W. Otto Friesen and Jonathon A. Friesen
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780195371833
- eISBN:
- 9780199865178
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195371833.003.0024
- Subject:
- Psychology, Cognitive Neuroscience
This chapter describes the method for numerical integration of the equations that underlie NeuroDynamix II models.
This chapter describes the method for numerical integration of the equations that underlie NeuroDynamix II models.
George Em Karniadakis and Spencer J. Sherwin
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198528692
- eISBN:
- 9780191713491
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528692.003.0002
- Subject:
- Mathematics, Numerical Analysis
This chapter discusses the fundamental concepts that are further developed in the remainder of the book, in the context of a one-dimensional formulation. In doing so, the principles and underlying ...
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This chapter discusses the fundamental concepts that are further developed in the remainder of the book, in the context of a one-dimensional formulation. In doing so, the principles and underlying theory behind the construction of the spectral/hp element method are illustrated. Topics covered include method of weighted residuals, Galerkin formulation, one-dimensional expansion bases, elemental operations, error estimates, and implementation of a one-dimensional spectral/hp element. The chapter ends with exercises aimed at developing a one-dimensional spectral element solver.Less
This chapter discusses the fundamental concepts that are further developed in the remainder of the book, in the context of a one-dimensional formulation. In doing so, the principles and underlying theory behind the construction of the spectral/hp element method are illustrated. Topics covered include method of weighted residuals, Galerkin formulation, one-dimensional expansion bases, elemental operations, error estimates, and implementation of a one-dimensional spectral/hp element. The chapter ends with exercises aimed at developing a one-dimensional spectral element solver.
Fred Campano and Dominick Salvatore
- Published in print:
- 2006
- Published Online:
- May 2006
- ISBN:
- 9780195300918
- eISBN:
- 9780199783441
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195300912.003.0006
- Subject:
- Economics and Finance, Development, Growth, and Environmental
Formulas for computing the most commonly used summary measure of income distribution are given. These include the mean, median, and mode. The methodology for computing shares of total income obtained ...
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Formulas for computing the most commonly used summary measure of income distribution are given. These include the mean, median, and mode. The methodology for computing shares of total income obtained by quantiles of income recipients, and the technique to graph the Lorenz curve and compute the Gini coefficient are described.Less
Formulas for computing the most commonly used summary measure of income distribution are given. These include the mean, median, and mode. The methodology for computing shares of total income obtained by quantiles of income recipients, and the technique to graph the Lorenz curve and compute the Gini coefficient are described.
Luc Bauwens, Michel Lubrano, and Jean-François Richard
- Published in print:
- 2000
- Published Online:
- September 2011
- ISBN:
- 9780198773122
- eISBN:
- 9780191695315
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198773122.001.0001
- Subject:
- Economics and Finance, Econometrics
This book contains an up-to-date coverage of the last twenty years of advances in Bayesian inference in econometrics, with an emphasis on dynamic models. It shows how to treat Bayesian inference in ...
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This book contains an up-to-date coverage of the last twenty years of advances in Bayesian inference in econometrics, with an emphasis on dynamic models. It shows how to treat Bayesian inference in non-linear models, by integrating the useful developments of numerical integration techniques based on simulations (such as Markov Chain Monte Carlo methods), and the long available analytical results of Bayesian inference for linear regression models. It thus covers a broad range of rather recent models for economic time series, such as non-linear models, autoregressive conditional heteroskedastic regressions, and cointegrated vector autoregressive models. It contains also an extensive chapter on unit root inference from the Bayesian viewpoint. Several examples illustrate the methods.Less
This book contains an up-to-date coverage of the last twenty years of advances in Bayesian inference in econometrics, with an emphasis on dynamic models. It shows how to treat Bayesian inference in non-linear models, by integrating the useful developments of numerical integration techniques based on simulations (such as Markov Chain Monte Carlo methods), and the long available analytical results of Bayesian inference for linear regression models. It thus covers a broad range of rather recent models for economic time series, such as non-linear models, autoregressive conditional heteroskedastic regressions, and cointegrated vector autoregressive models. It contains also an extensive chapter on unit root inference from the Bayesian viewpoint. Several examples illustrate the methods.
ANDRÉ AUTHIER
- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198528920
- eISBN:
- 9780191713125
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528920.003.0014
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter concerns highly deformed crystals where the Eikonal approximation is no longer valid. An expression is given for the limit of validity of this approximation. Takagi's equations are ...
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This chapter concerns highly deformed crystals where the Eikonal approximation is no longer valid. An expression is given for the limit of validity of this approximation. Takagi's equations are extended so as to apply to highly deformed crystals. Their resolution is the discussed and the principle of their numerical integration in an inverted Borrmann triangle given. The ray concept is generalized to the case of strong deformations by noting that new wavefields are generated in the highly strained regions; this is known as the interbranch scattering effect. The last part of the chapter is devoted to an account of the statistical dynamical theories for highly imperfect crystals, with emphasis on Kato's statistical theories. Examples of experimental test of the dynamical theory are also given.Less
This chapter concerns highly deformed crystals where the Eikonal approximation is no longer valid. An expression is given for the limit of validity of this approximation. Takagi's equations are extended so as to apply to highly deformed crystals. Their resolution is the discussed and the principle of their numerical integration in an inverted Borrmann triangle given. The ray concept is generalized to the case of strong deformations by noting that new wavefields are generated in the highly strained regions; this is known as the interbranch scattering effect. The last part of the chapter is devoted to an account of the statistical dynamical theories for highly imperfect crystals, with emphasis on Kato's statistical theories. Examples of experimental test of the dynamical theory are also given.
Heiner Igel
- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198717409
- eISBN:
- 9780191835070
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198717409.003.0007
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics
The spectral-element method is introduced as a finite-element method with high-order Lagrange polynomials as interpolating functions. The concept of numerical integration is introduced and the ...
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The spectral-element method is introduced as a finite-element method with high-order Lagrange polynomials as interpolating functions. The concept of numerical integration is introduced and the Gauss–Lobatto–Legendre approach is presented as a way to obtain a diagonal mass matrix. The calculation of the mass and stiffness matrices is presented first at an elemental level. The synthesis of the global system of equations is discussed with the result that the diagonal mass matrix allows for a fully explicit time-extrapolation scheme. The method is presented with examples of wave propagation in homogeneous and heterogeneous media.Less
The spectral-element method is introduced as a finite-element method with high-order Lagrange polynomials as interpolating functions. The concept of numerical integration is introduced and the Gauss–Lobatto–Legendre approach is presented as a way to obtain a diagonal mass matrix. The calculation of the mass and stiffness matrices is presented first at an elemental level. The synthesis of the global system of equations is discussed with the result that the diagonal mass matrix allows for a fully explicit time-extrapolation scheme. The method is presented with examples of wave propagation in homogeneous and heterogeneous media.
Olle Eriksson, Anders Bergman, Lars Bergqvist, and Johan Hellsvik
- Published in print:
- 2017
- Published Online:
- May 2017
- ISBN:
- 9780198788669
- eISBN:
- 9780191830747
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198788669.003.0007
- Subject:
- Physics, Atomic, Laser, and Optical Physics
In this chapter, we will present the technical aspects of atomistic spin dynamics, in particular how the method can be implemented in an actual computer software. This involves calculation of ...
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In this chapter, we will present the technical aspects of atomistic spin dynamics, in particular how the method can be implemented in an actual computer software. This involves calculation of effective field and creation of neighbour lists for setting up the geometry of the system of interest as well as choosing a suitable integrator scheme for the SLL (or SLLG) equation. We also give examples of extraction and processing of relevant observables that are common output from simulations. Atomistic spin dynamics simulations could be a computationally heavy tool but it is also very well adapted for modern computer architectures like massive parallel computing and/or graphics processing units and we provide examples how to utilize these architectures in an efficient manner. We use our own developed software UppASD as example, but the discussion could be applied to any other atomistic spin dynamics software.Less
In this chapter, we will present the technical aspects of atomistic spin dynamics, in particular how the method can be implemented in an actual computer software. This involves calculation of effective field and creation of neighbour lists for setting up the geometry of the system of interest as well as choosing a suitable integrator scheme for the SLL (or SLLG) equation. We also give examples of extraction and processing of relevant observables that are common output from simulations. Atomistic spin dynamics simulations could be a computationally heavy tool but it is also very well adapted for modern computer architectures like massive parallel computing and/or graphics processing units and we provide examples how to utilize these architectures in an efficient manner. We use our own developed software UppASD as example, but the discussion could be applied to any other atomistic spin dynamics software.
Robert E. Criss
- Published in print:
- 1999
- Published Online:
- November 2020
- ISBN:
- 9780195117752
- eISBN:
- 9780197561195
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195117752.003.0006
- Subject:
- Earth Sciences and Geography, Geochemistry
At the Earth’s surface, isotopic disequilibrium is far more common than isotopic equilibrium. Although isotopic equilibrium is approached in certain instances, ...
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At the Earth’s surface, isotopic disequilibrium is far more common than isotopic equilibrium. Although isotopic equilibrium is approached in certain instances, numerous constituents of the lithosphere, hydrosphere, atmosphere, and biosphere are simply not in mutual isotopic equilibrium. This condition is consistent with the complex and dynamic conditions typical of the Earth’s surface, particularly the large material fluxes, the rapid changes in temperature, and the biological mediation of chemical systems. Fortunately, several aspects of isotopic disequilibrium may be understood in terms of elementary physical laws. For homogeneous phases such as gases or well-stirred liquids, or for cases where spatial gradients in isotopic contents are not of primary interest, then the principles of elementary kinetics can be applied. For cases where isotopic gradients are important, the laws of diffusion are applicable. If two phases are out of isotopic equilibrium, they will progressively tend to approach the equilibrium state with the passage of time. This phenomenon occurs by the process of isotopic exchange, and its rate may be understood by examining isotopic exchange reactions from the viewpoint of elementary kinetic theory. In particular, consider the generalized exchange reaction where A and B are two phases that share a common major element, and A* and B* represent the same phases in which the trace isotope of that element is present. The present analysis is simplified if the exchange reaction is written so that only one atom is exchanged, in which case the stoichiometric coefficients are all unity. For reaction 4.1, kinetic principles assert that the forward and reverse reactions do not, in general, proceed at identical rates, but rather at the rates indicated by the quantities kα and k written by the arrows, multiplied by the appropriate concentrations terms. Assuming that the reaction is first order, then the reaction progress, represented by the quantity dA*/dt, may be expressed by the difference between these forward and reverse rates, as follows: . . . dA*/dt = kα(A)(B*) − k(A*)(B) (4.2) . . . In order to evaluate the exchange process more completely, is important to carefully chose a consistent set of concentrations for substitution equation 4.2.
Less
At the Earth’s surface, isotopic disequilibrium is far more common than isotopic equilibrium. Although isotopic equilibrium is approached in certain instances, numerous constituents of the lithosphere, hydrosphere, atmosphere, and biosphere are simply not in mutual isotopic equilibrium. This condition is consistent with the complex and dynamic conditions typical of the Earth’s surface, particularly the large material fluxes, the rapid changes in temperature, and the biological mediation of chemical systems. Fortunately, several aspects of isotopic disequilibrium may be understood in terms of elementary physical laws. For homogeneous phases such as gases or well-stirred liquids, or for cases where spatial gradients in isotopic contents are not of primary interest, then the principles of elementary kinetics can be applied. For cases where isotopic gradients are important, the laws of diffusion are applicable. If two phases are out of isotopic equilibrium, they will progressively tend to approach the equilibrium state with the passage of time. This phenomenon occurs by the process of isotopic exchange, and its rate may be understood by examining isotopic exchange reactions from the viewpoint of elementary kinetic theory. In particular, consider the generalized exchange reaction where A and B are two phases that share a common major element, and A* and B* represent the same phases in which the trace isotope of that element is present. The present analysis is simplified if the exchange reaction is written so that only one atom is exchanged, in which case the stoichiometric coefficients are all unity. For reaction 4.1, kinetic principles assert that the forward and reverse reactions do not, in general, proceed at identical rates, but rather at the rates indicated by the quantities kα and k written by the arrows, multiplied by the appropriate concentrations terms. Assuming that the reaction is first order, then the reaction progress, represented by the quantity dA*/dt, may be expressed by the difference between these forward and reverse rates, as follows: . . . dA*/dt = kα(A)(B*) − k(A*)(B) (4.2) . . . In order to evaluate the exchange process more completely, is important to carefully chose a consistent set of concentrations for substitution equation 4.2.
