Notes on Forcing Axioms


Author: Stevo Todorcevic
Publisher: World Scientific
ISBN: 9814571598
Category: Mathematics
Page: 236
View: 5871

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In the mathematical practice, the Baire category method is a tool for establishing the existence of a rich array of generic structures. However, in mathematics, the Baire category method is also behind a number of fundamental results such as the Open Mapping Theorem or the Banach–Steinhaus Boundedness Principle. This volume brings the Baire category method to another level of sophistication via the internal version of the set-theoretic forcing technique. It is the first systematic account of applications of the higher forcing axioms with the stress on the technique of building forcing notions rather than on the relationship between different forcing axioms or their consistency strengths. Contents:Baire Category Theorem and the Baire Category NumbersCoding Sets by the Real NumbersConsequences in Descriptive Set TheoryConsequences in Measure TheoryVariations on the Souslin HypothesisThe S-Spaces and the L-SpacesThe Side-condition MethodIdeal DichotomiesCoherent and Lipschitz TreesApplications to the S-Space Problem and the von Neumann ProblemBiorthogonal SystemsStructure of Compact SpacesRamsey Theory on OrdinalsFive Cofinal TypesFive Linear OrderingsCardinal Arithmetic and mmReflection PrinciplesAppendices:Basic NotionsPreserving Stationary SetsHistorical and Other Comments Readership: Graduate students and researchers in logic, set theory and related fields. Key Features:This is a first systematic exposition of the unified approach for building proper, semi-proper, and stationary preserving forcing notions through the method of using elementary submodels as side conditionsThe books starts from the classical applications of Martin's axioms and ends with some of the most sophisticated applications of the Proper Forcing Axioms. In this way, the reader is led into a natural process of understanding the combinatorics hidden behind the methodKeywords:Set Theory;Forcing Axioms

Slicing the Truth

On the Computable and Reverse Mathematics of Combinatorial Principles
Author: Denis R Hirschfeldt
Publisher: World Scientific
ISBN: 9814612634
Category: Mathematics
Page: 232
View: 5967

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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

Combinatorial And Toric Homotopy: Introductory Lectures


Author: Darby Alastair,Grbic Jelena,Lu Zhi
Publisher: World Scientific
ISBN: 9813226587
Category: Mathematics
Page: 448
View: 7054

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This volume consists of introductory lectures on the topics in the new and rapidly developing area of toric homotopy theory, and its applications to the current research in configuration spaces and braids, as well as to more applicable mathematics such as fr-codes and robot motion planning. The book starts intertwining homotopy theoretical and combinatorial ideas within the remits of toric topology and illustrates an attempt to classify in a combinatorial way polytopes known as fullerenes, which are important objects in quantum physics, quantum chemistry and nanotechnology. Toric homotopy theory is then introduced as a further development of toric topology, which describes properties of Davis–Januszkiewicz spaces, moment-angle complexes and their generalizations to polyhedral products. The book also displays the current research on configuration spaces, braids, the theory of limits over the category of presentations and the theory of fr-codes. As an application to robotics, the book surveys topological problems relevant to the motion planning problem of robotics and includes new results and constructions, which enrich the emerging area of topological robotics. The book is at research entry level addressing the core components in homotopy theory and their important applications in the sciences and thus suitable for advanced undergraduate and graduate students. Contents: Toric Homotopy Theory (Stephen Theriault)Fullerenes, Polytopes and Toric Topology (Victor M Buchstaber and Nikolay Yu Erokhovets)Around Braids (Vladimir Vershinin)Higher Limits, Homology Theories and fr-Codes (Sergei O Ivanov and Roman Mikhailov)Configuration Spaces and Robot Motion Planning Algorithms (Michael Farber)Cellular Stratified Spaces (Dai Tamaki) Readership: Advanced undergraduate and graduate students as well as researchers interested in homotopy theory and its applications in the sciences. Keywords: Toric Topology;Toric Homotopy;Configuration Space;Stratified Spaces;Braid Group;Fullerene;Polytope;Virtual Braid Group;Thompson Group;Robotics;Motion PlanningReview: Key Features: The first book in the area of toric homotopy theory consisting of introductory lectures on the topics and their applications to fr-codes and robot motion planning

Forcing, Iterated Ultrapowers, and Turing Degrees


Author: Chitat Chong,Qi Feng,Theodore A Slaman,W Hugh Woodin,Yue Yang
Publisher: World Scientific
ISBN: 9814699969
Category: Mathematics
Page: 184
View: 5493

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This volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2010 and 2011 Asian Initiative for Infinity Logic Summer Schools. The major topics covered set theory and recursion theory, with particular emphasis on forcing, inner model theory and Turing degrees, offering a wide overview of ideas and techniques introduced in contemporary research in the field of mathematical logic. Contents:Prikry-Type Forcings and a Forcing with Short Extenders (Moti Gitik)The Turing Degrees: An Introduction (Richard A Shore)An Introduction to Iterated Ultrapowers (John Steel) Readership: Graduate students in mathematics, and researchers in logic, set theory and computability theory. Key Features:These are notes based on short courses given by three leading experts in set theory, recursion theory and their applicationsKeywords:Logic;Set Theory;Forcing;Recursion Theory;Computability Theory;Turing Degrees;C*-algebra

E-Recursion, Forcing and C*-Algebras


Author: Chitat Chong,Qi Feng,Theodore A Slaman,W Hugh Woodin,Yue Yang
Publisher: World Scientific
ISBN: 9814602655
Category: Mathematics
Page: 228
View: 6994

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This volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2012 Asian Initiative for Infinity Logic Summer School. The major topics cover set-theoretic forcing, higher recursion theory, and applications of set theory to C*-algebra. This volume offers a wide spectrum of ideas and techniques introduced in contemporary research in the field of mathematical logic to students, researchers and mathematicians. Contents:Selected Applications of Logic to Classification Problem for C*-Algebras (Ilijas Farah)Subcomplete Forcing and L-Forcing (Ronald Jensen)E-Recursion (Gerald E Sacks) Readership: Mathematics graduate students, researchers in logic, set theory and related areas. Key Features:These are notes based on short courses given by three leading experts in set theory, recursion theory and their applicationsKeywords:Logic;Set Theory;Forcing;E-recursion;C*-Algebra;Recursion Theory;Computability Theory

Computational Prospects of Infinity: Presented talks


Author: Chi-Tat Chong
Publisher: World Scientific
ISBN: 9812796541
Category: Computers
Page: 420
View: 6357

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This volume is a collection of written versions of the talks given at the Workshop on Computational Prospects of Infinity, held at the Institute for Mathematical Sciences from 18 June to 15 August 2005. It consists of contributions from many of the leading experts in recursion theory (computability theory) and set theory. Topics covered include the structure theory of various notions of degrees of unsolvability, algorithmic randomness, reverse mathematics, forcing, large cardinals and inner model theory, and many others.

Liebe und Mathematik

Im Herzen einer verborgenen Wirklichkeit
Author: Edward Frenkel
Publisher: Springer-Verlag
ISBN: 3662434210
Category: Mathematics
Page: 317
View: 6312

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Mathematical works


Author: Helmut Wielandt
Publisher: Walter de Gruyter
ISBN: 9783110124538
Category: Mathematics
Page: 802
View: 5727

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Contains all the mathematical works of the 20th-century German mathematician except those on group theory, which comprise the first volume. Most are on matrix theory. Among them are 24 research papers arranged chronologically, many with commentary by specialists; lecture notes on the analytic theory of matrix groups; his series of contributions to the mathematical treatment of complex eigenvalue problems prepared during the Second World War; and 1973 lecture notes on selected topics of permutation groups. About half of the pieces are in German. No index. Annotation copyrighted by Book News, Inc., Portland, OR

Differentialgeometrie, Topologie und Physik


Author: Mikio Nakahara
Publisher: Springer-Verlag
ISBN: 3662453002
Category: Science
Page: 597
View: 2560

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Differentialgeometrie und Topologie sind wichtige Werkzeuge für die Theoretische Physik. Insbesondere finden sie Anwendung in den Gebieten der Astrophysik, der Teilchen- und Festkörperphysik. Das vorliegende beliebte Buch, das nun erstmals ins Deutsche übersetzt wurde, ist eine ideale Einführung für Masterstudenten und Forscher im Bereich der theoretischen und mathematischen Physik. - Im ersten Kapitel bietet das Buch einen Überblick über die Pfadintegralmethode und Eichtheorien. - Kapitel 2 beschäftigt sich mit den mathematischen Grundlagen von Abbildungen, Vektorräumen und der Topologie. - Die folgenden Kapitel beschäftigen sich mit fortgeschritteneren Konzepten der Geometrie und Topologie und diskutieren auch deren Anwendungen im Bereich der Flüssigkristalle, bei suprafluidem Helium, in der ART und der bosonischen Stringtheorie. - Daran anschließend findet eine Zusammenführung von Geometrie und Topologie statt: es geht um Faserbündel, characteristische Klassen und Indextheoreme (u.a. in Anwendung auf die supersymmetrische Quantenmechanik). - Die letzten beiden Kapitel widmen sich der spannendsten Anwendung von Geometrie und Topologie in der modernen Physik, nämlich den Eichfeldtheorien und der Analyse der Polakov'schen bosonischen Stringtheorie aus einer gemetrischen Perspektive. Mikio Nakahara studierte an der Universität Kyoto und am King’s in London Physik sowie klassische und Quantengravitationstheorie. Heute ist er Physikprofessor an der Kinki-Universität in Osaka (Japan), wo er u. a. über topologische Quantencomputer forscht. Diese Buch entstand aus einer Vorlesung, die er während Forschungsaufenthalten an der University of Sussex und an der Helsinki University of Sussex gehalten hat.

Poincarés Vermutung

die Geschichte eines mathematischen Abenteuers
Author: Donal O'Shea
Publisher: N.A
ISBN: 9783596176632
Category:
Page: 376
View: 4007

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Grundlagen der Computerlinguistik

Mensch-Maschine-Kommunikation in natürlicher Sprache
Author: Roland R. Hausser
Publisher: Springer-Verlag
ISBN: 3642573061
Category: Computers
Page: 572
View: 8023

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Die zentrale Aufgabe einer zukunftsorientierten Computerlinguistik ist die Entwicklung kognitiver Maschinen, mit denen Menschen in ihrer jeweiligen Sprache frei reden können. Langfristig umfaßt diese Zielsetzung eine funktional ausgerichtete Theoriebildung, eine objektive Verifikationsmethode und eine Fülle praktischer Anwendungen. Für die natürlichsprachliche Kommunikation wird nicht nur Sprachverarbeitung, sondern auch nichtsprachliche Wahrnehmung und Handlung benötigt. Deshalb ist der Inhalt dieses Lehrbuchs als Sprachtheorie für die Konstruktion sprechender Roboter organisiert. Sein zentrales Thema ist die Kommunikationsmechanik natürlicher Sprachen - beim Sprecher und beim Hörer. Der Inhalt ist in folgende vier Teile mit je sechs Kapiteln gegliedert: Sprachtheorie; Formale Grammatik; Morphologie und Syntax; Semantik und Pragmatik. Insgesamt 772 Übungsaufgaben dienen der Verständniskontrolle und -vertiefung.

Wolfgang Krull: Gesammelte Abhandlungen / Collected Papers. Volume 1+2


Author: Wolfgang Krull
Publisher: Walter de Gruyter
ISBN: 3110801027
Category: Mathematics
Page: 1750
View: 7360

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Wolfgang Krull (1899-1971) is famous for his contributions to the area of maths now known as abstract commutative algebra. These two volumes contain all his papers published in mathematical periodicals. Volume 1 covers the period 1921 to 1938, Volume 2 covers the period 1939 to 1973.

Der Pauli-Jung-Dialog und seine Bedeutung für die moderne Wissenschaft


Author: Harald Atmanspacher,Hans Primas,Eva Wertenschlag-Birkhäuser
Publisher: Springer-Verlag
ISBN: 3642793231
Category: Science
Page: 356
View: 8747

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Der vorliegende Band enthält eine Sammlung von Beiträgen zum Problem der Wechselwirkung zwischen Geist und Materie, einem der zentralen Probleme europäischer Geistesgeschichte. Die Blickwinkel, die dabei eingenommen werden, sind vorrangig die der Physik und der Psychologie. Die Wechselwirkung dieser Gebiete wird so deutlich wie nie zuvor im Dialog zwischen zwei Forscherpersön lichkeiten dieses Jahrhunderts sichtbar: dem Physiker Wolfgang Pauli (1900- 1958) und dem Psychologen Carl Gustav Jung (1875-1961). In zahlreichen Brie fen und Manuskripten Paulis, die erst in den letzten Jahren allgemein zugänglich wurden, finden sich bemerkenswerte und wichtige Beiträge zu diesem Dialog, die das Verständnis des Zusammenhanges von Geist und Materie in einem neuen Licht erscheinen lassen. Um den durch Pauli und Jung begonnenen Dialog fortzusetzen und weiter fruchtbar zu machen, ist das interdisziplinäre Gespräch zwischen Physikern und Psychologen nötig. Diesem Zweck diente eine von der Eidgenössischen Techni sehen Hochschule Zürich (der Hochschule, an der Pauli tätig war) und dem C.G.Jung-Institut Zürich (das Jung gegründet hat) gemeinsam veranstaltete Tagung im Centro Stefano Franscini (Monte Verita, Ascona) vom 13. bis 18. Juni 1993. Sie stand unter dem Thema Das Irrationale in den Naturwissenschaften: Wolf gang Paulis Begegnung mit dem Geist der Materie und wurde von Pier Luigi Luisi initiiert und organisiert. Als Berater fungierten Paul Brutsehe, Hans Primas und Eva Wertenschlag-Birkhäuser. Berichte und Kommentare zu dieser Tagung wurden in Heft 4/1993 der Zeitschrift Gaia veröffentlicht.