Srinivasa Rao
- Published in print:
- 2011
- Published Online:
- September 2012
- ISBN:
- 9780198079811
- eISBN:
- 9780199081707
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198079811.003.0019
- Subject:
- Philosophy, General
The term Advaita which literally means non-duality, is a denial only of the ultimacy of duality and difference. It only means that there is no difference which is so radical, ultimate and fundamental ...
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The term Advaita which literally means non-duality, is a denial only of the ultimacy of duality and difference. It only means that there is no difference which is so radical, ultimate and fundamental that we will have to concede that there is something else or someone that is truly “other” than ourselves. In other words, it is sarvātmabhāva. While we may always have enough basis to say about anything: “It looks or appears to be different”, we never have sufficient basis to say: “It is different; it is really, absolutely another”Less
The term Advaita which literally means non-duality, is a denial only of the ultimacy of duality and difference. It only means that there is no difference which is so radical, ultimate and fundamental that we will have to concede that there is something else or someone that is truly “other” than ourselves. In other words, it is sarvātmabhāva. While we may always have enough basis to say about anything: “It looks or appears to be different”, we never have sufficient basis to say: “It is different; it is really, absolutely another”
Sonya Stephens
- Published in print:
- 1999
- Published Online:
- October 2011
- ISBN:
- 9780198158776
- eISBN:
- 9780191673351
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198158776.001.0001
- Subject:
- Literature, European Literature, Poetry
The aim of this book is to offer a new reading of Baudelaire's Petits Poèmes en Prose that demonstrates the significance of ironic otherness for the theory and functioning of the work and for the ...
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The aim of this book is to offer a new reading of Baudelaire's Petits Poèmes en Prose that demonstrates the significance of ironic otherness for the theory and functioning of the work and for the genre of the prose poem itself. The book considers Baudelaire's choice of this genre and the way in which he seeks to define it, both paratextually and textually. It examines the ways in which the prose poem depends on dualities and déboublements as forms of lyrical and narrative difference which, in their turn, reveal ideological otherness and declare the oppositionality of the prose poem. Finally, the book demonstrates a relationship between these forms of otherness and Baudelaire's theory of the popular comic arts and, in doing so, proposes that the prose poems should be read as literary caricature.Less
The aim of this book is to offer a new reading of Baudelaire's Petits Poèmes en Prose that demonstrates the significance of ironic otherness for the theory and functioning of the work and for the genre of the prose poem itself. The book considers Baudelaire's choice of this genre and the way in which he seeks to define it, both paratextually and textually. It examines the ways in which the prose poem depends on dualities and déboublements as forms of lyrical and narrative difference which, in their turn, reveal ideological otherness and declare the oppositionality of the prose poem. Finally, the book demonstrates a relationship between these forms of otherness and Baudelaire's theory of the popular comic arts and, in doing so, proposes that the prose poems should be read as literary caricature.
Srinivasa Rao
- Published in print:
- 2011
- Published Online:
- September 2012
- ISBN:
- 9780198079811
- eISBN:
- 9780199081707
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198079811.001.0001
- Subject:
- Philosophy, General
The book proposes a contemporary framework for critiquing Advaita and formulating its basic thesis in a more logical and convincing way. Any proper theory in philosophy and science has to follow from ...
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The book proposes a contemporary framework for critiquing Advaita and formulating its basic thesis in a more logical and convincing way. Any proper theory in philosophy and science has to follow from accepted assumptions. Hence the book begins by identifying basic presuppositions required for Advaita and determining the different cognitive possibilities arising out of them. After thus determining what is logically and conceptually possible and impossible in Advaita, the new framework is used to assess whether or not the traditionally held Advaitic concepts and theories are satisfactory and acceptable. This is done in many chapters covering discussions of the notions of not-Self (anātman), cosmic ignorance (māyā), individual ignorance (avidyā), illusoriness (mithyātva), sublation (bādha), entities that are different from the real and the unreal (sadasadvilaksana) and so on. The book argues that all these concepts, as specifically formulated and defended in traditional Advaita for centuries after Śankara, are simply faulty and untenable both individually and as related clusters of concepts. Traditional Advaita has also defended an elaborate ontology of experiences like mistaking a rope-for a snake. It has also heavily defended the metaphysical thesis of the empirical world of our experience being a total illusion. The logical faults and conceptual inadequacies of this ontology and metaphysics are also discussed in great detail, offering absolutely new criticisms of them. Despite this almost totally negative portrayal of traditional Advaita, the book is also quite positive in showing that any belief in non-duality is still very much philosophically possible and also necessary.Less
The book proposes a contemporary framework for critiquing Advaita and formulating its basic thesis in a more logical and convincing way. Any proper theory in philosophy and science has to follow from accepted assumptions. Hence the book begins by identifying basic presuppositions required for Advaita and determining the different cognitive possibilities arising out of them. After thus determining what is logically and conceptually possible and impossible in Advaita, the new framework is used to assess whether or not the traditionally held Advaitic concepts and theories are satisfactory and acceptable. This is done in many chapters covering discussions of the notions of not-Self (anātman), cosmic ignorance (māyā), individual ignorance (avidyā), illusoriness (mithyātva), sublation (bādha), entities that are different from the real and the unreal (sadasadvilaksana) and so on. The book argues that all these concepts, as specifically formulated and defended in traditional Advaita for centuries after Śankara, are simply faulty and untenable both individually and as related clusters of concepts. Traditional Advaita has also defended an elaborate ontology of experiences like mistaking a rope-for a snake. It has also heavily defended the metaphysical thesis of the empirical world of our experience being a total illusion. The logical faults and conceptual inadequacies of this ontology and metaphysics are also discussed in great detail, offering absolutely new criticisms of them. Despite this almost totally negative portrayal of traditional Advaita, the book is also quite positive in showing that any belief in non-duality is still very much philosophically possible and also necessary.
Srinivasa Rao
- Published in print:
- 2011
- Published Online:
- September 2012
- ISBN:
- 9780198079811
- eISBN:
- 9780199081707
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198079811.003.0008
- Subject:
- Philosophy, General
No one believes that he does not know himself or the world around him because everyone is familiar with both these. Therefore the question naturally arises as to what exactly is meant by the ...
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No one believes that he does not know himself or the world around him because everyone is familiar with both these. Therefore the question naturally arises as to what exactly is meant by the Advaitin’s claim that no one really “knows the Self” and hence everyone must attain Self-knowledge. What is this “Self-knowledge” which everyone is said to be ignorant of? It is argued in this chapter that according to the non-duality thesis, there cannot really be “another” besides the non-dual Self and therefore not knowing this is itself ignorance. If so, naturally, knowing that whatever exists is the Self alone and nothing else (sarvātmabhāva) must truly constitute “Self-knowledge”. Therefore ignorance must truly be not knowing that whatever is, is the Self itself and nothing else. Then, obviously, not only the jīva but the world too is the Self.Less
No one believes that he does not know himself or the world around him because everyone is familiar with both these. Therefore the question naturally arises as to what exactly is meant by the Advaitin’s claim that no one really “knows the Self” and hence everyone must attain Self-knowledge. What is this “Self-knowledge” which everyone is said to be ignorant of? It is argued in this chapter that according to the non-duality thesis, there cannot really be “another” besides the non-dual Self and therefore not knowing this is itself ignorance. If so, naturally, knowing that whatever exists is the Self alone and nothing else (sarvātmabhāva) must truly constitute “Self-knowledge”. Therefore ignorance must truly be not knowing that whatever is, is the Self itself and nothing else. Then, obviously, not only the jīva but the world too is the Self.
Srinivasa Rao
- Published in print:
- 2011
- Published Online:
- September 2012
- ISBN:
- 9780198079811
- eISBN:
- 9780199081707
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198079811.003.0011
- Subject:
- Philosophy, General
If our perceptions are different, the objects we perceive in them are also not necessarily different because the very same object may come to be perceived differently, for example, a rope as a snake. ...
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If our perceptions are different, the objects we perceive in them are also not necessarily different because the very same object may come to be perceived differently, for example, a rope as a snake. Likewise, the very same things seen as things of the world, come to be seen as the Self when one attains true knowledge of everything as the Self (sarvātmabhāva). Here, the ways of seeing are different but what is seen also is not different on that account. What was earlier being seen as anātman or various forms of duality is itself now perceived to be non-different from the non-dual Self. Upon such seeing, the anātman does not disappear or get sublated; rather it appears in a new light altogether, as nothing different from the Self itself.Less
If our perceptions are different, the objects we perceive in them are also not necessarily different because the very same object may come to be perceived differently, for example, a rope as a snake. Likewise, the very same things seen as things of the world, come to be seen as the Self when one attains true knowledge of everything as the Self (sarvātmabhāva). Here, the ways of seeing are different but what is seen also is not different on that account. What was earlier being seen as anātman or various forms of duality is itself now perceived to be non-different from the non-dual Self. Upon such seeing, the anātman does not disappear or get sublated; rather it appears in a new light altogether, as nothing different from the Self itself.
Srinivasa Rao
- Published in print:
- 2011
- Published Online:
- September 2012
- ISBN:
- 9780198079811
- eISBN:
- 9780199081707
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198079811.003.0014
- Subject:
- Philosophy, General
The world is described in Advaita as anātman which is sadasadvilaksana in nature. But as a positive entity different in its basic nature from the Self, it leads to a logically unavoidable dualism ...
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The world is described in Advaita as anātman which is sadasadvilaksana in nature. But as a positive entity different in its basic nature from the Self, it leads to a logically unavoidable dualism that clearly contradicts true non-duality. Saying that it exists at a very different level and is then sublated does not really help because its existence at any level and at any time is contradictory of non-duality. To say that it never really exists at all nullifies all earlier discourse on the sadasadvilaksana and makes Advaita even more muddled. First attributing real existence to the world and later on completely withdrawing this existential status retrospectively means one is not really very sure of the definite nature of the world.Less
The world is described in Advaita as anātman which is sadasadvilaksana in nature. But as a positive entity different in its basic nature from the Self, it leads to a logically unavoidable dualism that clearly contradicts true non-duality. Saying that it exists at a very different level and is then sublated does not really help because its existence at any level and at any time is contradictory of non-duality. To say that it never really exists at all nullifies all earlier discourse on the sadasadvilaksana and makes Advaita even more muddled. First attributing real existence to the world and later on completely withdrawing this existential status retrospectively means one is not really very sure of the definite nature of the world.
D. Huybrechts
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199296866
- eISBN:
- 9780191711329
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296866.003.0003
- Subject:
- Mathematics, Geometry / Topology
The discussion of the previous chapter is applied to the derived category of the abelian category of coherent sheaves. The Serre functor is introduced, and particular spanning classes are ...
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The discussion of the previous chapter is applied to the derived category of the abelian category of coherent sheaves. The Serre functor is introduced, and particular spanning classes are constructed. The usual geometric functors, direct and inverse image, tensor product, and global sections, are derived and extended to functors between derived categories. The compatibilities between them are reviewed. The final section focuses on the Grothendieck-Verdier duality.Less
The discussion of the previous chapter is applied to the derived category of the abelian category of coherent sheaves. The Serre functor is introduced, and particular spanning classes are constructed. The usual geometric functors, direct and inverse image, tensor product, and global sections, are derived and extended to functors between derived categories. The compatibilities between them are reviewed. The final section focuses on the Grothendieck-Verdier duality.
Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0003
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter shows that the boxing and circling decorations of the BZL patterns are in a sense dual to each other. The circling and boxing rules seem quite different from each other, but they are ...
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This chapter shows that the boxing and circling decorations of the BZL patterns are in a sense dual to each other. The circling and boxing rules seem quite different from each other, but they are actually closely related, and the involution also sheds light on this fact. The chapter describes a natural box-circle duality: a bijection between the bᵢ and the lᵢ in which bᵢ is circled if and only if the corresponding lᵢ is boxed. It also considers a striking property of the crystal graph and proceeds by obtaining two BZL patterns in which circled entries in one correspond to boxed entries in the other, and illustrates this with an example.Less
This chapter shows that the boxing and circling decorations of the BZL patterns are in a sense dual to each other. The circling and boxing rules seem quite different from each other, but they are actually closely related, and the involution also sheds light on this fact. The chapter describes a natural box-circle duality: a bijection between the bᵢ and the lᵢ in which bᵢ is circled if and only if the corresponding lᵢ is boxed. It also considers a striking property of the crystal graph and proceeds by obtaining two BZL patterns in which circled entries in one correspond to boxed entries in the other, and illustrates this with an example.
Nicholas M. Katz
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0014
- Subject:
- Mathematics, Number Theory
This chapter takes up the proofs of Theorems 13.1–13.5. Suppose we are given some number r ≤ 2 of objects N₁; N₂, Nᵣ in Garith of some common “dimension” d ≤ 1. Suppose they are all ι-pure of weight ...
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This chapter takes up the proofs of Theorems 13.1–13.5. Suppose we are given some number r ≤ 2 of objects N₁; N₂, Nᵣ in Garith of some common “dimension” d ≤ 1. Suppose they are all ι-pure of weight zero, geometrically irreducible, and arithmetically self-dual, all with the same sign of duality.Less
This chapter takes up the proofs of Theorems 13.1–13.5. Suppose we are given some number r ≤ 2 of objects N₁; N₂, Nᵣ in Garith of some common “dimension” d ≤ 1. Suppose they are all ι-pure of weight zero, geometrically irreducible, and arithmetically self-dual, all with the same sign of duality.
Gregory B. Graybill
- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199589487
- eISBN:
- 9780191594588
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199589487.003.0005
- Subject:
- Religion, Church History
The Wittenberg Unrest of 1521–2 caused Melanchthon to emphasize civil freedom, although he still maintained the spiritual bondage of the will. This new emphasis occurred in tandem with the ...
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The Wittenberg Unrest of 1521–2 caused Melanchthon to emphasize civil freedom, although he still maintained the spiritual bondage of the will. This new emphasis occurred in tandem with the development of Luther's political theology of various dualities in reality. Meanwhile, Melanchthon strongly sided with Luther in his dispute with Erasmus over the freedom of the will.Less
The Wittenberg Unrest of 1521–2 caused Melanchthon to emphasize civil freedom, although he still maintained the spiritual bondage of the will. This new emphasis occurred in tandem with the development of Luther's political theology of various dualities in reality. Meanwhile, Melanchthon strongly sided with Luther in his dispute with Erasmus over the freedom of the will.
Pavol Hell and Jaroslav Nešetřil
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198528173
- eISBN:
- 9780191713644
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528173.003.0003
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter considers the order homomorphisms induce on the set of all cores; this order is rich enough to represent all countable partial orders. Antichains in the homomorphism order are discussed, ...
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This chapter considers the order homomorphisms induce on the set of all cores; this order is rich enough to represent all countable partial orders. Antichains in the homomorphism order are discussed, which are collections of incomparable graphs (graphs without homomorphisms between any two of them). Of particular interest are finite maximal antichains, and their structure turns out to be surprisingly revealing. Graphs only have trivial finite maximal antichains, while digraphs have many such antichains of all possible sizes, arising from duality relationships. This chapter also contains the (probabilistic) proof of the Sparse Incomparability Lemma, of the fact that asymptotically almost all graphs on $n$ vertices are cores, and of the fact that the number of incomparable graphs on $n$ vertices differs little (asymptotically) from the total number of non-isomorphic graphs on $n$ vertices. The density of the homomorphism order is related to duality, revealing an unexpected connection between these two seemingly unrelated concepts. Finally, it is shown that one can gain interesting insights into many traditional graph topics, such as Hadwiger’s conjecture, when interpreting them as statements about the homomorphism order.Less
This chapter considers the order homomorphisms induce on the set of all cores; this order is rich enough to represent all countable partial orders. Antichains in the homomorphism order are discussed, which are collections of incomparable graphs (graphs without homomorphisms between any two of them). Of particular interest are finite maximal antichains, and their structure turns out to be surprisingly revealing. Graphs only have trivial finite maximal antichains, while digraphs have many such antichains of all possible sizes, arising from duality relationships. This chapter also contains the (probabilistic) proof of the Sparse Incomparability Lemma, of the fact that asymptotically almost all graphs on $n$ vertices are cores, and of the fact that the number of incomparable graphs on $n$ vertices differs little (asymptotically) from the total number of non-isomorphic graphs on $n$ vertices. The density of the homomorphism order is related to duality, revealing an unexpected connection between these two seemingly unrelated concepts. Finally, it is shown that one can gain interesting insights into many traditional graph topics, such as Hadwiger’s conjecture, when interpreting them as statements about the homomorphism order.
Steve Awodey
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.003.0003
- Subject:
- Mathematics, Algebra
Examples of definitions and statements which exhibit a kind of ‘duality’, such as initial and terminal object and epimorphisms and monomorphisms have been shown. This chapter now considers this ...
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Examples of definitions and statements which exhibit a kind of ‘duality’, such as initial and terminal object and epimorphisms and monomorphisms have been shown. This chapter now considers this duality more systematically. Despite its rather trivial first impression, it is indeed a deep and powerful aspect of the categorical approach. Topics discussed include the duality principle, coproducts, equalizers, and coequalizers. Exercises are provided at the end of the chapter.Less
Examples of definitions and statements which exhibit a kind of ‘duality’, such as initial and terminal object and epimorphisms and monomorphisms have been shown. This chapter now considers this duality more systematically. Despite its rather trivial first impression, it is indeed a deep and powerful aspect of the categorical approach. Topics discussed include the duality principle, coproducts, equalizers, and coequalizers. Exercises are provided at the end of the chapter.
Steve Awodey
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.003.0007
- Subject:
- Mathematics, Algebra
This chapter develops a general theory for functors. Topics discussed include category of categories, representable structure, stone duality, naturality, examples of natural transformations, ...
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This chapter develops a general theory for functors. Topics discussed include category of categories, representable structure, stone duality, naturality, examples of natural transformations, exponentials of categories, functor categories, and equivalence of categories. The chapter ends with some exercises.Less
This chapter develops a general theory for functors. Topics discussed include category of categories, representable structure, stone duality, naturality, examples of natural transformations, exponentials of categories, functor categories, and equivalence of categories. The chapter ends with some exercises.
D. A. Bini, G. Latouche, and B. Meini
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.003.0005
- Subject:
- Mathematics, Numerical Analysis
In this chapter a series of processes with a variety of transition structures are considered and their analysis is presented in a unifying manner. These processes are grouped under the generic name ...
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In this chapter a series of processes with a variety of transition structures are considered and their analysis is presented in a unifying manner. These processes are grouped under the generic name of Phase-type queues, they include G/M/1-type Markov chains, QBD processes, Markov chains with Toeplitz-like transitions and limited displacements (non-skip-free), and tree-like processes. A duality property between M/G/1 and G/M/1 Markov chains is described and a reduction of M/G/1 and G/M/1 Markov chains to QBD is analysed.Less
In this chapter a series of processes with a variety of transition structures are considered and their analysis is presented in a unifying manner. These processes are grouped under the generic name of Phase-type queues, they include G/M/1-type Markov chains, QBD processes, Markov chains with Toeplitz-like transitions and limited displacements (non-skip-free), and tree-like processes. A duality property between M/G/1 and G/M/1 Markov chains is described and a reduction of M/G/1 and G/M/1 Markov chains to QBD is analysed.
S. N. Afriat
- Published in print:
- 1987
- Published Online:
- November 2003
- ISBN:
- 9780198284611
- eISBN:
- 9780191595844
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198284616.003.0026
- Subject:
- Economics and Finance, Microeconomics
This is the third of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It discusses linear programming, which might appear to ...
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This is the third of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It discusses linear programming, which might appear to be a special case of convex programming, but is more substantial, and is really an embodiment of the theory of systems of linear inequalities (as reflected here). This chapter initiates the subject with reference to systems of linear inequalities and natural questions about them, and all LP (linear programming) theorems are encountered simply in pursuing those. Theorems about linear inequalities that have uses directly on their own are also derived (and are illustrated in many places in this book). The eight sections of the chapter are: linear inequalities; separation theorems; theorems of alternatives; polyhedra and polytopes; LP Duality Theorem; the pivot operation; the Simplex Algorithm; and BASIC program.Less
This is the third of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It discusses linear programming, which might appear to be a special case of convex programming, but is more substantial, and is really an embodiment of the theory of systems of linear inequalities (as reflected here). This chapter initiates the subject with reference to systems of linear inequalities and natural questions about them, and all LP (linear programming) theorems are encountered simply in pursuing those. Theorems about linear inequalities that have uses directly on their own are also derived (and are illustrated in many places in this book). The eight sections of the chapter are: linear inequalities; separation theorems; theorems of alternatives; polyhedra and polytopes; LP Duality Theorem; the pivot operation; the Simplex Algorithm; and BASIC program.
Ken Binmore
- Published in print:
- 2007
- Published Online:
- May 2007
- ISBN:
- 9780195300574
- eISBN:
- 9780199783748
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195300574.003.0007
- Subject:
- Economics and Finance, Microeconomics
This chapter describes the theory of two-person, zero-sum games invented by John Von Neumann in 1928. It begins with an application to the computation of economic shadow prices. It shows that a ...
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This chapter describes the theory of two-person, zero-sum games invented by John Von Neumann in 1928. It begins with an application to the computation of economic shadow prices. It shows that a two-person game is strictly competitive if, and only if, it has a zero-sum representation. Such a game can be represented using only the first player's payoff matrix. The minimax and maximin values of the matrix are defined and linked to the concept of a saddle point. The ideas are then related to a player's security level in a game. An inductive proof of Von Neumann's minimax theorem is offered. The connexion between the minimax theorem and the duality theorem of linear programming is explained. The method of solving certain two-person, zero-sum games geometrically with the help of the theorem of the separating hyperplane is introduced. The Hide-and-Seek Game is used as a non-trivial example.Less
This chapter describes the theory of two-person, zero-sum games invented by John Von Neumann in 1928. It begins with an application to the computation of economic shadow prices. It shows that a two-person game is strictly competitive if, and only if, it has a zero-sum representation. Such a game can be represented using only the first player's payoff matrix. The minimax and maximin values of the matrix are defined and linked to the concept of a saddle point. The ideas are then related to a player's security level in a game. An inductive proof of Von Neumann's minimax theorem is offered. The connexion between the minimax theorem and the duality theorem of linear programming is explained. The method of solving certain two-person, zero-sum games geometrically with the help of the theorem of the separating hyperplane is introduced. The Hide-and-Seek Game is used as a non-trivial example.
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/9780198526599.003.0001
- Subject:
- Mathematics, Pure Mathematics
This chapter presents some background results about operator spaces and establishes some notations which will be used throughout this book. It can serve as a mini-course on the basics of operator ...
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This chapter presents some background results about operator spaces and establishes some notations which will be used throughout this book. It can serve as a mini-course on the basics of operator space theory for readers with less mathematical maturity. The lengthy proofs usually belong to very well-known results (such as Ruan's theorem, or the extension/characterization theorems for completely positive or completely bounded maps). Topics covered include notation and conventions, completely positive maps, operator space duality, operator space tensor products, and duality and tensor products. Notes and historical remarks are presented at the end of the chapter.Less
This chapter presents some background results about operator spaces and establishes some notations which will be used throughout this book. It can serve as a mini-course on the basics of operator space theory for readers with less mathematical maturity. The lengthy proofs usually belong to very well-known results (such as Ruan's theorem, or the extension/characterization theorems for completely positive or completely bounded maps). Topics covered include notation and conventions, completely positive maps, operator space duality, operator space tensor products, and duality and tensor products. Notes and historical remarks are presented at the end of the chapter.
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/9780198526599.003.0004
- Subject:
- Mathematics, Pure Mathematics
Many problems in functional analysis are best addressed via extreme points. Extreme points, particularly in the guise of the Choquet or Shilov boundaries, play a substantial role in the theory of ...
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Many problems in functional analysis are best addressed via extreme points. Extreme points, particularly in the guise of the Choquet or Shilov boundaries, play a substantial role in the theory of uniform algebras. This theory and its many applications are developed in this chapter. Topics covered include the Choquet boundary and boundary representations, the injective envelope, the C*-envelope, the multiplier algebra of an operator space, and multipliers and ‘characterization theorems’, and multipliers and duality.Less
Many problems in functional analysis are best addressed via extreme points. Extreme points, particularly in the guise of the Choquet or Shilov boundaries, play a substantial role in the theory of uniform algebras. This theory and its many applications are developed in this chapter. Topics covered include the Choquet boundary and boundary representations, the injective envelope, the C*-envelope, the multiplier algebra of an operator space, and multipliers and ‘characterization theorems’, and multipliers and duality.
Andrew Ranicki
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509240
- eISBN:
- 9780191708725
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509240.003.0004
- Subject:
- Mathematics, Geometry / Topology
Poincaré duality is an isomorphism between the homology and cohomology of a manifold. The chapter introduces the basic duality and the universal Poincaré duality for non-simply-connected manifolds ...
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Poincaré duality is an isomorphism between the homology and cohomology of a manifold. The chapter introduces the basic duality and the universal Poincaré duality for non-simply-connected manifolds using the involution on the fundamental group ring. Poincaré duality governs the homological effect of a surgery. The example of surgery on surfaces is given.Less
Poincaré duality is an isomorphism between the homology and cohomology of a manifold. The chapter introduces the basic duality and the universal Poincaré duality for non-simply-connected manifolds using the involution on the fundamental group ring. Poincaré duality governs the homological effect of a surgery. The example of surgery on surfaces is given.
Andrew Ranicki
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509240
- eISBN:
- 9780191708725
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509240.003.0009
- Subject:
- Mathematics, Geometry / Topology
A Poincaré complex is a space with Poincaré duality. Spherical fibrations are the homotopy theoretic analogues of vector bundles. The Spivak normal fibration is the homotopy theoretic analogue of the ...
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A Poincaré complex is a space with Poincaré duality. Spherical fibrations are the homotopy theoretic analogues of vector bundles. The Spivak normal fibration is the homotopy theoretic analogue of the normal bundle of a manifold.Less
A Poincaré complex is a space with Poincaré duality. Spherical fibrations are the homotopy theoretic analogues of vector bundles. The Spivak normal fibration is the homotopy theoretic analogue of the normal bundle of a manifold.
