Geometric Methods in the Algebraic Theory of Quadratic Forms

2004 ◽  
Author(s):  
Oleg T. Izhboldin ◽  
Bruno Kahn ◽  
Nikita A. Karpenko ◽  
Alexander Vishik
Keyword(s):  
Quadratic Forms ◽  
Algebraic Theory ◽  
Geometric Methods

scholarly journals A class of finite commutative rings constructed from Witt rings

2006 ◽  
Vol 73 (1) ◽  
pp. 47-64 ◽  
Author(s):  
Thomas Craven ◽  
Monika Vo
Keyword(s):  
Quadratic Forms ◽  
Special Class ◽  
Algebraic Theory ◽  
Commutative Rings ◽  
Group Rings ◽  
Integral Group ◽  
Witt Rings ◽  
Integral Group Rings ◽  
Quotient Rings ◽  
Finite Commutative Rings

Motivated by constructions of Witt rings in the algebraic theory of quadratic forms, the authors construct new classes of finite commutative rings and explore some of their properties. These rings are constructed as quotient rings of a special class of integral group rings for which the group is an elementary 2-group. The new constructions are compared to other rings in the literature.

scholarly journals Abstract theory of semiorderings

2005 ◽  
Vol 72 (2) ◽  
pp. 225-250
Author(s):  
Thomas C. Craven ◽  
Tara L. Smith
Keyword(s):  
Quadratic Forms ◽  
Algebraic Theory ◽  
Covering Number ◽  
Finitely Generated ◽  
Abstract Theory ◽  
Witt Ring ◽  
Witt Rings ◽  
Covering Numbers ◽  
Spaces Of Orderings

Marshall's abstract theory of spaces of orderings is a powerful tool in the algebraic theory of quadratic forms. We develop an abstract theory for semiorderings, developing a notion of a space of semiorderings which is a prespace of orderings. It is shown how to construct all finitely generated spaces of semiorderings. The morphisms between such spaces are studied, generalising the extension of valuations for fields into this context. An important invariant for studying Witt rings is the covering number of a preordering. Covering numbers are defined for abstract preorderings and related to other invariants of the Witt ring.

scholarly journals Algebraic Theory of Molecules

1995 ◽  
Author(s):  
F. Iachello ◽  
R. D. Levine
Keyword(s):  
Quantum Mechanics ◽  
Molecular Spectroscopy ◽  
Algebraic Theory ◽  
Algebraic Approach ◽  
Geometric Approach ◽  
Wave Functions ◽  
New Approach ◽  
Geometric Methods ◽  
The Relationship ◽  
Basic Level

Algebraic Theory of Molecules presents a fresh look at the mathematics of wave functions that provide the theoretical underpinnings of molecular spectroscopy. Written by renowned authorities in the field, the book demonstrates the advantages of algebraic theory over the more conventional geometric approach to developing the formal quantum mechanics inherent in molecular spectroscopy. Many examples are provided that compare the algebraic and geometric methods, illustrating the relationship between the algebraic approach and current experiments. The authors develop their presentation from a basic level so as to enable newcomers to enter the field while providing enough details and concrete examples to serve as a reference for the expert. Chemical physicists, physical chemists, and spectroscopists will want to read this exciting new approach to molecular spectroscopy.

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