David P. Blecher and Christian Le Merdy
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198526599
- eISBN:
- 9780191712159
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526599.001.0001
- Subject:
- Mathematics, Pure Mathematics
This book presents the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles ...
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This book presents the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles the basic concepts, theory, and methodologies. A major trend in modern mathematics, inspired largely by physics, is toward ‘noncommutative’ or ‘quantized’ phenomena. In functional analysis, this has appeared notably under the name of ‘operator spaces’, which is a variant of Banach spaces which is particularly appropriate for solving problems concerning spaces or algebras of operators on Hilbert space arising in ‘noncommutative mathematics’. The category of operator spaces includes operator algebras, selfadjoint (that is, C*-algebras) or otherwise. Also, most of the important modules over operator algebras are operator spaces. A common treatment of the subjects of C*-algebras, nonselfadjoint operator algebras, and modules over such algebras (such as Hilbert C*-modules), together under the umbrella of operator space theory, is the main topic of the book. A general theory of operator algebras and their modules naturally develops out of the operator space methodology. Indeed, operator space theory is a sensitive enough medium to reflect accurately many important noncommutative phenomena. Using recent advances in the field, the book shows how the underlying operator space structure captures, very precisely, the profound relations between the algebraic and the functional analytic structures involved. The rich interplay between spectral theory, operator theory, C*-algebra and von Neumann algebra techniques, and the influx of important ideas from related disciplines, such as pure algebra, Banach space theory, Banach algebras, and abstract function theory is highlighted. Each chapter ends with a section of notes containing additional information.Less
This book presents the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles the basic concepts, theory, and methodologies. A major trend in modern mathematics, inspired largely by physics, is toward ‘noncommutative’ or ‘quantized’ phenomena. In functional analysis, this has appeared notably under the name of ‘operator spaces’, which is a variant of Banach spaces which is particularly appropriate for solving problems concerning spaces or algebras of operators on Hilbert space arising in ‘noncommutative mathematics’. The category of operator spaces includes operator algebras, selfadjoint (that is, C*-algebras) or otherwise. Also, most of the important modules over operator algebras are operator spaces. A common treatment of the subjects of C*-algebras, nonselfadjoint operator algebras, and modules over such algebras (such as Hilbert C*-modules), together under the umbrella of operator space theory, is the main topic of the book. A general theory of operator algebras and their modules naturally develops out of the operator space methodology. Indeed, operator space theory is a sensitive enough medium to reflect accurately many important noncommutative phenomena. Using recent advances in the field, the book shows how the underlying operator space structure captures, very precisely, the profound relations between the algebraic and the functional analytic structures involved. The rich interplay between spectral theory, operator theory, C*-algebra and von Neumann algebra techniques, and the influx of important ideas from related disciplines, such as pure algebra, Banach space theory, Banach algebras, and abstract function theory is highlighted. Each chapter ends with a section of notes containing additional information.
Marcus Giaquinto
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199285945
- eISBN:
- 9780191713811
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199285945.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, ...
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Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? This book argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. The book explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. It shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. This book reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences.Less
Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? This book argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. The book explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. It shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. This book reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences.
Dov M. Gabbay and Larisa Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This book focuses on interpolation and definability. This notion is not only central in pure logic, but has significant meaning and applicability in all areas where logic itself is applied, ...
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This book focuses on interpolation and definability. This notion is not only central in pure logic, but has significant meaning and applicability in all areas where logic itself is applied, especially in computer science, artificial intelligence, logic programming, philosophy of science, and natural language. The book provides basic knowledge on interpolation and definability in logic, and contains a systematic account of material which has been presented in many papers. A variety of methods and results are presented beginning with the famous Beth's and Craig's theorems in classical predicate logic (1953-57), and to the most valuable achievements in non-classical topics on logic, mainly intuitionistic and modal logic. Together with semantical and proof-theoretic methods, close interrelations between logic and universal algebra are established and exploited.Less
This book focuses on interpolation and definability. This notion is not only central in pure logic, but has significant meaning and applicability in all areas where logic itself is applied, especially in computer science, artificial intelligence, logic programming, philosophy of science, and natural language. The book provides basic knowledge on interpolation and definability in logic, and contains a systematic account of material which has been presented in many papers. A variety of methods and results are presented beginning with the famous Beth's and Craig's theorems in classical predicate logic (1953-57), and to the most valuable achievements in non-classical topics on logic, mainly intuitionistic and modal logic. Together with semantical and proof-theoretic methods, close interrelations between logic and universal algebra are established and exploited.
John L. Bell
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568520
- eISBN:
- 9780191717581
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568520.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This is the third edition of a well-known graduate textbook on Boolean-valued models of set theory. The aim of the first and second editions was to provide a systematic and adequately motivated ...
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This is the third edition of a well-known graduate textbook on Boolean-valued models of set theory. The aim of the first and second editions was to provide a systematic and adequately motivated exposition of the theory of Boolean-valued models as developed by Scott and Solovay in the 1960s, deriving along the way the central set theoretic independence proofs of Cohen and others in the particularly elegant form that the Boolean-valued approach enables them to assume. In this edition, the background material has been augmented to include an introduction to Heyting algebras. It includes chapters on Boolean-valued analysis and Heyting-algebra-valued models of intuitionistic set theory.Less
This is the third edition of a well-known graduate textbook on Boolean-valued models of set theory. The aim of the first and second editions was to provide a systematic and adequately motivated exposition of the theory of Boolean-valued models as developed by Scott and Solovay in the 1960s, deriving along the way the central set theoretic independence proofs of Cohen and others in the particularly elegant form that the Boolean-valued approach enables them to assume. In this edition, the background material has been augmented to include an introduction to Heyting algebras. It includes chapters on Boolean-valued analysis and Heyting-algebra-valued models of intuitionistic set theory.
Jacqueline A. Stedall
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198526025
- eISBN:
- 9780191712364
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526025.001.0001
- Subject:
- Mathematics, History of Mathematics
This book casts new light on the work of Thomas Harriot (c.1560-1621), an innovative thinker and practitioner in several branches of the mathematical sciences, including navigation, astronomy, ...
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This book casts new light on the work of Thomas Harriot (c.1560-1621), an innovative thinker and practitioner in several branches of the mathematical sciences, including navigation, astronomy, optics, geometry, and algebra. On his death Harriot left behind over 4,000 manuscript sheets, but most of his work still remains unpublished. This book focuses on 140 of those sheets, those concerned with the structure and solution of equations. The original material has been carefully ordered, translated, and annotated to provide the first complete edition of Harriot's treatise, and an extended introduction provides the reader with a lucid background to the work. Illustrations from the manuscripts provide additional interest. The appendices discuss correlations between Harriot's manuscripts and those of this contemporaries, Viète, Warner, and Torporley.Less
This book casts new light on the work of Thomas Harriot (c.1560-1621), an innovative thinker and practitioner in several branches of the mathematical sciences, including navigation, astronomy, optics, geometry, and algebra. On his death Harriot left behind over 4,000 manuscript sheets, but most of his work still remains unpublished. This book focuses on 140 of those sheets, those concerned with the structure and solution of equations. The original material has been carefully ordered, translated, and annotated to provide the first complete edition of Harriot's treatise, and an extended introduction provides the reader with a lucid background to the work. Illustrations from the manuscripts provide additional interest. The appendices discuss correlations between Harriot's manuscripts and those of this contemporaries, Viète, Warner, and Torporley.
Alexander A. Ivanov
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198527596
- eISBN:
- 9780191713163
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527596.001.0001
- Subject:
- Mathematics, Pure Mathematics
This book illustrates how different methods of finite group theory including representation theory, cohomology theory, combinatorial group theory, and local analysis, are combined to construct one of ...
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This book illustrates how different methods of finite group theory including representation theory, cohomology theory, combinatorial group theory, and local analysis, are combined to construct one of the last of the sporadic finite simple groups — the fourth Janko group J4. This book's approach is based on analysis of group amalgams and the geometry of the complexes of these amalgams with emphasis on the underlying theory.Less
This book illustrates how different methods of finite group theory including representation theory, cohomology theory, combinatorial group theory, and local analysis, are combined to construct one of the last of the sporadic finite simple groups — the fourth Janko group J4. This book's approach is based on analysis of group amalgams and the geometry of the complexes of these amalgams with emphasis on the underlying theory.
David P. Blecher and Christian Le Merdy
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198526599
- eISBN:
- 9780191712159
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526599.003.0006
- Subject:
- Mathematics, Pure Mathematics
Tensor products and C*-norms play a prominent role in the theory of C*-algebras, in particular in the study of nuclear C*-algebras and semidiscrete (or injective) von Neumann algebras. This chapter ...
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Tensor products and C*-norms play a prominent role in the theory of C*-algebras, in particular in the study of nuclear C*-algebras and semidiscrete (or injective) von Neumann algebras. This chapter extends part of that theory to nonselfadjoint operator algebras, and gives some applications. Topics covered include maximal and normal tensor products, joint dilations and the disc algebra, tenser products with triangular algebras, Pisier's delta norm, factorization through matrix spaces, and nuclearity and semidiscreteness for linear operators. Notes and historical remarks are presented at the end of the chapter.Less
Tensor products and C*-norms play a prominent role in the theory of C*-algebras, in particular in the study of nuclear C*-algebras and semidiscrete (or injective) von Neumann algebras. This chapter extends part of that theory to nonselfadjoint operator algebras, and gives some applications. Topics covered include maximal and normal tensor products, joint dilations and the disc algebra, tenser products with triangular algebras, Pisier's delta norm, factorization through matrix spaces, and nuclearity and semidiscreteness for linear operators. Notes and historical remarks are presented at the end of the chapter.
Alessio Corti
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198570615
- eISBN:
- 9780191717703
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570615.003.0002
- Subject:
- Mathematics, Geometry / Topology
This chapter gives a concise, complete, and pedagogical proof of existence of 3-fold flips according to Shokurov. In particular, the foundation of the theory of b-divisors, algebras of rational ...
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This chapter gives a concise, complete, and pedagogical proof of existence of 3-fold flips according to Shokurov. In particular, the foundation of the theory of b-divisors, algebras of rational functions, and Shokurov's asymptotic saturation property are developed systematically from first principles.Less
This chapter gives a concise, complete, and pedagogical proof of existence of 3-fold flips according to Shokurov. In particular, the foundation of the theory of b-divisors, algebras of rational functions, and Shokurov's asymptotic saturation property are developed systematically from first principles.
Patrick R. Laughlin
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691147918
- eISBN:
- 9781400836673
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147918.001.0001
- Subject:
- Psychology, Social Psychology
Experimental research by social and cognitive psychologists has established that cooperative groups solve a wide range of problems better than individuals. Cooperative problem solving groups of ...
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Experimental research by social and cognitive psychologists has established that cooperative groups solve a wide range of problems better than individuals. Cooperative problem solving groups of scientific researchers, auditors, financial analysts, air crash investigators, and forensic art experts are increasingly important in our complex and interdependent society. This comprehensive textbook presents important theories and experimental research about group problem solving. The book focuses on tasks that have demonstrably correct solutions within mathematical, logical, scientific, or verbal systems, including algebra problems, analogies, vocabulary, and logical reasoning problems. It explores basic concepts in group problem solving, social combination models, group memory, group ability and world knowledge tasks, rule induction problems, letters-to-numbers problems, evidence for positive group-to-individual transfer, and social choice theory. The conclusion proposes ten generalizations that are supported by the theory and research on group problem solving. The book is an essential resource for decision-making research in social and cognitive psychology, but also extremely relevant to multidisciplinary and multicultural problem-solving teams in organizational behavior, business administration, management, and behavioral economics.Less
Experimental research by social and cognitive psychologists has established that cooperative groups solve a wide range of problems better than individuals. Cooperative problem solving groups of scientific researchers, auditors, financial analysts, air crash investigators, and forensic art experts are increasingly important in our complex and interdependent society. This comprehensive textbook presents important theories and experimental research about group problem solving. The book focuses on tasks that have demonstrably correct solutions within mathematical, logical, scientific, or verbal systems, including algebra problems, analogies, vocabulary, and logical reasoning problems. It explores basic concepts in group problem solving, social combination models, group memory, group ability and world knowledge tasks, rule induction problems, letters-to-numbers problems, evidence for positive group-to-individual transfer, and social choice theory. The conclusion proposes ten generalizations that are supported by the theory and research on group problem solving. The book is an essential resource for decision-making research in social and cognitive psychology, but also extremely relevant to multidisciplinary and multicultural problem-solving teams in organizational behavior, business administration, management, and behavioral economics.
Jacqueline A. Stedall
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198524953
- eISBN:
- 9780191711886
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198524953.001.0001
- Subject:
- Mathematics, History of Mathematics
This book provides an accessible account of the rise of algebra in England from the medieval period to the later years of the 17th century. The book includes new research and is the most detailed ...
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This book provides an accessible account of the rise of algebra in England from the medieval period to the later years of the 17th century. The book includes new research and is the most detailed study to date of early modern English algebra. In its structure and content this book builds on a much earlier history of algebra, A treatise of algebra, published in 1685 by John Wallis (Savilian Professor of Geometry at Oxford). This book both analyses Wallis' text and moves beyond it. Thus, it explores the reception and dissemination of important ideas from continental Europe up to the end of the 16th century, and the subsequent revolution in English mathematics in the 17th century. In particular, the book includes chapters on the work of Thomas Harriot, William Oughtred, John Pell, and William Brouncker, as well as of Wallis himself.Less
This book provides an accessible account of the rise of algebra in England from the medieval period to the later years of the 17th century. The book includes new research and is the most detailed study to date of early modern English algebra. In its structure and content this book builds on a much earlier history of algebra, A treatise of algebra, published in 1685 by John Wallis (Savilian Professor of Geometry at Oxford). This book both analyses Wallis' text and moves beyond it. Thus, it explores the reception and dissemination of important ideas from continental Europe up to the end of the 16th century, and the subsequent revolution in English mathematics in the 17th century. In particular, the book includes chapters on the work of Thomas Harriot, William Oughtred, John Pell, and William Brouncker, as well as of Wallis himself.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0002
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter presents basic definitions and known results related to modal and superintuitionistic logics. Well-known Hilbert-style axiomatizations of the most known modal systems K, D, T, K4, D4, ...
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This chapter presents basic definitions and known results related to modal and superintuitionistic logics. Well-known Hilbert-style axiomatizations of the most known modal systems K, D, T, K4, D4, S4, S5 and also of the intuitionistic logic are given. For each of the calculi, its relational semantics is considered. The algebraic semantics and its connection with relational semantics via representation theorems are presented.Less
This chapter presents basic definitions and known results related to modal and superintuitionistic logics. Well-known Hilbert-style axiomatizations of the most known modal systems K, D, T, K4, D4, S4, S5 and also of the intuitionistic logic are given. For each of the calculi, its relational semantics is considered. The algebraic semantics and its connection with relational semantics via representation theorems are presented.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0003
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
It is well known that the intuitionistic logic Int may be translated into S4. This translation was introduced by Gödel to give an interpretation of Int in the classical logic with an additional ...
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It is well known that the intuitionistic logic Int may be translated into S4. This translation was introduced by Gödel to give an interpretation of Int in the classical logic with an additional provability operator. This chapter pays special attention to extensions of the intuitionistic logic and of the modal logic S4. An algebraic semantics and Kripke semantics for these logics are presented in more detail. The main interrelations of the family of superintuitionistic logics and of the family of normal extensions of S4 induced by Gödel's translation are explained. In particular, any intermediate logic L has an infinite family of its modal companions over S4, which have L as their superintuitionistic fragment.Less
It is well known that the intuitionistic logic Int may be translated into S4. This translation was introduced by Gödel to give an interpretation of Int in the classical logic with an additional provability operator. This chapter pays special attention to extensions of the intuitionistic logic and of the modal logic S4. An algebraic semantics and Kripke semantics for these logics are presented in more detail. The main interrelations of the family of superintuitionistic logics and of the family of normal extensions of S4 induced by Gödel's translation are explained. In particular, any intermediate logic L has an infinite family of its modal companions over S4, which have L as their superintuitionistic fragment.
Jacqueline A. Stedall
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198524953
- eISBN:
- 9780191711886
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198524953.003.0002
- Subject:
- Mathematics, History of Mathematics
This chapter discusses the development of algebra in Europe. Topics covered include the origins from which European algebra began to evolve and from which it took its name; The Liber abbaci (1202) of ...
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This chapter discusses the development of algebra in Europe. Topics covered include the origins from which European algebra began to evolve and from which it took its name; The Liber abbaci (1202) of Leonardo of Pisa, which was the first major European mathematical text of any kind; the Ars magna, one of the great mathematical texts of all time; and Wallis' account of early algebra.Less
This chapter discusses the development of algebra in Europe. Topics covered include the origins from which European algebra began to evolve and from which it took its name; The Liber abbaci (1202) of Leonardo of Pisa, which was the first major European mathematical text of any kind; the Ars magna, one of the great mathematical texts of all time; and Wallis' account of early algebra.
Jacqueline A. Stedall
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198524953
- eISBN:
- 9780191711886
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198524953.003.0008
- Subject:
- Mathematics, History of Mathematics
This final chapter looks at some of the reactions to A treatise of algebra immediately after its publication and since, and draws together a few observations on Wallis' perspective on algebra and on ...
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This final chapter looks at some of the reactions to A treatise of algebra immediately after its publication and since, and draws together a few observations on Wallis' perspective on algebra and on history. In general Wallis' history drew more attention than his mathematics. That there were a number of criticisms in this respect can be inferred from revisions he made when the book was translated into Latin in 1693.Less
This final chapter looks at some of the reactions to A treatise of algebra immediately after its publication and since, and draws together a few observations on Wallis' perspective on algebra and on history. In general Wallis' history drew more attention than his mathematics. That there were a number of criticisms in this respect can be inferred from revisions he made when the book was translated into Latin in 1693.
David P. Blecher and Christian Le Merdy
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198526599
- eISBN:
- 9780191712159
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526599.003.0005
- Subject:
- Mathematics, Pure Mathematics
This chapter studies operator algebras ‘up to isomorphism’ or ‘up to complete isomorphism’. Topics covered include homomorphisms of operator algebras, completely bounded characterizations, examples ...
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This chapter studies operator algebras ‘up to isomorphism’ or ‘up to complete isomorphism’. Topics covered include homomorphisms of operator algebras, completely bounded characterizations, examples of operator algebra structures, Q-algebras, and applications to isomorphic theory. Notes and historical remarks are presented at the end of the chapter.Less
This chapter studies operator algebras ‘up to isomorphism’ or ‘up to complete isomorphism’. Topics covered include homomorphisms of operator algebras, completely bounded characterizations, examples of operator algebra structures, Q-algebras, and applications to isomorphic theory. Notes and historical remarks are presented at the end of the chapter.
John L. Bell
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568520
- eISBN:
- 9780191717581
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568520.003.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter provides a brief account of the theory of Boolean and Heyting algebras, including the basic representation theorems and their connections with logic.
This chapter provides a brief account of the theory of Boolean and Heyting algebras, including the basic representation theorems and their connections with logic.
John L. Bell
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568520
- eISBN:
- 9780191717581
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568520.003.0008
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter gives a brief account of real analysis in Boolean-valued models arising from measure algebras and algebras of projections on Hilbert space. The latter is applied, following Martin Davis, ...
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This chapter gives a brief account of real analysis in Boolean-valued models arising from measure algebras and algebras of projections on Hilbert space. The latter is applied, following Martin Davis, to provide a novel interpretation of the formalism of quantum theory.Less
This chapter gives a brief account of real analysis in Boolean-valued models arising from measure algebras and algebras of projections on Hilbert space. The latter is applied, following Martin Davis, to provide a novel interpretation of the formalism of quantum theory.
Victor J. Katz and Karen Hunger Parshall
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691149059
- eISBN:
- 9781400850525
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149059.001.0001
- Subject:
- Mathematics, History of Mathematics
What is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and ...
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What is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. This book considers how these two seemingly different types of algebra evolved and how they relate. The book explores the history of algebra, from its roots in the ancient civilizations of Egypt, Mesopotamia, Greece, China, and India, through its development in the medieval Islamic world and medieval and early modern Europe, to its modern form in the early twentieth century. Defining algebra originally as a collection of techniques for determining unknowns, the book traces the development of these techniques from geometric beginnings in ancient Egypt and Mesopotamia and classical Greece. It shows how similar problems were tackled in Alexandrian Greece, in China, and in India, then looks at how medieval Islamic scholars shifted to an algorithmic stage, which was further developed by medieval and early modern European mathematicians. With the introduction of a flexible and operative symbolism in the sixteenth and seventeenth centuries, algebra entered into a dynamic period characterized by the analytic geometry that could evaluate curves represented by equations in two variables, thereby solving problems in the physics of motion. This new symbolism freed mathematicians to study equations of degrees higher than two and three, ultimately leading to the present abstract era. The book follows algebra's remarkable growth through different epochs around the globe.Less
What is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. This book considers how these two seemingly different types of algebra evolved and how they relate. The book explores the history of algebra, from its roots in the ancient civilizations of Egypt, Mesopotamia, Greece, China, and India, through its development in the medieval Islamic world and medieval and early modern Europe, to its modern form in the early twentieth century. Defining algebra originally as a collection of techniques for determining unknowns, the book traces the development of these techniques from geometric beginnings in ancient Egypt and Mesopotamia and classical Greece. It shows how similar problems were tackled in Alexandrian Greece, in China, and in India, then looks at how medieval Islamic scholars shifted to an algorithmic stage, which was further developed by medieval and early modern European mathematicians. With the introduction of a flexible and operative symbolism in the sixteenth and seventeenth centuries, algebra entered into a dynamic period characterized by the analytic geometry that could evaluate curves represented by equations in two variables, thereby solving problems in the physics of motion. This new symbolism freed mathematicians to study equations of degrees higher than two and three, ultimately leading to the present abstract era. The book follows algebra's remarkable growth through different epochs around the globe.
Hartry Field
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199230747
- eISBN:
- 9780191710933
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230747.003.0011
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter examines the simplest supervaluationist fixed points, and the ‘internal’ fixed-point theories based on them. These are not gap theories; they are ‘weakly classical’ theories in that they ...
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This chapter examines the simplest supervaluationist fixed points, and the ‘internal’ fixed-point theories based on them. These are not gap theories; they are ‘weakly classical’ theories in that they preserve the implications of classical logic but not certain meta-rules. Most dramatically, they do not allow for reasoning by cases. Two notions of validity are distinguished (strong and weak), and the framework of deMorgan semantics and Boolean semantics is introduced and discussed.Less
This chapter examines the simplest supervaluationist fixed points, and the ‘internal’ fixed-point theories based on them. These are not gap theories; they are ‘weakly classical’ theories in that they preserve the implications of classical logic but not certain meta-rules. Most dramatically, they do not allow for reasoning by cases. Two notions of validity are distinguished (strong and weak), and the framework of deMorgan semantics and Boolean semantics is introduced and discussed.
Robert Alicki and Mark Fannes
- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This introductory chapter outlines the basic ideas of the book using two simple examples: the quantum harmonic oscillator as a prototype of an integrable system versus the quantum Arnold cat map ...
More
This introductory chapter outlines the basic ideas of the book using two simple examples: the quantum harmonic oscillator as a prototype of an integrable system versus the quantum Arnold cat map leading to quantum chaos and the irrational rotation algebra.Less
This introductory chapter outlines the basic ideas of the book using two simple examples: the quantum harmonic oscillator as a prototype of an integrable system versus the quantum Arnold cat map leading to quantum chaos and the irrational rotation algebra.
