O Minimal Structures
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Tame Topology And O Minimal Structures
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Author : Lou Van den Dries
language : en
Publisher:
Release Date : 1998
Tame Topology And O Minimal Structures written by Lou Van den Dries and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with Ordered algebraic structures categories.
These notes give a self-contained treatment of the theory of o-minimal structures.
O Minimal Structures
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Author : Mário J. Edmundo
language : en
Publisher: Cuvillier Verlag
Release Date : 2005
O Minimal Structures written by Mário J. Edmundo and has been published by Cuvillier Verlag this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with categories.
Ultraproducts Of O Minimal Structures
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Author : Alex Rennet
language : en
Publisher:
Release Date : 2012
Ultraproducts Of O Minimal Structures written by Alex Rennet and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.
There are three main parts to this thesis, all centred around ultraproducts of o-minimal structures. In the first part we investigate (for a fixed first-order language L) what we call the L-theory of o-minimality. It is the theory consisting of those L-sentences true in all o-minimal L-structures. We find that when L expands the language of real closed fields by at least one new function or relation symbol, the L-theory of o-minimality is not recursively axiomatizable. In particular, for any recursive list of axioms A which is consistent with the L-theory of o-minimality, we find that there are locally o-minimal, definably complete structures satisfying A which are not elementarily equivalent to an ultraproduct of o-minimal structures. We call the latter sort of structures pseudo-o-minimal. In the second part we investigate uniform finiteness and cell decomposition in the pseudo-o-minimal setting. To do this, we introduce the notion of a pseudo-o-minimal structure tallying a discrete definable set. Investigating this notion, we answer some questions of uniqueness and existence. Finally, we show that under certain assumptions about the discrete definable sets that a given pseudo-o-minimal structure can tally, we have a version of uniform finiteness, at least in the planar case. This is the first step towards a cell decomposition theorem in this setting. In the final section, we look into two classes of examples of ultraproducts of o-minimal structures. For the first class, we note the o-minimality of a certain subset of these structures, and show the non-o-minimality of another. In particular, we derive the o-minimality of a new structure related to the real field with the exponential function. The second class is relatively intractable, but we discuss its relation to an important open problem in o-minimality.
Lecture Notes On O Minimal Structures And Real Analytic Geometry
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Author : Chris Miller
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-09-14
Lecture Notes On O Minimal Structures And Real Analytic Geometry written by Chris Miller and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-09-14 with Mathematics categories.
This volume was produced in conjunction with the Thematic Program in o-Minimal Structures and Real Analytic Geometry, held from January to June of 2009 at the Fields Institute. Five of the six contributions consist of notes from graduate courses associated with the program: Felipe Cano on a new proof of resolution of singularities for planar analytic vector fields; Chris Miller on o-minimality and Hardy fields; Jean-Philippe Rolin on the construction of o-minimal structures from quasianalytic classes; Fernando Sanz on non-oscillatory trajectories of vector fields; and Patrick Speissegger on pfaffian sets. The sixth contribution, by Antongiulio Fornasiero and Tamara Servi, is an adaptation to the nonstandard setting of A.J. Wilkie's construction of o-minimal structures from infinitely differentiable functions. Most of this material is either unavailable elsewhere or spread across many different sources such as research papers, conference proceedings and PhD theses. This book will be a useful tool for graduate students or researchers from related fields who want to learn about expansions of o-minimal structures by solutions, or images thereof, of definable systems of differential equations.
O Minimality And Diophantine Geometry
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Author : G. O. Jones
language : en
Publisher: Cambridge University Press
Release Date : 2015-08-20
O Minimality And Diophantine Geometry written by G. O. Jones and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-08-20 with Mathematics categories.
This collection of articles, originating from a short course held at the University of Manchester, explores the ideas behind Pila's proof of the Andre–Oort conjecture for products of modular curves. The basic strategy has three main ingredients: the Pila–Wilkie theorem, bounds on Galois orbits, and functional transcendence results. All of these topics are covered in this volume, making it ideal for researchers wishing to keep up to date with the latest developments in the field. Original papers are combined with background articles in both the number theoretic and model theoretic aspects of the subject. These include Martin Orr's survey of abelian varieties, Christopher Daw's introduction to Shimura varieties, and Jacob Tsimerman's proof via o-minimality of Ax's theorem on the functional case of Schanuel's conjecture.
A Guide To Classical And Modern Model Theory
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Author : Annalisa Marcja
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-09-10
A Guide To Classical And Modern Model Theory written by Annalisa Marcja and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-09-10 with Philosophy categories.
Since its birth, Model Theory has been developing a number of methods and concepts that have their intrinsic relevance, but also provide fruitful and notable applications in various fields of Mathematics. It is a lively and fertile research area which deserves the attention of the mathematical world. This volume: -is easily accessible to young people and mathematicians unfamiliar with logic; -gives a terse historical picture of Model Theory; -introduces the latest developments in the area; -provides 'hands-on' proofs of elimination of quantifiers, elimination of imaginaries and other relevant matters. A Guide to Classical and Modern Model Theory is for trainees and professional model theorists, mathematicians working in Algebra and Geometry and young people with a basic knowledge of logic.
Mathematical Logic In Asia
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Author : S. S. Goncharov
language : en
Publisher: World Scientific
Release Date : 2006
Mathematical Logic In Asia written by S. S. Goncharov and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.
This volume is devoted to the main areas of mathematical logic and applications to computer science. There are articles on weakly o-minimal theories, algorithmic complexity of relations, models within the computable model theory, hierarchies of randomness tests, computable numberings, and complexity problems of minimal unsatisfiable formulas. The problems of characterization of the deduction-detachment theorem, o 1 -induction, completeness of Leoniewski''s systems, and reduction calculus for the satisfiability problem are also discussed. The coverage includes the answer to Kanovei''s question about the upper bound for the complexity of equivalence relations by convergence at infinity for continuous functions. The volume also gives some applications to computer science such as solving the problems of inductive interference of languages from the full collection of positive examples and some negative data, the effects of random negative data, methods of formal specification and verification on the basis of model theory and multiple-valued logics, interval fuzzy algebraic systems, the problems of information exchange among agents on the base topological structures, and the predictions provided by inductive theories. Sample Chapter(s). Chapter 1: Another Characterization of the Deduction-Detachment Theorem (535 KB). Contents: Another Characterization of the Deduction-Detachment Theorem (S V Babyonyshev); On Behavior of 2-Formulas in Weakly o-Minimal Theories (B S Baizhanov & B Sh Kulpeshov); Arithmetic Turing Degrees and Categorical Theories of Computable Models (E Fokina); Negative Data in Learning Languages (S Jain & E Kinber); Effective Cardinals in the Nonstandard Universe (V Kanovei & M Reeken); Model-Theoretic Methods of Analysis of Computer Arithmetic (S P Kovalyov); The Functional Completeness of Leoniewski''s Systems (F Lepage); Hierarchies of Randomness Tests (J Reimann & F Stephan); Intransitive Linear Temporal Logic Based on Integer Numbers, Decidability, Admissible Logical Consecutions (V V Rybakov); The Logic of Prediction (E Vityaev); Conceptual Semantic Systems Theory and Applications (K E Wolff); Complexity Results on Minimal Unsatisfiable Formulas (X Zhao); and other papers. Readership: Researchers in mathematical logic and algebra, computer scientists in artificial intelligence and fuzzy logic."
Finite Model Theory And Its Applications
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Author : Erich Grädel
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-04-24
Finite Model Theory And Its Applications written by Erich Grädel and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-04-24 with Computers categories.
Finite model theory,as understoodhere, is an areaof mathematicallogic that has developed in close connection with applications to computer science, in particular the theory of computational complexity and database theory. One of the fundamental insights of mathematical logic is that our understanding of mathematical phenomena is enriched by elevating the languages we use to describe mathematical structures to objects of explicit study. If mathematics is the science of patterns, then the media through which we discern patterns, as well as the structures in which we discern them, command our attention. It isthis aspect oflogicwhichis mostprominentin model theory,“thebranchof mathematical logic which deals with the relation between a formal language and its interpretations”. No wonder, then, that mathematical logic, and ?nite model theory in particular, should ?nd manifold applications in computer science: from specifying programs to querying databases, computer science is rife with phenomena whose understanding requires close attention to the interaction between language and structure. This volume gives a broadoverviewof some central themes of ?nite model theory: expressive power, descriptive complexity, and zero–one laws, together with selected applications to database theory and arti?cial intelligence, es- cially constraint databases and constraint satisfaction problems. The ?nal chapter provides a concise modern introduction to modal logic,which emp- sizes the continuity in spirit and technique with ?nite model theory.
Model Theory Algebra And Geometry
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Author : Deirdre Haskell
language : en
Publisher: Cambridge University Press
Release Date : 2000-07-03
Model Theory Algebra And Geometry written by Deirdre Haskell and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-07-03 with Mathematics categories.
Model theory has made substantial contributions to semialgebraic, subanalytic, p-adic, rigid and diophantine geometry. These applications range from a proof of the rationality of certain Poincare series associated to varieties over p-adic fields, to a proof of the Mordell-Lang conjecture for function fields in positive characteristic. In some cases (such as the latter) it is the most abstract aspects of model theory which are relevant. This book, originally published in 2000, arising from a series of introductory lectures for graduate students, provides the necessary background to understanding both the model theory and the mathematics behind these applications. The book is unique in that the whole spectrum of contemporary model theory (stability, simplicity, o-minimality and variations) is covered and diverse areas of geometry (algebraic, diophantine, real analytic, p-adic, and rigid) are introduced and discussed, all by leading experts in their fields.
Tame Topology And O Minimal Structures
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Author : Lou Van den Dries
language : en
Publisher: Cambridge University Press
Release Date : 1998-05-07
Tame Topology And O Minimal Structures written by Lou Van den Dries and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-05-07 with Mathematics categories.
These notes give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. This book should be of interest to model theorists, analytic geometers and topologists.