General Topology-I

This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups It aims to provide the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields Involutions are viewed as twisted forms of similarity classes of hermitian or bilinear forms, leading to new developments on the model of the algebraic theory of quadratic forms Besides classical groups, phenomena related to triality are also discussed, as well as groups of type F_4 or G_2 arising from exceptional Jordan or composition algebras Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type D_4 For research mathematicians and graduate students working in central simple algebras, algebraic groups, nonabelian Galois cohomology or Jordan algebras

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The Book of Involutions

Table of basic properties Notation and basic definitions Preface 1. What is the integral closure 2. Integral closure of rings 3. Separability 4. Noetherian rings 5. Rees algebras 6. Valuations 7. Derivations 8. Reductions 9. Analytically unramified rings 10. Rees valuations 11. Multiplicity and integral closure 12. The conductor 13. The Briancon-Skoda theorem 14. Two-dimensional regular local rings 15. Computing the integral closure 16. Integral dependence of modules 17. Joint reductions 18. Adjoints of ideals 19. Normal homomorphisms Appendix A. Some background material Appendix B. Height and dimension formulas References Index.

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Integral closure of ideals, rings, and modules

The asymptotic "merit factor," i.e, the ratio of central to sidelobe energy of extremely long, optimally Iow autocorrelation sequences, formerly calculated as 2e^{2}=14.778 \cdots with the use of an ergodicity hypothesis and a convenient, but faulty, approximation, is recalculated without that approximation and is established at 12.32 \cdots .

The merit factor of long low autocorrelation binary sequences (Corresp.)

In this paper we present the weight distribution of all 2^26 cosets of the (32,6) first-order Reed-Muller code. The code is invariant under the complete affine group, of order 32 \times 31 \times 30 \times 28 \times 24 \times 16. In the Appendix we show (by hand computations) that this group partitions the 2^26 cosets into only 48 equivalence classes, and we obtain the number of cosets in each class. A simple computer program then enumerated the weights of the 32 vectors ih each of the 48 cosets. These coset enumerations also answer this equivalent problem: how well are the 2^32 Boolean functions of five variables approximated by the 2^5 linear functions and their complements?

Weight distributions of the cosets of the (32,6) Reed-Muller code

I Galois Theory.- 1 Field Extensions.- 2 Automorphisms.- 3 Normal Extensions.- 4 Separable and Inseparable Extensions.- 5 The Fundamental Theorem of Galois Theory.- II Some Galois Extensions.- 6 Finite Fields.- 7 Cyclotomic Extensions.- 8 Norms and Traces.- 9 Cyclic Extensions.- 10 Hubert Theorem 90 and Group Cohomology.- 11 Kummer Extensions.- III Applications of Galois Theory.- 12 Discriminants.- 13 Polynomials of Degree 3 and 4.- 14 The Transcendence of ? and e.- 15 Ruler and Compass Constructions.- 16 Solvability by Radicals.- IV Infinite Algebraic Extensions.- 17 Infinite Galois Extensions.- 18 Some Infinite Galois Extensions.- V Transcendental Extensions.- 19 Transcendence Bases.- 20 Linear Disjointness.- 21 Algebraic Varieties.- 22 Algebraic Function Fields.- 23 Derivations and Differentials.- Appendix A Ring Theory.- 1 Prime and Maximal Ideals.- 2 Unique Factorization Domains.- 3 Polynomials over a Field.- 4 Factorization in Polynomial Rings.- 5 Irreducibility Tests.- Appendix B Set Theory.- 1 Zorn's Lemma.- 2 Cardinality and Cardinal Arithmetic.- Appendix C Group Theory.- 1 Fundamentals of Finite Groups.- 2 The Sylow Theorems.- 3 Solvable Groups.- 4 Profinite Groups.- Appendix D Vector Spaces.- 1 Bases and Dimension.- 2 Linear Transformations.- 3 Systems of Linear Equations and Determinants.- 4 Tensor Products.- Appendix E Topology.- 1 Topological Spaces.- 2 Topological Properties.- References.

Field and Galois theory

Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces

The identification number schemes we discussed in the previous chapter give us the ability to determine if an error has been made in recording or transmitting information. However, they are limited in two ways. First, the types of errors detected are fairly restrictive, e.g. single digit errors or interchanging digits. Second, they provide no way to recover the intended information. Some more sophisticated ideas and mathematical concepts enable methods to encoding and transmit information in ways that allow both detection and correction of errors. There are many applications of these so-called error correcting codes, among them transmission of digital images from planetary probes and playing compact discs and DVD movies.

Error Correcting Codes

On the Tensor Product of Composition Algebras

An extension of de Vries duality to completely regular spaces and compactifications

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