Series: Progress in Nonlinear Differential Equations and Their Applications , Vol. 74

2008, XVI, 176 p., 10 illus., Hardcover

ISBN: 978-0-8176-4680-6

About this book

Many partial differential equations (PDEs) that arise in physics can be viewed as infinite-dimensional Hamiltonian systems. This monograph presents recent existence results of nonlinear oscillations of Hamiltonian PDEs, particularly of periodic solutions for completely resonant nonlinear wave equations.

After introducing the reader to classical finite-dimensional dynamical system theory, including the Weinstein?Moser and Fadell?Rabinowitz bifurcation results, the analogous theory for nonlinear wave equations is developed. The theory and applications of the Nash?Moser theorem to a class of nonlinear wave equations is also discussed together with other basic notions of Hamiltonian PDEs and number theory. The main examples of Hamiltonian PDEs presented include: the nonlinear wave equation, the nonlinear Schrodinger equation, beam equations, and the Euler equations of hydrodynamics.

This text serves as an introduction to research in this fascinating and rapidly growing field. Graduate students and researchers interested in variational techniques and nonlinear analysis applied to Hamiltonian PDEs will find inspiration in the book.

Written for:

Graduate students and researchers interested in variational techniques and nonlinear analysis applied to Hamiltonian PDEs

Keywords:

Nash-Moser theorem

bifurcations

critical point theory

Table of contents

Introduction.- Finite dimension.-Infinite dimension.- A tutorial in Nash?Moser theory.- Application to the NLW.- Forced vibrations.- Appendix.- Index.

Series: Progress in Mathematics , Vol. 264

2008, Approx. 200 p., Hardcover

ISBN: 978-3-7643-8534-7

About this book

Winner of the Ferran Sunyer i Balaguer Prize 2007

Gives new insights in open problems in liaison theory and Hilbert schemes

Publishes new results on determinantal ideals and gives an overview of recent developments

Determinantal ideals are a central topic in both commutative algebra and algebraic geometry. In this book, three problems are addressed: CI-liaison class and G-liaison class of standard determinantal ideals; the multiplicity conjecture for standard determinantal ideals; unobstructedness and dimension of families of standard determinantal ideals.

Winner of the Ferran Sunyer i Balaguer Prize 2007.

Written for:

Graduates, postgraduates and researchers

Keywords:

algebraic geometry

commutative algebra

determinantal ideal

liaison class

Table of contents

Introduction.- 1. Background.- 2. CI-Liaison and G-Liaison of Standard Determinantal Ideals.- 3. Multiplicity Conjecture for Standard Determinantal Ideals.- 4. Unobstructedness and Dimension of Families of Standard Determinantal Ideals.- 5. Determinantal Ideals, Symmetric Determinantal Ideals and Open Problems.- Bibliography.- Index.

Series: Studies in Universal Logic

2008, Approx. 180 p., Softcover

ISBN: 978-3-7643-8517-0

About this book

The book develops the theory of one of the most important notions in the methodology of formal systems. Particularly, completeness plays an important role in propositional logic where many variants of the notion have been defined. Global variants of the notion mean the possibility of getting all correct and reliable schemata of inference. Its local variants refer to the notion of truth given by some semantics. A uniform theory of completeness in its general and local meaning is carried out and it generalizes and systematizes some variety of the notion of completeness such as Post-completeness, structural completeness and many others. This approach allows also for a more profound view upon some essential properties (e.g. two-valuedness) of propositional systems. For these purposes, the theory of logical matrices, and the theory of consequence operations is exploited.

Written for:

Graduates, postgraduates, and researchers, computer scientists, mathematicians, philosophers, linguists

Keywords:

Completeness

Consequence operation

Logical matrix

Post-completeness

Structural completeness

Universal Logic

Table of contents

Introduction.- 1. Basic notions: Propositional languages.- Abstract algebras.- Preliminary lattice-theoretical notions.- Propositional logics.- Brief exposition of the most important propositional logics.- 2. Semantic methods in propositional logic: Preordered sets.- Preordered algebras.- Logical matrices.- Adequacy.- Propositional logic and lattice theory.- 3. Completeness of propositional logic: Generalized completeness.- Post-completeness.- The problem of uniqueness of Lindenbaum extensions.- Some related concepts.- 4. Characterization of propositional connectives: Cn-definitions.- The system (D).- Variants.- The system (I).- Classical logic.- Appendix: The fundamental metatheorem for the classical propositional logic.- A proof system for the classical logic

Dedicated to the Unity of Science

Series: Studies in Universal Logic

2008, Approx. 260 p., Softcover

ISBN: 978-3-7643-8505-7

About this book

The three main themes of this book are (i) universal logic and the question of what logic is, (ii) universal algebraic logic and duality theories between the world of logics and the world of algebra, and (iii) algebraic logic proper including algebras of relations of various ranks, Tarski's cylindric algebras, relation algebras, Halmos' polyadic algebras and other kinds of algebras of logic. Besides Tarskian algebraizations of logics, category theoretical perspectives are also touched upon. Following the Tarskian tradition, besides the connections between logic and algebra, related logical connections with geometry and eventually spacetime geometry leading up to relativity are also part of the perspective of the book. An introductory chapter contains the necessary algebraic basics, this can be used in its own right as a quick introduction to universal algebra.

Table of contents

1. Introduction.- 2. Notation, elementary concepts: 2.1 Sets, classes, tuples, simple operations on sets.- 2.2 Binary relations, equivalence relations, functions.-

2.3 Orderings, ordinals, cardinals.- 2.4 Sequences.- 2.5 Direct product of families of sets.- 2.6 Relations of higher ranks.- 2.7 First-order logic (FOL).-

3. Basics from universal algebra: 3.1 Examples for algebras.- 3.2 Building new algebras from old ones (operations on algebras): 3.2.1 Subalgebra.-

3.2.2 Homomorphic image.- 3.2.3 Cartesian product, direct decomposition.- 3.2.4 Subdirect decomposition.- 3.2.5 Ultraproduct, reduced product.-

3.3 Variety characterization, quasi-variety characterization.- 3.4 Discriminator varieties.- 3.5 Boolean algebras.- 3.6 Boolean algebras with operators.- 4. General framework for studying logics: 4.1 Defining the framework.- 4.2 Concrete logics in the new framework.- 4.2.1 Distinguished logics.- 4.2.2 The finite variable fragment of FOL. Finite model theory.- 4.2.3 Summary.- 4.2.4 Solutions for some exercises of subsections 4.2.1 and 4.2.2.- 5. Bridge between the world of logics and the world of algebras: 5.1 Fine-tuning the framework.- 5.2 Algebraic characterizations of completeness and compactness properties.- 6. Generalizations.- 7. Further equivalence results (definability, interpolation, omitting types) 8. New kinds of logics 9. The guarded fragment and its finite variable hierarchy.- 10. Distinguished algebras of logic: 10.1 Cylindric algebras.- 10.2 Relation algebras, substitution cylindric algebras, polyadic algebras.- 10.3 Relativization. The non-square approach to ``taming logics''.- 10.4 Brief return to guarded fragment and related positive results.- 11. Applications of the algebraic results to logic (via our bridge).- 12. Connections with set theory and foundational thinking.- 13. Connections with abstract model theory.- 14. Extending the bridge to including logic of spacetime, connections with relativity.- 15. On the problems in the books Henkin-Monk-Tarski: Cylindric Algebras Parts I,II and Henkin-Monk-Tarski-Andreka-Nemeti.- Some further problems.