Alessio Corti (ed.)
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198570615
- eISBN:
- 9780191717703
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570615.001.0001
- Subject:
- Mathematics, Geometry / Topology
The minimal model program in algebraic geometry is a conjectural sequence of algebraic surgery operations that simplifies any algebraic variety to a point where it can be decomposed into pieces with ...
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The minimal model program in algebraic geometry is a conjectural sequence of algebraic surgery operations that simplifies any algebraic variety to a point where it can be decomposed into pieces with negative, zero, and positive curvature, in a similar vein as the geometrization program in topology decomposes a three-manifold into pieces with a standard geometry. The last few years have seen dramatic advances in the minimal model program for higher dimensional algebraic varieties, with the proof of the existence of minimal models under appropriate conditions, and the prospect within a few years of having a complete generalization of the minimal model program and the classification of varieties in all dimensions, comparable to the known results for surfaces and 3-folds. This edited collection of chapters, authored by leading experts, provides a complete and self-contained construction of 3-fold and 4-fold flips, and n-dimensional flips assuming minimal models in dimension n-1. A large part of the text is an elaboration of the work of Shokurov, and a complete and pedagogical proof of the existence of 3-fold flips is presented. The book contains a self-contained treatment of many topics that could only be found, with difficulty, in the specialized literature. The text includes a ten-page glossary.Less
The minimal model program in algebraic geometry is a conjectural sequence of algebraic surgery operations that simplifies any algebraic variety to a point where it can be decomposed into pieces with negative, zero, and positive curvature, in a similar vein as the geometrization program in topology decomposes a three-manifold into pieces with a standard geometry. The last few years have seen dramatic advances in the minimal model program for higher dimensional algebraic varieties, with the proof of the existence of minimal models under appropriate conditions, and the prospect within a few years of having a complete generalization of the minimal model program and the classification of varieties in all dimensions, comparable to the known results for surfaces and 3-folds. This edited collection of chapters, authored by leading experts, provides a complete and self-contained construction of 3-fold and 4-fold flips, and n-dimensional flips assuming minimal models in dimension n-1. A large part of the text is an elaboration of the work of Shokurov, and a complete and pedagogical proof of the existence of 3-fold flips is presented. The book contains a self-contained treatment of many topics that could only be found, with difficulty, in the specialized literature. The text includes a ten-page glossary.
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.
Reinhard B. Neder and Thomas Proffen
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199233694
- eISBN:
- 9780191715563
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233694.003.0006
- Subject:
- Physics, Crystallography: Physics
This chapter discusses modulated structures. Modulations lead to the appearance of sharp satellite reflections in the diffraction pattern. The examples provide an introduction to density and ...
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This chapter discusses modulated structures. Modulations lead to the appearance of sharp satellite reflections in the diffraction pattern. The examples provide an introduction to density and displacement waves, and explain the relationship between modulation wave parameters and the location and intensity of the resulting satellite reflections. The simulation of finite waves is described, since modulations are often not propagating throughout the entire crystal.Less
This chapter discusses modulated structures. Modulations lead to the appearance of sharp satellite reflections in the diffraction pattern. The examples provide an introduction to density and displacement waves, and explain the relationship between modulation wave parameters and the location and intensity of the resulting satellite reflections. The simulation of finite waves is described, since modulations are often not propagating throughout the entire crystal.
Juan Luis Vázquez
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199202973
- eISBN:
- 9780191707919
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199202973.001.0001
- Subject:
- Mathematics, Applied Mathematics
This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as ...
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This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of physics, chemistry, biology, and engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis. Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity (equations of porous medium type), the aim of this book is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity.Less
This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of physics, chemistry, biology, and engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis. Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity (equations of porous medium type), the aim of this book is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity.
Bettelou Los
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780199274765
- eISBN:
- 9780191705885
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199274765.001.0001
- Subject:
- Linguistics, Historical Linguistics
This book describes the historical emergence and spread of the to-infinitive in English. It shows that to + infinitive emerged from a reanalysis of the preposition to plus a deverbal nominalization, ...
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This book describes the historical emergence and spread of the to-infinitive in English. It shows that to + infinitive emerged from a reanalysis of the preposition to plus a deverbal nominalization, which spread first to purpose clauses, then to other non-finite environments. The book challenges the traditional reasoning that infinitives must have been nouns in Old English because they inflected for dative case and can follow prepositions. In fact, as early as Old English, the to-infinitive was established in most of the environments in which it is found today, and its syntactic behaviour clearly shows that it is already a clause rather than a phrase at this early date. Its spread was largely due to competition with finite subjunctive that-clauses, which it gradually replaced. Later chapters consider Middle English developments. The book provides a measured evaluation of the evidence that the infinitive marker to undergoes a period of degrammaticalization. It concludes that the extent to which to gains syntactic freedom in Middle English is due to the fact that speakers began to equate it with the modal verbs, and therefore to treat it syntactically as a modal verb. The rise of to-infinitival Exceptional Case-Marking constructions is a Middle English innovation, triggered by changes in information structure that were in turn caused by the loss of verb-second.Less
This book describes the historical emergence and spread of the to-infinitive in English. It shows that to + infinitive emerged from a reanalysis of the preposition to plus a deverbal nominalization, which spread first to purpose clauses, then to other non-finite environments. The book challenges the traditional reasoning that infinitives must have been nouns in Old English because they inflected for dative case and can follow prepositions. In fact, as early as Old English, the to-infinitive was established in most of the environments in which it is found today, and its syntactic behaviour clearly shows that it is already a clause rather than a phrase at this early date. Its spread was largely due to competition with finite subjunctive that-clauses, which it gradually replaced. Later chapters consider Middle English developments. The book provides a measured evaluation of the evidence that the infinitive marker to undergoes a period of degrammaticalization. It concludes that the extent to which to gains syntactic freedom in Middle English is due to the fact that speakers began to equate it with the modal verbs, and therefore to treat it syntactically as a modal verb. The rise of to-infinitival Exceptional Case-Marking constructions is a Middle English innovation, triggered by changes in information structure that were in turn caused by the loss of verb-second.
Aman Ullah
- Published in print:
- 2004
- Published Online:
- August 2004
- ISBN:
- 9780198774471
- eISBN:
- 9780191601347
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774478.001.0001
- Subject:
- Economics and Finance, Econometrics
This book presents a comprehensive and unified treatment of finite sample theory, and its application to estimators and test statistics used in various econometric models. Time series, cross section, ...
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This book presents a comprehensive and unified treatment of finite sample theory, and its application to estimators and test statistics used in various econometric models. Time series, cross section, and panel data models are considered. The results are explored for linear and nonlinear models, as well as models with normal and nonnormal errors. The book contains seven chapters. Chapter 1 presents an introduction to finite sample econometrics. Chapter 2 gives methods of obtaining the moments of econometric statistics. Chapter 3 provides methods for analysing distributions. Finite sample results for various econometric models are discussed in Chapters 4-7.Less
This book presents a comprehensive and unified treatment of finite sample theory, and its application to estimators and test statistics used in various econometric models. Time series, cross section, and panel data models are considered. The results are explored for linear and nonlinear models, as well as models with normal and nonnormal errors. The book contains seven chapters. Chapter 1 presents an introduction to finite sample econometrics. Chapter 2 gives methods of obtaining the moments of econometric statistics. Chapter 3 provides methods for analysing distributions. Finite sample results for various econometric models are discussed in Chapters 4-7.
Jeff Jordan
- Published in print:
- 2006
- Published Online:
- January 2007
- ISBN:
- 9780199291328
- eISBN:
- 9780191710698
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199291328.003.0004
- Subject:
- Philosophy, Philosophy of Religion
Pascal's Wager employed infinite utilities, but standard decision theory is incompatible with infinite utilities. In this chapter, the consequences of restricting the Wager to finite utilities are ...
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Pascal's Wager employed infinite utilities, but standard decision theory is incompatible with infinite utilities. In this chapter, the consequences of restricting the Wager to finite utilities are explored. The St. Petersburg paradox is also examined.Less
Pascal's Wager employed infinite utilities, but standard decision theory is incompatible with infinite utilities. In this chapter, the consequences of restricting the Wager to finite utilities are explored. The St. Petersburg paradox is also examined.
Theodore Markopoulos
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780199539857
- eISBN:
- 9780191716317
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199539857.001.0001
- Subject:
- Linguistics, Syntax and Morphology, Historical Linguistics
This book investigates the development of three future‐referring constructions in Greek, namely “μέλλω / œχω / θέλω + Infinitive / complement clause” in the classical (5th–4th c. BC), the ...
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This book investigates the development of three future‐referring constructions in Greek, namely “μέλλω / œχω / θέλω + Infinitive / complement clause” in the classical (5th–4th c. BC), the Hellenistic–Roman (3rd c. BC–4th c. AD), the Early Medieval (5th–10th c. AD), and the Late Medieval period (11th–15th c. AD). Despite their co‐occurrence in all these periods, it is shown for the first time that these constructions were increasingly differentiated in terms of their semantic, syntactic, and sociolinguistic properties. The analysis sheds new light on these developments, since large parts are based on hitherto unknown material, drawn especially from papyri and non‐literary documents. The investigation is based on the functional–typological perspective of grammaticalization, and it pays particular attention to a variety of—often neglected—factors, such as language contact. The typological predictions concerning future‐referring forms are found lacking in some respects, and various modifications are proposed accordingly.Less
This book investigates the development of three future‐referring constructions in Greek, namely “μέλλω / œχω / θέλω + Infinitive / complement clause” in the classical (5th–4th c. BC), the Hellenistic–Roman (3rd c. BC–4th c. AD), the Early Medieval (5th–10th c. AD), and the Late Medieval period (11th–15th c. AD). Despite their co‐occurrence in all these periods, it is shown for the first time that these constructions were increasingly differentiated in terms of their semantic, syntactic, and sociolinguistic properties. The analysis sheds new light on these developments, since large parts are based on hitherto unknown material, drawn especially from papyri and non‐literary documents. The investigation is based on the functional–typological perspective of grammaticalization, and it pays particular attention to a variety of—often neglected—factors, such as language contact. The typological predictions concerning future‐referring forms are found lacking in some respects, and various modifications are proposed accordingly.
J. N. Reddy
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198525295
- eISBN:
- 9780191711671
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525295.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book presents the theory and computer implementation of the finite element method as applied to nonlinear problems of heat transfer and similar field problems, fluid mechanics (flows of ...
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This book presents the theory and computer implementation of the finite element method as applied to nonlinear problems of heat transfer and similar field problems, fluid mechanics (flows of incompressible fluids), and solid mechanics (elasticity, beams and plates). Both geometric as well as material nonlinearities are considered, and static and transient (that is, time-dependent) responses are studied. Although there exist a number of books on nonlinear finite elements that serve as good references for engineers who are familiar with the subject and wish to learn advanced topics or the latest developments, there is currently no book that is suitable as a textbook for a first course on nonlinear finite element analysis. This book fills the void in the market, providing a clear understanding of the concepts of nonlinear finite element analyses through detailed theoretical formulations and computer implementation steps, examples and exercises, in a tutorial, user-oriented style. In addition, the book is a useful reference for industrial engineers and scientists, and serves as a prelude to more advanced books on the subject.Less
This book presents the theory and computer implementation of the finite element method as applied to nonlinear problems of heat transfer and similar field problems, fluid mechanics (flows of incompressible fluids), and solid mechanics (elasticity, beams and plates). Both geometric as well as material nonlinearities are considered, and static and transient (that is, time-dependent) responses are studied. Although there exist a number of books on nonlinear finite elements that serve as good references for engineers who are familiar with the subject and wish to learn advanced topics or the latest developments, there is currently no book that is suitable as a textbook for a first course on nonlinear finite element analysis. This book fills the void in the market, providing a clear understanding of the concepts of nonlinear finite element analyses through detailed theoretical formulations and computer implementation steps, examples and exercises, in a tutorial, user-oriented style. In addition, the book is a useful reference for industrial engineers and scientists, and serves as a prelude to more advanced books on the subject.
Nicholas M. Katz
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.001.0001
- Subject:
- Mathematics, Number Theory
This book explores an important aspect of number theory—the theory of exponential sums over finite fields and their Mellin transforms—from a new, categorical point of view. The book presents ...
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This book explores an important aspect of number theory—the theory of exponential sums over finite fields and their Mellin transforms—from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods. By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.Less
This book explores an important aspect of number theory—the theory of exponential sums over finite fields and their Mellin transforms—from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods. By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.
Paula Tretkoff
- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. It emphasizes those finite coverings that ...
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This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. It emphasizes those finite coverings that are free quotients of the complex 2-ball. The book also includes a background on the classical Gauss hypergeometric function of one variable, and a chapter on the Appell two-variable F1 hypergeometric function. The book began as a set of lecture notes, taken by the author, of a course given by Friedrich Hirzebruch at ETH Zürich in 1996. The lecture notes were then considerably expanded over a number of years. In this book, the author has expanded those notes even further, still stressing examples offered by finite covers of line arrangements. The book is largely self-contained and foundational material is introduced and explained as needed, but not treated in full detail. References to omitted material are provided for interested readers. Aimed at graduate students and researchers, this is an accessible account of a highly informative area of complex geometry.Less
This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. It emphasizes those finite coverings that are free quotients of the complex 2-ball. The book also includes a background on the classical Gauss hypergeometric function of one variable, and a chapter on the Appell two-variable F1 hypergeometric function. The book began as a set of lecture notes, taken by the author, of a course given by Friedrich Hirzebruch at ETH Zürich in 1996. The lecture notes were then considerably expanded over a number of years. In this book, the author has expanded those notes even further, still stressing examples offered by finite covers of line arrangements. The book is largely self-contained and foundational material is introduced and explained as needed, but not treated in full detail. References to omitted material are provided for interested readers. Aimed at graduate students and researchers, this is an accessible account of a highly informative area of complex geometry.
Bijan Mohammadi and Olivier Pironneau
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199546909
- eISBN:
- 9780191720482
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199546909.003.0004
- Subject:
- Mathematics, Mathematical Physics
This chapter deals with the numerical methods that will be used for the flow solvers. As in most commercial and industrial packages, unstructured meshes with automatic mesh generation and adaptation ...
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This chapter deals with the numerical methods that will be used for the flow solvers. As in most commercial and industrial packages, unstructured meshes with automatic mesh generation and adaptation are used together with finite volume or finite element discretization for these complex geometries. The iterative solvers and the flux functions for upwinding are also presented. Main algorithms for compressible and incompressible fluid flow calculations are given.Less
This chapter deals with the numerical methods that will be used for the flow solvers. As in most commercial and industrial packages, unstructured meshes with automatic mesh generation and adaptation are used together with finite volume or finite element discretization for these complex geometries. The iterative solvers and the flux functions for upwinding are also presented. Main algorithms for compressible and incompressible fluid flow calculations are given.
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/9780691153308.003.0003
- Subject:
- Mathematics, Number Theory
Let k be a finite field, q its cardinality, p its characteristic, 𝓁 ≠ p a prime number, and 𝓖/k a smooth commutative group scheme which over k¯ becomes isomorphic to 𝔾ₘ/k¯. This chapter is concerned ...
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Let k be a finite field, q its cardinality, p its characteristic, 𝓁 ≠ p a prime number, and 𝓖/k a smooth commutative group scheme which over k¯ becomes isomorphic to 𝔾ₘ/k¯. This chapter is concerned with perverse sheaves on 𝓖/k and on 𝓖/k¯.Less
Let k be a finite field, q its cardinality, p its characteristic, 𝓁 ≠ p a prime number, and 𝓖/k a smooth commutative group scheme which over k¯ becomes isomorphic to 𝔾ₘ/k¯. This chapter is concerned with perverse sheaves on 𝓖/k and on 𝓖/k¯.
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/9780691153308.003.0025
- Subject:
- Mathematics, Number Theory
This chapter works over a finite field k of odd characteristic. Fix an even integer 2n ≤ 4 and a monic polynomial f(x) ɛ k[x] of degree 2n, f(x)=∑i=02nAixi, A2n=1. The following three assumptions are ...
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This chapter works over a finite field k of odd characteristic. Fix an even integer 2n ≤ 4 and a monic polynomial f(x) ɛ k[x] of degree 2n, f(x)=∑i=02nAixi, A2n=1. The following three assumptions are made about f: (1) f has 2n distinct roots in k¯, and A₀ = −1; gcd{i¦Aᵢ ≠ = 0} = 1; and (3) f is antipalindromic, i.e., for fpal(x) := x²ⁿf(1/x), we have fpal(x) = −f(x).Less
This chapter works over a finite field k of odd characteristic. Fix an even integer 2n ≤ 4 and a monic polynomial f(x) ɛ k[x] of degree 2n, f(x)=∑i=02nAixi, A2n=1. The following three assumptions are made about f: (1) f has 2n distinct roots in k¯, and A₀ = −1; gcd{i¦Aᵢ ≠ = 0} = 1; and (3) f is antipalindromic, i.e., for fpal(x) := x²ⁿf(1/x), we have fpal(x) = −f(x).
Marcus Giaquinto
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199285945
- eISBN:
- 9780191713811
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199285945.003.0011
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter investigates the nature of our grasp of structures by means of visual experience. Structures, which are properties of structured sets of things, are abstract. Does this mean that we can ...
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This chapter investigates the nature of our grasp of structures by means of visual experience. Structures, which are properties of structured sets of things, are abstract. Does this mean that we can only know a structure under some theoretical description, as the structure of models of this or that theory? Or can there be some more direct mode of comprehension, one that involves visuo-spatial experience? The case for the latter possibility is presented.Less
This chapter investigates the nature of our grasp of structures by means of visual experience. Structures, which are properties of structured sets of things, are abstract. Does this mean that we can only know a structure under some theoretical description, as the structure of models of this or that theory? Or can there be some more direct mode of comprehension, one that involves visuo-spatial experience? The case for the latter possibility is presented.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0002
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter describes a numerical method for solving equations of thermal convection on a computer. It begins by introducing the vorticity-streamfunction formulation as a means of conserving mass. ...
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This chapter describes a numerical method for solving equations of thermal convection on a computer. It begins by introducing the vorticity-streamfunction formulation as a means of conserving mass. The approach involves updating for the vorticity first and then solving for the fluid velocity each time step. The chapter continues with a discussion of two very different spatial discretizations, whereby the vertical derivatives are approximated with a finite-difference method and the horizontal derivatives with a spectral method. The nonlinear terms are computed in spectral space. The chapter also considers the Adams-Bashforth time integration scheme and explains how the Poisson equation can be solved at each time step for the updated streamfunction given the updated vorticity.Less
This chapter describes a numerical method for solving equations of thermal convection on a computer. It begins by introducing the vorticity-streamfunction formulation as a means of conserving mass. The approach involves updating for the vorticity first and then solving for the fluid velocity each time step. The chapter continues with a discussion of two very different spatial discretizations, whereby the vertical derivatives are approximated with a finite-difference method and the horizontal derivatives with a spectral method. The nonlinear terms are computed in spectral space. The chapter also considers the Adams-Bashforth time integration scheme and explains how the Poisson equation can be solved at each time step for the updated streamfunction given the updated vorticity.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter modifies the numerical code by adding the nonlinear terms to produce finite-amplitude simulations. The nonlinear terms are calculated using a Galerkin method in spectral space. After ...
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This chapter modifies the numerical code by adding the nonlinear terms to produce finite-amplitude simulations. The nonlinear terms are calculated using a Galerkin method in spectral space. After explaining the modifications to the linear model, the chapter shows how to add the nonlinear terms to the code. It also discusses the Galerkin method, the strategy of computing the contribution to the nonlinear terms for each mode due to the binary interactions of many other modes. The Galerkin method works fine as far as calculating the nonlinear terms is concerned because of the simple geometry and convenient boundary conditions. The chapter concludes by showing how to construct a nonlinear code and performing nonlinear simulations.Less
This chapter modifies the numerical code by adding the nonlinear terms to produce finite-amplitude simulations. The nonlinear terms are calculated using a Galerkin method in spectral space. After explaining the modifications to the linear model, the chapter shows how to add the nonlinear terms to the code. It also discusses the Galerkin method, the strategy of computing the contribution to the nonlinear terms for each mode due to the binary interactions of many other modes. The Galerkin method works fine as far as calculating the nonlinear terms is concerned because of the simple geometry and convenient boundary conditions. The chapter concludes by showing how to construct a nonlinear code and performing nonlinear simulations.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0009
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It ...
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This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It first provides an overview of nonuniform grids before discussing coordinate mapping as an alternative way of achieving spatial discretization. It then describes an approach for treating both the vertical and horizontal directions with simple finite-difference methods: defining a streamfunction, which automatically satisfies mass conservation, and solving for vorticity via the curl of the momentum conservation equation. It also explains the use of the Chebyshev–Fourier method to simulate the convection or gravity wave problem by employing spectral methods in both the horizontal and vertical directions. Finally, it looks at the basic ideas and some issues that need to be addressed with respect to parallel processing as well as choices that need to be made when designing a parallel code.Less
This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It first provides an overview of nonuniform grids before discussing coordinate mapping as an alternative way of achieving spatial discretization. It then describes an approach for treating both the vertical and horizontal directions with simple finite-difference methods: defining a streamfunction, which automatically satisfies mass conservation, and solving for vorticity via the curl of the momentum conservation equation. It also explains the use of the Chebyshev–Fourier method to simulate the convection or gravity wave problem by employing spectral methods in both the horizontal and vertical directions. Finally, it looks at the basic ideas and some issues that need to be addressed with respect to parallel processing as well as choices that need to be made when designing a parallel code.
Aman Ullah
- Published in print:
- 2004
- Published Online:
- August 2004
- ISBN:
- 9780198774471
- eISBN:
- 9780191601347
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774478.003.0001
- Subject:
- Economics and Finance, Econometrics
This introductory chapter begins with a brief discussion on the importance of finite sample econometrics. The developments in this area are described. An overview of the chapters in this volume is ...
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This introductory chapter begins with a brief discussion on the importance of finite sample econometrics. The developments in this area are described. An overview of the chapters in this volume is then presented.Less
This introductory chapter begins with a brief discussion on the importance of finite sample econometrics. The developments in this area are described. An overview of the chapters in this volume is then presented.
Sophie Repp
- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780199543601
- eISBN:
- 9780191715587
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199543601.003.0005
- Subject:
- Linguistics, Syntax and Morphology, Theoretical Linguistics
Chapter 5 delivers a new semantic definition of gapping: it is the referential anchoring of a sentence which is elided in the second conjunct of a gapping sentence. It shows that gapping does not ...
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Chapter 5 delivers a new semantic definition of gapping: it is the referential anchoring of a sentence which is elided in the second conjunct of a gapping sentence. It shows that gapping does not necessarily involve the elision of the finite verb but can also occur in non‐finite structures.Less
Chapter 5 delivers a new semantic definition of gapping: it is the referential anchoring of a sentence which is elided in the second conjunct of a gapping sentence. It shows that gapping does not necessarily involve the elision of the finite verb but can also occur in non‐finite structures.
