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/9780198509783.003.0005
- Subject:
- Mathematics, Mathematical Physics
In order to deal with the nonlinear Radon transform, this chapter introduces a new type of Fourier transform: the nonlinear Fourier transform. This can be used to analyze various nonlinear Radon ...
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In order to deal with the nonlinear Radon transform, this chapter introduces a new type of Fourier transform: the nonlinear Fourier transform. This can be used to analyze various nonlinear Radon transforms. It is also useful in giving a new approach to wave front sets. (The quadratic case is similar to the FBI transform.)Less
In order to deal with the nonlinear Radon transform, this chapter introduces a new type of Fourier transform: the nonlinear Fourier transform. This can be used to analyze various nonlinear Radon transforms. It is also useful in giving a new approach to wave front sets. (The quadratic case is similar to the FBI transform.)
Rob H. Bisseling
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198529392
- eISBN:
- 9780191712869
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198529392.001.0001
- Subject:
- Mathematics, Applied Mathematics
This book explains the use of the bulk synchronous parallel (BSP) model and the BSPlib communication library in parallel algorithm design and parallel programming. The main topics treated in the book ...
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This book explains the use of the bulk synchronous parallel (BSP) model and the BSPlib communication library in parallel algorithm design and parallel programming. The main topics treated in the book are central to the area of scientific computation: solving dense linear systems by Gaussian elimination, computing fast Fourier transforms, and solving sparse linear systems by iterative methods based on sparse matrix-vector multiplication. Each topic is treated in depth, starting from the problem formulation and a sequential algorithm, through a parallel algorithm and its cost analysis, to a complete parallel program written in C and BSPlib, and experimental results obtained using this program on a parallel computer. Throughout the book, emphasis is placed on analyzing the cost of the parallel algorithms developed, expressed in three terms: computation cost, communication cost, and synchronization cost. The book contains five example programs written in BSPlib, which illustrate the methods taught. These programs are freely available as the package BSPedupack. An appendix on the message-passing interface (MPI) discusses how to program in a structured, bulk synchronous parallel style using the MPI communication library, and presents MPI equivalents of all the programs in the book.Less
This book explains the use of the bulk synchronous parallel (BSP) model and the BSPlib communication library in parallel algorithm design and parallel programming. The main topics treated in the book are central to the area of scientific computation: solving dense linear systems by Gaussian elimination, computing fast Fourier transforms, and solving sparse linear systems by iterative methods based on sparse matrix-vector multiplication. Each topic is treated in depth, starting from the problem formulation and a sequential algorithm, through a parallel algorithm and its cost analysis, to a complete parallel program written in C and BSPlib, and experimental results obtained using this program on a parallel computer. Throughout the book, emphasis is placed on analyzing the cost of the parallel algorithms developed, expressed in three terms: computation cost, communication cost, and synchronization cost. The book contains five example programs written in BSPlib, which illustrate the methods taught. These programs are freely available as the package BSPedupack. An appendix on the message-passing interface (MPI) discusses how to program in a structured, bulk synchronous parallel style using the MPI communication library, and presents MPI equivalents of all the programs in the book.
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/9780199559046.003.0005
- Subject:
- Physics, Crystallography: Physics
This chapter revisits the essential mathematics of integral calculus and introduces the important concept of the Fourier transform, the properties of which are elegantly demonstrated by sketching the ...
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This chapter revisits the essential mathematics of integral calculus and introduces the important concept of the Fourier transform, the properties of which are elegantly demonstrated by sketching the curves of various functions. It demonstrates the essential principles of diffraction by determining the Fourier transforms of regularly repeating patterns which can be represented mathematically by Dirac delta functions — the very important concept of Fourier space (or reciprocal space) follows from this discussion. This section leads into a description of another highly important mathematical concept, the convolution. Convolutions allow two functions to be combined and provide an extremely elegant mathematical description of the crystalline state as well as an insight into one of crystallography's most important structure-solving tools, the Patterson function.Less
This chapter revisits the essential mathematics of integral calculus and introduces the important concept of the Fourier transform, the properties of which are elegantly demonstrated by sketching the curves of various functions. It demonstrates the essential principles of diffraction by determining the Fourier transforms of regularly repeating patterns which can be represented mathematically by Dirac delta functions — the very important concept of Fourier space (or reciprocal space) follows from this discussion. This section leads into a description of another highly important mathematical concept, the convolution. Convolutions allow two functions to be combined and provide an extremely elegant mathematical description of the crystalline state as well as an insight into one of crystallography's most important structure-solving tools, the Patterson function.
Dennis Sherwood and Jon Cooper
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199559046
- eISBN:
- 9780191595028
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199559046.001.0001
- Subject:
- Physics, Crystallography: Physics
This book presents a complete account of the theory of the diffraction of X-rays by crystals with particular reference to the processes of determining the structures of protein molecules. The book ...
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This book presents a complete account of the theory of the diffraction of X-rays by crystals with particular reference to the processes of determining the structures of protein molecules. The book develops from first principles all relevant mathematics, diffraction, and wave theory. The practical aspects of sample preparation and X-ray data collection using both laboratory and synchrotron sources are covered along with data analysis at both the theoretical and practical levels. The important role played by the Patterson function in structure analysis by both molecular replacement and experimental phasing approaches is covered, as are methods for improving the resulting electron density map. The theoretical basis of methods used in refinement of protein crystal structures are then covered in depth along with the crucial task of defining the binding sites of ligands and drug molecules. The complementary roles of other diffraction methods which reveal further detail of great functional importance in a crystal structure are outlined.Less
This book presents a complete account of the theory of the diffraction of X-rays by crystals with particular reference to the processes of determining the structures of protein molecules. The book develops from first principles all relevant mathematics, diffraction, and wave theory. The practical aspects of sample preparation and X-ray data collection using both laboratory and synchrotron sources are covered along with data analysis at both the theoretical and practical levels. The important role played by the Patterson function in structure analysis by both molecular replacement and experimental phasing approaches is covered, as are methods for improving the resulting electron density map. The theoretical basis of methods used in refinement of protein crystal structures are then covered in depth along with the crucial task of defining the binding sites of ligands and drug molecules. The complementary roles of other diffraction methods which reveal further detail of great functional importance in a crystal structure are outlined.
D. A. Bini, G. Latouche, and B. Meini
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.003.0002
- Subject:
- Mathematics, Numerical Analysis
This chapter describes structural and computational properties which are the basis of the design and analysis of fast algorithms for the numerical solution of structured Markov chains. Toeplitz ...
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This chapter describes structural and computational properties which are the basis of the design and analysis of fast algorithms for the numerical solution of structured Markov chains. Toeplitz matrices, circulant and z-circulant matrices with their block analogs are introduced together with the fast algorithms for their manipulation. The concept of discrete Fourier transform is recalled with the FFT algorithm for its computation. Attention is given to the concept of displacement operator and of displacement rank needed to design efficient algorithms for a wide class of matrices related to Toeplitz matrices.Less
This chapter describes structural and computational properties which are the basis of the design and analysis of fast algorithms for the numerical solution of structured Markov chains. Toeplitz matrices, circulant and z-circulant matrices with their block analogs are introduced together with the fast algorithms for their manipulation. The concept of discrete Fourier transform is recalled with the FFT algorithm for its computation. Attention is given to the concept of displacement operator and of displacement rank needed to design efficient algorithms for a wide class of matrices related to Toeplitz matrices.
Chun Wa Wong
- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199641390
- eISBN:
- 9780191747786
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641390.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The Fourier expansion of an arbitrary piecewise-smooth function in the finite interval —π 〈 x 〈 π as an infinite sum of terms (a 0; a n ...
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The Fourier expansion of an arbitrary piecewise-smooth function in the finite interval —π 〈 x 〈 π as an infinite sum of terms (a 0; a n cos nx, b n sin nx, integer n = l,…,∞) is described. The complex exponent functions e inx , integer —∞ 〈 n 〈 ∞, (or another set of orthogonal functions,) in the interval (a, b) can also be used. A special limiting case for the interval (—∞,∞) yields the Fourier integral transform. It is used to solve inhomogeneous differential equations with the help of Dirac delta functions and Green's functions. The convergence and completeness of Fourier-series representation are discussed. Fourier transforms in multidimensional space-time give access to Fourier spaces of wave vector and frequency. In these Fourier spaces, Maxwell's equations simplify to simple algebraic equations. The Helmholtz decomposition theorem is derived.Less
The Fourier expansion of an arbitrary piecewise-smooth function in the finite interval —π 〈 x 〈 π as an infinite sum of terms (a 0; a n cos nx, b n sin nx, integer n = l,…,∞) is described. The complex exponent functions e inx , integer —∞ 〈 n 〈 ∞, (or another set of orthogonal functions,) in the interval (a, b) can also be used. A special limiting case for the interval (—∞,∞) yields the Fourier integral transform. It is used to solve inhomogeneous differential equations with the help of Dirac delta functions and Green's functions. The convergence and completeness of Fourier-series representation are discussed. Fourier transforms in multidimensional space-time give access to Fourier spaces of wave vector and frequency. In these Fourier spaces, Maxwell's equations simplify to simple algebraic equations. The Helmholtz decomposition theorem is derived.
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/9780199559046.003.0006
- Subject:
- Physics, Crystallography: Physics
This chapter covers the principles of the interaction of waves with obstacles and the manner in which information on the nature of an obstacle is contained within the resulting wave forms. The ...
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This chapter covers the principles of the interaction of waves with obstacles and the manner in which information on the nature of an obstacle is contained within the resulting wave forms. The essential mathematics of the interaction of a wave with a perturbing obstacle are derived from first principles and the key significance of the Fourier transform is described in depth — essentially the diffraction pattern of an object is given by the Fourier transform of that object and, most importantly, an image of an object can be obtained by calculating an inverse Fourier transform of its diffraction pattern. The chapter then introduces one of the key experimental limitations of diffraction analysis — the inability to measure the phases of diffracted waves and hence the need for complex computational methods to determine the phases which are outlined in the final chapters of the book.Less
This chapter covers the principles of the interaction of waves with obstacles and the manner in which information on the nature of an obstacle is contained within the resulting wave forms. The essential mathematics of the interaction of a wave with a perturbing obstacle are derived from first principles and the key significance of the Fourier transform is described in depth — essentially the diffraction pattern of an object is given by the Fourier transform of that object and, most importantly, an image of an object can be obtained by calculating an inverse Fourier transform of its diffraction pattern. The chapter then introduces one of the key experimental limitations of diffraction analysis — the inability to measure the phases of diffracted waves and hence the need for complex computational methods to determine the phases which are outlined in the final chapters of the book.
Reinhard B. Neder and Thomas Proffen
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199233694
- eISBN:
- 9780191715563
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233694.003.0004
- Subject:
- Physics, Crystallography: Physics
In most cases, simulations of disordered materials are performed to understand experimental observations, in this case diffraction data. This chapter discusses the calculation of several experimental ...
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In most cases, simulations of disordered materials are performed to understand experimental observations, in this case diffraction data. This chapter discusses the calculation of several experimental quantities: single crystal diffuse scattering, powder diffraction, and the atomic pair distribution function (PDF). Since diffraction data are obtained via a Fourier transform, the finite size of the model crystal as well as issues concerning coherence are discussed in detail. The PDF is basically calculated from the atomic structure directly. Different ways to incorporate thermal motion are illustrated.Less
In most cases, simulations of disordered materials are performed to understand experimental observations, in this case diffraction data. This chapter discusses the calculation of several experimental quantities: single crystal diffuse scattering, powder diffraction, and the atomic pair distribution function (PDF). Since diffraction data are obtained via a Fourier transform, the finite size of the model crystal as well as issues concerning coherence are discussed in detail. The PDF is basically calculated from the atomic structure directly. Different ways to incorporate thermal motion are illustrated.
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.
Xiaodong Zou, Sven Hovmöller, and Peter Oleynikov
- Published in print:
- 2011
- Published Online:
- January 2012
- ISBN:
- 9780199580200
- eISBN:
- 9780191731211
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199580200.003.0007
- Subject:
- Physics, Crystallography: Physics
A fundamentally important feature of electron crystallography is its power in solving crystal structures directly from HRTEM images, using crystallographic image processing. With the appropriate ...
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A fundamentally important feature of electron crystallography is its power in solving crystal structures directly from HRTEM images, using crystallographic image processing. With the appropriate formulas, the important factors in how to do this in practice are described, including Fourier transformation and lattice refinement, symmetry determination and origin refinement, leading up to imposing the symmetry on the experimentally observed structure factor amplitudes and phases in HRTEM images. Determination of and correcting for defocus and astigmatism and crystal misalignment, and the reconstruction of the projected potential map from a single or a few HRTEM image(s) are described.Less
A fundamentally important feature of electron crystallography is its power in solving crystal structures directly from HRTEM images, using crystallographic image processing. With the appropriate formulas, the important factors in how to do this in practice are described, including Fourier transformation and lattice refinement, symmetry determination and origin refinement, leading up to imposing the symmetry on the experimentally observed structure factor amplitudes and phases in HRTEM images. Determination of and correcting for defocus and astigmatism and crystal misalignment, and the reconstruction of the projected potential map from a single or a few HRTEM image(s) are described.
B. D. Guenther
- Published in print:
- 2015
- Published Online:
- January 2016
- ISBN:
- 9780198738770
- eISBN:
- 9780191801983
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198738770.003.0006
- Subject:
- Physics, Atomic, Laser, and Optical Physics
The techniques developed by Fourier and Dirichlet state that a Fourier series, a sum of sinusoidal functions, can be used to describe any periodic functions and that the Fourier transform, an ...
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The techniques developed by Fourier and Dirichlet state that a Fourier series, a sum of sinusoidal functions, can be used to describe any periodic functions and that the Fourier transform, an integral transform, can be used to describe aperiodic functions. The measurement process used to obtain information about continuous functions found in nature is accomplished by making discrete measurements (called samples) of the functions. The development of the Fourier theory presents the opportunity to justify the experimental approach, by examining the processes of replication and sampling. The concepts, developed for electrical communication systems, that depend upon the use of the Fourier method are used for analysis of optical imaging systems and will be needed in the Fresnel formulation of diffraction in Chapters 9 and 10.Less
The techniques developed by Fourier and Dirichlet state that a Fourier series, a sum of sinusoidal functions, can be used to describe any periodic functions and that the Fourier transform, an integral transform, can be used to describe aperiodic functions. The measurement process used to obtain information about continuous functions found in nature is accomplished by making discrete measurements (called samples) of the functions. The development of the Fourier theory presents the opportunity to justify the experimental approach, by examining the processes of replication and sampling. The concepts, developed for electrical communication systems, that depend upon the use of the Fourier method are used for analysis of optical imaging systems and will be needed in the Fresnel formulation of diffraction in Chapters 9 and 10.
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/9780199559046.003.0012
- Subject:
- Physics, Crystallography: Physics
This chapter demonstrates that the experimental observable in diffraction analysis is the intensity of each diffraction spot which provides only the amplitude of the corresponding structure factor ...
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This chapter demonstrates that the experimental observable in diffraction analysis is the intensity of each diffraction spot which provides only the amplitude of the corresponding structure factor and not its phase — the fundamental phase problem in crystallography. To obtain an image of the molecule forming a crystal we need to calculate a Fourier transform, which requires that we know both the amplitude and phase of each structure factor. Nevertheless, very important information can be derived by calculating a ‘phase-less’ Fourier transform of the intensities alone, which is known as the Patterson function. Although this function is inherently more complex than an electron density map since it displays all inter-atomic vectors, certain sections of the Patterson function, known as Harker sections, can yield information on the positions of the most electron-rich atoms within the crystal. The Patterson function is exploited in most methods of the solving the phase problem for proteins and simple rules for the interpretation of a Patterson function are derived.Less
This chapter demonstrates that the experimental observable in diffraction analysis is the intensity of each diffraction spot which provides only the amplitude of the corresponding structure factor and not its phase — the fundamental phase problem in crystallography. To obtain an image of the molecule forming a crystal we need to calculate a Fourier transform, which requires that we know both the amplitude and phase of each structure factor. Nevertheless, very important information can be derived by calculating a ‘phase-less’ Fourier transform of the intensities alone, which is known as the Patterson function. Although this function is inherently more complex than an electron density map since it displays all inter-atomic vectors, certain sections of the Patterson function, known as Harker sections, can yield information on the positions of the most electron-rich atoms within the crystal. The Patterson function is exploited in most methods of the solving the phase problem for proteins and simple rules for the interpretation of a Patterson function are derived.
Peter Main
- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199219469
- eISBN:
- 9780191722516
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219469.003.0001
- Subject:
- Physics, Crystallography: Physics
This introductory chapter provides information on some fundamental aspects of crystal structures and their diffraction of X-rays as a basis for the rest of the book. It describes electrons, atoms, ...
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This introductory chapter provides information on some fundamental aspects of crystal structures and their diffraction of X-rays as a basis for the rest of the book. It describes electrons, atoms, molecules, and crystals scatter X-rays, leading to the observed diffraction pattern, and introduces concepts such as the reciprocal lattice, structure factors, Fourier transforms, Bragg's law for the geometry of diffraction, the phase problem encountered in crystallography, and the meaning of resolution and how it is related to the extent of the measured diffraction pattern.Less
This introductory chapter provides information on some fundamental aspects of crystal structures and their diffraction of X-rays as a basis for the rest of the book. It describes electrons, atoms, molecules, and crystals scatter X-rays, leading to the observed diffraction pattern, and introduces concepts such as the reciprocal lattice, structure factors, Fourier transforms, Bragg's law for the geometry of diffraction, the phase problem encountered in crystallography, and the meaning of resolution and how it is related to the extent of the measured diffraction pattern.
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/9780691141725.003.0005
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and ...
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This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and store results in a file during the computer simulation, assuming the Fourier transforms to x-space are done within the main computational code during the simulation. It then describes the postprocessing code for reading these files and displaying the various fields, along with the use of graphics software packages that provide additional, more sophisticated visualizations of the scalar and vector data. It also discusses the computer analysis of several additional properties of the solution, focusing on measurements of nonlinear convection such as Rayleigh number, Nusselt number, Reynolds number, and kinetic energy spectrum.Less
This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and store results in a file during the computer simulation, assuming the Fourier transforms to x-space are done within the main computational code during the simulation. It then describes the postprocessing code for reading these files and displaying the various fields, along with the use of graphics software packages that provide additional, more sophisticated visualizations of the scalar and vector data. It also discusses the computer analysis of several additional properties of the solution, focusing on measurements of nonlinear convection such as Rayleigh number, Nusselt number, Reynolds number, and kinetic energy spectrum.
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/9780198717409.003.0005
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics
The concepts of Fourier series and the Fourier transforms are introduced as a way to approximate arbitrary functions. The interpolation properties on regular grids (collocation points, cardinal ...
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The concepts of Fourier series and the Fourier transforms are introduced as a way to approximate arbitrary functions. The interpolation properties on regular grids (collocation points, cardinal functions) are presented along with the possibility of calculating highly accurate space derivatives. Spectral derivates are used to solve the acoustic and elastic wave equations. Fourier transform-based truncated finite-difference operators are presented as an alternative for high-order derivative calculations. Chebyshev polynomials are introduced as an alternative to approximate functions in limited areas. The elastic wave equation is solved using the Chebyshev pseudospectral methods.Less
The concepts of Fourier series and the Fourier transforms are introduced as a way to approximate arbitrary functions. The interpolation properties on regular grids (collocation points, cardinal functions) are presented along with the possibility of calculating highly accurate space derivatives. Spectral derivates are used to solve the acoustic and elastic wave equations. Fourier transform-based truncated finite-difference operators are presented as an alternative for high-order derivative calculations. Chebyshev polynomials are introduced as an alternative to approximate functions in limited areas. The elastic wave equation is solved using the Chebyshev pseudospectral methods.
Vlatko Vedral
- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780199215706
- eISBN:
- 9780191706783
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199215706.003.0011
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
As computers get faster and faster, the size of the circuitry imprinted onto silicon chips decreases. The size of the circuitry becomes so small that its behavior is governed by the laws of quantum ...
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As computers get faster and faster, the size of the circuitry imprinted onto silicon chips decreases. The size of the circuitry becomes so small that its behavior is governed by the laws of quantum mechanics. Such a computer, whose computations would be fully quantum mechanical, is called a quantum computer. Any computational task such as addition, multiplication, displaying graphics, or updating databases is performed by a computer according to an algorithm — an abstract set of instructions. Quantum computers would accomplish tasks by performing quantum algorithms. A quantum algorithm is a sequence of unitary evolutions carried out on a quantum string made up of qubits, which can exist as a superposition of classical strings. This chapter discusses the computational complexity of a quantum algorithm, Deutsch's algorithm and its efficiency as quantified by the Holevo bound, Oracles, Grover's search algorithm, quantum factorisation, quantum Fourier transform, and phase estimation.Less
As computers get faster and faster, the size of the circuitry imprinted onto silicon chips decreases. The size of the circuitry becomes so small that its behavior is governed by the laws of quantum mechanics. Such a computer, whose computations would be fully quantum mechanical, is called a quantum computer. Any computational task such as addition, multiplication, displaying graphics, or updating databases is performed by a computer according to an algorithm — an abstract set of instructions. Quantum computers would accomplish tasks by performing quantum algorithms. A quantum algorithm is a sequence of unitary evolutions carried out on a quantum string made up of qubits, which can exist as a superposition of classical strings. This chapter discusses the computational complexity of a quantum algorithm, Deutsch's algorithm and its efficiency as quantified by the Holevo bound, Oracles, Grover's search algorithm, quantum factorisation, quantum Fourier transform, and phase estimation.
Ali Taheri
- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198733133
- eISBN:
- 9780191797712
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198733133.003.0011
- Subject:
- Mathematics, Analysis
This chapter begins with an introduction of Fourier transform on ℝn and then covers various summability methods including Abel-Poisson, Gauss-Weierstrass and later Bochner-Riesz. It also discusses ...
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This chapter begins with an introduction of Fourier transform on ℝn and then covers various summability methods including Abel-Poisson, Gauss-Weierstrass and later Bochner-Riesz. It also discusses the Calderon-Zygmund decomposition and Fefferman’s ball multiplier theorem.Less
This chapter begins with an introduction of Fourier transform on ℝn and then covers various summability methods including Abel-Poisson, Gauss-Weierstrass and later Bochner-Riesz. It also discusses the Calderon-Zygmund decomposition and Fefferman’s ball multiplier theorem.
P. A. Davidson
- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780198722588
- eISBN:
- 9780191789298
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198722588.003.0008
- Subject:
- Mathematics, Applied Mathematics, Mathematical Physics
Chapter 8 recasts the dynamical properties of turbulence in spectral space. The chapter starts by redeveloping the kinematics of turbulence in Fourier space, introducing the ideas of the spectral ...
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Chapter 8 recasts the dynamical properties of turbulence in spectral space. The chapter starts by redeveloping the kinematics of turbulence in Fourier space, introducing the ideas of the spectral tensor, energy spectrum, one-dimensional spectra, and spectral singularities. The Karman–Howarth equation is then rewritten in spectral space and the associated dynamical processes discussed. Two-point spectral closure models are outlined, along with their deficiencies.Less
Chapter 8 recasts the dynamical properties of turbulence in spectral space. The chapter starts by redeveloping the kinematics of turbulence in Fourier space, introducing the ideas of the spectral tensor, energy spectrum, one-dimensional spectra, and spectral singularities. The Karman–Howarth equation is then rewritten in spectral space and the associated dynamical processes discussed. Two-point spectral closure models are outlined, along with their deficiencies.
Max A. Little
- Published in print:
- 2019
- Published Online:
- October 2019
- ISBN:
- 9780198714934
- eISBN:
- 9780191879180
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198714934.003.0007
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy, Mathematical Physics
Linear systems theory, based on the mathematics of vector spaces, is the backbone of all “classical” DSP and a large part of statistical machine learning. The basic idea -- that linear algebra ...
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Linear systems theory, based on the mathematics of vector spaces, is the backbone of all “classical” DSP and a large part of statistical machine learning. The basic idea -- that linear algebra applied to a signal can of substantial practical value -- has counterparts in many areas of science and technology. In other areas of science and engineering, linear algebra is often justified by the fact that it is often an excellent model for real-world systems. For example, in acoustics the theory of (linear) wave propagation emerges from the concept of linearization of small pressure disturbances about the equilibrium pressure in classical fluid dynamics. Similarly, the theory of electromagnetic waves is also linear. Except when a signal emerges from a justifiably linear system, in DSP and machine learning we do not have any particular correspondence to reality to back up the choice of linearity. However, the mathematics of vector spaces, particularly when applied to systems which are time-invariant and jointly Gaussian, is highly tractable, elegant and immensely useful.Less
Linear systems theory, based on the mathematics of vector spaces, is the backbone of all “classical” DSP and a large part of statistical machine learning. The basic idea -- that linear algebra applied to a signal can of substantial practical value -- has counterparts in many areas of science and technology. In other areas of science and engineering, linear algebra is often justified by the fact that it is often an excellent model for real-world systems. For example, in acoustics the theory of (linear) wave propagation emerges from the concept of linearization of small pressure disturbances about the equilibrium pressure in classical fluid dynamics. Similarly, the theory of electromagnetic waves is also linear. Except when a signal emerges from a justifiably linear system, in DSP and machine learning we do not have any particular correspondence to reality to back up the choice of linearity. However, the mathematics of vector spaces, particularly when applied to systems which are time-invariant and jointly Gaussian, is highly tractable, elegant and immensely useful.
Peter B. Moore
- Published in print:
- 2012
- Published Online:
- May 2015
- ISBN:
- 9780199767090
- eISBN:
- 9780190267841
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:osobl/9780199767090.003.0002
- Subject:
- Biology, Biochemistry / Molecular Biology
This chapter explores the mathematical relationships existing between known molecular structures from Chapter 1 and the structure factors, particularly dealing with the Fourier transforms and Fourier ...
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This chapter explores the mathematical relationships existing between known molecular structures from Chapter 1 and the structure factors, particularly dealing with the Fourier transforms and Fourier series. Structure factors are derived from the previous knowledge on molecular structures. in this regard, the chapter explains how to recover the other bits of scientific bases such as the three-dimensional function, shift theorem, scaling theorem, Dirac delta function, convolution theorem, instrument transfer function, autocorrelation theorem, and Rayleigh's theorem. Sample problems are also provided at the end of the chapter.Less
This chapter explores the mathematical relationships existing between known molecular structures from Chapter 1 and the structure factors, particularly dealing with the Fourier transforms and Fourier series. Structure factors are derived from the previous knowledge on molecular structures. in this regard, the chapter explains how to recover the other bits of scientific bases such as the three-dimensional function, shift theorem, scaling theorem, Dirac delta function, convolution theorem, instrument transfer function, autocorrelation theorem, and Rayleigh's theorem. Sample problems are also provided at the end of the chapter.
