*Gary A. Glatzmaier*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.001.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This book provides readers with the skills they need to write computer codes that simulate convection, internal gravity waves, and magnetic field generation in the interiors and atmospheres of ...
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This book provides readers with the skills they need to write computer codes that simulate convection, internal gravity waves, and magnetic field generation in the interiors and atmospheres of rotating planets and stars. Using a teaching method perfected in the classroom, the book begins by offering a step-by-step guide on how to design codes for simulating nonlinear time-dependent thermal convection in a 2D box using Fourier expansions in the horizontal direction and finite differences in the vertical direction. It then describes how to implement more efficient a nd accurate numerical methods and more realistic geometries in two and three dimensions. The third part of the book demonstrates how to incorporate more sophisticated physics, including the effects of magnetic field, density stratification, and rotation. The book features numerous exercises throughout, and is an ideal textbook for students and an essential resource for researchers. It explains how to create codes that simulate the internal dynamics of planets and stars, and builds on basic concepts and simple methods. The book shows how to improve the efficiency and accuracy of the numerical methods. It considers more relevant geometries and boundary conditions.Less

This book provides readers with the skills they need to write computer codes that simulate convection, internal gravity waves, and magnetic field generation in the interiors and atmospheres of rotating planets and stars. Using a teaching method perfected in the classroom, the book begins by offering a step-by-step guide on how to design codes for simulating nonlinear time-dependent thermal convection in a 2D box using Fourier expansions in the horizontal direction and finite differences in the vertical direction. It then describes how to implement more efficient a nd accurate numerical methods and more realistic geometries in two and three dimensions. The third part of the book demonstrates how to incorporate more sophisticated physics, including the effects of magnetic field, density stratification, and rotation. The book features numerous exercises throughout, and is an ideal textbook for students and an essential resource for researchers. It explains how to create codes that simulate the internal dynamics of planets and stars, and builds on basic concepts and simple methods. The book shows how to improve the efficiency and accuracy of the numerical methods. It considers more relevant geometries and boundary conditions.

*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/9780198528920.003.0010
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter is the first of two dealing with the dynamical diffraction of incident spherical waves. It makes use of Kato's theory, which is based on a Fourier expansion of the spherical wave. The ...
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This chapter is the first of two dealing with the dynamical diffraction of incident spherical waves. It makes use of Kato's theory, which is based on a Fourier expansion of the spherical wave. The transmission and reflection geometries are handled separately. Two methods of integration are given — direct integration and integration by the stationary phase method. The amplitude and intensity distributions of the reflected and refracted waves on the exit surface are calculated. It is shown that equal-intensity fringes are formed within the Borrmann triangle (Pendellösung fringes) that can be interpreted as due to interferences between the waves associated with the two branches of the dispersion surface. The integrated intensity is calculated and the influence of the polarization of the incident wave discussed. The last section describes the diffraction of ultra-short pulses of plane-wave X-rays such as those emitted by a free-electron laser and which can be handled by considering their Fourier expansion in frequency space.Less

This chapter is the first of two dealing with the dynamical diffraction of incident spherical waves. It makes use of Kato's theory, which is based on a Fourier expansion of the spherical wave. The transmission and reflection geometries are handled separately. Two methods of integration are given — direct integration and integration by the stationary phase method. The amplitude and intensity distributions of the reflected and refracted waves on the exit surface are calculated. It is shown that equal-intensity fringes are formed within the Borrmann triangle (*Pendellösung* fringes) that can be interpreted as due to interferences between the waves associated with the two branches of the dispersion surface. The integrated intensity is calculated and the influence of the polarization of the incident wave discussed. The last section describes the diffraction of ultra-short pulses of plane-wave X-rays such as those emitted by a free-electron laser and which can be handled by considering their Fourier expansion in frequency space.

*Kai-Wen Lan*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691156545
- eISBN:
- 9781400846016
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691156545.003.0007
- Subject:
- Mathematics, Geometry / Topology

This chapter first studies the automorphic forms that are defined as global sections of certain invertible sheaves on the toroidal compactifications. The local structures of toroidal ...
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This chapter first studies the automorphic forms that are defined as global sections of certain invertible sheaves on the toroidal compactifications. The local structures of toroidal compactifications lead naturally to the theory of Fourier–Jacobi expansions and the Fourier–Jacobi expansion principle. The chapter also obtains the algebraic construction of arithmetic minimal compactifications (of the coarse moduli associated with moduli problems), which are projective normal schemes defined over the same integral bases as the moduli problems are. As a by-product of codimension counting, we obtain Koecher's principle for arithmetic automorphic forms (of naive parallel weights). Furthermore, this chapter shows the projectivity of a large class of arithmetic toroidal compactifications by realizing them as normalizations of blowups of the corresponding minimal compactifications.Less

This chapter first studies the automorphic forms that are defined as global sections of certain invertible sheaves on the toroidal compactifications. The local structures of toroidal compactifications lead naturally to the theory of Fourier–Jacobi expansions and the Fourier–Jacobi expansion principle. The chapter also obtains the algebraic construction of arithmetic minimal compactifications (of the coarse moduli associated with moduli problems), which are projective normal schemes defined over the same integral bases as the moduli problems are. As a by-product of codimension counting, we obtain Koecher's principle for arithmetic automorphic forms (of naive parallel weights). Furthermore, this chapter shows the projectivity of a large class of arithmetic toroidal compactifications by realizing them as normalizations of blowups of the corresponding minimal compactifications.

*Fon-Che Liu*

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198790426
- eISBN:
- 9780191831676
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198790426.003.0005
- Subject:
- Mathematics, Analysis

This chapter on the basic principles of linear analysis presents elements of functional analysis that are frequently used in setting up spaces on which certain operators intimately related to the ...
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This chapter on the basic principles of linear analysis presents elements of functional analysis that are frequently used in setting up spaces on which certain operators intimately related to the problems at issue can be studied with a firm base. The most fundamental are the Baire category theorem and the separation principle. The Baire category theorem manifests itself in the uniform boundedness principle, open mapping theorem, and closed graph theorem; while the separation principle is applied in the name of Hahn–Banach. Our treatment of the separation principle is more geometrical than usual. In the latter part of the chapter, emphasis is put on geometric aspects by introducing Hilbert space in which a concept of orthogonal projection plays a leading role. The Riesz representation for bounded linear functionals and Fourier expansion with respect to an orthonormal basis are the main components of the theory addressed.Less

This chapter on the basic principles of linear analysis presents elements of functional analysis that are frequently used in setting up spaces on which certain operators intimately related to the problems at issue can be studied with a firm base. The most fundamental are the Baire category theorem and the separation principle. The Baire category theorem manifests itself in the uniform boundedness principle, open mapping theorem, and closed graph theorem; while the separation principle is applied in the name of Hahn–Banach. Our treatment of the separation principle is more geometrical than usual. In the latter part of the chapter, emphasis is put on geometric aspects by introducing Hilbert space in which a concept of orthogonal projection plays a leading role. The Riesz representation for bounded linear functionals and Fourier expansion with respect to an orthonormal basis are the main components of the theory addressed.