Lecture Notes by Oliver Knill
2003. 140 pages. Softcover
ISBN 3-7643-2185-7
English
Lectures in Mathematics - ETH Zurich
These lecture notes describe the Aubry-Mather-Theory
within the
calculus of variations. The text consists
of the translated
original lectures of Jurgen Moser and a bibliographic
appendix
with comments on the current state-of-the-art
in this field of
interest. Students will find a rapid introduction
to the calculus
of variations, leading to modern dynamical
systems theory.
Differential geometric applications are discussed,
in particular
billiards and minimal geodesics on the two-dimensional
torus.
Many exercises and open questions make this
book a valuable
resource for both teaching and research.
Table of contents
Introduction (.-) 1. One-dimensional variational
problems (.-) 2.
Extremal fields and global minimal (.-) 3.
Discrete Systems,
Applications (.-) Bibliography (.-) Remarks
on the literature
(.-) Additional Bibliography
2003. 224 pages. Hardcover
ISBN 3-7643-0084-1
English
Operator Theory,vol.144
This monograph is devoted to the study of
functional equations
with the transformed argument on the real
line and on the unit
circle. Such equations systematically arise
in dynamical systems,
differential equations, probabilities, singularities
of smooth
mappings, and other areas. The purpose of
the book is to present
modern methods and new results in the subject,
with an emphasis
on a connection between local and global
solvability. The general
concepts developed in the book are applicable
to multidimensional
functional equations.
Some of the methods are presented for the
first time in the
monograph literature, in particular, a functional
parametrization
of local mappings, the gluing of local solutions,
and a
decomposition method.
The book is addressed to graduates and researchers
interested in
dynamical systems, differential equations,
operator theory, or
the theory of functions and their applications.
Table of contents
Preface .- 1. Implicit Functions .- 2. Classification
of One-dimensional
Mappings .- 3. Generalized Abel Equation
.- 4. Equations with
Several Transformations of Argument .- 5.
Linear Equations .-
Bibliography .- Index
February 2004 | Hardback | 350 pages 140
line diagrams 7 half-tones
120 exercises | ISBN: 0-521-82650-0
February 2004 | Paperback | 350 pages 140
line diagrams 7 half-tones
120 exercises | ISBN: 0-521-53391-0
This refreshing, introductory textbook covers
both standard
techniques for solving ordinary differential
equations, as well
as introducing students to qualitative methods
such as phase-plane
analysis. The presentation is concise, informal
yet rigorous; it
can be used either for 1-term or 1-semester
courses. Topics such
as Euler’s method, difference equations,
the dynamics of the
logistic map, and the Lorenz equations, demonstrate
the vitality
of the subject, and provide pointers to further
study. The author
also encourages a graphical approach to the
equations and their
solutions, and to that end the book is profusely
illustrated. The
files to produce the figures using MATLAB
are all provided in an
accompanying website. Numerous worked examples
provide motivation
for and illustration of key ideas and show
how to make the
transition from theory to practice. Exercises
are also provided
to test and extend understanding: solutions
for these are
available for teachers.
Contents
Introduction; Part I. First Order Differential
Equations: 1.
Radioactive decay and carbon dating; 2. Integration
variables; 3.
Classification of differential equations;
4. Graphical
representation of solutions using MATLAB;
5. ‘Trivial’
differential equations; 6. Existence and
uniqueness of solutions;
7. Scalar autonomous ODEs; 8. Separable equations;
9. First order
linear equations and the integrating factor;
10. Two ‘tricks’
for nonlinear equations; Part II. Second
Order Linear Equations
With Constant Coefficients: 11. Second order
linear equations:
general theory; 12. Homogeneous 2nd order
linear ODEs; 13.
Oscillations; 14. Inhomogeneous 2nd order
linear equations; 15.
Resonance; 16. Higher order linear equations;
Part III. Linear
Second Order Equations With Variable Coefficients:
17. Reduction
of order; 18. The variation of constants
formula; 19. Cauchy-Euler
equations; 20. Series solutions of second
order linear equations;
Part IV. Numerical Methods and Difference
Equations: 21. Euler’s
method; 22. Difference equations; 23. Nonlinear
first order
difference equations; 24. The logistic map;
Part V. Coupled
Linear Equations: 25. Vector first order
equations and higher
order equations; 26. Explicit solutions of
coupled linear
systems; 27. Eigenvalues and eigenvectors;
28. Distinct real
eigenvalues; 29. Complex eigenvalues; 30.
A repeated real
eigenvalue; 31. Summary of phase portraits
for linear equations;
Part VI. Coupled Nonlinear Equations: 32.
Coupled nonlinear
equations; 33. Ecological models; 34. Newtonian
dynamics; 35. The
‘real’ pendulum; 36. Periodic orbits; 37.
The Lorenz
equations; 38. What next?.
September 2003 | Hardback | 528 pages 84
line diagrams 55
tables 227 exercises | ISBN: 0-521-77251-6
September 2003 | Paperback| 528 pages 84
line diagrams 55 tables
227 exercises | ISBN: 0-521-77590-6
This textbook offers a systematic, self-contained
account of the
main contributions of modern game theory
and its applications to
economics. Starting with a detailed description
of how to model
strategic situations, the discussion proceeds
by studying basic
solution concepts, their main refinements,
games played under
incomplete information, and repeated games.
For each of these
theoretical developments, there is a companion
set of
applications that cover the most representative
instances of game-theoretic
analysis in economics, e.g. oligopolistic
competition, public
goods, coordination failures, bargaining,
insurance markets,
implementation theory, signaling and auctions.
The theory and
applications covered in the first part of
the book fall under the
so-called ‘classical’ approach to game
theory, which is
founded on the paradigm of players’ unlimited
rationality. The
second part shifts towards topics that no
longer abide by that
paradigm. This leads to the study of topics
such as the interplay
between evolution and rationality.
Contents
Part I. Theoretical Framework: 1.1 Introduction
and examples; 1.2
The representation of a game in extensive
form; 1.3 The
representation of a game in strategic form;
1.4 The mixed
extension of a game; 1.5 Mixed and behavioral
strategies; 1.6
Representation in coalition form; Summary;
Exercises; Part II.
Strategic-Form Analysis: Theory: 2.1 Dominance
and iterative
dominance; 2.2 Nash equilibrium; 2.3 Zero-sum
bilateral games; 2.4
Nash equilibrium: formal existence results;
2.5 Strong and
coalition-proof equilibrium; 2.6 Correlated
equilibrium; 2.7
Rationalizability; Summary; Exercises; Part
III. Strategic-Form
Analysis:Applications: 3.1 Oligopoly (I):
static models; 3.2
Mechanism design (I): efficient allocation
of public goods; 3.3
Mechanism design (II): Nash implementation;
3.4 Markets (I):
macroeconomic coordination failures; Summary;
Exercises; Part IV.
Refinements of Nash Equilibrium: Theory:
4.1 Introduction; 4.2
Refinements excluding ‘incredible threats’:
examples; 4.3
Subgame-perfect equilibrium; 4.4 Weak-perfect
Bayesian
equilibrium; 4.5 Refinements excluding ‘untenable
beliefs’:
examples; 4.6 Sequential equilibrium; 4.7
Perfect and proper
equilibria; 4.8 Strategic-form refinements;
Summary; Exercises;
Part V. Refinements of Nash Equilibrium:
Applications; 5.1
Oligopoly (II): sequential moves; 5.2 Markets
(II): decentralized
price formation; 5.3 Oligopoly (III): differentiated
products; 5.4
Mechanism design (III): efficient allocation
of an indivisible
object; Summary, Exercises; Part VI. Incomplete
Information:
Theory: 6.1 Introduction and examples; 6.2
Bayesian games; 6.3
Bayes-Nash equilibrium; 6.4 Signalling games;
6.5 Mixed
strategies revisited: a purification approach;
6.6 Forward
induction; Summary; Exercises; Part VII..
Incomplete Information:
Applications: 7.1 Markets (III): signalling
in the labor market;
7.2 Markets (IV): insurance markets and adverse
selection; 7.3
Mechanism design (IV): one-sided auctions;
7.4 Mechanism design (V):
buyer-seller trade; Summary; Exercises; Part
VIII.. Repeated
Interaction: Theory: 8.1 Introduction and
examples; 8.2 Repeated
games: basic theoretical framework; 8.3 Folk
theorems: Nash
equilibrium; 8.4 Reputation and ‘irrationality’:
informal
discussion; 8.5 Folk theorems: subgame=perfect
equilibrium; 8.6
Reputation and ‘irrationality’: formal
analysis; Summary;
Exercises; Part IX. Repeated Interaction:
Applications: 9.1
Oligopoly (IV): intertemporal collusion in
a Cournot scenario; 9.2
Oligopoly (V): intertemporal collusion in
a Bertrand scenario; 9.3
Markets (V): efficiency, wages and unemployment;
Summary;
Exercises; Part X. Evolutionary Foundations
of Equilibrium: 10.1
Introduction; 10.2 Static analysis; 10.3
Basic dynamic analysis;
10.4 Evolution in social environments; 10.5
The evolution of
cooperation: an example; Summary; Exercises;
Part XI. Learning to
Play: 11.1 Introduction; 11.2 Reinforcement
learning; 11.3 Static
perceptions and Nash equilibrium; 11.4 Memory,
expectations and
foresight; Summary; Exercises; Part XII.
Social Learning and
Equilibrium Selection; 12.1 Introduction;
12.2 Evolutionary games:
theoretical framework; 12.3 Evolutionary
games: alternative
scenarios; 12.4 Stochastic stability and
equilibrium selection;
12.5 Experimental evidence; 12.6 Perturbed
Markov processes:
basic concepts and techniques; 12.7 Reinforcement
learning with
flexible aspirations; Summary; Exercises.