Studies in Logic and the Foundations of Mathematics, vol.151.
Description
The book is meant to serve two purposes. The first and more
obvious one is to present state of the art results in algebraic
research into residuated structures related to substructural
logics. The second, less obvious but equally important, is to
provide a reasonably gentle introduction to algebraic logic. At
the beginning, the second objective is predominant. Thus, in the
first few chapters the reader will find a primer of universal
algebra for logicians, a crash course in nonclassical logics for
algebraists, an introduction to residuated structures, an outline
of Gentzen-style calculi as well as some titbits of proof theory
- the celebrated Hauptsatz, or cut elimination theorem, among
them. These lead naturally to a discussion of interconnections
between logic and algebra, where we try to demonstrate how they
form two sides of the same coin. We envisage that the initial
chapters could be used as a textbook for a graduate course,
perhaps entitled Algebra and Substructural Logics. As the book
progresses the first objective gains predominance over the second.
Although the precise point of equilibrium would be difficult to
specify, it is safe to say that we enter the technical part with
the discussion of various completions of residuated structures.
These include Dedekind-McNeille completions and canonical
extensions. Completions are used later in investigating several
finiteness properties such as the finite model property,
generation of varieties by their finite members, and finite
embeddability. The algebraic analysis of cut elimination that
follows, also takes recourse to completions. Decidability of
logics, equational and quasi-equational theories comes next,
where we show how proof theoretical methods like cut elimination
are preferable for small logics/theories, but semantic tools like
Rabin's theorem work better for big ones. Then we turn to
Glivenko's theorem, which says that a formula is an
intuitionistic tautology if and only if its double negation is a
classical one. We generalise it to the substructural setting,
identifying for each substructural logic its Glivenko equivalence
class with smallest and largest element. This is also where we
begin investigating lattices of logics and varieties, rather than
particular examples. We continue in this vein by presenting a
number of results concerning minimal varieties/maximal logics. A
typical theorem there says that for some given well-known variety
its subvariety lattice has precisely such-and-such number of
minimal members (where values for such-and-such include, but are
not limited to, continuum, countably many and two). In the last
two chapters we focus on the lattice of varieties corresponding
to logics without contraction. In one we prove a negative result:
that there are no nontrivial splittings in that variety. In the
other, we prove a positive one: that semisimple varieties
coincide with discriminator ones. Within the second, more
technical part of the book another transition process may be
traced. Namely, we begin with logically inclined technicalities
and end with algebraically inclined ones. Here, perhaps,
algebraic rendering of Glivenko theorems marks the equilibrium
point, at least in the sense that finiteness properties,
decidability and Glivenko theorems are of clear interest to
logicians, whereas semisimplicity and discriminator varieties are
universal algebra par exellence. It is for the reader to judge
whether we succeeded in weaving these threads into a seamless
fabric.
Audience
This book is intended for: Research mathematicians and graduate
students and: Computer scientists
Contents
Contents List of Figures List of Tables Introduction Chapter 1.
Getting started Chapter 2. Substructural logics and residuated
lattices Chapter 3. Residuation and structure theory Chapter 4.
Decidability Chapter 5. Logical and algebraic properties Chapter
6. completions and finite embeddability Chapter 7. Algebraic
aspects of cut elimination Chapter 8. Glivenko theorems Chapter 9.
Lattices of logics and varieties Chapter 10. Splittings Chapter
11. Semisimplicity Bibliography Index
Hardbound, 532 pages, publication date: APR-2007
ISBN-13: 978-0-444-52141-5
ISBN-10: 0-444-52141-0
Series: Cambridge Monographs on Mathematical Physics
Paperback (ISBN-13: 9780521036702)
This book provides a thorough introduction to the theory of
classical integrable systems, discussing the various approaches
to the subject and explaining their interrelations. The book
begins by introducing the central ideas of the theory of
integrable systems, based on Lax representations, loop groups and
Riemann surfaces. These ideas are then illustrated with detailed
studies of model systems. The connection between isomonodromic
deformation and integrability is discussed, and integrable field
theories are covered in detail. The KP, KdV and Toda hierarchies
are explained using the notion of Grassmannian, vertex operators
and pseudo-differential operators. A chapter is devoted to the
inverse scattering method and three complementary chapters cover
the necessary mathematical tools from symplectic geometry,
Riemann surfaces and Lie algebras. The book contains many worked
examples and is suitable for use as a textbook on graduate
courses. It also provides a comprehensive reference for
researchers already working in the field.
* Discusses all approaches to the subject and emphasises their
unity
* Written in a clear and pedagogical style
* Includes developments of importance for the study of quantum
systems
Contents
1. Introduction; 2. Integrable dynamical systems; 3. Synopsis of
integrable systems; 4. Algebraic methods; 5. Analytical methods;
6. The closed Toda chain; 7. The Calogero-Moser model; 8.
Isomonodromic deformations; 9. Grassmannian and integrable
hierarchies; 10. The KP hierarchy; 11. The KdV hierarchy; 12. The
Toda field theories; 13. Classical inverse scattering method; 14.
Symplectic geometry; 15. Riemann surfaces; 16. Lie algebras;
Index.
Hardback (ISBN-13: 9780521874069)
In this book, a new approach is pioneered in providing a unified
theory in continuum mechanics. General Continuum Mechanics is
intended for the beginner, but it develops advanced material
covering interdisciplinary subjects. With applications of
convective, Lagrangian, and Eulerian coordinates and the first
and second laws of thermodynamics, the first-year graduate
student will learn solid mechanics and fluid mechanics as an
integrated subject. Electromagnetic continuum and relativistic
continuum are included. The conservational properties of mass,
momentum, and energy on earth and in the universe constitute the
ingredients of this book. They are the monumental contributions
of Newton, Maxwell, and Einstein, a panorama of beauty of
universal laws that evolved over the last four centuries. No
boundaries are needed to separate them, but rather we integrate
them in harmony and place them in perspective. This is the book
for interdisciplinary studies to carry out the modern scientific
projects in which engineering, physics, and applied mathematics
must be combined.
* This is for a broad audience including engineers, physicists,
and applied mathematicians
* Integrated thermodynamics treatments for solids and fluids
* Electromagnetic continuum and relativistic continuum are
included
Contents
Part I. Basic Topics: 1. Introduction; 2. Kinematics; 3.
Kinetics; 4. Linear elasticity; 5. Newtonian fluid mechanics;
Part II. Special Topics: 6. Curvilinear continuum; 7. Nonlinear
continuum; 8. Electromagnetic continuum; 9. Differential geometry
continuum; Epilogue.
Series: Cambridge Mathematical Library
Paperback (ISBN-13: 9780521705127)
This classic work continues to offer a comprehensive treatment of
the theory of univariate and tensor-product splines. It will be
of interest to researchers and students working in applied
analysis, numerical analysis, computer science, and engineering.
The material covered provides the reader with the necessary tools
for understanding the many applications of splines in such
diverse areas as approximation theory, computer-aided geometric
design, curve and surface design and fitting, image processing,
numerical solution of differential equations, and increasingly in
business and the biosciences. This new edition includes a
supplement outlining some of the major advances in the theory
since 1981, and some 250 new references. It can be used as the
main or supplementary text for courses in splines, approximation
theory or numerical analysis.
* Comprehensive reference, with preparatory material on
polynomials, Tchebycheff systems etc, plus historical notes and
comments, and comprehensive list of references
* Includes efficent algorithms for evaluating B-splines, treats
generalized splines, gives full account of approximation
properties of splines
* New supplement helps keep the book up to date
Contents
1. Introduction; 2. Preliminaries; 3. Polynomials; 4. Polynomial
splines; 5. Computational methods; 6. Approximation power of
splines; 7. Approximation power of splines (free knots); 8. Other
spaces of polynomial splines; 9. Tchebycheffian splines; 10. L-Splines;
11. Generalized splines; 12. Tensor-product splines; 13. Some
multidimensional tools; Supplement; References; New references;
Index.
Series: Encyclopedia of Mathematics and its Applications
Hardback (ISBN-13: 9780521832588)
Designed to work as a reference and as a supplement to an
advanced course on dynamical systems, this book presents a self-contained
and comprehensive account of modern smooth ergodic theory. Among
other things, this provides a rigorous mathematical foundation
for the phenomenon known as deterministic chaos - the appearance
of echaoticf motions in pure deterministic dynamical systems.
A sufficiently complete description of topological and ergodic
properties of systems exhibiting deterministic chaos can be
deduced from relatively weak requirements on their local behavior
known as nonuniform hyperbolicity conditions. Nonuniform
hyperbolicity theory is an important part of the general theory
of dynamical systems. Its core is the study of dynamical systems
with nonzero Lyapunov exponents both conservative and
dissipative, in addition to cocycles and group actions. The
results of this theory are widely used in geometry (e.g.,
geodesic flows and Teichmuller flows), in rigidity theory, in the
study of some partial differential equations (e.g., the
Schrodinger equation), in the theory of billiards, as well as in
applications to physics, biology, engineering, and other fields.
* The book summarizes and unifies results of smooth ergodic
theory, which is one of the core parts of the general dynamical
system theory
* Describes the theory of deterministic chaos
* The book can be used as supporting material for an advanced
course on dynamical systems
Contents
Part I. Linear Theory: 1. The concept of nonuniform
hyperbolicity; 2. Lyapunov exponents for linear extensions; 3.
Regularity of cocycles; 4. Methods for estimating exponents; 5.
The derivative cocycle; Part II. Examples and Foundations of the
Nonlinear Theory: 6. Examples of systems with hyperbolic
behavior; 7. Stable manifold theory; 8. Basic properties of
stable and unstable manifolds; Part III. Ergodic Theory of Smooth
and SRB Measures: 9. Smooth measures; 10. Measure-Theoretic
entropy and Lyapunov exponents; 11. Stable ergodicity and
Lyapunov exponents; 12, Geodesic flows; 13. SRB measures; Part IV.
General Hyperbolic Measures: 14. Hyperbolic measures: entropy and
dimension; 15. Hyperbolic measures: topological properties.