n^{(1)}$. They define intertwining operators so that the direct sum of the four modules forms a VOPA. This work includes the bosonic analog of the fermionic construction of a vertex operator superalgebra from the four level 1 irreducible modules of type $D_n^{(1)}$ given by Feingold, Frenkel, and Ries. While they get only a VOPA when $n = 4$ using classical triality, the techniques in this work apply to any $n \geq 1$.