Steve Awodey
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568612
- eISBN:
- 9780191717567
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568612.001.0001
- Subject:
- Mathematics, Algebra
This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible ...
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This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.Less
This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.
Thanh V. Tran
- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780195325089
- eISBN:
- 9780199864515
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195325089.001.0001
- Subject:
- Social Work, Research and Evaluation
This book provides an overview of issues and techniques relevant to the development of cross-cultural measures and provides a step-by-step approach to the assessment of cross-cultural equivalence of ...
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This book provides an overview of issues and techniques relevant to the development of cross-cultural measures and provides a step-by-step approach to the assessment of cross-cultural equivalence of measurement properties. The readers are introduced to an overview of the definitions of culture and a brief discussion of cross-cultural anthropology, psychology, sociology, and political science and the influences of these fields on social work. The book describes the process of cross-cultural instrument development, from formulating the research aims to the assessments of cross-cultural measurement properties. There are guides and recommendations for building a cross-cultural research support team for various critical tasks. The book addresses the issues of adopting and adapting existing research instruments. The processes and issues of cross-cultural translation and assessments are presented and discussed in detail. The book offers a discussion of the foundation of measurement theories and the entire process of instrument development from the definitions of abstract concepts, the construction of observed indicators, and assessment of the validity and reliability of the new instruments. The book demonstrates the application of item distribution analysis, internal consistency analysis, and exploratory factor analysis as a preliminary assessment of cross-cultural equivalence of research instruments. In addition, the book explains and illustrates the application of confirmatory factor analysis and multisample confirmatory factor analysis to evaluate the factor structure and testing of cross-cultural measurement invariance.Less
This book provides an overview of issues and techniques relevant to the development of cross-cultural measures and provides a step-by-step approach to the assessment of cross-cultural equivalence of measurement properties. The readers are introduced to an overview of the definitions of culture and a brief discussion of cross-cultural anthropology, psychology, sociology, and political science and the influences of these fields on social work. The book describes the process of cross-cultural instrument development, from formulating the research aims to the assessments of cross-cultural measurement properties. There are guides and recommendations for building a cross-cultural research support team for various critical tasks. The book addresses the issues of adopting and adapting existing research instruments. The processes and issues of cross-cultural translation and assessments are presented and discussed in detail. The book offers a discussion of the foundation of measurement theories and the entire process of instrument development from the definitions of abstract concepts, the construction of observed indicators, and assessment of the validity and reliability of the new instruments. The book demonstrates the application of item distribution analysis, internal consistency analysis, and exploratory factor analysis as a preliminary assessment of cross-cultural equivalence of research instruments. In addition, the book explains and illustrates the application of confirmatory factor analysis and multisample confirmatory factor analysis to evaluate the factor structure and testing of cross-cultural measurement invariance.
Flavio M. Menezes and Paulo K. Monteiro
- Published in print:
- 2004
- Published Online:
- April 2005
- ISBN:
- 9780199275984
- eISBN:
- 9780191602214
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/019927598X.001.0001
- Subject:
- Economics and Finance, Microeconomics
This book presents an in-depth discussion of the auction theory. It introduces the concept of Bayesian Nash equilibrium and the idea of studying auctions as games. Private, common, and affiliated ...
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This book presents an in-depth discussion of the auction theory. It introduces the concept of Bayesian Nash equilibrium and the idea of studying auctions as games. Private, common, and affiliated values models and multi-object auction models are described. A general version of the Revenue Equivalence Theorem is derived and the optimal auction is characterized to relate the field of mechanism design to auction theory.Less
This book presents an in-depth discussion of the auction theory. It introduces the concept of Bayesian Nash equilibrium and the idea of studying auctions as games. Private, common, and affiliated values models and multi-object auction models are described. A general version of the Revenue Equivalence Theorem is derived and the optimal auction is characterized to relate the field of mechanism design to auction theory.
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/9780199296866.003.0011
- Subject:
- Mathematics, Geometry / Topology
This chapter presents a series of results that elucidate the relation between D-equivalence and K-equivalence. Derived categories are first studied. These results can then be used to show that the ...
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This chapter presents a series of results that elucidate the relation between D-equivalence and K-equivalence. Derived categories are first studied. These results can then be used to show that the derived category does not change under the standard flop (Bondal and Orlov), one of the simplest birational correspondences. More recently, Bridgeland was able to show that the derived category is invariant under general three-dimensional flops. The situation is slightly more complicated for a Mukai flop, another classical birational correspondence. In this case, the birational correspondence itself does not define a derived equivalence, but Kawamata and Namikawa were able to prove that one can nevertheless find another Fourier-Mukai kernel that does.Less
This chapter presents a series of results that elucidate the relation between D-equivalence and K-equivalence. Derived categories are first studied. These results can then be used to show that the derived category does not change under the standard flop (Bondal and Orlov), one of the simplest birational correspondences. More recently, Bridgeland was able to show that the derived category is invariant under general three-dimensional flops. The situation is slightly more complicated for a Mukai flop, another classical birational correspondence. In this case, the birational correspondence itself does not define a derived equivalence, but Kawamata and Namikawa were able to prove that one can nevertheless find another Fourier-Mukai kernel that does.
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/9780199296866.003.0001
- Subject:
- Mathematics, Geometry / Topology
Reviewing the basic notions of additive and abelian categories, left and right adjoint functors, and Serre functors, this chapter is mainly devoted to triangulated categories. In particular, criteria ...
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Reviewing the basic notions of additive and abelian categories, left and right adjoint functors, and Serre functors, this chapter is mainly devoted to triangulated categories. In particular, criteria are established which decide when a given functor is fully-faithful or an equivalence. This is formulated in terms of spanning classes. The last section discusses exceptional objects in triangulated categories which lead naturally to the notion of orthogonal decompositions of categories.Less
Reviewing the basic notions of additive and abelian categories, left and right adjoint functors, and Serre functors, this chapter is mainly devoted to triangulated categories. In particular, criteria are established which decide when a given functor is fully-faithful or an equivalence. This is formulated in terms of spanning classes. The last section discusses exceptional objects in triangulated categories which lead naturally to the notion of orthogonal decompositions of categories.
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/9780199296866.003.0006
- Subject:
- Mathematics, Geometry / Topology
Based on the work of Orlov, Kawamata, and others, this chapter shows that the (numerical) Kodaira dimension and the canonical ring are preserved under derived equivalence. The same techniques can be ...
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Based on the work of Orlov, Kawamata, and others, this chapter shows that the (numerical) Kodaira dimension and the canonical ring are preserved under derived equivalence. The same techniques can be used to derive the invariance of Hochschild cohomology under derived equivalence. Going one step further, it is shown that the nefness of the canonical bundle is detected by the derived category. The chapter also studies the relation between derived and birational (or rather K-) equivalence. The special case of a central conjecture predicts that two birational Calabi-Yau varieties have equivalent derived categories.Less
Based on the work of Orlov, Kawamata, and others, this chapter shows that the (numerical) Kodaira dimension and the canonical ring are preserved under derived equivalence. The same techniques can be used to derive the invariance of Hochschild cohomology under derived equivalence. Going one step further, it is shown that the nefness of the canonical bundle is detected by the derived category. The chapter also studies the relation between derived and birational (or rather K-) equivalence. The special case of a central conjecture predicts that two birational Calabi-Yau varieties have equivalent derived categories.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.001.0001
- Subject:
- Mathematics, Geometry / Topology
Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a ...
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Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.Less
Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.
Harvey R. Brown
- Published in print:
- 2005
- Published Online:
- September 2006
- ISBN:
- 9780199275830
- eISBN:
- 9780191603914
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199275831.003.0009
- Subject:
- Philosophy, Philosophy of Science
In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the ...
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In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the four-dimensional space-time interval ds, whose meaning is the usual one associated with the readings of physical rods and clocks and possibly light rays. The other less familiar chain of reasoning ‘binds the physical manifestations of the energy tensor and the interval; it passes from matter as now defined by the energy-tensor to the interval regarded as the result of measurements made with this matter.’ This chapter takes up the challenge of outlining this second chain of reasoning. In developing this reasoning, it argues that the dynamical underpinning of relativistic kinematics that has been defended in this book is consistent with the structure and logic of GR.Less
In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the four-dimensional space-time interval ds, whose meaning is the usual one associated with the readings of physical rods and clocks and possibly light rays. The other less familiar chain of reasoning ‘binds the physical manifestations of the energy tensor and the interval; it passes from matter as now defined by the energy-tensor to the interval regarded as the result of measurements made with this matter.’ This chapter takes up the challenge of outlining this second chain of reasoning. In developing this reasoning, it argues that the dynamical underpinning of relativistic kinematics that has been defended in this book is consistent with the structure and logic of GR.
Anthony B. Atkinson (ed.)
- Published in print:
- 2004
- Published Online:
- January 2005
- ISBN:
- 9780199278558
- eISBN:
- 9780191601590
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199278555.003.0002
- Subject:
- Economics and Finance, Development, Growth, and Environmental
This chapter, together with chs. 10 to 11, approaches the question of development funding in a theoretical way, rather than by examining individual proposals for sources. Chapter 2 itself deals with ...
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This chapter, together with chs. 10 to 11, approaches the question of development funding in a theoretical way, rather than by examining individual proposals for sources. Chapter 2 itself deals with over‐arching issues and aims to set out a number of the common questions that arise in considering sources of new revenue for development finance. These concern the precise specification of the proposal, its relation with Official Development Assistance (ODA), and the administrative architecture. The key questions addressed in the different sections are: the role of the new sources; their fiscal architecture (flexibility in country involvement, institutional arrangements, fiscal federalism); the marginal cost of development funds (the leaky bucket argument) and the double dividend argument; the transfer problem (the possible effects of transfers on the economies of the donor and recipient countries); and the equivalent effects of different economic measures/policy instruments.Less
This chapter, together with chs. 10 to 11, approaches the question of development funding in a theoretical way, rather than by examining individual proposals for sources. Chapter 2 itself deals with over‐arching issues and aims to set out a number of the common questions that arise in considering sources of new revenue for development finance. These concern the precise specification of the proposal, its relation with Official Development Assistance (ODA), and the administrative architecture. The key questions addressed in the different sections are: the role of the new sources; their fiscal architecture (flexibility in country involvement, institutional arrangements, fiscal federalism); the marginal cost of development funds (the leaky bucket argument) and the double dividend argument; the transfer problem (the possible effects of transfers on the economies of the donor and recipient countries); and the equivalent effects of different economic measures/policy instruments.
Thanh V. Tran
- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780195325089
- eISBN:
- 9780199864515
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195325089.003.0006
- Subject:
- Social Work, Research and Evaluation
In comparative research — whether it is cross-cultural, cross-national, or multigroup comparison — the assumption of measurement equivalence is crucially important. If nonequivalent measures were ...
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In comparative research — whether it is cross-cultural, cross-national, or multigroup comparison — the assumption of measurement equivalence is crucially important. If nonequivalent measures were used, the outcomes would be seriously biased. Equivalence is the fundamental issue in cross-cultural research and evaluation. A cross-cultural comparison can be misleading for two reasons: (1) comparison is made using different attributes and (2) comparison is made using different scale units. But even when the problems of equivalence in attributes and scale units are resolved, it does not warrant a valid cross-cultural comparison. In every step of the research process, the researcher must ensure that equivalence in concept, operationalization, methods, analysis, and interpretation receive the same attention.Less
In comparative research — whether it is cross-cultural, cross-national, or multigroup comparison — the assumption of measurement equivalence is crucially important. If nonequivalent measures were used, the outcomes would be seriously biased. Equivalence is the fundamental issue in cross-cultural research and evaluation. A cross-cultural comparison can be misleading for two reasons: (1) comparison is made using different attributes and (2) comparison is made using different scale units. But even when the problems of equivalence in attributes and scale units are resolved, it does not warrant a valid cross-cultural comparison. In every step of the research process, the researcher must ensure that equivalence in concept, operationalization, methods, analysis, and interpretation receive the same attention.
Thanh V. Tran
- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780195325089
- eISBN:
- 9780199864515
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195325089.003.0007
- Subject:
- Social Work, Research and Evaluation
There has been a general consensus that multigroup (sample) confirmatory factor analysis offers a strong approach to evaluate cross-cultural equivalence of measurement properties. Researchers have ...
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There has been a general consensus that multigroup (sample) confirmatory factor analysis offers a strong approach to evaluate cross-cultural equivalence of measurement properties. Researchers have proposed different procedural steps in the testing of measurement equivalence hypotheses, including the equivalence of the covariance matrices of the observed indicators and the equivalence of factor means among groups. Joreskog and Sorbom (2001) recommend the testing of five general hypotheses, including equivalence of covariance matrices of observed indicators of a scale or an instrument; equivalence of factor patterns of observed indicators; equivalence of factor loadings of observed indicators on their respective factors; equivalence of measurement errors of observed indicators; and equivalence of factor variances and covariance across groups. Invariance of factor pattern and factor loadings is sufficient to determine whether a construct can be measured across different cultural, national, or racial groups.Less
There has been a general consensus that multigroup (sample) confirmatory factor analysis offers a strong approach to evaluate cross-cultural equivalence of measurement properties. Researchers have proposed different procedural steps in the testing of measurement equivalence hypotheses, including the equivalence of the covariance matrices of the observed indicators and the equivalence of factor means among groups. Joreskog and Sorbom (2001) recommend the testing of five general hypotheses, including equivalence of covariance matrices of observed indicators of a scale or an instrument; equivalence of factor patterns of observed indicators; equivalence of factor loadings of observed indicators on their respective factors; equivalence of measurement errors of observed indicators; and equivalence of factor variances and covariance across groups. Invariance of factor pattern and factor loadings is sufficient to determine whether a construct can be measured across different cultural, national, or racial groups.
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/9780691150659.003.0017
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter presents the tools to prove that the proof of Theorem 1.2 is reduced to Statement G. It begins with the lemma stating that the cardinality of each Γ-pack or Δ-pack is a power of 2. It ...
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This chapter presents the tools to prove that the proof of Theorem 1.2 is reduced to Statement G. It begins with the lemma stating that the cardinality of each Γ-pack or Δ-pack is a power of 2. It then introduces a proposition in which (σ, Σ) is an origin for a Γ-equivalence class. It is easy to see that a Γ-swap does not change α, Γ(σ), while it decreases κΓ(σ) by 1. To finish the proof of the theorem, the bijection is constructed, where the number of divisibility conditions is necessarily constant when the bijection obtains.Less
This chapter presents the tools to prove that the proof of Theorem 1.2 is reduced to Statement G. It begins with the lemma stating that the cardinality of each Γ-pack or Δ-pack is a power of 2. It then introduces a proposition in which (σ, Σ) is an origin for a Γ-equivalence class. It is easy to see that a Γ-swap does not change α, Γ(σ), while it decreases κΓ(σ) by 1. To finish the proof of the theorem, the bijection is constructed, where the number of divisibility conditions is necessarily constant when the bijection obtains.
Philip Burton
- Published in print:
- 2000
- Published Online:
- November 2003
- ISBN:
- 9780198269885
- eISBN:
- 9780191600449
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198269889.003.0005
- Subject:
- Religion, Early Christian Studies
An analysis of the conceptual framework for analysing the translation technique of the Old Latin Gospels. The terminology found in Cicero, Augustine, and Jerome is summarized, along with modern terms ...
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An analysis of the conceptual framework for analysing the translation technique of the Old Latin Gospels. The terminology found in Cicero, Augustine, and Jerome is summarized, along with modern terms such as ’formal equivalence’ and ’dynamic equivalence’. An approach is adopted based on modern language‐contact studies.Less
An analysis of the conceptual framework for analysing the translation technique of the Old Latin Gospels. The terminology found in Cicero, Augustine, and Jerome is summarized, along with modern terms such as ’formal equivalence’ and ’dynamic equivalence’. An approach is adopted based on modern language‐contact studies.
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.
Henry E. Allison
- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199647033
- eISBN:
- 9780191741166
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199647033.003.0010
- Subject:
- Philosophy, History of Philosophy, Metaphysics/Epistemology
This essay defends Kant's thesis that there is only a single categorical imperative. This needs a defense because in the Groundwork Kant presents a number of formulations of this imperative (the ...
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This essay defends Kant's thesis that there is only a single categorical imperative. This needs a defense because in the Groundwork Kant presents a number of formulations of this imperative (the precise number itself being subject to dispute), which are not obviously equivalent. Its strategy is to show that these formulations result from Kant's attempt to provide a complete construction of the concept of the categorical imperative by linking them with various steps in a progressive account of rational agency. By the complete construction of the concept is understood an account of the necessary and sufficient conditions of the possibility of the categorical imperative. Kant's analysis of rational agency is termed progressive because each step, which is correlated with a distinct formula, adds a fresh dimension to the conception of such agency. The claim is that this makes it possible to view these formulas as extensionally but not intentionally equivalent.Less
This essay defends Kant's thesis that there is only a single categorical imperative. This needs a defense because in the Groundwork Kant presents a number of formulations of this imperative (the precise number itself being subject to dispute), which are not obviously equivalent. Its strategy is to show that these formulations result from Kant's attempt to provide a complete construction of the concept of the categorical imperative by linking them with various steps in a progressive account of rational agency. By the complete construction of the concept is understood an account of the necessary and sufficient conditions of the possibility of the categorical imperative. Kant's analysis of rational agency is termed progressive because each step, which is correlated with a distinct formula, adds a fresh dimension to the conception of such agency. The claim is that this makes it possible to view these formulas as extensionally but not intentionally equivalent.
Javier Díaz‐Giménez
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780199248278
- eISBN:
- 9780191596605
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199248273.003.0002
- Subject:
- Economics and Finance, Macro- and Monetary Economics
This is a brief introduction to dynamic programming and the method of using linear quadratic (LQ) approximations to the return function; the method is an approximation because it computes the ...
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This is a brief introduction to dynamic programming and the method of using linear quadratic (LQ) approximations to the return function; the method is an approximation because it computes the solution to a quadratic expansion of the utility function about the steady state or the stable growth path of model economies. The main purpose of the chapter is to review the theoretical basis for the LQ approximation and to illustrate its use with a detailed example (social planning). The author demonstrates that, using the LQ approximation approach and the certainty equivalence principle, solving for the value function is a relatively easy task. The different sections of the chapter describe the standard neoclassical growth model, present a social planner problem that can be used to solve the model, give a recursive formulation of the social planner's problem, and describe an LQ approximation to this problem. Exercises are included throughout, and an appendix presents a MATLAB program to illustrate the LQ method.Less
This is a brief introduction to dynamic programming and the method of using linear quadratic (LQ) approximations to the return function; the method is an approximation because it computes the solution to a quadratic expansion of the utility function about the steady state or the stable growth path of model economies. The main purpose of the chapter is to review the theoretical basis for the LQ approximation and to illustrate its use with a detailed example (social planning). The author demonstrates that, using the LQ approximation approach and the certainty equivalence principle, solving for the value function is a relatively easy task. The different sections of the chapter describe the standard neoclassical growth model, present a social planner problem that can be used to solve the model, give a recursive formulation of the social planner's problem, and describe an LQ approximation to this problem. Exercises are included throughout, and an appendix presents a MATLAB program to illustrate the LQ method.
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/9780195300574.003.0021
- Subject:
- Economics and Finance, Microeconomics
The most successful area of game theory is auction design. This chapter offers a short review of the subject. Various traditional auctions are described. After showing that most of these generate the ...
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The most successful area of game theory is auction design. This chapter offers a short review of the subject. Various traditional auctions are described. After showing that most of these generate the same expected revenue to the seller under appropriate conditions, a version of the revenue equivalence theorem is proved. Reserve prices are briefly considered. An extended example of the use of the principles of mechanism design in constructing an optimal auction is offered, and the case of common-value auctions is reviewed. The chapter ends with a discussion of the important case of multiunit auctions, and concludes that treasury bonds could be sold on a more scientific basis than at present.Less
The most successful area of game theory is auction design. This chapter offers a short review of the subject. Various traditional auctions are described. After showing that most of these generate the same expected revenue to the seller under appropriate conditions, a version of the revenue equivalence theorem is proved. Reserve prices are briefly considered. An extended example of the use of the principles of mechanism design in constructing an optimal auction is offered, and the case of common-value auctions is reviewed. The chapter ends with a discussion of the important case of multiunit auctions, and concludes that treasury bonds could be sold on a more scientific basis than at present.
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.
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/9780198568520.003.0006
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter discusses the following topics: the collapsing of a cardinal to a smaller one in a Boolean-valued model; infinitary equivalence of structures as Boolean isomorphism; and the use of ...
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This chapter discusses the following topics: the collapsing of a cardinal to a smaller one in a Boolean-valued model; infinitary equivalence of structures as Boolean isomorphism; and the use of Boolean-valued models in proving theorems about complete Boolean algebras.Less
This chapter discusses the following topics: the collapsing of a cardinal to a smaller one in a Boolean-valued model; infinitary equivalence of structures as Boolean isomorphism; and the use of Boolean-valued models in proving theorems about complete Boolean algebras.
Michael Potter
- Published in print:
- 2002
- Published Online:
- May 2007
- ISBN:
- 9780199252619
- eISBN:
- 9780191712647
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199252619.003.0005
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
In the Grundlagen, Frege sketched a proof of the basic properties of natural numbers from the numerical equivalence and showed how that equivalence could be proved if numbers were explicitly defined ...
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In the Grundlagen, Frege sketched a proof of the basic properties of natural numbers from the numerical equivalence and showed how that equivalence could be proved if numbers were explicitly defined as extensions of certain second-level concepts. Two tasks were left to complete: to provide in the formal system of his concept script a derivation of the basic properties of numbers from the explicit definition in terms of extensions; and to show how extensions are given to us as logical objects independent of intuition. Frege did not delineate either task with any great clarity in the Grundlagen, but in Grundgesetze der Arithmetik he set about completing both tasks. The first involved little that was essentially new: Frege had sketched what amounts to a proof of Peano's axioms in the Grundlagen and by the time he came to write Grundgesetze, he had available to him Dedekind's derivation from them of the existence of functions satisfying Grassmann's recursion equations for addition and multiplication. The reason that it was necessary to convert both of these into detailed proofs was just to confirm that they made no illicit appeal to intuition for their justification. This chapter concentrates on the second task, that of accounting for the existence and uniqueness, independent of intuition, of extensions of concepts, i.e. objects satisfying what called the extensional equivalence.Less
In the Grundlagen, Frege sketched a proof of the basic properties of natural numbers from the numerical equivalence and showed how that equivalence could be proved if numbers were explicitly defined as extensions of certain second-level concepts. Two tasks were left to complete: to provide in the formal system of his concept script a derivation of the basic properties of numbers from the explicit definition in terms of extensions; and to show how extensions are given to us as logical objects independent of intuition. Frege did not delineate either task with any great clarity in the Grundlagen, but in Grundgesetze der Arithmetik he set about completing both tasks. The first involved little that was essentially new: Frege had sketched what amounts to a proof of Peano's axioms in the Grundlagen and by the time he came to write Grundgesetze, he had available to him Dedekind's derivation from them of the existence of functions satisfying Grassmann's recursion equations for addition and multiplication. The reason that it was necessary to convert both of these into detailed proofs was just to confirm that they made no illicit appeal to intuition for their justification. This chapter concentrates on the second task, that of accounting for the existence and uniqueness, independent of intuition, of extensions of concepts, i.e. objects satisfying what called the extensional equivalence.
