Helmut Hofmann
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198504016
- eISBN:
- 9780191708480
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504016.003.0026
- Subject:
- Physics, Nuclear and Plasma Physics
This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, ...
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This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.Less
This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.
Mary Orr
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780199258581
- eISBN:
- 9780191718083
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199258581.003.0007
- Subject:
- Literature, World Literature, 19th-century and Victorian Literature
Devil's rides into space are a well‐worked topos in literature, but this chapter points out for the first time their literal realities in the Montgolfier balloons and Garnarin's parachute that ...
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Devil's rides into space are a well‐worked topos in literature, but this chapter points out for the first time their literal realities in the Montgolfier balloons and Garnarin's parachute that constitute the 19th‐century ‘transports’ of Antoine's literary‐scientific imagination. The chapter then offers further appraisal of what the Devil ‘shows’ Antoine in space, namely (1) the (19th‐century) heliocentric solar system with the new planets, Uranus and Neptune discovered through understanding of gravitational pull, and (2) the huge literary‐scientific joke behind the Devil's transformations as the Norman mathematician Laplace's famous ‘demon’. The chapter ends by rethinking the genesis of the Tentation through the modern mystères of Le Poittevin's Bélial and Byron's Cain as among Flaubert's personal demons.Less
Devil's rides into space are a well‐worked topos in literature, but this chapter points out for the first time their literal realities in the Montgolfier balloons and Garnarin's parachute that constitute the 19th‐century ‘transports’ of Antoine's literary‐scientific imagination. The chapter then offers further appraisal of what the Devil ‘shows’ Antoine in space, namely (1) the (19th‐century) heliocentric solar system with the new planets, Uranus and Neptune discovered through understanding of gravitational pull, and (2) the huge literary‐scientific joke behind the Devil's transformations as the Norman mathematician Laplace's famous ‘demon’. The chapter ends by rethinking the genesis of the Tentation through the modern mystères of Le Poittevin's Bélial and Byron's Cain as among Flaubert's personal demons.
Martin Schöneld
- Published in print:
- 2000
- Published Online:
- May 2006
- ISBN:
- 9780195132182
- eISBN:
- 9780199786336
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195132181.003.0006
- Subject:
- Philosophy, History of Philosophy
This chapter explores Kant’s second book, Universal Natural History and Theory of the Heavens (1755). Section 1 describes the context of the book and Kant’s critique of static and anthropocentric ...
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This chapter explores Kant’s second book, Universal Natural History and Theory of the Heavens (1755). Section 1 describes the context of the book and Kant’s critique of static and anthropocentric conceptions of nature by the Pietists, Physico-Theologians, Newton, and Wolff. Section 2 describes the goal of Kant’s teleology, its naturalized thrust toward well-ordered complexity or “relative perfection.” Section 3 examines the means of Kant”s teleology, the dynamic interplay of attractive and repulsive forces. Section 4 analyzes the application of teleology to cosmic phenomena such as the solar system, Wright’s earlier stipulation, Laplace’s later conjecture, and the eventual confirmation of Kant’s nebular hypothesis. Section 5 explores Kant’s arguments for life, humanity, and reason as products of cosmic evolution. Section 6 discusses Kant’s “static law” — that the mean planetary density determines the biospherical potential of reason — and its incongruity with the racism in Physical Geography (1756-60) and Beautiful and Sublime (1764). Section 7 describes Kant’s dynamic cosmology, explicates his “phoenix”-symbol, and discusses his various scientific aperçus.Less
This chapter explores Kant’s second book, Universal Natural History and Theory of the Heavens (1755). Section 1 describes the context of the book and Kant’s critique of static and anthropocentric conceptions of nature by the Pietists, Physico-Theologians, Newton, and Wolff. Section 2 describes the goal of Kant’s teleology, its naturalized thrust toward well-ordered complexity or “relative perfection.” Section 3 examines the means of Kant”s teleology, the dynamic interplay of attractive and repulsive forces. Section 4 analyzes the application of teleology to cosmic phenomena such as the solar system, Wright’s earlier stipulation, Laplace’s later conjecture, and the eventual confirmation of Kant’s nebular hypothesis. Section 5 explores Kant’s arguments for life, humanity, and reason as products of cosmic evolution. Section 6 discusses Kant’s “static law” — that the mean planetary density determines the biospherical potential of reason — and its incongruity with the racism in Physical Geography (1756-60) and Beautiful and Sublime (1764). Section 7 describes Kant’s dynamic cosmology, explicates his “phoenix”-symbol, and discusses his various scientific aperçus.
Paul Baird and John C. Wood
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.001.0001
- Subject:
- Mathematics, Pure Mathematics
Harmonic morphisms are maps which preserve Laplace's equation. More explicitly, a map between Riemannian manifolds is called a harmonic morphism if its composition with any locally defined harmonic ...
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Harmonic morphisms are maps which preserve Laplace's equation. More explicitly, a map between Riemannian manifolds is called a harmonic morphism if its composition with any locally defined harmonic function on the codomain is a harmonic function on the domain; it thus ‘pulls back’ germs of harmonic functions to germs of harmonic functions. Harmonic morphisms can be characterized as harmonic maps satisfying a condition dual to weak conformality called ‘horizontal weak conformality’ or ‘semiconformality’. Examples include harmonic functions, conformal mappings in the plane, holomorphic mappings with values in a Riemann surface, and certain submersions arising from Killing fields and geodesic fields. The study of harmonic morphisms involves many different branches of mathematics: the book includes discussion on aspects of the theory of foliations, polynomials induced by Clifford systems and orthogonal multiplications, twistor and mini-twistor spaces, and Hermitian structures. Relations with topology are discussed, including Seifert fibre spaces and circle actions, also relations with isoparametric functions and the Beltrami fields equation of hydrodynamics.Less
Harmonic morphisms are maps which preserve Laplace's equation. More explicitly, a map between Riemannian manifolds is called a harmonic morphism if its composition with any locally defined harmonic function on the codomain is a harmonic function on the domain; it thus ‘pulls back’ germs of harmonic functions to germs of harmonic functions. Harmonic morphisms can be characterized as harmonic maps satisfying a condition dual to weak conformality called ‘horizontal weak conformality’ or ‘semiconformality’. Examples include harmonic functions, conformal mappings in the plane, holomorphic mappings with values in a Riemann surface, and certain submersions arising from Killing fields and geodesic fields. The study of harmonic morphisms involves many different branches of mathematics: the book includes discussion on aspects of the theory of foliations, polynomials induced by Clifford systems and orthogonal multiplications, twistor and mini-twistor spaces, and Hermitian structures. Relations with topology are discussed, including Seifert fibre spaces and circle actions, also relations with isoparametric functions and the Beltrami fields equation of hydrodynamics.
Leon Ehrenpreis
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198509783
- eISBN:
- 9780191709166
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509783.003.0003
- Subject:
- Mathematics, Mathematical Physics
This chapter looks at a variation of “harmonic function” which was introduced by Chevalley. Ordinary harmonic functions are solutions of the Laplace equation; Chevalley harmonic functions are ...
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This chapter looks at a variation of “harmonic function” which was introduced by Chevalley. Ordinary harmonic functions are solutions of the Laplace equation; Chevalley harmonic functions are solutions of certain systems of partial differential equations. The equations and the harmonic functions combine to give a tensor product decomposition of the space of functions. This theory is indispensable for extending the concept of Radon transform to algebraic varieties. The geometry related to harmonic functions is discussed in detail.Less
This chapter looks at a variation of “harmonic function” which was introduced by Chevalley. Ordinary harmonic functions are solutions of the Laplace equation; Chevalley harmonic functions are solutions of certain systems of partial differential equations. The equations and the harmonic functions combine to give a tensor product decomposition of the space of functions. This theory is indispensable for extending the concept of Radon transform to algebraic varieties. The geometry related to harmonic functions is discussed in detail.
Jan Modersitzki
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528418
- eISBN:
- 9780191713583
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528418.003.0011
- Subject:
- Mathematics, Applied Mathematics
A gradient-based regularization for registration is introduced, and a fast and stable implementation is developed. In contrast to the physically motivated elastic and fluid registrations, the ...
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A gradient-based regularization for registration is introduced, and a fast and stable implementation is developed. In contrast to the physically motivated elastic and fluid registrations, the diffusion regularizer is motivated by smoothing properties of the displacement. Another important motivation is that a registration step can be performed in linear complexity of the number of given data. The main tool is the so-called additive operator splitting scheme (AOS). The idea is to split the original problem into a number of simpler problems which allow for a fast numerical solution. A new proof for the accuracy of AOS is given, which is based purely on matrix analysis. Thus, the result also applies to more general situations. Thirion's demons registration is discussed.Less
A gradient-based regularization for registration is introduced, and a fast and stable implementation is developed. In contrast to the physically motivated elastic and fluid registrations, the diffusion regularizer is motivated by smoothing properties of the displacement. Another important motivation is that a registration step can be performed in linear complexity of the number of given data. The main tool is the so-called additive operator splitting scheme (AOS). The idea is to split the original problem into a number of simpler problems which allow for a fast numerical solution. A new proof for the accuracy of AOS is given, which is based purely on matrix analysis. Thus, the result also applies to more general situations. Thirion's demons registration is discussed.
Jon Williamson
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198530794
- eISBN:
- 9780191712982
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198530794.003.0005
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
Objective Bayesianism imposes two norms on degrees of belief: degrees of belief should be constrained by empirical information and they should otherwise be as equivocal as possible. The origins of ...
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Objective Bayesianism imposes two norms on degrees of belief: degrees of belief should be constrained by empirical information and they should otherwise be as equivocal as possible. The origins of objective Bayesianism are explained, with the work of Jakob Bernoulli and Laplace presented at some length. A contemporary reading of the two norms is developed in detail. Then, it is shown how Bayesian nets can be used to represent objective Bayesian degrees of belief (this leads to what are now called objective Bayesian nets).Less
Objective Bayesianism imposes two norms on degrees of belief: degrees of belief should be constrained by empirical information and they should otherwise be as equivocal as possible. The origins of objective Bayesianism are explained, with the work of Jakob Bernoulli and Laplace presented at some length. A contemporary reading of the two norms is developed in detail. Then, it is shown how Bayesian nets can be used to represent objective Bayesian degrees of belief (this leads to what are now called objective Bayesian nets).
Shoutir Kishore Chatterjee
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198525318
- eISBN:
- 9780191711657
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525318.003.0007
- Subject:
- Mathematics, Probability / Statistics
Around the middle of the 18th century, Bayes conceived the idea of treating an unknown parameter as a subjective random variable distributed according to a prior, and inferring about it from its ...
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Around the middle of the 18th century, Bayes conceived the idea of treating an unknown parameter as a subjective random variable distributed according to a prior, and inferring about it from its conditional (posterior) distribution given the observations. He considered the particular case of a binomial parameter subject to a uniform prior, and following the pro-subjective approach used the posterior to derive an interval estimate. Later, Laplace stated the result in its general form and employed it extensively for pro-subjective inference of various types in different situations, often basing his computation on the asymptotic normality of the posterior distribution. In a novel application, Laplace used pro-subjective reasoning and the data from a sample survey to estimate the size of the population of France.Less
Around the middle of the 18th century, Bayes conceived the idea of treating an unknown parameter as a subjective random variable distributed according to a prior, and inferring about it from its conditional (posterior) distribution given the observations. He considered the particular case of a binomial parameter subject to a uniform prior, and following the pro-subjective approach used the posterior to derive an interval estimate. Later, Laplace stated the result in its general form and employed it extensively for pro-subjective inference of various types in different situations, often basing his computation on the asymptotic normality of the posterior distribution. In a novel application, Laplace used pro-subjective reasoning and the data from a sample survey to estimate the size of the population of France.
Shoutir Kishore Chatterjee
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198525318
- eISBN:
- 9780191711657
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525318.003.0008
- Subject:
- Mathematics, Probability / Statistics
In the first half of the 19th century, Laplace, Gauss, and a number of other contributors enriched statistical thought. The quest for a suitable model for the distribution of observational errors led ...
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In the first half of the 19th century, Laplace, Gauss, and a number of other contributors enriched statistical thought. The quest for a suitable model for the distribution of observational errors led to Gauss’s derivation of the normal model from the ‘A. M. postulate’. Laplace’s derivation of the Central Limit Theorem gave further support to the model. Different methods of curve fitting, of which the Least Squares method — which was heuristically proposed by Legendre and to which Gauss provided first a Bayesian and then a sampling theory justification — was the most important. During this period, Laplace worked on the large sample sampling theory approach to inference, and both he and Gauss introduced the idea of relative efficiency of estimates in the context of particular problems. In fact, the seeds of some later concepts like that of sufficiency, variance component models, and diffusion processes can be found in works carried out at this time.Less
In the first half of the 19th century, Laplace, Gauss, and a number of other contributors enriched statistical thought. The quest for a suitable model for the distribution of observational errors led to Gauss’s derivation of the normal model from the ‘A. M. postulate’. Laplace’s derivation of the Central Limit Theorem gave further support to the model. Different methods of curve fitting, of which the Least Squares method — which was heuristically proposed by Legendre and to which Gauss provided first a Bayesian and then a sampling theory justification — was the most important. During this period, Laplace worked on the large sample sampling theory approach to inference, and both he and Gauss introduced the idea of relative efficiency of estimates in the context of particular problems. In fact, the seeds of some later concepts like that of sufficiency, variance component models, and diffusion processes can be found in works carried out at this time.
Athol Fitzgibbons
- Published in print:
- 1990
- Published Online:
- November 2003
- ISBN:
- 9780198283201
- eISBN:
- 9780191596254
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198283202.003.0002
- Subject:
- Economics and Finance, History of Economic Thought
Introduces Keynes's philosophy of probability, including his critiques of David Hume and Laplace, and the reason for his emphasis on non‐quantitative probabilities.
Introduces Keynes's philosophy of probability, including his critiques of David Hume and Laplace, and the reason for his emphasis on non‐quantitative probabilities.
William L. Harper
- Published in print:
- 2011
- Published Online:
- May 2012
- ISBN:
- 9780199570409
- eISBN:
- 9780191728679
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199570409.003.0001
- Subject:
- Philosophy, History of Philosophy, Philosophy of Science
This chapter includes some historical background on Astronomy, mechanics and physical causes, as well as an overview of Newton’s background framework and his theoretical concept of a centripetal ...
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This chapter includes some historical background on Astronomy, mechanics and physical causes, as well as an overview of Newton’s background framework and his theoretical concept of a centripetal force. It introduces Newton’s Rules for reasoning in natural philosophy and gives an overview of Newton’s argument for universal gravity and its application to the solar system. A comparison with a passage from Huygens on hypothetico-deductive confirmation helps inform Newton’s classic hypotheses non-fingo passage. Lessons on scientific method also include an informative contrast with Laplace’s search for solar system stability, and a contrast between theory acceptance guided by empirical success and mere assignment of high probability.Less
This chapter includes some historical background on Astronomy, mechanics and physical causes, as well as an overview of Newton’s background framework and his theoretical concept of a centripetal force. It introduces Newton’s Rules for reasoning in natural philosophy and gives an overview of Newton’s argument for universal gravity and its application to the solar system. A comparison with a passage from Huygens on hypothetico-deductive confirmation helps inform Newton’s classic hypotheses non-fingo passage. Lessons on scientific method also include an informative contrast with Laplace’s search for solar system stability, and a contrast between theory acceptance guided by empirical success and mere assignment of high probability.
William L. Harper
- Published in print:
- 2011
- Published Online:
- May 2012
- ISBN:
- 9780199570409
- eISBN:
- 9780191728679
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199570409.003.0010
- Subject:
- Philosophy, History of Philosophy, Philosophy of Science
Part I. Distinctive features of Newton’s method: Successively more accurate approximations and increasing empirical support from measurements. Part II. The Mercury perihelion problem: A proposal to ...
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Part I. Distinctive features of Newton’s method: Successively more accurate approximations and increasing empirical support from measurements. Part II. The Mercury perihelion problem: A proposal to alter the inverse-square law ruled out by a more precise measurement. Einstein’s theory accounts for the extra precession and recovers the successful measurements of Newton’s theory. An alternative to general relativity that would answer a new challenge from Mercury is ruled out by a more precise measurement. Part III. Newton does not require or endorse scientific progress as progress toward Laplace’s ideal limit of a final theory. Part IV. Newton’s conception of scientific progress through successively more accurate approximations is not undermined by the classic argument against convergent realism. Part V: Agreeing measurements from diverse phenomena play a decisive role of in transforming dark energy from a dubious hypothesis into part of the accepted background framework guiding empirical research in cosmology today.Less
Part I. Distinctive features of Newton’s method: Successively more accurate approximations and increasing empirical support from measurements. Part II. The Mercury perihelion problem: A proposal to alter the inverse-square law ruled out by a more precise measurement. Einstein’s theory accounts for the extra precession and recovers the successful measurements of Newton’s theory. An alternative to general relativity that would answer a new challenge from Mercury is ruled out by a more precise measurement. Part III. Newton does not require or endorse scientific progress as progress toward Laplace’s ideal limit of a final theory. Part IV. Newton’s conception of scientific progress through successively more accurate approximations is not undermined by the classic argument against convergent realism. Part V: Agreeing measurements from diverse phenomena play a decisive role of in transforming dark energy from a dubious hypothesis into part of the accepted background framework guiding empirical research in cosmology today.
E. Brian Davies
- Published in print:
- 2007
- Published Online:
- September 2008
- ISBN:
- 9780199219186
- eISBN:
- 9780191711695
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219186.003.0006
- Subject:
- Physics, History of Physics
This chapter considers two related topics. The first is the development of astronomy in the 16th and 17th centuries, culminating in Newton's publication of his laws of motion in 1687; the second ...
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This chapter considers two related topics. The first is the development of astronomy in the 16th and 17th centuries, culminating in Newton's publication of his laws of motion in 1687; the second concerns the subsequent history of these laws. Observations confirmed the predictions of Newton's theory, and after about 1750 nobody had any doubt that his theory of gravitation provided a true description of the world. However, in the first decades of the 20th century, it was discovered that this certainty was a chimera. Einstein dethroned Newton, and physics moved into a period of flux which has continued ever since. The fact that such a well-established theory could eventually be superseded poses a severe challenge to any theory of scientific knowledge. The chapter recounts the story of the period, selecting the aspects which are most relevant to this matter.Less
This chapter considers two related topics. The first is the development of astronomy in the 16th and 17th centuries, culminating in Newton's publication of his laws of motion in 1687; the second concerns the subsequent history of these laws. Observations confirmed the predictions of Newton's theory, and after about 1750 nobody had any doubt that his theory of gravitation provided a true description of the world. However, in the first decades of the 20th century, it was discovered that this certainty was a chimera. Einstein dethroned Newton, and physics moved into a period of flux which has continued ever since. The fact that such a well-established theory could eventually be superseded poses a severe challenge to any theory of scientific knowledge. The chapter recounts the story of the period, selecting the aspects which are most relevant to this matter.
Kyösti Kontturi, Lasse Murtomäki, and José A. Manzanares
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199533817
- eISBN:
- 9780191714825
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199533817.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter is concerned with transport in the vicinity of electrodes, and hence on the coupling between Faradaic electrode processes and mass transport. It covers transport in stationary and ...
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This chapter is concerned with transport in the vicinity of electrodes, and hence on the coupling between Faradaic electrode processes and mass transport. It covers transport in stationary and transient condition, planar and spherical geometries, presence and absence of supporting electrolytes, as well as convective transport in hydrodynamic electrodes. Some common electrochemical techniques are also discussed, and the solutions of the corresponding transient transport problems are worked out in detail.Less
This chapter is concerned with transport in the vicinity of electrodes, and hence on the coupling between Faradaic electrode processes and mass transport. It covers transport in stationary and transient condition, planar and spherical geometries, presence and absence of supporting electrolytes, as well as convective transport in hydrodynamic electrodes. Some common electrochemical techniques are also discussed, and the solutions of the corresponding transient transport problems are worked out in detail.
Paul C. Gutjahr
- Published in print:
- 2011
- Published Online:
- May 2011
- ISBN:
- 9780199740420
- eISBN:
- 9780199894703
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199740420.003.0056
- Subject:
- Religion, Church History
Chapter fifty-six takes a close look at Hodge’s lifelong commitment and interest in science. As many as twenty percent of Repertory articles concerned themselves with science, and Hodge remained ...
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Chapter fifty-six takes a close look at Hodge’s lifelong commitment and interest in science. As many as twenty percent of Repertory articles concerned themselves with science, and Hodge remained committed throughout his life to seeing a close connection between scientific and religious inquiry. His last book, What is Darwinism?, functioned largely as a defense of the complementary nature of science and religion. Hodge believed Darwinism to be atheistic because it posited a theory of world development that had no place for a Divine Designer. Darwin’s theories implied a randomness that had no place in Hodge’s views of Divine Soveriegnty.Less
Chapter fifty-six takes a close look at Hodge’s lifelong commitment and interest in science. As many as twenty percent of Repertory articles concerned themselves with science, and Hodge remained committed throughout his life to seeing a close connection between scientific and religious inquiry. His last book, What is Darwinism?, functioned largely as a defense of the complementary nature of science and religion. Hodge believed Darwinism to be atheistic because it posited a theory of world development that had no place for a Divine Designer. Darwin’s theories implied a randomness that had no place in Hodge’s views of Divine Soveriegnty.
Christopher D. Sogge
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160757
- eISBN:
- 9781400850549
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160757.001.0001
- Subject:
- Mathematics, Numerical Analysis
Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl ...
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Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace–Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. The book shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. It begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. The book avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. It also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, the book demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.Less
Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace–Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. The book shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. It begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. The book avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. It also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, the book demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.
C. A. J. Coady
- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198235514
- eISBN:
- 9780191597220
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198235518.003.0011
- Subject:
- Philosophy, Metaphysics/Epistemology
The puzzle addressed in this chapter is most explicitly raised by Laplace, though foreshadowed by Locke, and is concerned with the supposed diminution in probative force of testimony that has passed ...
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The puzzle addressed in this chapter is most explicitly raised by Laplace, though foreshadowed by Locke, and is concerned with the supposed diminution in probative force of testimony that has passed through too many hands (or tongues). Laplace suggests that lengthy transmission chains must ‘enfeeble’ the probability of historical facts, but this disappearance of history thesis (DOHT) has some very counter‐intuitive consequences. It argues against various forms of the DOHT by exploring the diverse types of testimony chains available in written and oral traditions.Less
The puzzle addressed in this chapter is most explicitly raised by Laplace, though foreshadowed by Locke, and is concerned with the supposed diminution in probative force of testimony that has passed through too many hands (or tongues). Laplace suggests that lengthy transmission chains must ‘enfeeble’ the probability of historical facts, but this disappearance of history thesis (DOHT) has some very counter‐intuitive consequences. It argues against various forms of the DOHT by exploring the diverse types of testimony chains available in written and oral traditions.
Charles L. Epstein and Rafe1 Mazzeo
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0008
- Subject:
- Mathematics, Probability / Statistics
This chapter presents the Hölder space estimates for Euclidean model problems. It first considers the homogeneous Cauchy problem and the inhomogeneous problem before defining the resolvent operator ...
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This chapter presents the Hölder space estimates for Euclidean model problems. It first considers the homogeneous Cauchy problem and the inhomogeneous problem before defining the resolvent operator as the Laplace transform of the heat kernel. It then describes the 1-dimensional kernel estimates that form essential components of the proofs of the Hölder estimates for the general model problems; these include basic kernel estimates, first derivative estimates, and second derivative estimates. The proofs of these estimates are elementary. The chapter concludes by proving estimates on the resolvent and investigating the off-diagonal behavior of the heat kernel in many variables.Less
This chapter presents the Hölder space estimates for Euclidean model problems. It first considers the homogeneous Cauchy problem and the inhomogeneous problem before defining the resolvent operator as the Laplace transform of the heat kernel. It then describes the 1-dimensional kernel estimates that form essential components of the proofs of the Hölder estimates for the general model problems; these include basic kernel estimates, first derivative estimates, and second derivative estimates. The proofs of these estimates are elementary. The chapter concludes by proving estimates on the resolvent and investigating the off-diagonal behavior of the heat kernel in many variables.
Robert Alicki and Mark Fannes
- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter presents the basic mathematical facts concerning Schrödinger's equation with constant and time-dependent Hamiltonians: Stone's theorem on strongly continuous one-parameter groups of ...
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This chapter presents the basic mathematical facts concerning Schrödinger's equation with constant and time-dependent Hamiltonians: Stone's theorem on strongly continuous one-parameter groups of unitaries, the Kato–Rellich criterion for self-adjointness, and the Dyson expansion. In particular, Floquet's theory for periodic perturbations is outlined and illustrated by examples of kicked systems: the kicked top and the baker map. Then classical mechanics is introduced as a limit of quantum theory using coherent states and mean-field limits. The formalism of classical differentiable dynamics is briefly described and the classical and quantum aspects of the motion of a free particle on a compact Riemannian manifold are discussed including Weyl's theorem characterizing spectra of generalized Laplacians such as Beltrami–Laplace operators.Less
This chapter presents the basic mathematical facts concerning Schrödinger's equation with constant and time-dependent Hamiltonians: Stone's theorem on strongly continuous one-parameter groups of unitaries, the Kato–Rellich criterion for self-adjointness, and the Dyson expansion. In particular, Floquet's theory for periodic perturbations is outlined and illustrated by examples of kicked systems: the kicked top and the baker map. Then classical mechanics is introduced as a limit of quantum theory using coherent states and mean-field limits. The formalism of classical differentiable dynamics is briefly described and the classical and quantum aspects of the motion of a free particle on a compact Riemannian manifold are discussed including Weyl's theorem characterizing spectra of generalized Laplacians such as Beltrami–Laplace operators.
Bernhard Blümich
- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198526766
- eISBN:
- 9780191709524
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526766.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials
Transformation, convolution, and correlation are used over and over again in nuclear magnetic resonance (NMR) spectroscopy and imaging in different contexts and sometimes with different meanings. The ...
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Transformation, convolution, and correlation are used over and over again in nuclear magnetic resonance (NMR) spectroscopy and imaging in different contexts and sometimes with different meanings. The transformation best known in NMR is the Fourier transformation in one or more dimensions. It is used to generate one- and multi-dimensional spectra from experimental data as well as ID, 2D, and 3D images. Furthermore, different types of multi-dimensional spectra are explicitly called correlation spectra. These are related to nonlinear correlation functions of excitation and response. This chapter discusses convolution in linear and nonlinear systems, along with the convolution theorem, linear system analysis, nonlinear cross-correlation, correlation theorem, Laplace transformation, Hankel transformation, Abel transformation, z transformation, Hadamard transformation, and wavelet transformation.Less
Transformation, convolution, and correlation are used over and over again in nuclear magnetic resonance (NMR) spectroscopy and imaging in different contexts and sometimes with different meanings. The transformation best known in NMR is the Fourier transformation in one or more dimensions. It is used to generate one- and multi-dimensional spectra from experimental data as well as ID, 2D, and 3D images. Furthermore, different types of multi-dimensional spectra are explicitly called correlation spectra. These are related to nonlinear correlation functions of excitation and response. This chapter discusses convolution in linear and nonlinear systems, along with the convolution theorem, linear system analysis, nonlinear cross-correlation, correlation theorem, Laplace transformation, Hankel transformation, Abel transformation, z transformation, Hadamard transformation, and wavelet transformation.
