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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.)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.