SMM Applied Proof Theory  Proof Interpretations and Their Use in Mathematics
Preface
This book gives an introduction to so-called proof interpretations, more specifically
various forms of realizability and functional interpretations, and their use in mathematics.
Whereas earlier treatments of these techniques (e.g. [366, 266, 122, 369, 7])
emphasize foundational and logical issues the focus of this book is on applications
of the methods to extract new effective information such as computable uniform
bounds from given (typically ineffective) proofs. This line of research, which has its
roots in G. Kreisel’s pioneering work on ‘unwinding of proofs’ from the 50’s, has
in more recent years developed into a field of mathematical logic which has been
called (suggested by D. Scott) ‘proof mining’. The areas where proof mining based
on proof interpretations has been applied most systematically are numerical analysis
and functional analysis and so the book concentrates on those. There are also some
extractions of effective information from proofs (guided by logic) in number theory
(G. Kreisel, H. Luckhardt, see e.g. [249, 268, 267, 122]) and algebra (G. Kreisel, C.
Delzell, H. Lombardi, T. Coquand and others, see e.g. [252, 84, 77, 74, 76]). However,
here mainly methods from structural proof theory such as Herbrand’s theorem,
ε -substitution and cut-elimination are used and we will refer to the literature for
more information on these results.
In this book two kinds of systems play an important role: those with full induction
and variants with induction for purely existential formulas (whose central role has
been singled out in the context of so-called reverse mathematics, [338]). Further
(still weaker) fragments are briefly discussed in comments and referred to in the
literature.
Modified realizability (due to G. Kreisel) and functional interpretation (due to K.
G¨odel) are both first developed in the framework of constructive (‘intuitionistic’)
arithmetic in higher types to which consecutively various non-constructive principles
are added.
After this, systems based on ordinary (‘classical’) logic are studied. It is shown
that the combination of G¨odel’s functional (‘Dialectica’) interpretation with the
so-called negative translation, which embeds certain classical theories into approximately
intuitionistic counterparts, can be used to unwind fully non-constructive
proofs. Since the main emphasis throughout this book is on ineffective proofs based
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Common Notations and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Unwinding proofs (‘Proof Mining’) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Introductory remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Informal treatment of ineffective proofs . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Herbrand’s theorem and the no-counterexample interpretation . . . . 22
2.4 Exercises, historical comments and suggested further reading . . . . 38
3 Intuitionistic and classical arithmetic in all finite types . . . . . . . . . . . . . 41
3.1 Intuitionistic and classical predicate logic . . . . . . . . . . . . . . . . . . . . . 41
3.2 Intuitionistic (‘Heyting’) arithmetic HA and Peano arithmetic PA . 44
3.3 Extensional intuitionistic (‘Heyting’) and classical (‘Peano’)
arithmetic in all finite types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Fragments of (W)E-HAω and (W)E-PAω . . . . . . . . . . . . . . . . . . . . . 52
3.5 Fragments corresponding to the Grzegorczyk hierarchy . . . . . . . . . . 54
3.6 Models of E-PAω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7 Exercises, historical comments and suggested further reading . . . . 73
4 RepresentationofPolish metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1 Representation of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Representation of complete separable metric (‘Polish’) spaces . . . . 81
4.3 Special representation of compact metric spaces . . . . . . . . . . . . . . . . 88
4.4 Fragments, exercises, historical comments and suggested further
reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


12 A non-standard principle of uniform boundedness . . . . . . . . . . . . . . . . . 223
12.1 The Σ 0
1 -boundedness principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.2 Applications of Σ 0
1 -boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
12.3 Remarks on the fragments E-GnAω . . . . . . . . . . . . . . . . . . . . . . . . . . 238
12.4 Exercises, historical comments and suggested further reading . . . . 241
13 Elimination of monotone Skolem functions . . . . . . . . . . . . . . . . . . . . . . . . 243
13
15.2 Applications to uniqueness proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
15.3 Applications to monotone convergence theorems . . . . . . . . . . . . . . . 291
15.4 Applications to proofs of contractivity . . . . . . . . . . . . . . . . . . . . . . . . 292
15.5 Remarks on fragments of T ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
15.6 Historical comments and suggested further reading . . . . . . . . . . . . . 295
16 Case study I: Uniqueness proofs in approximation theory . . . . . . . . . . . 297
16.1 Uniqueness proofs in best approximation theory. . . . . . . . . . . . . . . . 297
16.2 Best Chebycheff approximation I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
16.3 Best Chebycheff approximation II . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
16.4 Best L1-approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
16.5 Exercises, historical comments and suggested further reading . . . . 376
17 Applications to analysis: general metatheorems II . . . . . . . . . . . . . . . . . 377
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
17.2 Main results in the metric and hyperbolic case . . . . . . . . . . . . . . . . . 391
17.3 The case
theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
18.4 Asymptotically nonexpansivemappings . . . . . . . . . . . . . . . . . . . . . . 496
18.5 Applications of proof mining in ergodic theory . . . . . . . . . . . . . . . . . 499
18.6 Exercises, historical comments and suggested further reading . . . . 501
19 Final comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
List of formal systems and term classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
List of axioms and rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

SMM Modern Methods in the Calculus of Variations  L^p spaces Fonseca G. Leoni
Preface
In recent years there has been renewed interest in the calculus of variations,
motivated in part by ongoing research in materials science and other disciplines.
Often, the study of certain material instabilities such as phase transitions,
formation of defects, the onset of microstructures in ordered materials,
fracture and damage, leads to the search for equilibria through a minimization
problem of the type
min {I (v) : v ∈ V},
where the class V of admissible functions v is a subset of some Banach space V .
This is the essence of the calculus of variations: the identification of necessary
and sufficient conditions on the functional I that guarantee the existence
of minimizers. These rest on certain growth, coercivity, and convexity conditions,
which often fail to be satisfied in the context of interesting applications,
thus requiring the relaxation of the energy. New ideas were needed, and the introduction
of innovative techniques has resulted in remarkable developments
in the subject over the past twenty years, somewhat scattered in articles,
preprints, books, or available only through oral communication, thus making
it difficult to educate young researchers in this area.
This is the first of two books in the calculus of variations and measure
theory in which many results, some now classical and others at the forefront
of research in the subject, are gathered in a unified, consistent way. A main
concern has been to use contemporary arguments throughout the text to revisit
and streamline well-known aspects of the theory, while providing novel
contributions.
The core of this book is the analysis of necessary and sufficient conditions
for sequential lower semicontinuity of functionals on Lp spaces, followed by
relaxation techniques. What sets this book apart from existing introductory
texts in the calculus of variations is twofold: Instead of laying down the theory
in the one-dimensional setting for integrands f = f(x, u, u
), we work in N
dimensions and no derivatives are present. In addition, it is self-contained in

Contents
Part I Measure Theory and Lp Spaces
1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Measures and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Measures and Outer Measures . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Radon and Borel Measures and Outer Measures . . . . . . . 22
1.1.3 Measurable Functions and Lebesgue Integration . . . . . . . 37
1.1.4 Comparison Between Measures . . . . . . . . . . . . . . . . . . . . . . 55
1.1.5 Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
1.1.6 Projection of Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . 83
1.2 Covering Theorems and Differentiation of Measures in RN . . . . 90
1.2.1 Covering Theorems in RN . . . . . . . . . . . . . . . . . . . . . . . . . . 90
1.2.2 Differentiation Between Radon Measures in RN . . . . . . . 103
1.3 Spaces of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
1.3.1 Signed Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
1.3.2 Signed Finitely Additive Measures . . . . . . . . . . . . . . . . . . . 119
1.3.3 Spaces of Measures as Dual Spaces . . . . . . . . . . . . . . . . . . 123
1.3.4 Weak Star Convergence of Measures . . . . . . . . . . . . . . . . . 129
2 Lp Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.1 Abstract Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.1.1 Definition and Main Properties . . . . . . . . . . . . . . . . . . . . . . 139
2.1.2 Strong Convergence in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . 148
2.1.3 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
2.1.4 Weak Convergence in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
2.1.5 Biting Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
2.2 Euclidean Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
2.2.1 Approximation by Regular Functions . . . . . . . . . . . . . . . . 190
2.2.2 Weak Convergence in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
2.2.3 Maximal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
2.3 Lp Spaces on Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.4 Sequential Lower Semicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . 436
6.4.1 Strong Convergence in Lp, 1 ≤ p < ∞. . . . . . . . . . . . . . . . 436
6.4.2 Strong Convergence in L∞ . . . . . . . . . . . . . . . . . . . . . . . . . 442
6.4.3 Weak Convergence in Lp, 1 ≤ p < ∞. . . . . . . . . . . . . . . . . 445
6.4.4 Weak Star Convergence in L∞ . . . . . . . . . . . . . . . . . . . . . . 448
6.4.5 Weak Star Convergence in the Sense of Measures . . . . . . 449
6.5 Integral Representation in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
6.6 Relaxation in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
6.6.1 Weak Convergence and Weak Star Convergence in Lp,
1 ≤ p≤∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
6.6.2 Weak Star Convergence in the Sense of Measures in L1 . 478
7 Integrands f = f (x, u, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
7.1 Convex Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
7.2 Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
7.3 Sequential Lower Semicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . 491
7.3.1 Strong–Strong Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 491
7.3.2 Strong–Weak Convergence 1 ≤ p, q < ∞ . . . . . . . . . . . . . 492
7.4 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
8 Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
8.1 The Fundamental Theorem for Young Measures . . . . . . . . . . . . . 518
8.2 Characterization of Young Measures . . . . . . . . . . . . . . . . . . . . . . . 532
8.2.1 The Homogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
8.2.2 The Inhomogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
8.3 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
Part IV Appendix
A Functional Analysis and Set Theory . . . . . . . . . . . . . . . . . . . . . . . 549
A.1 Some Results from Functional Analysis . . . . . . . . . . . . . . . . . . . . . 549
A.1.1 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
A.1.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
A.1.3 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 554
A.1.4 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
A.1.5 Weak Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
A.1.6 Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
A.1.7 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
A.2 Wellorderings, Ordinals, and Cardinals . . . . . . . . . . . . . . . . . . . . . 567
B Notes and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
Notation and List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

SMM Methods in Nonlinear Analysis(Kung-Ching Chang)
Preface
Nonlinear analysis is a new area that was born and has matured from abundant
research developed in studying nonlinear problems. In the past thirty
years, nonlinear analysis has undergone rapid growth; it has become part of
the mainstream research fields in contemporary mathematical analysis.
Many nonlinear analysis problems have their roots in geometry, astronomy,
fluid and elastic mechanics, physics, chemistry, biology, control theory, image
processing and economics. The theories and methods in nonlinear analysis
stem from many areas of mathematics: Ordinary differential equations, partial
differential equations, the calculus of variations, dynamical systems, differential
geometry, Lie groups, algebraic topology, linear and nonlinear functional
analysis, measure theory, harmonic analysis, convex analysis, game theory,
optimization theory, etc. Amidst solving these problems, many branches are
intertwined, thereby advancing each other.
The author has been offering a course on nonlinear analysis to graduate
students at Peking University and other universities every two or three
years over the past two decades. Facing an enormous amount of material,
vast numbers of references, diversities of disciplines, and tremendously different
backgrounds of students in the audience, the author is always concerned
with how much an individual can truly learn, internalize and benefit from a
mere semester course in this subject.
The author’s approach is to emphasize and to demonstrate the most fundamental
principles and methods through important and interesting examples
from various problems in different branches of mathematics. However, there
are technical difficulties: Not only do most interesting problems require background
knowledge in other branches of mathematics, but also, in order to solve
these problems, many details in argument and in computation should be included.
In this case, we have to get around the real problem, and deal with a
simpler one, such that the application of the method is understandable. The
author does not always pursue each theory in its broadest generality; instead,
he stresses the motivation, the success in applications and its limitations.
Contents
1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Differential Calculus in Banach Spaces . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Frechet Derivatives and Gateaux Derivatives . . . . . . . . . . 2
1.1.2 Nemytscki Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.3 High-Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Implicit Function Theorem and Continuity Method . . . . . . . . . . 12
1.2.1 Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.3 Continuity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 Lyapunov–Schmidt Reduction and Bifurcation . . . . . . . . . . . . . . 30
1.3.1 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.2 Lyapunov–Schmidt Reduction . . . . . . . . . . . . . . . . . . . . . . . 33
1.3.3 A Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.3.4 Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3.5 Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.4 Hard Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.4.1 The Small Divisor Problem . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.4.2 Nash–Moser Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2 Fixed-Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.1 Order Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.2 Convex Function and Its Subdifferentials . . . . . . . . . . . . . . . . . . . 80
2.2.1 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.2.2 Subdifferentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.3 Convexity and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.4 Nonexpansive Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.5 Monotone Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.6 Maximal Monotone Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.6 Free Discontinuous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
4.6.1 Γ-convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
4.6.2 A Phase Transition Problem . . . . . . . . . . . . . . . . . . . . . . . . 280
4.6.3 Segmentation and Mumford–Shah Problem . . . . . . . . . . . 284
4.7 Concentration Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
4.7.1 Concentration Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
4.7.2 The Critical Sobolev Exponent and the Best Constants 295
4.8 Minimax Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
4.8.1 Ekeland Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 301
4.8.2 Minimax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
4.8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
5 Topological and Variational Methods . . . . . . . . . . . . . . . . . . . . . . 315
5.1 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
5.1.2 Deformation Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
5.1.3 Critical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
5.1.4 Global Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
5.1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
5.2 Minimax Principles (Revisited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
5.2.1 A Minimax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
5.2.2 Category and Ljusternik–Schnirelmann
Multiplicity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
5.2.3 Cap Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
5.2.4 Index Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
5.2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
5.3 Periodic Orbits for Hamiltonian System
and Weinstein Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
5.3.1 Hamiltonian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
5.3.2 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
5.3.3 Weinstein Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
5.4 Prescribing Gaussian Curvature Problem on S2 . . . . . . . . . . . . . 380
5.4.1 The Conformal Group and the Best Constant . . . . . . . . . 380
5.4.2 The Palais–Smale Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 387
5.4.3 Morse Theory for the Prescribing Gaussian Curvature
Equation on S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
5.5 Conley Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
5.5.1 Isolated Invariant Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
5.5.2 Index Pair and Conley Index. . . . . . . . . . . . . . . . . . . . . . . . 397
5.5.3 Morse Decomposition on Compact Invariant Sets
and Its Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

SMM Normal Forms and Unfoldings for Local Dynamical Systems
Preface
The subject of local dynamical systems is concerned with the following two
questions:
1. Given an n × n matrix A, describe the behavior, in a neighborhood
of the origin, of the solutions of all systems of differential equations
having a rest point at the origin with linear part Ax, that is, all
systems of the form
x˙ = Ax + · · · ,
where x ∈ Rn and the dots denote terms of quadratic and higher
order.
2. Describe the behavior (near the origin) of all systems close to a system
of the type just described.
To answer these questions, the following steps are employed:
1. A normal form is obtained for the general system with linear part
Ax. The normal form is intended to be the simplest form into which
any system of the intended type can be transformed by changing the

Contents
Preface v
1 Two Examples 1
1.1 The (Single) Nonlinear Center . . . . . . . . . . . . . . . 1
1.2 The Nonsemisimple Double-Zero Eigenvalue . . . . . . . 21
2 The Splitting Problem for Linear Operators 27
2.1 The Splitting Problem in the Semisimple Case . . . . . . 28
2.2 Splitting by Means of an Inner Product . . . . . . . . . . 32
2.3 Nilpotent Operators . . . . . . . . . . . . . . . . . . . . . 34
2.4 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 * An Introduction to sl(2) Representation Theory . . . . 47
2.6 * Algorithms for the sl(2) Splittings . . . . . . . . . . . . 57
2.7 * Obtaining sl(2) Triads . . . . . . . . . . . . . . . . . . 63
3 Linear Normal Forms 69
3.1 Perturbations of Matrices . . . . . . . . . . . . . . . . . . 69
3.2 An Introduction to the Five Formats . . . . . . . . . . . 74
3.3 Normal and Hypernormal Forms When A0 Is Semisimple 87
3.4 Inner Product and Simplified Normal Forms . . . . . . . 99
3.5 * The sl(2) Normal Form . . . . . . . . . . . . . . . . . . 118
3.6 Lie Theory and the Generated Formats . . . . . . . . . . 129
3.7 Metanormal Forms and k-Determined Hyperbolicity . . . 142
xviii Contents
4 Nonlinear Normal Forms 157
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.2 Settings for Nonlinear Normal Forms . . . . . . . . . . . 160
4.3 The Direct Formats (1a and 1b) . . . . . . . . . . . . . . 164
4.4 Lie Theory and the Generated Formats . . . . . . . . . . 174
4.5 The Semisimple Normal Form . . . . . . . . . . . . . . . 190
4.6 The Inner Product and Simplified Normal Forms . . . . 221
4.7 The Module Structure of Inner Product and
Simplified Normal Forms . . . . . . . . . . . . . . . . . . 242
4.8 * The sl(2) Normal Form . . . . . . . . . . . . . . . . . . 265
4.9 The Hamiltonian Case . . . . . . . . . . . . . . . . . . . 271
4.10 Hypernormal Forms for Vector Fields . . . . . . . . . . . 283
5 Geometrical Structures in Normal Forms 295
5.1 Preserved Structures in Truncated Normal Forms . . . . 297
5.2 Geometrical Structures in the Full System . . . . . . . . 316
5.3 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . 323
5.4 The Nilpotent Center Manifold Case . . . . . . . . . . . 335
6 Selected Topics in Local Bifurcation Theory 339
6.1 Bifurcations from a Single-Zero Eigenvalue:
A “Neoclassical” Approach . . . . . . . . . . . . . . . . . 341
6.2 Bifurcations from a Single-Zero Eigenvalue:
A “Modern” Approach . . . . . . . . . . . . . . . . . . . 356
6.3 Unfolding the Single-Zero Eigenvalue . . . . . . . . . . . 366
6.4 Unfolding in the Presence of
Generic Quadratic Terms . . . . . . . . . . . . . . . . . . 371
6.5 Bifurcations from a Single Center (Hopf and
Degenerate Hopf Bifurcations) . . . . . . . . . . . . . . . 382
6.6 Bifurcations from the Nonsemisimple Double-Zero
Eigenvalue (Takens–Bogdanov Bifurcations) . . . . . . . 389
6.7 Unfoldings of Mode Interactions . . . . . . . . . . . . . . 396
A Rings 405
A.1 Rings, Ideals, and Division . . . . . . . . . . . . . . . . . 405
A.2 Monomials and Monomial Ideals . . . . . . . . . . . . . . 412
A.3 Flat Functions and Formal Power Series . . . . . . . . . 421
A.4 Orderings of Monomials . . . . . . . . . . . . . . . . . . 424
A.5 Division in Polynomial Rings; Gr¨obner Bases . . . . . . . 427
A.6 Division in Power Series Rings; Standard Bases . . . . . 438
A.7 Division in the Ring of Germs . . . . . . . . . . . . . . . 444
B Modules 447
B.1 Submodules of Zn . . . . . . . . . . . . . . . . . . . . . . 447
B.2 Modules of Vector Fields . . . . . . . . . . . . . . . . . . 449
C Format 2b: Generated Recursive (Hori) 451
C.1 Format 2b, Linear Case (for Chapter 3) . . . . . . . . . . 451
C.2 Format 2b, Nonlinear Case (for Chapter 4) . . . . . . . . 457
D Format 2c: Generated Recursive (Deprit) 463
D.1 Format 2c, Linear Case (for Chapter 3) . . . . . . . . . . 463
D.2 Format 2c, Nonlinear Case (for Chapter 4) . . . . . . . . 471
E On Some Algorithms in Linear Algebra 477
References 481
Index 489
coordinates in a prescribed manner.
2. An unfolding of the normal form is obtained. This is intended to be

the simplest form into which all systems close to the original system
can be transformed. It will contain parameters, called unfolding
parameters, that are not present in the normal form found in step 1.

SMM Lie Algebras And Algebraic Groups (2005) Tauvel Yu
Preface
The theory of groups and Lie algebras is interesting for many reasons. In
the mathematical viewpoint, it employs at the same time algebra, analysis
and geometry. On the other hand, it intervenes in other areas of science, in
particular in different branches of physics and chemistry. It is an active domain
of current research.
One of the difficulties that graduate students or mathematicians interested
in the theory come across, is the fact that the theory has very much advanced,
and consequently, they need to read a vast amount of books and articles before
they could tackle interesting problems.
One of the goals we wish to achieve with this book is to assemble in
a single volume the basis of the algebraic aspects of the theory of groups
and Lie algebras. More precisely, we have presented the foundation of the
study of finite-dimensional Lie algebras over an algebraically closed field of
characteristic zero.
Here, the geometrical aspect is fundamental, and consequently, we need to
use the notion of algebraic groups. One of the main differences between this
book and many other books on the subject is that we give complete proofs
for the relationships between algebraic groups and Lie algebras, instead of
admitting them.
We have also given the proofs of certain results on commutative algebra
and algebraic geometry that we needed so as to make this book as selfcontained
as possible. We believe that in this way, the book can be useful for
both graduate students and mathematicians working in this area.
Let us give a brief description of the material treated in this book.
As we have stated earlier, our goal is to study Lie algebras over an algebraically
closed field of characteristic zero. This allows us to avoid, in considering
questions concerning algebraic geometry, the notion of separability, which
simplifies considerably our presentation. In fact, under certain conditions of
separability, the correspondence between Lie algebras and algebraic groups
described in chapter 24 has a very nice generalization when the algebraically
closed base field has prime characteristic.

Contents
1 Results on topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Irreducible sets and spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Noetherian spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Constructible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Gluing topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Prime and maximal ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Rings of fractions and localization. . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Localizations of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
24 Factorial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Unique factorization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Principal ideal domains and Euclidean domains . . . . . . . . . . . . . 43
4.4 Polynomials and factorial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5 Symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 Resultant and discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

16.5 Smooth points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
17 Normal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
17.1 Normal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
17.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
17.3 Products of normal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
17.4 Properties of normal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
18 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
18.1 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
18.2 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
18.3 Root systems and bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . 238
18.4 Passage to the field of real numbers . . . . . . . . . . . . . . . . . . . . . . . 239
18.5 Relations between two roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
18.6 Examples of root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
18.7 Base of a root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
18.8 Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
18.9 Highest root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
18.10 Closed subsets of roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
18.11Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
20 Semisimple and reductive Lie algebras . . . . . . . . . . . . . . . . . . . . . 299
20.1 Semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
20.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
20.3 Semisimplicity of representations . . . . . . . . . . . . . . . . . . . . . . . . . . 302
20.4 Semisimple and nilpotent elements . . . . . . . . . . . . . . . . . . . . . . . . 305
20.5 Reductive Lie algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

38 Semisimple symmetric Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 561
38.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
38.2 Iwasawa decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
38.7 Symmetric invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
38.8 Double centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
38.9 Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
38.10 Distinguished elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
39 Sheets of Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
39.1 Jordan classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
39.2 Topology of Jordan classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
39.3 Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
39.4 Dixmier sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
39.5 Jordan classes in the symmetric case . . . . . . . . . . . . . . . . . . . . . . 605
39.6 Sheets in the symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
40 Index and linear forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
40.1 Stable linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
40.2 Index of a representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
40.3 Some useful inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
40.4 Index and semi-direct products . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
40.5 Heisenberg algebras in semisimple Lie algebras . . . . . . . . . . . . . 621
40.6 Index of Lie subalgebras of Borel subalgebras . . . . . . . . . . . . . . . 625
40.7 Seaweed Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
40.8 An upper bound for the index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
40.9 Cases where the bound is exact . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
40.10 On the index of parabolic subalgebras . . . . . . . . . . . . . . . . . . . . . 638
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
List of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

SMM Random Fields and Geometry
Preface
Since the term “random field’’ has a variety of different connotations, ranging from
agriculture to statistical mechanics, let us start by clarifying that, in this book, a
random field is a stochastic process, usually taking values in a Euclidean space, and
defined over a parameter space of dimensionality at least 1.
Consequently, random processes defined on countable parameter spaces will not
appear here. Indeed, even processes on R1 will make only rare appearances and,
from the point of view of this book, are almost trivial. The parameter spaces we like
best are manifolds, although for much of the time we shall require no more than that
they be pseudometric spaces.
With this clarification in hand, the next thing that you should know is that this
book will have a sequel dealing primarily with applications.
In fact, as we complete this book, we have already started, together with KW
(KeithWorsley), on a companion volume [8] tentatively entitled RFG-A, or Random
Fields and Geometry: Applications. The current volume—RFG—concentrates on
the theory and mathematical background of random fields, while RFG-A is intended
to do precisely what its title promises. Once the companion volume is published,
you will find there not only applications of the theory of this book, but of (smooth)
random fields in general.
Making a clear split between theory and practice has both advantages and disadvantages.
It certainly eased the pressure on us to attempt the almost impossible goal
of writing in a style that would be accessible to all. It also, to a large extent, eases the
load on you, the reader, since you can now choose the volume closer to your interests
and so avoid either “irrelevant’’mathematical detail or the “real world,’’depending on
your outlook and tastes. However, these are small gains when compared to the major
loss of creating an apparent dichotomy between two things that should, in principle,
go hand-in-hand: theory and application. What is true in principle is particularly true
of the topic at hand, and, to explain why, we shall indulge ourselves in a paragraph
or two of history.
The precusor to both of the current volumes was the 1981 monograph The Geometry
of Random Fields (GRF) which grew out of RJA’s (i.e., Robert Adler’s) Ph.D.
thesis under Michael Hasofer. The problem that gave birth to the thesis was an applied
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Part I Gaussian Processes
1 Gaussian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Gaussian Variables and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Boundedness and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 Fields on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.2 Differentiability on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.3 The Brownian Family of Processes . . . . . . . . . . . . . . . . . . . . 24
1.4.4 Generalized Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4.5 Set-Indexed Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.4.6 Non-Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5 Majorizing Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 Gaussian Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.1 Borell–TIS Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 Comparison Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Orthogonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1 The General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 The Karhunen–Loève Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Excursion Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1 Entropy Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Processes with a Unique Point of Maximal Variance . . . . . . . . . . . . . 86
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
13 Mean Intrinsic Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
13.1 Crofton’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
13.2 Mean Intrinsic Volumes: The Isotropic Case . . . . . . . . . . . . . . . . . . . 333
13.3 A Gaussian Crofton Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
13.4 Mean Intrinsic Volumes: The General Case . . . . . . . . . . . . . . . . . . . . 342
13.5 Two Gaussian Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
14 Excursion Probabilities for Smooth Fields . . . . . . . . . . . . . . . . . . . . . . . . 349
14.1 On Global Suprema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
14.1.1 A First Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
14.1.2 The Problem with the First Representation . . . . . . . . . . . . . . 354
14.1.3 A Second Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
14.1.4 Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
14.1.5 Suprema and Euler Characteristics . . . . . . . . . . . . . . . . . . . . 362
14.2 Some Fine Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
14.3 Gaussian Fields with Constant Variance . . . . . . . . . . . . . . . . . . . . . . . 368
14.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
14.4.1 Stationary Processes on [0, T ] . . . . . . . . . . . . . . . . . . . . . . . . 372
14.4.2 Isotropic Fields with Monotone Covariance . . . . . . . . . . . . . 374
14.4.3 A Geometric Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
14.4.4 The Cosine Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
15 Non-Gaussian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
15.1 A Plan of Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
15.2 A Representation for Mean Intrinsic Volumes . . . . . . . . . . . . . . . . . . 391
15.3 Proof of the Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
15.4 Poincaré’s Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
15.5 Kinematic Fundamental Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
15.5.1 The KFF on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
15.5.2 The KFF on Sλ(Rn). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
15.6 A Model Process on the l-Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
15.6.1 The Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
15.6.2 Mean Curvatures for the Model Process . . . . . . . . . . . . . . . . 404
15.7 The Canonical Gaussian Field on the l-Sphere . . . . . . . . . . . . . . . . . . 410
15.7.1 Mean Curvatures for Excursion Sets . . . . . . . . . . . . . . . . . . . 411
15.7.2 Implications for More General Fields . . . . . . . . . . . . . . . . . . 415
15.8 Warped Products of Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . 416
15.8.1 Warped Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
15.8.2 A Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . . . 419
15.9 Non-Gaussian Mean Intrinsic Volumes . . . . . . . . . . . . . . . . . . . . . . . . 421

SMM Variational and Potential Methods for a Class of Linear Hyperbolic Evolutionary Processes
Preface
Variational and boundary integral equation techniques are two of the most
useful methods for solving time-dependent problems described by systems of
equations of the form
∂2u
∂t2 = Au,
where u = u(x, t) is a vector-valued function, x is a point in a domain in R2 or
R3, and A is a linear elliptic differential operator. To facilitate a better understanding
of these two types of methods, below we propose to illustrate their
mechanisms in action on a specific mathematical model rather than in a more
impersonal abstract setting. For this purpose, we have chosen the hyperbolic
system of partial differential equations governing the nonstationary bending
of elastic plates with transverse shear deformation. The reason for our choice
is twofold. On the one hand, in a certain sense this is a “hybrid” system, consisting
of three equations for three unknown functions in only two independent
variables, which makes it more unusual—and thereby more interesting to the
analyst—than other systems arising in solid mechanics. On the other hand,
this particular plate model has received very little attention compared to the
so-called classical one, based on Kirchhoff’s simplifying hypotheses, although,
as acknowledged by practitioners, it represents a substantial refinement of the
latter and therefore needs a rigorous discussion of the existence, uniqueness,
and continuous dependence of its solution on the data before any construction
of numerical approximation algorithms can be contemplated.
The first part of our analysis is conducted by means of a procedure that
is close in both nature and detail to a variational method, and which, for this
reason, we also call variational. Once the results have been established in the
general setting of Sobolev spaces, we carry out the second part of the study by
seeking useful, closed-form integral representations of the solutions in terms
of dynamic (retarded) plate potentials.

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Formulation of the Problems and Their Nonstationary
Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . 1
1.2 A Matrix of Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Time-dependent Plate Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Nonstationary Boundary Integral Equations . . . . . . . . . . . . . . . 16
2 Problems with Dirichlet Boundary Conditions . . . . . . . . . . . . 19
2.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Solvability of the Transformed Problems . . . . . . . . . . . . . . . . . . 21
2.3 Solvability of the Time-dependent Problems . . . . . . . . . . . . . . . 28
3 Problems with Neumann Boundary Conditions . . . . . . . . . . . 37
3.1 The Poincar&acute;e–Steklov Operators . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Solvability of the Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Boundary Integral Equations for Problems with Dirichlet
and Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Time-dependent Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Nonstationary Boundary Integral Equations . . . . . . . . . . . . . . . 51
4.3 The Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Transmission Problems and Multiply Connected Plates . . . 57
5.1 Infinite Plate with a Finite Inclusion . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Multiply Connected Finite Plate . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Finite Plate with an Inclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

XII Contents
6 Plate Weakened by a Crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.1 Formulation and Solvability of the Problems . . . . . . . . . . . . . . . 81
6.2 The Poincar&acute;e–Steklov Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3 Time-dependent Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.4 Infinite Plate with a Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.5 Finite Plate with a Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7 Initial-Boundary Value Problems with Other Types of
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.1 Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2 Combined Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3 Elastic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8 Boundary Integral Equations for Plates on a Generalized
Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.1 Formulation and Solvability of the Problems . . . . . . . . . . . . . . . 119
8.2 A Matrix of Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . 121
8.3 Properties of the Boundary Operators. . . . . . . . . . . . . . . . . . . . . 126
8.4 Solvability of the Boundary Equations . . . . . . . . . . . . . . . . . . . . 127
9 Problems with Nonhomogeneous Equations
and Nonhomogeneous Initial Conditions . . . . . . . . . . . . . . . . . 129
9.1 The Time-dependent Area Potential . . . . . . . . . . . . . . . . . . . . . . 129
9.2 The Nonhomogeneous Equation of Motion . . . . . . . . . . . . . . . . . 131
9.3 Initial Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A The Fourier and Laplace Transforms of Distributions . . . . . 139
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

SMM Valued fields (Engler A. Prestel A. )
Preface
The purpose of this book is to give a self-contained and comprehensive introduction
to the theory of general valuations, in contrast to classical absolute
values. In particular, we present some applications of the general theory going
beyond the use of absolute values. The book does not aim for an encyclopaedic
presentation, but rather prefers a streamlined style, leading eventually to deep
results of recent research.
While the classical theory of absolute values can be found in many books,
in particular those on number theory, there are few textbooks devoted to the
general theory of valuations. To our knowledge, these are O. Schilling (1950,
[27]), P. Ribenboim (1965, [23]), and O. Endler (1972, [6]). Besides those, one
can find, however, chapters on general valuation theory in several books, such
as in O. Zariski – P. Samuel (1960, [33]) or Y. Ershov (2001, [8]). Concerning
the history of valuation theory, the reader is referred to P. Roquette [25].
Both authors of this book have been deeply influenced by the late Otto
Endler – the first author as a student, the second as a colleague. It was at the
IMPA in Rio de Janeiro where we all met in the mid seventies. Since then we
became followers of Krull’s development of henselian valued fields, and since
then we have tried to convince other mathematicians of the beauty of this
theory.
The book is based on courses given by the first author in Brazil, and
by the second author in Pisa, Freiburg and Konstanz. We are grateful to
K. Becher, M. Illengo, I. Klep, J. Koenigsmann, J. Schmid and T. Unger for
reading parts of the book and making many valuable comments. We are also
grateful to C. N. Delzell for checking the layout and the use of the English
language.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 Absolute Values – Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Archimedean Complete Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Non-Archimedean Complete Fields . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Ordered Abelian Groups – Valuations . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Constructions of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Rational Function Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Ordered Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.3 Rigid Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Dependent Valuations – Induced Topology . . . . . . . . . . . . . . . . . . 42
2.4 Approximation – Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Extension of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 Chevalley’s Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Algebraic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 The Fundamental Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4 Transcendental Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1 Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 p-Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3 Ordered Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 The Canonical Henselian Valuation . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5 Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1 Infinite Galois Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Unramified Extensions – First Exact Sequence . . . . . . . . . . . . . . 120
5.3 Ramified Extensions – Second Exact Sequence . . . . . . . . . . . . . . 126
5.4 Galois Characterization of Henselian Fields . . . . . . . . . . . . . . . . . 136
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6 Applications of Valuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.1 Artin’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 p-Adically Closed Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3 A Local-Global Principle for Quadratic Forms . . . . . . . . . . . . . . . 163
A Ultraproducts of Valued Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B Classification of V -Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Standard Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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