List of contributors
Preface
Quadratic Modules Over Polynomial Rings
Introduction
1. Definitions, and Background of the Problem
2. Reduction by the Cancellation Theorem to Low Ranks
3. Quadratic Spaces of Low Rank
4. Proof of the Cancellation Theorem
References
The Action of the Universal Modular Group on Certain Boundary Points
Text
References
Differentially Closed Fields: A Model-Theoretic Tour
Introduction
1. Background
2. Model Completions
3. Main Results
4. Some Properties of Differential Closures and Some Immediate Consequences
5. Notes
Appendix: Details of Uniqueness Proof Filled in
References
On Finite Projective Groups
1. Introduction
2. Preliminaries
3. Finite Strongly Irreducible Linear Groups
4. Case I
5. Case II
6. Quasi-Primitive Linear Groups
References
Unipotent Differential Algebraic Groups
Introduction of Subject Matter and Notation
1. The Structure of Unipotent Differential Algebraic Groups
2. Commutative Linear Unipotent Differential Algebraic Groups
3. Differential Rational Cohomology
4. Extensions of Differential Algebraic Groups
5. The Groups HΔ2(Ga, Ga) and Cent ExtΔ(Ga, Ga)
References
Solutions in the General Solution
1. Introduction
2. Differentially Closed Places
3. Preliminary Criteria
4. Algebraic Extensions
5. Final Criteria
6. Several Indeterminates
References
Folk Theorems on Elliptic Equations
Text
References
Limit Properties of Stochastic Matrices
1. Preliminaries
2. Shrinking Matrices
3. Analysis of Supports
4. The Case d = 1
5. The General Case
A Fixed-Point Characterization of Linearly Reductive Groups
1. Introduction
2. Proof of the Theorem
References
Orthogonal and Unitary Invariants of Families of Subspaces
1. Pairs of Subspaces
2. Systems of Lines
3. Representations of H*-Algebras
4. Systems of Subspaces
References
The Macdonald-Kac Formulas as a Consequence of the Euler-Poincare Principle
Introduction
1. The Lie Algebras of Kac and Moody
2. Quasisimple Modules
3. Homology Associated with F-parabolic Subalgebras
4. Combinatorial Identities
References
The Characters of Reductive P-Adic Groups
Text
References
Basic Constructions in Group Extension Theory
1. Introduction
2. Preliminaries
3. Group Extensions
4. Crossed Homomorphisms
5. Transgression
6. Inflation
7. Abelian Kernels
8. Cohomology
On the Hyperalgebra of a Semisimple Algebraic Group
1. The Hyperalgebra
2. The Algebra u, and its Representations
3. Tensor Products
4. Completely Reducible G-Modules
5. Mumford's Conjecture
References
A Notion of Regularity for Differential Local Algebras
Introduction
1. Some Preliminary Facts About Kahler Differentials
2. Regular Differential Algebras
3. A Sufficient Condition for Regularity
4. Regular Graded Modules
5. The Openness of the Set of Regular Points
6. Regularity for Δ-Fields
References
The Engel-Kolchin Theorem Revisited
1. Introduction
2. A Second Unification of Kolchin's Theorem and Levitzki's Theorem
3. Two More Theorems
4. Lie and Jordan Analogues
5. The Infinite Case
References
Prime Differential Ideals in Differential Rings
1. Special Differential Rings
2. The Prime Differential Spectrum of a Differential Ring
References
Constrained Cohomology
Introduction
Notation
1. Constrained Cohomology Groups
2. Galois Cohomology
3. Change of Group
4. F-Cohomology
5. Change of Field
6. The Hochschild-Serre Sequence
References
The Integrability Condition of Deformations of CR Structures
Introduction
1. The Integrability Condition of CR Structures
2. The Integrability Condition of Deformations
References
Noetherian Rings with Many Derivations
Text
References
Hopf Maps and Quadratic Forms Over Z
1. Hurwitz Triple and the Map ƒ
2. The Map ƒ and the Map h
3. The Family se
4. Hopf Fibration S15 → S8 Over Z
References
Families of Subgroup Schemes of Formal Groups
1. Strict Families of Finite Subgroup Schemes of Formal Groups
2. Liftability of Abelian Varieties
References
An Effective Lower Bound on the "Diophantine" Approximation of Algebraic Functions by Rational Functions (II)
Text
References
On Elementary, Generalized Elementary, and Liouvillian Extension Fields
Text
References
Derivations and Valuation Rings
Text
References
On Theorems of Lie-Kolchin, Borel, and Lang
Text
References
A Differential-Algebraic Study of the Intrusion of Logarithms into Asymptotic Expansions
A. Introduction
B. Some Basic Notations, Procedures, and Lemmas for the Adjunction Rank-Rise Problem
C. Instability Ladders for Differential Polynomials of Class (V, r)
D. Instability Ladders for R-Normal Differential Polynomials
E. Instability Ladders for Asymptotically Nonsingular Differential Polynomials
F. Rank-Rise Results for the General First-Order Equation
G. Other Rank-Rise Results
Appendix
References
A "Theorem of Lie-Kolchin" for Trees
Introduction
1. Trees
2. Various Fixed-Point Properties
3. Solvable Groups
4. Algebraic Simple Groups of Relative Rank ≥ 2
5. The Rank 1 Case
References
Regular Elements in Anisotropic Tori
Introduction
1.
2.
3. Classical Groups
4. Anisotropic Tori in Classical Groups
5. Results for the classical Groups
6. Type 1Al
7. Type 1Bl
8. Type 1Cl
9. Type 1Dl
10. Type 2Al
11. Type 2Dl
12. The Suzuki Groups 2C2(q2)
13. Groups of Type 3D4
14. Conjugacy Classes in the Weyl Group of F4
15. Groups of Type 1F4
16. The Ree Groups 2F4
17. Groups of Type 1G2
18. Groups of Type 2G2
19. Endomorphisms σ with Cyclic Centralizer in W
20. Groups of Type 1E6
21. Groups of Type 2E6
22. Groups of Type 1Ε7
23. Groups of Type 1E8
References