Many of the preceding chapters involved optimization formulations: linear least squares, Procrustes, low-rank approximation, multidimensional scaling. All these have analytical solutions, like the pseudoinverse for minimum-norm least squares problems and the truncated singular value decomposition for low-rank approximation. But often we need iterative optimization algorithms, for example if no closed-form minimizer exists, or if the analytical solution requires too much computation and/or memory (e.g., singular value decomposition for large problems. To solve an optimization problem via an iterative method, we start with some initial guess and then the algorithm produces a sequence that hopefully converges to a minimizer. This chapter describes the basics of gradient-based iterative optimization algorithms, including preconditioned gradient descent (PGD) for the linear LS problem. PGD uses a fixed step size, whereas preconditioned steepest descent uses a line search to determine the step size. The chapter then considers gradient descent and accelerated versions for general smooth convex functions. It applies gradient descent to the machine learning application of binary classification via logistic regression. Finally, it summarizes stochastic gradient descent.

The primary assumptions and formulations for single-phase flow regimes are reviewed in this chapter. This includes the governing partial differential equations for general fluid dynamics (mass, momentum, energy, and species), equations of state and associated flow regimes, rotational effects and the stream function for incompressible flow, and viscous effects with the Reynolds number, including flow instability mechanisms.

In the preceding chapters, we have discussed definitions and identification techniques for observed material barriers to the transport of fluid particles, inertial particles and passive scalar fields. All these barriers are directly observable in flow visualizations based on their impact on tracers carried by the flow. However, the transport of several important physical quantities, such as the energy, momentum, angular momentum, vorticity and enstrophy, is also broadly studied but allows no direct experimental visualization. These important scalar and vector quantities are dynamically active fields, i.e., functions of the velocity field and its derivatives.We will collectively refer to barriers to the transport of such fieldsas dynamically active transport barriers. This chapter will be devoted to the development ofan objective notion of active barriers in 3D unsteady velocity data. Additionally, 2D velocity fields can also be handled via this approach by treating them as 3D flows with a symmetry.

We learn about unbound states and find that the energies are no longer quantized. We learn about momentum eigenstates and superposing momentum eigenstates in a wave packet. We apply unbound states to the problem of scattering from potential wells and barriers in one dimension.

This chapter describes methods based on gradient information that achieve faster rates than basic algorithms such as those described in Chapter 3. These accelerated gradient methods, most notably the heavy-ball method and Nesterov’s optimal method, use the concept of momentum which means that each step combines information from recent gradient values but also earlier steps. These methods are described and analyzed using an analysis based on Lyapunov functions. The cases of convex and strongly convex functions are analyzed separately. We motivate these methods using continuous-time limits, which link gradient methods to dynamical systems described by differential equations. We mention also the conjugate gradient method, which was developed separately from the other method but which also makes use of momentum. Finally, we discuss the concept of lower bounds on algorithmic complexity, introducing a function on which no method based on gradients can attain convergence faster than a certain given rate.

The basic quantitative tools to study the hydrodynamics, and any suspended and dissolved matter in water, of semi-enclosed basins are the conservation equations. Conservation of momentum and conservation of mass are used for water motion, while conservation of salt and conservation of heat address their distribution in space and time. An equation of state, the Thermodynamic Equation of Seawater, relates temperature (heat content in the water column), salinity, and pressure to water density, which plays a dynamic role in the conservation of momentum. The conservation of suspended or dissolved matter has an advective contribution that depends on water motion, a diffusive contribution, and a source/sink contribution. The source/sink contribution represents the greatest uncertainty in water-related studies.

Having described motion, we can now explain it. We introduce the conserved 4-momentum, and with it the ideas of energy-momentum, conserved mass, and scattering.

Understanding the purely legal point of view – which Holmes ultimately personified as the“bad man” – depends on appreciating the temporal dimensions of existence. Working off James Willard Hurst’s Justice Holmes on Legal History, this part attempts to lay the basis for a three-dimensional view of law. Hurst’s ideas of sequence and context lead to a greater appreciation of law’s horizontal and vertical dimensions, and provide the basis for appreciating its depth and vitality. A three-dimensional view of law brings law to present purpose when tied to what Holmes called the law’s fundamental theorem: the philosophical idea that any possible conduct is either lawful or unlawful. From this starting point it might one day be possible to create a metric of the law, one that would materially expand the law’s capacity to be known.

Chapter 4 introduces basic differential equations and boundary conditions for gravity waves propagating along a water surface. Assuming low wave amplitudes, equations are linearised. Then a quantitative discussion is given for harmonical (sinusoidal) waves propagating either on deep water, or otherwise on water of constant depth. Phase and group velocities are introduced, and then formulas are derived for the potential energy and the kinetic energy associated with a water wave. A closely related result is an important formula for the wave-power level, which equals the wave’s group velocity multiplied by the wave’s stored – kinetic + potential – energy per unit of sea surface. An additional subject is the wave’s momentum density. A section concerns real sea waves. Further, circular waves are mathematically described. Two sections of the chapter concern mathematical tools to be applied in Chapters 5–8 of the book. A final section considers water waves analysed in the time domain.

This chapter summarises the key aerodynamic theory of horizontal-axis wind turbine rotors. The actuator disc concept leads to the relationships between induced velocity, axial thrust, and power extraction. The theory is extended to multiple streamtubes, which, combined with 2D wing theory, establish the basis of blade-element momentum (BEM) theory. A straightforward mathematical treatment of BEM theory is included, with an iterative procedure suitable for coding. Measurements from a full-scale rotor illustrate the applicability of BEM theory but also its fundamental limitations: the latter are described, and measures are outlined to compensate for them in practical BEM codes. Simple relationships are given for the axial and tangential load distributions on an optimal HAWT blade. The structure of the rotor wake is described, leading into a description of vortex-wake theory, which provides a more physically realistic description of the airflow. Vortex wake codes are described in non-mathematical terms. The chapter includes wake measurements from full-scale wind turbines and small models. Vorticity maps from the latter verify the underlying mathematical model of a helical vortex wake.