Marcus Giaquinto
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199285945
- eISBN:
- 9780191713811
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199285945.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, ...
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Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? This book argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. The book explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. It shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. This book reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences.Less
Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? This book argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. The book explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. It shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. This book reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences.
Michael Potter
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780199215836
- eISBN:
- 9780191721243
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199215836.003.0018
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
From a single proposition p we form its negation not-p. From two propositions p and q we form various further propositions, e.g., p or q, p and q, etc.; similarly (though less often) with three or ...
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From a single proposition p we form its negation not-p. From two propositions p and q we form various further propositions, e.g., p or q, p and q, etc.; similarly (though less often) with three or more propositions. For each such method of forming propositions it is common to speak of a certain truth-function as corresponding to it. On the one hand, there is a function which takes senses of propositions as inputs and returns the sense of a proposition as output. On the other hand, there is a truth-function, i.e., a function taking some finite number of truth-values as arguments and giving a single truth-value as value, which is representable by a truth-table or truth-diagram. This chapter discusses the relationship between the two.Less
From a single proposition p we form its negation not-p. From two propositions p and q we form various further propositions, e.g., p or q, p and q, etc.; similarly (though less often) with three or more propositions. For each such method of forming propositions it is common to speak of a certain truth-function as corresponding to it. On the one hand, there is a function which takes senses of propositions as inputs and returns the sense of a proposition as output. On the other hand, there is a truth-function, i.e., a function taking some finite number of truth-values as arguments and giving a single truth-value as value, which is representable by a truth-table or truth-diagram. This chapter discusses the relationship between the two.
Naomi E. Chayen, John R. Helliwell, and Edward H. Snell
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199213252
- eISBN:
- 9780191707575
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213252.003.0002
- Subject:
- Physics, Crystallography: Physics
A basic understanding of crystallization theory exists through the phase‐diagram description. More detailed theories are available but no one theory can currently predict crystallization behaviour. ...
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A basic understanding of crystallization theory exists through the phase‐diagram description. More detailed theories are available but no one theory can currently predict crystallization behaviour. In this chapter we discuss the basic theory and present a typical phase diagram.Less
A basic understanding of crystallization theory exists through the phase‐diagram description. More detailed theories are available but no one theory can currently predict crystallization behaviour. In this chapter we discuss the basic theory and present a typical phase diagram.
Dov-Ber Kerler
- Published in print:
- 1993
- Published Online:
- October 2011
- ISBN:
- 9780198151661
- eISBN:
- 9780191672798
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198151661.003.0023
- Subject:
- Literature, European Literature
This chapter uses a schematic diagram to illustrate the scope of Linguistic Easternization. The parameters used are morphology and syntax; these are deemed to be representationally sufficient for the ...
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This chapter uses a schematic diagram to illustrate the scope of Linguistic Easternization. The parameters used are morphology and syntax; these are deemed to be representationally sufficient for the analysis of the transition from the Old to the Modern Yiddish literature. The premise is that the more Easternized in form the material is, the further is the relationship between morphology and syntax. Suffice to say that none of the late eighteenth-century works achieved the perfect state of Easternization. The later versions had a more modern and smooth feel to them, due to the advancement in syntactic changes. However, even in these, there can be found minimal resistance in form.Less
This chapter uses a schematic diagram to illustrate the scope of Linguistic Easternization. The parameters used are morphology and syntax; these are deemed to be representationally sufficient for the analysis of the transition from the Old to the Modern Yiddish literature. The premise is that the more Easternized in form the material is, the further is the relationship between morphology and syntax. Suffice to say that none of the late eighteenth-century works achieved the perfect state of Easternization. The later versions had a more modern and smooth feel to them, due to the advancement in syntactic changes. However, even in these, there can be found minimal resistance in form.
Antônio M. Figueiredo Neto and Silvio R. A. Salinas
- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198525509
- eISBN:
- 9780191711756
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525509.001.0001
- Subject:
- Physics, Crystallography: Physics
Mixtures of surfactant or amphiphilic molecules and solvents are known to display a large number of lyotropic mesophases. Although the physics of thermotropic liquid crystals has been vastly ...
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Mixtures of surfactant or amphiphilic molecules and solvents are known to display a large number of lyotropic mesophases. Although the physics of thermotropic liquid crystals has been vastly discussed in the literature, lyotropic mesophases have been much less explored. This book gives a unifying presentation of the structural and physical properties of lyotropic liquid crystalline systems, including a set of experimental results and a discussion of the characterization of different structures and the corresponding phase transitions. The initial chapters contain a description of the main experimental results and techniques associated with the characterization of lyotropic mixtures, and an overview of some theoretical approaches to account for the phase transitions and phase diagrams of these mesophases. This book includes discussions on periodically organized lyotropic systems, micellar and bicontinuous structures, modulated and cholesteric structures, and a number of related developments (magnetic colloids, microemulsions, films, and lyo-banana mesophases).Less
Mixtures of surfactant or amphiphilic molecules and solvents are known to display a large number of lyotropic mesophases. Although the physics of thermotropic liquid crystals has been vastly discussed in the literature, lyotropic mesophases have been much less explored. This book gives a unifying presentation of the structural and physical properties of lyotropic liquid crystalline systems, including a set of experimental results and a discussion of the characterization of different structures and the corresponding phase transitions. The initial chapters contain a description of the main experimental results and techniques associated with the characterization of lyotropic mixtures, and an overview of some theoretical approaches to account for the phase transitions and phase diagrams of these mesophases. This book includes discussions on periodically organized lyotropic systems, micellar and bicontinuous structures, modulated and cholesteric structures, and a number of related developments (magnetic colloids, microemulsions, films, and lyo-banana mesophases).
Ian Budge and Hans Keman
- Published in print:
- 1993
- Published Online:
- November 2003
- ISBN:
- 9780198279259
- eISBN:
- 9780191598883
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198279256.003.0005
- Subject:
- Political Science, Comparative Politics
The policy goals sought by parties in the various ideological ‘families’ are specified and flow diagrams created of how relevant ministries should be distributed among parties in government under ...
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The policy goals sought by parties in the various ideological ‘families’ are specified and flow diagrams created of how relevant ministries should be distributed among parties in government under various coalition combinations. The comparative data is matched to these to see how far they work. An overall predictive success rate of 0.75 is obtained.Less
The policy goals sought by parties in the various ideological ‘families’ are specified and flow diagrams created of how relevant ministries should be distributed among parties in government under various coalition combinations. The comparative data is matched to these to see how far they work. An overall predictive success rate of 0.75 is obtained.
A. Townsend Peterson, Jorge Soberón, Richard G. Pearson, Robert P. Anderson, Enrique Martínez-Meyer, Miguel Nakamura, and Miguel Bastos Araújo
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691136868
- eISBN:
- 9781400840670
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691136868.003.0016
- Subject:
- Biology, Ecology
This book has described a comprehensive framework for thinking about the geography and ecology of species distributions, arguing that such a framework is critical to further progress in the field of ...
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This book has described a comprehensive framework for thinking about the geography and ecology of species distributions, arguing that such a framework is critical to further progress in the field of ecological niches and distributions. To develop this framework, traditional concepts in ecology have been radically reworked. In this conclusion, some of the challenges for future work regarding ecological niche modeling are discussed, such as fully integrating the BAM diagram with central concepts of population biology and statistical theory; clarifying the notion of niche conservatism versus niche evolution as regards scenopoetic versus bionomic environmental dimensions; and improving the link between correlational and mechanistic approaches to estimating and understanding ecological niches. The book argues that careful conceptual thinking must be combined with detailed empirical exploration in order to address each of these challenges.Less
This book has described a comprehensive framework for thinking about the geography and ecology of species distributions, arguing that such a framework is critical to further progress in the field of ecological niches and distributions. To develop this framework, traditional concepts in ecology have been radically reworked. In this conclusion, some of the challenges for future work regarding ecological niche modeling are discussed, such as fully integrating the BAM diagram with central concepts of population biology and statistical theory; clarifying the notion of niche conservatism versus niche evolution as regards scenopoetic versus bionomic environmental dimensions; and improving the link between correlational and mechanistic approaches to estimating and understanding ecological niches. The book argues that careful conceptual thinking must be combined with detailed empirical exploration in order to address each of these challenges.
John F. Wilson and Andrew Thomson
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199261581
- eISBN:
- 9780191718588
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199261581.003.0003
- Subject:
- Business and Management, Business History
This chapter describes a period when in spite of the scale and nature of British business activities and the emergence of big business (mostly through intense merger activity), there was a ...
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This chapter describes a period when in spite of the scale and nature of British business activities and the emergence of big business (mostly through intense merger activity), there was a persistence of personal and proprietorial capitalism, as well as the extent to which firms operated in dense industrial districts that afforded extensive opportunities to externalize a wide range of activities. An analysis of the drivers, using a force-field diagram, demonstrates how the British environment was not conducive to the development of managerial capitalism.Less
This chapter describes a period when in spite of the scale and nature of British business activities and the emergence of big business (mostly through intense merger activity), there was a persistence of personal and proprietorial capitalism, as well as the extent to which firms operated in dense industrial districts that afforded extensive opportunities to externalize a wide range of activities. An analysis of the drivers, using a force-field diagram, demonstrates how the British environment was not conducive to the development of managerial capitalism.
Nikolai B. Kopnin
- Published in print:
- 2001
- Published Online:
- January 2010
- ISBN:
- 9780198507888
- eISBN:
- 9780191709722
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507888.003.04
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter explains how to incorporate scattering by random impurity atoms into the general Green function formalism of the theory of superconductivity. The cross-diagram technique based on the ...
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This chapter explains how to incorporate scattering by random impurity atoms into the general Green function formalism of the theory of superconductivity. The cross-diagram technique based on the averaging over random impurity positions is derived using the Born approximation for the scattering amplitude. Impurity self-energy is derived. Homogeneous state of an s-wave superconductor is considered.Less
This chapter explains how to incorporate scattering by random impurity atoms into the general Green function formalism of the theory of superconductivity. The cross-diagram technique based on the averaging over random impurity positions is derived using the Born approximation for the scattering amplitude. Impurity self-energy is derived. Homogeneous state of an s-wave superconductor is considered.
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/9780199285945.003.0004
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter shows how, using basic beliefs, one can go on to make a geometrical discovery by visual means in a non-empirical manner. It focuses on a simple example in order to illustrate the general ...
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This chapter shows how, using basic beliefs, one can go on to make a geometrical discovery by visual means in a non-empirical manner. It focuses on a simple example in order to illustrate the general possibility of what Kant would call synthetic a priori judgements in geometry. It attempts to show how such a judgement can be knowledge. It is commonly asserted that diagrams have no non-redundant role in a proof, even in a geometric proof.Less
This chapter shows how, using basic beliefs, one can go on to make a geometrical discovery by visual means in a non-empirical manner. It focuses on a simple example in order to illustrate the general possibility of what Kant would call synthetic a priori judgements in geometry. It attempts to show how such a judgement can be knowledge. It is commonly asserted that diagrams have no non-redundant role in a proof, even in a geometric proof.
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/9780199285945.003.0005
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Presentations of geometric proofs are often accompanied by diagrams for quick and easy comprehension. But to many people, it seems clear that diagrammatic reasoning cannot be a part of the argument ...
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Presentations of geometric proofs are often accompanied by diagrams for quick and easy comprehension. But to many people, it seems clear that diagrammatic reasoning cannot be a part of the argument itself, otherwise it would be prey to the very insecurity that we want to eliminate — insecurity from visual thinking — and so the argument would not be able to justify its conclusion; it would not be a proof. This is the line of thought that most strongly supports the widespread belief that diagrams can have no epistemological role in proof. The main aim of this chapter is to investigate this negative view and the argument for it presented in this book.Less
Presentations of geometric proofs are often accompanied by diagrams for quick and easy comprehension. But to many people, it seems clear that diagrammatic reasoning cannot be a part of the argument itself, otherwise it would be prey to the very insecurity that we want to eliminate — insecurity from visual thinking — and so the argument would not be able to justify its conclusion; it would not be a proof. This is the line of thought that most strongly supports the widespread belief that diagrams can have no epistemological role in proof. The main aim of this chapter is to investigate this negative view and the argument for it presented in this book.
A. A. Ivanov
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198527596
- eISBN:
- 9780191713163
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527596.003.0009
- Subject:
- Mathematics, Pure Mathematics
This chapter surveys the history of discovery, construction, and characterization of J4. It shows when and how the crucial ingredients of the proof of the Main Theorem emerged. Topics covered include ...
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This chapter surveys the history of discovery, construction, and characterization of J4. It shows when and how the crucial ingredients of the proof of the Main Theorem emerged. Topics covered include Janko's discovery, Ronan-Smith geometry, Cambridge five, uniqueness of J4, Lempken's construction, the Rowley-Walker diagram, and locally projective graphs.Less
This chapter surveys the history of discovery, construction, and characterization of J4. It shows when and how the crucial ingredients of the proof of the Main Theorem emerged. Topics covered include Janko's discovery, Ronan-Smith geometry, Cambridge five, uniqueness of J4, Lempken's construction, the Rowley-Walker diagram, and locally projective graphs.
Kenneth Manders
- Published in print:
- 2008
- Published Online:
- February 2010
- ISBN:
- 9780199296453
- eISBN:
- 9780191711961
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296453.003.0005
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is ...
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This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed.Less
This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed.
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/9780198568612.003.0008
- Subject:
- Mathematics, Algebra
This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it ...
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This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it is a straightforward generalization of facts that are fairly easily shown in relation to monoids and posets. The topics discussed include set-valued functor categories, Yoneda embedding, limits in categories of diagrams, colimits in categories of diagrams, exponentials in categories of diagrams, and Topoi. Exercises are provided in the last part of the chapter.Less
This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it is a straightforward generalization of facts that are fairly easily shown in relation to monoids and posets. The topics discussed include set-valued functor categories, Yoneda embedding, limits in categories of diagrams, colimits in categories of diagrams, exponentials in categories of diagrams, and Topoi. Exercises are provided in the last part of the chapter.
Jie W Weiss and David J Weiss
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195322989
- eISBN:
- 9780199869206
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195322989.003.0010
- Subject:
- Psychology, Cognitive Psychology
Bayes Nets (BNs) and Influence Diagrams (IDs), new tools that use graphic user interfaces to facilitate representation of complex inference and decision structures, it are the core elements of new ...
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Bayes Nets (BNs) and Influence Diagrams (IDs), new tools that use graphic user interfaces to facilitate representation of complex inference and decision structures, it are the core elements of new computer technologies that make the 21st century the Century of Bayes. BNs are a way of representing a set of related uncertainties. They facilitate Bayesian inference by separating structural information from parameters. Hailfinder is a BN that predicts severe summer weather in Eastern Colorado. Its design led to a number of novel ideas about how to build such BNs. This chapter argues that psychologists should care about these new tools because the rapid development of BN and ID programs will create a market for skilled elicitors and for ways of teaching domain experts how to make appropriate judgments. Psychologists should care about these new tools if they would prefer not to be omitted from the sellers in that market. Psychologists should also care because these new tools constitute a new, normatively appropriate way of performing important intellectual tasks. Psychologists who want to explain how people perform intellectual tasks should not be indifferent to the development of tools that will help people to do well in three tasks that, if we are to believe a generation of researchers in cognitive psychology, they now do poorly: evaluation, inference, and decision.Less
Bayes Nets (BNs) and Influence Diagrams (IDs), new tools that use graphic user interfaces to facilitate representation of complex inference and decision structures, it are the core elements of new computer technologies that make the 21st century the Century of Bayes. BNs are a way of representing a set of related uncertainties. They facilitate Bayesian inference by separating structural information from parameters. Hailfinder is a BN that predicts severe summer weather in Eastern Colorado. Its design led to a number of novel ideas about how to build such BNs. This chapter argues that psychologists should care about these new tools because the rapid development of BN and ID programs will create a market for skilled elicitors and for ways of teaching domain experts how to make appropriate judgments. Psychologists should care about these new tools if they would prefer not to be omitted from the sellers in that market. Psychologists should also care because these new tools constitute a new, normatively appropriate way of performing important intellectual tasks. Psychologists who want to explain how people perform intellectual tasks should not be indifferent to the development of tools that will help people to do well in three tasks that, if we are to believe a generation of researchers in cognitive psychology, they now do poorly: evaluation, inference, and decision.
Wai-Kee Li, Gong-Du Zhou, and Thomas Chung Wai Mak
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199216949
- eISBN:
- 9780191711992
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199216949.003.0008
- Subject:
- Physics, Crystallography: Physics
This chapter covers the electronic structure of coordination complexes, using the language of group theory throughout. The bonding of these complexes ranges from ionic (crystal field theory, or CFT) ...
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This chapter covers the electronic structure of coordination complexes, using the language of group theory throughout. The bonding of these complexes ranges from ionic (crystal field theory, or CFT) to covalent (molecular orbital theory). In CFT, group theory is employed to derive the terms split from a given atomic electronic state under the influence of a set of ligands. Other topics discussed include Orgel diagrams, Tanabe-Sugano diagrams, correlation of weak and strong field approximations, spin-orbit interaction (double group), etc. Regarding molecular orbital theory, techniques used in Chapters 3 and 7 are applied to coordination complexes, with the metal d-orbitals now playing a central role. These theories are then used to interpret a number of electronic spectra of complexes selected from the literature. The chapter concludes with sections on vibronic interaction in complexes and on the f-orbitals and their crystal field splitting patterns.Less
This chapter covers the electronic structure of coordination complexes, using the language of group theory throughout. The bonding of these complexes ranges from ionic (crystal field theory, or CFT) to covalent (molecular orbital theory). In CFT, group theory is employed to derive the terms split from a given atomic electronic state under the influence of a set of ligands. Other topics discussed include Orgel diagrams, Tanabe-Sugano diagrams, correlation of weak and strong field approximations, spin-orbit interaction (double group), etc. Regarding molecular orbital theory, techniques used in Chapters 3 and 7 are applied to coordination complexes, with the metal d-orbitals now playing a central role. These theories are then used to interpret a number of electronic spectra of complexes selected from the literature. The chapter concludes with sections on vibronic interaction in complexes and on the f-orbitals and their crystal field splitting patterns.
Peter Monk
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.003.0005
- Subject:
- Mathematics, Numerical Analysis
The finite element method is based on a geometric decomposition of the domain of Maxwell’s equations into simple elements. This chapter is devoted to tetrahedral elements, which are very common in ...
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The finite element method is based on a geometric decomposition of the domain of Maxwell’s equations into simple elements. This chapter is devoted to tetrahedral elements, which are very common in practice. Details of the constructions of scalar and vector finite elements of all orders are presented. The vector elements are due to Nedelec. In particular, the curl-conforming elements of this chapter are the widely used ‘edge-elements’ whereas the corresponding divergence-conforming elements are often termed ‘face elements’ (they are extensions to 3D of the Raviart-Thomas elements). The appropriate conforming and unisolvence properties of the elements are proven, and the important discrete de Rham diagram relating the interpolation operators for these finite elements with the divergence, gradient, and curl operators are verified; this is used heavily in later theory. Interpolation error estimates under mesh refinement are derived (h-version of the finite element method). A convenient basis for linear and quadratic finite elements is presented, and spaces of elements on boundaries of the domain are briefly discussed.Less
The finite element method is based on a geometric decomposition of the domain of Maxwell’s equations into simple elements. This chapter is devoted to tetrahedral elements, which are very common in practice. Details of the constructions of scalar and vector finite elements of all orders are presented. The vector elements are due to Nedelec. In particular, the curl-conforming elements of this chapter are the widely used ‘edge-elements’ whereas the corresponding divergence-conforming elements are often termed ‘face elements’ (they are extensions to 3D of the Raviart-Thomas elements). The appropriate conforming and unisolvence properties of the elements are proven, and the important discrete de Rham diagram relating the interpolation operators for these finite elements with the divergence, gradient, and curl operators are verified; this is used heavily in later theory. Interpolation error estimates under mesh refinement are derived (h-version of the finite element method). A convenient basis for linear and quadratic finite elements is presented, and spaces of elements on boundaries of the domain are briefly discussed.
Peter Monk
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.003.0006
- Subject:
- Mathematics, Numerical Analysis
An alternative to the tetrahedral elements discussed in the previous chapter is to use finite elements based on cubes, or more generally, hexahedra. Hexahedral elements have been used in several ...
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An alternative to the tetrahedral elements discussed in the previous chapter is to use finite elements based on cubes, or more generally, hexahedra. Hexahedral elements have been used in several important codes. This chapter concerns Nedelec’s family of edge and face elements on a hexahedral mesh with edges parallel to the coordinate axis. Conformance and unisolvence are proven, and h-error estimates are derived. The appropriate discrete de Rham diagram is shown to hold in this case, and boundary spaces are discussed briefly.Less
An alternative to the tetrahedral elements discussed in the previous chapter is to use finite elements based on cubes, or more generally, hexahedra. Hexahedral elements have been used in several important codes. This chapter concerns Nedelec’s family of edge and face elements on a hexahedral mesh with edges parallel to the coordinate axis. Conformance and unisolvence are proven, and h-error estimates are derived. The appropriate discrete de Rham diagram is shown to hold in this case, and boundary spaces are discussed briefly.
Lucy Donkin and Hanna Vorholt (eds)
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780197265048
- eISBN:
- 9780191754159
- Item type:
- book
- Publisher:
- British Academy
- DOI:
- 10.5871/bacad/9780197265048.001.0001
- Subject:
- History, European Medieval History
Jerusalem was the object of intense study and devotion throughout the Middle Ages. This book illuminates ways in which the city was represented by Christians in Western Europe, from the 600s the ...
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Jerusalem was the object of intense study and devotion throughout the Middle Ages. This book illuminates ways in which the city was represented by Christians in Western Europe, from the 600s the 1500s. Focusing on maps in illuminated manuscripts and early printed books, it also considers views and architectural replicas, and treats depictions of the Temple and the Church of the Holy Sepulchre alongside those of Jerusalem as a whole. The chapters draw on new research and a range of disciplinary perspectives to show how such depictions responded to developments in the West, as well as to the shifting political circumstances of Jerusalem and its wider region. One central theme is the relationship between text, image and manuscript context, including discussion of images as scriptural exegesis and the place of schematic diagrams and plans in the presentation of knowledge. Another is the impact of trends in learning, such as the reception of Jewish scholarship, the move from monastic to university education, and the creation of yet wider audiences through mendicant preaching and the development of printing. The book also examines the role of changing liturgical and devotional practices, including imagined pilgrimage and the mapping of Jerusalem onto European cities and local landscapes. Finally, it seeks to elucidate how two- and three-dimensional representations of the city both resulted from and prompted processes of mental visualization. In this way, the book is conceived as a contribution to manuscript studies, the history of cartography, visual studies and the history of ideas.Less
Jerusalem was the object of intense study and devotion throughout the Middle Ages. This book illuminates ways in which the city was represented by Christians in Western Europe, from the 600s the 1500s. Focusing on maps in illuminated manuscripts and early printed books, it also considers views and architectural replicas, and treats depictions of the Temple and the Church of the Holy Sepulchre alongside those of Jerusalem as a whole. The chapters draw on new research and a range of disciplinary perspectives to show how such depictions responded to developments in the West, as well as to the shifting political circumstances of Jerusalem and its wider region. One central theme is the relationship between text, image and manuscript context, including discussion of images as scriptural exegesis and the place of schematic diagrams and plans in the presentation of knowledge. Another is the impact of trends in learning, such as the reception of Jewish scholarship, the move from monastic to university education, and the creation of yet wider audiences through mendicant preaching and the development of printing. The book also examines the role of changing liturgical and devotional practices, including imagined pilgrimage and the mapping of Jerusalem onto European cities and local landscapes. Finally, it seeks to elucidate how two- and three-dimensional representations of the city both resulted from and prompted processes of mental visualization. In this way, the book is conceived as a contribution to manuscript studies, the history of cartography, visual studies and the history of ideas.
A. Townsend Peterson, Jorge Soberón, Richard G. Pearson, Robert P. Anderson, Enrique Martínez-Meyer, Miguel Nakamura, and Miguel Bastos Araújo
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691136868
- eISBN:
- 9781400840670
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691136868.003.0003
- Subject:
- Biology, Ecology
This chapter considers a concept of niche that emphasizes multidimensional spaces of scenopoetic variables and provides a natural connection to the study of geographic distributions of species. It ...
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This chapter considers a concept of niche that emphasizes multidimensional spaces of scenopoetic variables and provides a natural connection to the study of geographic distributions of species. It first explains the relations between environmental and geographic spaces before discussing the use of equations to link spatially explicit population growth patterns to variation in the ecological characteristics of species. It then describes the BAM diagram, a Venn diagram that displays the joint fulfillment in geographic space of three sets of conditions that together determine species distribution: biotic conditions, abiotic conditions, and movement of the species. The chapter also explores the spatial resolution of scenopoetic variables, estimation of the fundamental and existing fundamental niches, the biotically reduced niche, and caveats about reducing Grinnellian niches and the Eltonian Noise Hypothesis. Finally, it shows how distributional areas and ecological niches can be estimated.Less
This chapter considers a concept of niche that emphasizes multidimensional spaces of scenopoetic variables and provides a natural connection to the study of geographic distributions of species. It first explains the relations between environmental and geographic spaces before discussing the use of equations to link spatially explicit population growth patterns to variation in the ecological characteristics of species. It then describes the BAM diagram, a Venn diagram that displays the joint fulfillment in geographic space of three sets of conditions that together determine species distribution: biotic conditions, abiotic conditions, and movement of the species. The chapter also explores the spatial resolution of scenopoetic variables, estimation of the fundamental and existing fundamental niches, the biotically reduced niche, and caveats about reducing Grinnellian niches and the Eltonian Noise Hypothesis. Finally, it shows how distributional areas and ecological niches can be estimated.
