Robert James Matthys
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198529712
- eISBN:
- 9780191712791
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198529712.003.0004
- Subject:
- Physics, History of Physics
To determine a pendulum's axis of rotation, a small piece of paper is temporarily mounted on the front of the pendulum rod at the rod's top end, so that the paper extends up an inch or two past the ...
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To determine a pendulum's axis of rotation, a small piece of paper is temporarily mounted on the front of the pendulum rod at the rod's top end, so that the paper extends up an inch or two past the suspension spring. With the pendulum stopped, two small dots (A and B) are marked on the paper about an inch directly above and another inch directly below the top end of the free unclamped part of the suspension spring. Although the one-inch dimensions are not critical, they accurately measure the actual distance between the two dots (A and B). The pendulum must be set swinging at its normal swing amplitude, and an accurate ruler (a six-inch machinist's scale calibrated in decimal inches is ideal) is used to measure the horizontal motion of each dot. The location of the axis of rotation changes slightly with the pendulum's swing amplitude.Less
To determine a pendulum's axis of rotation, a small piece of paper is temporarily mounted on the front of the pendulum rod at the rod's top end, so that the paper extends up an inch or two past the suspension spring. With the pendulum stopped, two small dots (A and B) are marked on the paper about an inch directly above and another inch directly below the top end of the free unclamped part of the suspension spring. Although the one-inch dimensions are not critical, they accurately measure the actual distance between the two dots (A and B). The pendulum must be set swinging at its normal swing amplitude, and an accurate ruler (a six-inch machinist's scale calibrated in decimal inches is ideal) is used to measure the horizontal motion of each dot. The location of the axis of rotation changes slightly with the pendulum's swing amplitude.
Robert James Matthys
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198529712
- eISBN:
- 9780191712791
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198529712.003.0005
- Subject:
- Physics, History of Physics
This chapter describes an experiment to determine whether a pendulum's axis of rotation moves when the swing amplitude is increased. This issue presupposes that the pendulum has a flat spring type of ...
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This chapter describes an experiment to determine whether a pendulum's axis of rotation moves when the swing amplitude is increased. This issue presupposes that the pendulum has a flat spring type of suspension. A flat suspension spring bends all along a section of its length, so there is no obvious reason to indicate that a pendulum's axis of rotation would remain fixed when the swing amplitude is increased. The axis of rotation is found by measuring the horizontal travel of two points on the pendulum rod, one point a little above the axis of rotation and the other a little below it. The axis of rotation does move downward slightly (down the suspension spring), as the pendulum's swing amplitude increases. This should make the pendulum speed up as the amplitude increases, but in reality a pendulum actually slows down with increasing amplitude.Less
This chapter describes an experiment to determine whether a pendulum's axis of rotation moves when the swing amplitude is increased. This issue presupposes that the pendulum has a flat spring type of suspension. A flat suspension spring bends all along a section of its length, so there is no obvious reason to indicate that a pendulum's axis of rotation would remain fixed when the swing amplitude is increased. The axis of rotation is found by measuring the horizontal travel of two points on the pendulum rod, one point a little above the axis of rotation and the other a little below it. The axis of rotation does move downward slightly (down the suspension spring), as the pendulum's swing amplitude increases. This should make the pendulum speed up as the amplitude increases, but in reality a pendulum actually slows down with increasing amplitude.
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.
Mark Tatham and Katherine Morton
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780199250677
- eISBN:
- 9780191719462
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199250677.003.0012
- Subject:
- Linguistics, Phonetics / Phonology
This chapter describes a variety of current synthesis systems, including formant and concatenative systems, and the problems associated with unit selection. The success of these systems is discussed ...
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This chapter describes a variety of current synthesis systems, including formant and concatenative systems, and the problems associated with unit selection. The success of these systems is discussed in terms of the standard parameterizable features of speech: acoustic formants, amplitude, timing and rate, fundamental frequency, etc. The cognitive constructs to which the physical parameters are related are described: voice quality, loudness, rhythm, duration, and pitch. Variability in speech, the obverse process, and automatic speech recognition are also discussed.Less
This chapter describes a variety of current synthesis systems, including formant and concatenative systems, and the problems associated with unit selection. The success of these systems is discussed in terms of the standard parameterizable features of speech: acoustic formants, amplitude, timing and rate, fundamental frequency, etc. The cognitive constructs to which the physical parameters are related are described: voice quality, loudness, rhythm, duration, and pitch. Variability in speech, the obverse process, and automatic speech recognition are also discussed.
Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this ...
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This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this chapter include potential scattering in infinite one-dimensional space; Lippmann-Schwinger equation; free Green function; reflection and transmission amplitudes; transfer matrix; T matrix; S matrix and its analytic structure; phase shifts and resonances from the analytic structure of S matrix in complex momentum and complex energy planes; parametrization of the matrices; combination of the S matrices for two scatterers in series; and invariant-imbedding approach for a one-dimensional disordered conductor.Less
This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this chapter include potential scattering in infinite one-dimensional space; Lippmann-Schwinger equation; free Green function; reflection and transmission amplitudes; transfer matrix; T matrix; S matrix and its analytic structure; phase shifts and resonances from the analytic structure of S matrix in complex momentum and complex energy planes; parametrization of the matrices; combination of the S matrices for two scatterers in series; and invariant-imbedding approach for a one-dimensional disordered conductor.
Russell L. De Valois and Karen K. De Valois
- Published in print:
- 1991
- Published Online:
- January 2008
- ISBN:
- 9780195066579
- eISBN:
- 9780199872220
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195066579.003.0011
- Subject:
- Psychology, Cognitive Neuroscience
This chapter discusses the nonlinear responses of the visual system. Topics covered include threshold nonlinearity, phase nonlinearity, amplitude nonlinearity, half-wave rectification, and ...
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This chapter discusses the nonlinear responses of the visual system. Topics covered include threshold nonlinearity, phase nonlinearity, amplitude nonlinearity, half-wave rectification, and interactions between different spatial frequencies.Less
This chapter discusses the nonlinear responses of the visual system. Topics covered include threshold nonlinearity, phase nonlinearity, amplitude nonlinearity, half-wave rectification, and interactions between different spatial frequencies.
R. C. O. Matthews, C. H. Feinstein, and J. C. Odling‐Smee
- Published in print:
- 1982
- Published Online:
- November 2003
- ISBN:
- 9780198284536
- eISBN:
- 9780191596629
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198284535.003.0010
- Subject:
- Economics and Finance, Economic History
Intermittent phases of high demand may affect the way demand keeps up with supply. The pressure of demand was obviously higher in 1951–73 than before 1914 and a fortiori higher than in the interwar ...
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Intermittent phases of high demand may affect the way demand keeps up with supply. The pressure of demand was obviously higher in 1951–73 than before 1914 and a fortiori higher than in the interwar period. Differences in cycle‐amplitude partly accounted for the differences between periods in the average pressure of demand. A Keynesian decomposition of the sources of demand pressure indicates that foreign trade was the main reason whythere was more excess supply in the interwar period than earlier. But investment, not the direct injection of demand by government, was what caused the high level of demand after World War II.Less
Intermittent phases of high demand may affect the way demand keeps up with supply. The pressure of demand was obviously higher in 1951–73 than before 1914 and a fortiori higher than in the interwar period. Differences in cycle‐amplitude partly accounted for the differences between periods in the average pressure of demand. A Keynesian decomposition of the sources of demand pressure indicates that foreign trade was the main reason whythere was more excess supply in the interwar period than earlier. But investment, not the direct injection of demand by government, was what caused the high level of demand after World War II.
J. C. Garrison and R. Y. Chiao
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198508861
- eISBN:
- 9780191708640
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508861.003.0007
- Subject:
- Physics, Atomic, Laser, and Optical Physics
A state of two distinguishable particles is “separable” if the wavefunction is a product of single-particle wavefunctions. Schrödinger introduced the term “entangled” to describe a two-particle state ...
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A state of two distinguishable particles is “separable” if the wavefunction is a product of single-particle wavefunctions. Schrödinger introduced the term “entangled” to describe a two-particle state that is not separable. This notion is extended to all pairs of distinguishable quantum systems by using tensor products of Hilbert spaces and the Schmidt decomposition. Equivalent operational definitions are expressed in terms of correlation functions representing measurements. A state of two indistinguishable particles is kinematically separable if the product function satisfies Bose or Fermi statistics, otherwise it is kinematically entangled. Alternatively, a two-particle state is dynamically separable if the wavefunction has the minimal form required by Bose or Fermi statistics, and dynamically entangled otherwise. For photons, the role of the missing position-space wave function is played by a detection amplitude directly related to counting rates.Less
A state of two distinguishable particles is “separable” if the wavefunction is a product of single-particle wavefunctions. Schrödinger introduced the term “entangled” to describe a two-particle state that is not separable. This notion is extended to all pairs of distinguishable quantum systems by using tensor products of Hilbert spaces and the Schmidt decomposition. Equivalent operational definitions are expressed in terms of correlation functions representing measurements. A state of two indistinguishable particles is kinematically separable if the product function satisfies Bose or Fermi statistics, otherwise it is kinematically entangled. Alternatively, a two-particle state is dynamically separable if the wavefunction has the minimal form required by Bose or Fermi statistics, and dynamically entangled otherwise. For photons, the role of the missing position-space wave function is played by a detection amplitude directly related to counting rates.
Nasr M. Ghoniem and Daniel D. Walgraef
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199298686
- eISBN:
- 9780191720222
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199298686.003.0009
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter begins with a discussion of reduced dynamics and amplitude equations. It then discusses the generic aspects of pattern selection and stability, the effect of external fields; group ...
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This chapter begins with a discussion of reduced dynamics and amplitude equations. It then discusses the generic aspects of pattern selection and stability, the effect of external fields; group velocity, convective, and absolute instabilities; and pattern formation and front propagation.Less
This chapter begins with a discussion of reduced dynamics and amplitude equations. It then discusses the generic aspects of pattern selection and stability, the effect of external fields; group velocity, convective, and absolute instabilities; and pattern formation and front propagation.
J. C. Garrison and R. Y. Chiao
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198508861
- eISBN:
- 9780191708640
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508861.003.0016
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter reviews theoretical and experimental aspects of non-classical states of light, e.g., squeezed states and number states. The definition of squeezed states and a review of their general ...
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This chapter reviews theoretical and experimental aspects of non-classical states of light, e.g., squeezed states and number states. The definition of squeezed states and a review of their general properties is followed by a discussion of special kinds of squeezing, e.g., squeezed coherent states, displaced squeezed states, and amplitude or phase squeezing. An overview of photon counting statistics and robustness of squeezed states is followed by a theoretical model of their generation and a description of their experimental production. The chapter ends with a brief discussion of the experimental generation of number states, including methods for generation of single-photon states on demand, which have applications to quantum information processing.Less
This chapter reviews theoretical and experimental aspects of non-classical states of light, e.g., squeezed states and number states. The definition of squeezed states and a review of their general properties is followed by a discussion of special kinds of squeezing, e.g., squeezed coherent states, displaced squeezed states, and amplitude or phase squeezing. An overview of photon counting statistics and robustness of squeezed states is followed by a theoretical model of their generation and a description of their experimental production. The chapter ends with a brief discussion of the experimental generation of number states, including methods for generation of single-photon states on demand, which have applications to quantum information processing.
C. Julian Chen
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199211500
- eISBN:
- 9780191705991
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199211500.003.0015
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter discusses atomic force microscopy (AFM), focusing on the methods for atomic force detection. Although the force detection always requires a cantilever, there are two types of modes: the ...
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This chapter discusses atomic force microscopy (AFM), focusing on the methods for atomic force detection. Although the force detection always requires a cantilever, there are two types of modes: the static mode and the dynamic mode. The general design and the typical method of manufacturing of the cantilevers are discussed. Two popular methods of static force detection are presented. The popular dynamic-force detection method, the tapping mode is described, especially the methods in liquids. The non-contact AFM, which has achieved atomic resolution in the weak attractive force regime, is discussed in detail. An elementary and transparent analysis of the principles, including the frequency shift, the second harmonics, and the average tunneling current, is presented. It requires only Newton's equation and Fourier analysis, and the final results are analyzed over the entire range of vibrational amplitude. The implementation is briefly discussed.Less
This chapter discusses atomic force microscopy (AFM), focusing on the methods for atomic force detection. Although the force detection always requires a cantilever, there are two types of modes: the static mode and the dynamic mode. The general design and the typical method of manufacturing of the cantilevers are discussed. Two popular methods of static force detection are presented. The popular dynamic-force detection method, the tapping mode is described, especially the methods in liquids. The non-contact AFM, which has achieved atomic resolution in the weak attractive force regime, is discussed in detail. An elementary and transparent analysis of the principles, including the frequency shift, the second harmonics, and the average tunneling current, is presented. It requires only Newton's equation and Fourier analysis, and the final results are analyzed over the entire range of vibrational amplitude. The implementation is briefly discussed.
C. Julian Chen
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199211500
- eISBN:
- 9780191705991
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199211500.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter discusses the imaging mechanism of STM and AFM at the atomic scale. Experimental facts show that at atomic resolution, tip electronic states play a key role. Analytic theoretical ...
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This chapter discusses the imaging mechanism of STM and AFM at the atomic scale. Experimental facts show that at atomic resolution, tip electronic states play a key role. Analytic theoretical treatments provide quantitative explanation of the effect of the tip electronic states. On transition-metal tips, first-principle studies unanimously show that d-type tip electronic states dominate the Fermi-level DOS. First-principle studies of the combined tip-sample systems show that for both STM and AFM, the p- and d-type tip electronic states are the keys to understanding the atomic-scale images. The case of spin-polarized STM and the chemical identification of surface atoms are also discussed in terms of tip electronic structure. The chapter concludes with discussions of experimental verifications of the reciprocity principle: at atomic resolution, the role of tip electronic states and the sample electronic states are interchangeable.Less
This chapter discusses the imaging mechanism of STM and AFM at the atomic scale. Experimental facts show that at atomic resolution, tip electronic states play a key role. Analytic theoretical treatments provide quantitative explanation of the effect of the tip electronic states. On transition-metal tips, first-principle studies unanimously show that d-type tip electronic states dominate the Fermi-level DOS. First-principle studies of the combined tip-sample systems show that for both STM and AFM, the p- and d-type tip electronic states are the keys to understanding the atomic-scale images. The case of spin-polarized STM and the chemical identification of surface atoms are also discussed in terms of tip electronic structure. The chapter concludes with discussions of experimental verifications of the reciprocity principle: at atomic resolution, the role of tip electronic states and the sample electronic states are interchangeable.
C. Julian Chen
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199211500
- eISBN:
- 9780191705991
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199211500.003.0008
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter discusses the effect of force and deformation of the tip apex and the sample surface in the operation and imaging mechanism of STM and AFM. Because the contact area is of atomic ...
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This chapter discusses the effect of force and deformation of the tip apex and the sample surface in the operation and imaging mechanism of STM and AFM. Because the contact area is of atomic dimension, a very small force and deformation would generate a large measurable effect. Three effects are discussed. First is the stability of the STM junction, which depends on the rigidity of the material. For soft materials, hysterisis is more likely. For rigid materials, the approaching and retraction cycles are continuous and reproducible. Second is the effect of force and deformation to the STM imaging mechanism. For soft material such as graphite, force and deformation can amplify the observed corrugation. For hard materials as most metals, force and deformation can decrease the observed corrugation. Finally, the effect of force and deformation on tunneling barrier height measurements is discussed.Less
This chapter discusses the effect of force and deformation of the tip apex and the sample surface in the operation and imaging mechanism of STM and AFM. Because the contact area is of atomic dimension, a very small force and deformation would generate a large measurable effect. Three effects are discussed. First is the stability of the STM junction, which depends on the rigidity of the material. For soft materials, hysterisis is more likely. For rigid materials, the approaching and retraction cycles are continuous and reproducible. Second is the effect of force and deformation to the STM imaging mechanism. For soft material such as graphite, force and deformation can amplify the observed corrugation. For hard materials as most metals, force and deformation can decrease the observed corrugation. Finally, the effect of force and deformation on tunneling barrier height measurements is discussed.
Marcos Mariño
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568490
- eISBN:
- 9780191717604
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568490.003.0005
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter discusses a particular class of Calabi-Yau geometries characterized by being non-compact, focusing on non-compact toric Calabi-Yau threefolds. These are threefolds that have the ...
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This chapter discusses a particular class of Calabi-Yau geometries characterized by being non-compact, focusing on non-compact toric Calabi-Yau threefolds. These are threefolds that have the structure of a fibration with torus fibres. The manifolds have the structure of a fibration of IR3 by T2 x IR. It turns out that the geometry of these threefolds can be packaged in a two-dimensional graph that encodes the information about the degeneration locus of the fibration. These graphs are called the toric diagrams of the corresponding Calabi-Yau manifolds. A general introduction to the construction of non-compact Calabi-Yau geometries is presented, and the toric approach is discussed. Examples of closed string amplitudes are given.Less
This chapter discusses a particular class of Calabi-Yau geometries characterized by being non-compact, focusing on non-compact toric Calabi-Yau threefolds. These are threefolds that have the structure of a fibration with torus fibres. The manifolds have the structure of a fibration of IR3 by T2 x IR. It turns out that the geometry of these threefolds can be packaged in a two-dimensional graph that encodes the information about the degeneration locus of the fibration. These graphs are called the toric diagrams of the corresponding Calabi-Yau manifolds. A general introduction to the construction of non-compact Calabi-Yau geometries is presented, and the toric approach is discussed. Examples of closed string amplitudes are given.
Marcos Mariño
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568490
- eISBN:
- 9780191717604
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568490.003.0009
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter explains the cut-and-paste approach to toric Calabi-Yau manifolds developed previously with the large-N duality relating Chern-Simons theory and topological strings, to find a building ...
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This chapter explains the cut-and-paste approach to toric Calabi-Yau manifolds developed previously with the large-N duality relating Chern-Simons theory and topological strings, to find a building block for topological string amplitudes on those geometries. This building block is an open string amplitude called the topological vertex. In order to understand topological vertex it is necessary to discuss one aspect of open string amplitudes: the framing ambiguity. Three gluing rules for the topological vertex are discussed: for a change of orientation in one edge, for the propagator, and for the matching of framings in the gluing. Some examples of computation of topological string amplitudes by using the topological vertex are presented.Less
This chapter explains the cut-and-paste approach to toric Calabi-Yau manifolds developed previously with the large-N duality relating Chern-Simons theory and topological strings, to find a building block for topological string amplitudes on those geometries. This building block is an open string amplitude called the topological vertex. In order to understand topological vertex it is necessary to discuss one aspect of open string amplitudes: the framing ambiguity. Three gluing rules for the topological vertex are discussed: for a change of orientation in one edge, for the propagator, and for the matching of framings in the gluing. Some examples of computation of topological string amplitudes by using the topological vertex are presented.
S. F. Edwards and D. R. Wilkinson
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198528531
- eISBN:
- 9780191713415
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528531.003.0022
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This paper examines the problem of the surface fluctuations in a settled granular material. A simple model is given which describes the process by which a particle settles and comes to rest on the ...
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This paper examines the problem of the surface fluctuations in a settled granular material. A simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing, and from this a set of Langevin equations for the Fourier modes of the surface are derived. These equations imply that the Fourier amplitudes behave like the velocities of a set of independent Brownian particles. This results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random, the divergence being removed by a more accurate description of the settling material, for example by having the granules fall through a sieve.Less
This paper examines the problem of the surface fluctuations in a settled granular material. A simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing, and from this a set of Langevin equations for the Fourier modes of the surface are derived. These equations imply that the Fourier amplitudes behave like the velocities of a set of independent Brownian particles. This results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random, the divergence being removed by a more accurate description of the settling material, for example by having the granules fall through a sieve.
Dennis Sherwood and Jon Cooper
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199559046
- eISBN:
- 9780191595028
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199559046.003.0004
- Subject:
- Physics, Crystallography: Physics
This chapter outlines the fundamental physical terms used to define wave motion such as wavelength, frequency, velocity, amplitude, wave vector, and period from first principles. It also introduces ...
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This chapter outlines the fundamental physical terms used to define wave motion such as wavelength, frequency, velocity, amplitude, wave vector, and period from first principles. It also introduces detailed mathematical functions which can be used to describe waves, and explains the interaction of waves with one another (superposition) in depth due to its importance in diffraction analysis. The chapter describes in depth the important concept of the phase of a wave, as is the physical basis of constructive and destructive interference. It emphasises the elegance of the complex exponential form in describing wave motion in diffraction analysis along with the concept of intensity and how it can be derived from the amplitude of the wave. Finally, the chapter describes the nature of electromagnetic radiation along with the physical basis underlying the scattering of electromagnetic radiation by sub-atomic particles.Less
This chapter outlines the fundamental physical terms used to define wave motion such as wavelength, frequency, velocity, amplitude, wave vector, and period from first principles. It also introduces detailed mathematical functions which can be used to describe waves, and explains the interaction of waves with one another (superposition) in depth due to its importance in diffraction analysis. The chapter describes in depth the important concept of the phase of a wave, as is the physical basis of constructive and destructive interference. It emphasises the elegance of the complex exponential form in describing wave motion in diffraction analysis along with the concept of intensity and how it can be derived from the amplitude of the wave. Finally, the chapter describes the nature of electromagnetic radiation along with the physical basis underlying the scattering of electromagnetic radiation by sub-atomic particles.
James H. Fuller
- Published in print:
- 1992
- Published Online:
- March 2012
- ISBN:
- 9780195068207
- eISBN:
- 9780199847198
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195068207.003.0013
- Subject:
- Neuroscience, Sensory and Motor Systems
In this chapter, a variety of saccadic eye-head movements evoked by visual and auditory stimuli are reviewed. Variation in head movement strategies resulting from methodology as well as the subject's ...
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In this chapter, a variety of saccadic eye-head movements evoked by visual and auditory stimuli are reviewed. Variation in head movement strategies resulting from methodology as well as the subject's own biases are considered alongside factors already known to affect eye-head movements. Search strategies are compared in different tasks. First, in the relatively simple situation in which the head is immobilized; second, in the more complicated situation when the head is free to move. The variables of movement amplitude and sensory modality are compared at the same time. In the studies of Guitton and Volle as well as Bizzi et al., saccadic latency was majorly affected by the predictability of the fixation-saccade interval and the saccade amplitude and direction.Less
In this chapter, a variety of saccadic eye-head movements evoked by visual and auditory stimuli are reviewed. Variation in head movement strategies resulting from methodology as well as the subject's own biases are considered alongside factors already known to affect eye-head movements. Search strategies are compared in different tasks. First, in the relatively simple situation in which the head is immobilized; second, in the more complicated situation when the head is free to move. The variables of movement amplitude and sensory modality are compared at the same time. In the studies of Guitton and Volle as well as Bizzi et al., saccadic latency was majorly affected by the predictability of the fixation-saccade interval and the saccade amplitude and direction.
William Clegg
- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199219469
- eISBN:
- 9780191722516
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219469.003.0008
- Subject:
- Physics, Crystallography: Physics
A diffraction pattern is the (forward) Fourier transform of a crystal structure, obtained physically in an experiment and mathematically from a known or model structure (providing both amplitudes and ...
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A diffraction pattern is the (forward) Fourier transform of a crystal structure, obtained physically in an experiment and mathematically from a known or model structure (providing both amplitudes and phases). The (reverse) Fourier transform of a diffraction pattern is an image of the electron density of the structure, unachievable physically, and mathematically, possible only if estimates are available for the missing reflection phases for combination with the observed amplitudes. This chapter considers computing aspects of the required calculations. Variants on the reverse Fourier transform arise from the use of different coefficients instead of the observed amplitudes: squared amplitudes, with no phases, give the Patterson function; ‘normalised’ amplitudes give an E-map in direct methods; differences between observed and calculated amplitudes give difference electron density maps, with applications at various stages of structure determination; weighted amplitudes emphasize or suppress particular features. The concepts are illustrated with a one-dimensional example based on a real structure.Less
A diffraction pattern is the (forward) Fourier transform of a crystal structure, obtained physically in an experiment and mathematically from a known or model structure (providing both amplitudes and phases). The (reverse) Fourier transform of a diffraction pattern is an image of the electron density of the structure, unachievable physically, and mathematically, possible only if estimates are available for the missing reflection phases for combination with the observed amplitudes. This chapter considers computing aspects of the required calculations. Variants on the reverse Fourier transform arise from the use of different coefficients instead of the observed amplitudes: squared amplitudes, with no phases, give the Patterson function; ‘normalised’ amplitudes give an E-map in direct methods; differences between observed and calculated amplitudes give difference electron density maps, with applications at various stages of structure determination; weighted amplitudes emphasize or suppress particular features. The concepts are illustrated with a one-dimensional example based on a real structure.
E. J. N. Wilson
- Published in print:
- 2001
- Published Online:
- January 2010
- ISBN:
- 9780198508298
- eISBN:
- 9780191706363
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508298.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
The lattice described in the chapter is the periodic pattern of focusing and bending magnets that a particle experiences on each turn. Hills Equation describes the vertical and horizontal transverse ...
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The lattice described in the chapter is the periodic pattern of focusing and bending magnets that a particle experiences on each turn. Hills Equation describes the vertical and horizontal transverse oscillations of the particle. Its solution is a modified form of the harmonic oscillator. The square of the local amplitude is the product of the beam emittance and the betatron function, beta, which depends on the lattice pattern alone. Beta links to the rate of phase advance of the motion. Two-by-two transport matrices for one turn of the ring have elements that depend on the Twiss parameters, which are the beta amplitude and its derivatives together with the phase advance. They may be computed numerically by multiplying a chain of individual matrices—one for each magnetic element in the ring. The trace of the matrix defines the number of oscillations per turn, Q, which must be real for stability.Less
The lattice described in the chapter is the periodic pattern of focusing and bending magnets that a particle experiences on each turn. Hills Equation describes the vertical and horizontal transverse oscillations of the particle. Its solution is a modified form of the harmonic oscillator. The square of the local amplitude is the product of the beam emittance and the betatron function, beta, which depends on the lattice pattern alone. Beta links to the rate of phase advance of the motion. Two-by-two transport matrices for one turn of the ring have elements that depend on the Twiss parameters, which are the beta amplitude and its derivatives together with the phase advance. They may be computed numerically by multiplying a chain of individual matrices—one for each magnetic element in the ring. The trace of the matrix defines the number of oscillations per turn, Q, which must be real for stability.
