Slicing The Truth

Author: Denis R Hirschfeldt
Publisher: World Scientific
ISBN: 9814612634
Size: 33.49 MB
Format: PDF, ePub
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This book is a brief and focused introduction to the reverse mathematics and computability theory of combinatorial principles, an area of research which has seen a particular surge of activity in the last few years. It provides an overview of some fundamental ideas and techniques, and enough context to make it possible for students with at least a basic knowledge of computability theory and proof theory to appreciate the exciting advances currently happening in the area, and perhaps make contributions of their own. It adopts a case-study approach, using the study of versions of Ramsey's Theorem (for colorings of tuples of natural numbers) and related principles as illustrations of various aspects of computability theoretic and reverse mathematical analysis. This book contains many exercises and open questions. Contents:Setting Off: An IntroductionGathering Our Tools: Basic Concepts and NotationFinding Our Path: König's Lemma and ComputabilityGauging Our Strength: Reverse MathematicsIn Defense of DisarrayAchieving Consensus: Ramsey's TheoremPreserving Our Power: ConservativityDrawing a Map: Five DiagramsExploring Our Surroundings: The World Below RT22Charging Ahead: Further TopicsLagniappe: A Proof of Liu's Theorem Readership: Graduates and researchers in mathematical logic. Key Features:This book assumes minimal background in mathematical logic and takes the reader all the way to current research in a highly active areaIt is the first detailed introduction to this particular approach to this area of researchThe combination of fully worked out arguments and exercises make this book well suited to self-study by graduate students and other researchers unfamiliar with the areaKeywords:Reverse Mathematics;Computability Theory;Computable Mathematics;Computable Combinatorics

Induction Bounding Weak Combinatorial Principles And The Homogeneous Model Theorem

Author: Denis R. Hirschfeldt
Publisher: American Mathematical Soc.
ISBN: 1470426579
Size: 27.45 MB
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Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of . Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model.

Computability And Complexity

Author: Adam Day
Publisher: Springer
ISBN: 3319500627
Size: 53.78 MB
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This Festschrift is published in honor of Rodney G. Downey, eminent logician and computer scientist, surfer and Scottish country dancer, on the occasion of his 60th birthday. The Festschrift contains papers and laudations that showcase the broad and important scientific, leadership and mentoring contributions made by Rod during his distinguished career. The volume contains 42 papers presenting original unpublished research, or expository and survey results in Turing degrees, computably enumerable sets, computable algebra, computable model theory, algorithmic randomness, reverse mathematics, and parameterized complexity, all areas in which Rod Downey has had significant interests and influence. The volume contains several surveys that make the various areas accessible to non-specialists while also including some proofs that illustrate the flavor of the fields.

Uncertainty Within Economic Models

Author: Hansen Lars Peter
Publisher: World Scientific
ISBN: 9814578134
Size: 70.43 MB
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Written by Lars Peter Hansen (Nobel Laureate in Economics, 2013) and Thomas Sargent (Nobel Laureate in Economics, 2011), Uncertainty within Economic Models includes articles adapting and applying robust control theory to problems in economics and finance. This book extends rational expectations models by including agents who doubt their models and adopt precautionary decisions designed to protect themselves from adverse consequences of model misspecification. This behavior has consequences for what are ordinarily interpreted as market prices of risk, but big parts of which should actually be interpreted as market prices of model uncertainty. The chapters discuss ways of calibrating agents' fears of model misspecification in quantitative contexts.

Algorithmic Randomness And Complexity

Author: Rodney G. Downey
Publisher: Springer Science & Business Media
ISBN: 0387684417
Size: 66.43 MB
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Computability and complexity theory are two central areas of research in theoretical computer science. This book provides a systematic, technical development of "algorithmic randomness" and complexity for scientists from diverse fields.

Combinatorial Set Theory

Author: Lorenz J. Halbeisen
Publisher: Springer Science & Business Media
ISBN: 9781447121732
Size: 34.39 MB
Format: PDF
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This book provides a self-contained introduction to modern set theory and also opens up some more advanced areas of current research in this field. The first part offers an overview of classical set theory wherein the focus lies on the axiom of choice and Ramsey theory. In the second part, the sophisticated technique of forcing, originally developed by Paul Cohen, is explained in great detail. With this technique, one can show that certain statements, like the continuum hypothesis, are neither provable nor disprovable from the axioms of set theory. In the last part, some topics of classical set theory are revisited and further developed in the light of forcing. The notes at the end of each chapter put the results in a historical context, and the numerous related results and the extensive list of references lead the reader to the frontier of research. This book will appeal to all mathematicians interested in the foundations of mathematics, but will be of particular use to graduates in this field.

Complex Quantum Systems

Author: Heinz Siedentop
Publisher: World Scientific
ISBN: 9814460168
Size: 40.82 MB
Format: PDF, ePub, Mobi
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This volume is based on lectures given during the program “Complex Quantum Systems” held at the National University of Singapore's Institute for Mathematical Sciences from 17 February to 27 March 2010. It guides the reader through two introductory expositions on large Coulomb systems to five of the most important developments in the field: derivation of mean field equations, derivation of effective Hamiltonians, alternative high precision methods in quantum chemistry, modern many-body methods originating from quantum information, and — the most complex — semirelativistic quantum electrodynamics. These introductions are written by leaders in their fields; amongst them are Volker Bach, Rafael Benguria, Thomas Chen, and Jan Philip Solovej. Together, they fill a gap between current textbooks and the vast modern literature on complex quantum systems. Contents:Stability of Matter (Rafael D Benguria and Benjamín A Loewe)Mathematical Density and Density Matrix Functional Theory (DFT and DMFT) (Volker Bach)On the Dynamics of a Fermi Gas in a Random Medium with Dynamical Hartree–Fock Interactions (Thomas Chen)On the Minimization of Hamiltonians over Pure Gaussian States (Jan Dereziński, Marcin Napiórkowski and Jan Philip Solovej)Variational Approach to Electronic Structure Calculations on Second-Order Reduced Density Matrices and the N-Representability Problem (Maho Nakata, Mituhiro Fukuda, and Katsuki Fujisawa)Fermionic Quantum Many-Body Systems: A Quantum Information Approach (Christina V Kraus)Hydrogen-Like Atoms in Relativistic QED (Martin Könenberg, Oliver Matte, and Edgardo Stockmeyer) Readership: Mathematicians and mathematical physicists, advanced graduate students and researchers in rigorous many body theory. Keywords:Analysis;Large Coulomb Systems;Stability;Correlation Inequalities;Effective Equations;Quantum Information;Matrix Product States;Quantum Electrodynamics

Self Organization And Emergence In Life Sciences

Author: Bernard Feltz
Publisher: Springer Science & Business Media
ISBN: 1402039174
Size: 56.13 MB
Format: PDF
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Self-organization constitutes one of the most important theoretical debates in contemporary life sciences. The present book explores the relevance of the concept of self-organization and its impact on such scientific fields as: immunology, neurosciences, ecology and theories of evolution. Historical aspects of the issue are also broached. Intuitions relative to self-organization can be found in the works of such key western philosophical figures as Aristotle, Leibniz and Kant. Interacting with more recent authors and cybernetics, self-organization represents a notion in keeping with the modern world's discovery of radical complexity. The themes of teleology and emergence are analyzed by philosophers of sciences with regards to the issues of modelization and scientific explanation. The implications of self-organization for life sciences are here approached from an interdisciplinary angle, revealing the notion as already rewarding and full of promise for the future.

On Fuzziness

Author: Rudolf Seising
Publisher: Springer
ISBN: 3642356419
Size: 18.35 MB
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The notion of Fuzziness stands as one of the really new concepts that have recently enriched the world of Science. Science grows not only through technical and formal advances on one side and useful applications on the other side, but also as consequence of the introduction and assimilation of new concepts in its corpus. These, in turn, produce new developments and applications. And this is what Fuzziness, one of the few new concepts arisen in the XX Century, has been doing so far. This book aims at paying homage to Professor Lotfi A. Zadeh, the “father of fuzzy logic” and also at giving credit to his exceptional work and personality. In a way, this is reflected in the variety of contributions collected in the book. In some of them the authors chose to speak of personal meetings with Lotfi; in others, they discussed how certain papers of Zadeh were able to open for them a new research horizon. Some contributions documented results obtained from the author/s after taking inspiration from a particular idea of Zadeh, thus implicitly acknowledging him. Finally, there are contributions of several “third generation fuzzysists or softies” who were firstly led into the world of Fuzziness by a disciple of Lotfi Zadeh, who, following his example, took care of opening for them a new road in science. Rudolf Seising is Adjoint Researcher at the European Centre for Soft Computing in Mieres, Asturias (Spain). Enric Trillas and Claudio Moraga are Emeritus Researchers at the European Centre for Soft Computing, Mieres, Asturias (Spain). Settimo Termini is Professor of Theoretical Computer Science at the University of Palermo, Italy and Affiliated Researcher at the European Centre for Soft Computing, Mieres, Asturias (Spain)