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Statistical Shape Theory

Wilfrid S. Kendall and Huiling Le

in New Perspectives in Stochastic Geometry

There are many variations on what one may regard as statistical shape, depending on the application in mind. The focus of this chapter is the statistical analysis of the shapes determined by finite ... More

There are many variations on what one may regard as statistical shape, depending on the application in mind. The focus of this chapter is the statistical analysis of the shapes determined by finite sequences of points in a Euclidean space. We shall draw together a range of ideas from statistical shape theory, including distributions, diffusions, estimations and computations, emphasizing the role played by the underlying geometry. Applications in selected areas of current interest will be discussed.Less

Complex-valued harmonic morphisms on three-dimensional Euclidean space

Paul Baird and John C. Wood

in Harmonic Morphisms Between Riemannian Manifolds

This chapter presents an introduction to the theory of harmonic morphisms for the case of maps from open subsets of Euclidean 3-space to the complex plane. Harmonic morphisms are characterized by ... More

This chapter presents an introduction to the theory of harmonic morphisms for the case of maps from open subsets of Euclidean 3-space to the complex plane. Harmonic morphisms are characterized by elementary means which involve only a little simple geometry. In subsequent chapters some of the results are generalized and interpreted the context of differential geometry. Some results apply to maps from open subsets of higher-dimensional Euclidean spaces to the complex plane, but many methods are special to Euclidean 3-space, in that case, giving us all harmonic morphisms both locally and globally.Less

THE CONSTRUCTIVE UNIQUENESS OF THE LOCALLY CONVEX TOPOLOGY ON ℝ N

Douglas Bridges and Luminita Vîtă

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

This chapter proves constructively that every n-dimensional real locally convex space is homeomorphic to ℝn. Although essentially nonconstructive theorems of real analysis are used in the customary ... More

This chapter proves constructively that every n-dimensional real locally convex space is homeomorphic to ℝn. Although essentially nonconstructive theorems of real analysis are used in the customary proofs of the uniqueness of the topology on the real Euclidean space of a fixed dimension, this chapter succeeds in avoiding any fragment of the law of excluded middle, at least when the given topology is locally convex. The proof uses barely more than elementary geometric reasoning combined with approximation techniques typical of constructive analysis, but contains an algorithm for computing the desired homeomorphism. This algorithm could, in principle, be extracted and implemented by any of the several systems designed for such a purpose.Less

Preliminaries to Main Results

Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav

in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of ... More

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of n-dimensional surfaces. The idea is to deform a flat surface passing through a point x (along which we imagine that a certain function f is almost affine) to a surface passing through a point witnessing that f is not Fréchet differentiable at x. This is done in such a way that certain “energy” associated to surfaces increases less than the “energy” of the functionf along the surface. The chapter also discusses linear operators and tensor products, various notions and notation related to Fréchet differentiability, and deformation of surfaces controlled by ω‎ⁿ. Finally, it proves some integral estimates of derivatives of Lipschitz maps between Euclidean spaces (not necessarily of the same dimension).Less

Coxeter Groups

Adam Piggott

in Office Hours with a Geometric Group Theorist

This chapter considers Coxeter groups and how to find a space on which a group acts by building a space using combinatorics from the group. It first describes groups generated by reflections, ... More

This chapter considers Coxeter groups and how to find a space on which a group acts by building a space using combinatorics from the group. It first describes groups generated by reflections, focusing on Euclidean spaces and showing that some natural, beautiful, and important subsets of Euclidean spaces have symmetric groups that are discrete and are generated by reflections. It then explores discrete groups generated by reflections, beginning with irreducible finite groups generated by reflections followed by infinite reflection groups. It also looks at relations in finite groups generated by reflections before concluding with an analysis of special subgroups of the Coxeter group and how to construct a geometric space for a Coxeter group. The discussion includes exercises and research projects.Less

The Standard Metric

Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss

in Descent in Buildings (AM-190)

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine ... More

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.Less

The Euclidean Functional Integrals

Laurent Baulieu, John Iliopoulos, and Roland Sénéor

in From Classical to Quantum Fields

The Wick rotation and the functional integral in Euclidean space. Some mathematical theorems. Perturbation theory and Feynman rules in Euclidean space.

The action functional and the harmonic oscillator

Jean-Michel Bismut

in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

This chapter solves explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean vector space. ... More

This chapter solves explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean vector space. It depends on the parameter b > 0. The behavior of the minimum values as well as of the minimizing trajectories is studied when b → 0 and when b → +∞. Finally, certain heat kernels are computed in terms of the minimum value of the action. The aforementioned variational problem has already been considered previously as a warm up to the more general problem on Riemannian manifolds, in order to study in detail the small time asymptotics of the hypoelliptic heat kernel.Less

Minkowski spacetime

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

This chapter presents the main features of the Minkowski spacetime, which is the geometrical framework in which the laws of relativistic dynamics are formulated. It is a very simple mathematical ... More

This chapter presents the main features of the Minkowski spacetime, which is the geometrical framework in which the laws of relativistic dynamics are formulated. It is a very simple mathematical extension of three-dimensional Euclidean space. In special relativity, ‘relative, apparent, and common’ (in the words of Newton) space and time are represented by a mathematical set of points called events, which constitute the Minkowski spacetime. This chapter also stresses the interpretation of the fourth dimension, which in special relativity is time. Here, time now loses the ‘universal’ and ‘absolute’ nature that it had in the Newtonian theory.Less

A review: The Laplacian and the d’Alembertian

Christopher D. Sogge

in Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

This chapter reviews the Laplacian and the d'Alembertian. It begins with a brief discussion on the solution of wave equation both in Euclidean space and on manifolds and how this knowledge can be ... More

This chapter reviews the Laplacian and the d'Alembertian. It begins with a brief discussion on the solution of wave equation both in Euclidean space and on manifolds and how this knowledge can be used to derive properties of eigenfunctions on Riemannian manifolds. A key step in understanding properties of solutions of wave equations on manifolds is to compute the types of distributions that include the fundamental solution of the wave operator in Minkowski space (d'Alembertian), with a specific function for the Euclidean Laplacian on Rn. The chapter also reviews another equation involving the Laplacian, before discussing the fundamental solutions of the d'Alembertian in R1+n.Less

Wright-Fisher Geometry

Charles L. Epstein and Rafe1 Mazzeo

in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

This chapter introduces the geometric preliminaries needed to analyze generalized Kimura diffusions, with particular emphasis on Wright–Fisher geometry. It begins with a discussion of the natural ... More

This chapter introduces the geometric preliminaries needed to analyze generalized Kimura diffusions, with particular emphasis on Wright–Fisher geometry. It begins with a discussion of the natural domains of definition for generalized Kimura diffusions: polyhedra in Euclidean space or, more generally, abstract manifolds with corners. Amongst the convex polyhedra, the chapter distinguishes the subclass of regular convex polyhedra P. P is a regular convex polyhedron if it is convex and if near any corner, P is the intersection of no more than N half-spaces with corresponding normal vectors that are linearly independent. These definitions establish that any regular convex polyhedron is a manifold with corners. The chapter concludes by defining the general class of elliptic Kimura operators on a manifold with corners P and shows that there is a local normal form for any operator L in this class.Less

Second Order and Gaussian Fields

Ulf Grenander and Michael I. Miller

in Pattern Theory: From representation to inference

This chapter studies second order and Gaussian fields on the background spaces which are the continuum limits of the finite graphs. For this random processes in ... More

This chapter studies second order and Gaussian fields on the background spaces which are the continuum limits of the finite graphs. For this random processes in Hilbert spaces are examined. Orthogonal expansions such as Karhunen–Loeve are examined, with spectral representations of the processes established. Gaussian processes induced by differential operators representing physical processes in the world are studied. Less

Intensionality

E. A. Ashcroft, A. A. Faustini, R. Jaggannathan, and W. W. Wadge

in Multidimensional Programming

The intensional programming language, Lucid, described in Chapter 1 is based directly on intensional logic, a family of mathematical formal systems that permit ... More

The intensional programming language, Lucid, described in Chapter 1 is based directly on intensional logic, a family of mathematical formal systems that permit expressions whose value depends on hidden contexts or indices. Our use of intensional logic is one in which the hidden contexts or indices are integers or tuples of integers. Intensional logic, as used to give semantics to natural language, uses a much more general notion of context or index. Of course, intensional logic is hardly the first example of a formal system of interest to both logicians and computer scientists. The language LISP (invented by McCarthy and others in the early sixties [34]) was originally intended to be an adaptation of the lambda calculus, although it diverged in its treatment of variable-binding and higher-order functions. Shortly after, however, Landin produced ISWIM, the first true functional language [30]. These “logical” programming languages such as ISWIM are in many respects vastly superior to the more conventional ones. They are much simpler and better defined and yet at the same time more regular and more powerful. These languages are notationally closer to ordinary mathematics and are much more problem-oriented. Finally, programs are still expressions in a formal system, and are still subject to the rules of the formal system. It is therefore much easier to reason formally about their correctness, or to apply meaningpreserving transformations. With these languages, programming really is a respectable branch of applied mathematical logic. These logic-based (or declarative) languages at first proved difficult to implement efficiently, and interest in declarative languages declined soon after the promising initial work of McCarthy and Landin. Fortunately, the advent of large scale integration and new compiling technology reawakened interest in declarative languages, and brought about a series of new “second generation” declarative languages, such as Prolog [12] and Miranda [44]. Lucid itself was one of these second generation declarative languages. Lucid is based not so much on classical logical systems as on the possible worlds approach to intensional logic—itself a relatively new branch of logic [43] which reached maturity during the period (1965-75) in which declarative programming languages were in eclipse. Less