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This chapter introduces non-parametric registrations. The idea behind this type of registration is to come up with an appropriate measure both for the similarity as well as for the likelihood of a non-parametric transformation. It sets up a general framework for the consideration of different registration techniques, which is based on a variational formulation of the registration problem; the numerical schemes to be considered are based on the Euler-Lagrange equations which characterize a minimizer.

This chapter discusses partial differential equation models. Partial differential equations can describe the dynamics of phenotype distributions of polymorphic populations, and they allow for a mathematically concise formulation from which some analytical insights can be obtained. It has been argued that because partial differential equations can describe polymorphic populations, results from such models are fundamentally different from those obtained using adaptive dynamics. In partial differential equation models, diversification manifests itself as pattern formation in phenotype distribution. More precisely, diversification occurs when phenotype distributions become multimodal, with the different modes corresponding to phenotypic clusters, or to species in sexual models. Such pattern formation occurs in partial differential equation models for competitive as well as for predator–prey interactions.

This chapter shows how the book's main theory allows for the treatment of extension problems for partial differential equations. The culmination of this is a solution of the asymptotic edge of the wedge theorem — a problem that was posed by Nirenberg and C. Fefferman.

This chapter looks at a variation of “harmonic function” which was introduced by Chevalley. Ordinary harmonic functions are solutions of the Laplace equation; Chevalley harmonic functions are solutions of certain systems of partial differential equations. The equations and the harmonic functions combine to give a tensor product decomposition of the space of functions. This theory is indispensable for extending the concept of Radon transform to algebraic varieties. The geometry related to harmonic functions is discussed in detail.

Introduces some of the methods and underlying ideas behind computational fluid dynamics—in particular, the use is discussed of finite‐difference methods for the simulation of dynamic economies. A standard stochastic dynamic programming model is considered of a macroeconomy. Finite‐difference methods are applied to this problem (model), resulting in a second‐order nonlinear partial differential equation that has some features in common with the governing equations of fluid dynamics; the idea is also introduced of ‘upwind’ or solution‐dependent differencing methods, and the stability of these is discussed through the analysis of model problems. An implicit solution to the nonlinear dynamic programming problem is then developed and tested, with the motivation of reducing the computer time required to solve it. Finally, the extension of the finite‐difference method to a two‐state dynamic programming problem is considered.

This chapter introduces elastic registration using the general framework introduced in Chapter 8. A physical motivation for this particular regularizer is given. The two images are viewed as two different observations of an elastic body, one before and one after deformation. The displacement (or transformation) of the elastic body is derived using a linear elasticity model. In order to introduce different boundary conditions and for later reference, the eigenvalues and eigenfunctions of the Navier-Lame operator are computed. It is shown that the Navier-Lame equations can be derived from the general framework. The continuous Euler-Lagrange equations are discretized and the discretized partial differential operator can be diagonalized using Fast-Fourier type techniques. The Moore-Penrose pseudo inverse of the discrete Navier-Lamé operator is explicitly given. Registration results for academic as well as real life problems are presented.

This chapter reviews chemical kinetics to illustrate the formulation of model equations for a given reaction mechanism. For spatially uniform systems, these model equations are usually ordinary differential equations; but coupling of chemical reactions to physical processes such as diffusion requires the formulation of partial differential equations to describe the spatiotemporal evolution of the system. Mathematical analysis of the dynamical models involves basic concepts from ordinary and partial differential equations. Computational methods, including stochastic simulations and sources of computer software programs available free on the internet are also summarized.

This chapter discusses the complete quadrilateral line arrangement, and especially its relationship with the space of regular points of the system of partial differential equations defining the Appell hypergeometric function. Appell introduced four series F1, F2, F3, F4 in two complex variables, each of which generalizes the classical Gauss hypergeometric series and satisfies its own system of two linear second order partial differential equations. The solution spaces of the systems corresponding to the series F2, F3, F4 all have dimension 4, whereas that of the system corresponding to the series F1 has dimension 3. This chapter focuses on the F1-system whose monodromy group, under certain conditions, acts on the complex 2-ball. It first considers the action of S5 on the blown-up projective plane before turning to Appell hypergeometric functions, arithmetic monodromy groups, and an invariant known as the signature.

Thermal noise and Gaussian noise sources combine to create a category of Markovian processes known as Fokker–Planck processes. This chapter discusses Fokker–Planck processes such as generation-recombination processes, linearly damped processes, Doob's theorem, and multivariable processes. The determination of the behaviour of a Markovian random variable is reduced to the solution of a partial differential equation for the probability that such variable will assume the value at a certain time. In this way, a problem of stochastic processes has been reduced to a more conventional mathematical problem, the solution of a partial differential equation. This chapter also examines drift vectors and diffusion coefficients, the average motion of a general random variable, generalised Fokker–Planck equation, characteristic function, path integral average, linear damping and homogeneous noise, backward equation, extension of the Fokker–Planck equation to many variables, and time reversal in the linear case.

Ordinary and partial differential equations appear in physics as equations of motion or of state. They are often linear differential equations for which a sum of solutions remains a solution. The solution of first- and second-order linear differential equations are obtained. The specification of linearly independent solutions using suitable boundary/initial conditions is discussed. Special methods of solution using Green's functions, separation of variables and eigenfunction expansions are described.

Population density methods have a rich history in theoretical and computational neuroscience. In earlier years, these methods were used in large part to study the statistics of spike trains. Starting in the 1990s population density function (PDF) methods have been used as an analytical and computational tool to study neural network dynamics. In this chapter, we discuss the motivation and theory underlying PDF methods and a few selected examples of computational and analytical applications in neural network modelling.

This chapter focuses on the mathematical structures underlying the notions of stability, bifurcation, and instabilities in complex nonlinear dynamical systems described by sets of ordinary or partial differential equations. It begins by presenting the basic ideas of stability analysis in systems described by ordinary differential equations, introducing Lyapunov functions and their utilization in stability analysis. The stability of systems described by partial differential equations is discussed, emphasizing some of the basic ideas that allow quantitative descriptions of patterns. Specific models illustrating the concepts behind Hopf and Turing instabilities are also discussed.

This chapter gives an outline of the history of wavelets, in particular with respect to their use to numerically solve partial differential equations. The scope of the book and its outline is described.

This chapter, which addresses the partial differential equations (PDEs) with the example of finding the speed and profile of a propagating impulse for a Hodgkin-Huxley-like cable equation, highlights a few properties of differential equations and concepts for understanding them. It mentions that the notions of stability are a crucial aspect of linear autonomous differential equations, and shows that linear autonomous systems have solutions which are sums of exponentials. The chapter suggests that PDEs are important when spatial differences matter—they require both initial and boundary conditions; and certain forms of solutions to PDEs can be reduced to ordinary differential equations (ODEs)—and reviews the methods for solving ODEs using a one-dimensional model.

This chapter presents, in a synthetic way, a series of recent works by X. Blanc, C. Le Bris, and P-L. Lions on homogenization of an elliptic partial differential equation under certain periodic or random assumptions. The coefficients are non-constant but are a stationary random deformation of a periodic set of coefficients; a limit is taken where the period (in d-space) of the periodicity shrinks to zero. The chapter also describes related work on average energies of nonperiodic infinite sets of points.

When physics tells its story of the world, it writes on spatial pages and we flip pages in the temporal directions. The present moment contains the seeds of what happens next. Relativity challenges many of our pre-theoretical thoughts about time, yet even this would-be destroyer of time adheres to the idea that production or determination runs along the set of temporal directions. We might think of this fact as one of the last remnants left of manifest time in physics. Is even this residue of manifest time safe from physics? Looking at the world sideways, can we march “initial” data from “east” to “west” as well as from earlier to later? Or put even more loosely: can physics tell its stories if we write on non-spatial pages and read in non-temporal directions?

The topics discussed are all related to basic fluid mechanics. In these introductory notes I highlight some of the main features of fluid flows and their mathematical characterization. There is much physical intuition encapsulated in the differential equations, and one of our goals is to gain more experience (i) understanding the governing equations and various related principles of kinematics, (ii) developing intuition with approximating the equations, (iii) applying the principles to a wide range of problems, which includes (iv) being able to rationalize scaling laws and quantitative trends, often without having a detailed solution in hand. Where possible we provide examples of the ideas with ‘soft interfaces’ in mind.

The pricing of fixed income securities follows the same general principles as the pricing of all other financial assets. This chapter explains some important general concepts and results in asset pricing theory that are applied in the rest of the book to the term structure of interest rate and the pricing of fixed income securities. The fundamental concepts discussed are arbitrage, state prices, risk-neutral probability measures, market prices of risk, market completeness, and representative agents. For the popular class of diffusion models, asset prices are shown to satisfy certain partial differential equations.