*Leon Ehrenpreis*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198509783
- eISBN:
- 9780191709166
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509783.001.0001
- Subject:
- Mathematics, Mathematical Physics

Radon showed how to write arbitrary functions in Rn in terms of the characteristic functions (delta functions) of hyperplanes. This idea leads to various generalizations. For example, R can be ...
More

Radon showed how to write arbitrary functions in Rn in terms of the characteristic functions (delta functions) of hyperplanes. This idea leads to various generalizations. For example, R can be replaced by a more general group and “plane” can be replaced by other types of geometric objects. All this is for the “nonparametric” Radon transform. For the parametric Radon transform, this book parametrizes the points of the geometric objects, leading to differential equations in the parameters because the Radon transform is overdetermined. Such equations were first studied by F. John. This book elaborates on them and puts them in a general framework.Less

Radon showed how to write arbitrary functions in R^{n} in terms of the characteristic functions (delta functions) of hyperplanes. This idea leads to various generalizations. For example, R can be replaced by a more general group and “plane” can be replaced by other types of geometric objects. All this is for the “nonparametric” Radon transform. For the parametric Radon transform, this book parametrizes the points of the geometric objects, leading to differential equations in the parameters because the Radon transform is overdetermined. Such equations were first studied by F. John. This book elaborates on them and puts them in a general framework.

*Matt Clay and Dan Margalit*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses the notion of space, first by explaining what it means for a group to be a group of symmetries of a geometric object. This is the idea of group action, and some examples are ...
More

This chapter discusses the notion of space, first by explaining what it means for a group to be a group of symmetries of a geometric object. This is the idea of group action, and some examples are given. The chapter proceeds by defining, for any group G, the Cayley graph of G and shows that the symmetric group of of this graph is precisely the group G. It then introduces metric spaces, which formalize the notion of a geometric object, and highlights numerous metric spaces that groups can act on. It also demonstrates that groups themselves are metric spaces; in other words, groups themselves can be thought of as geometric objects. The chapter concludes by using these ideas to frame the motivating questions of geometric group theory. Exercises relevant to each idea are included.Less

This chapter discusses the notion of space, first by explaining what it means for a group to be a group of symmetries of a geometric object. This is the idea of group action, and some examples are given. The chapter proceeds by defining, for any group *G*, the Cayley graph of *G* and shows that the symmetric group of of this graph is precisely the group *G*. It then introduces metric spaces, which formalize the notion of a geometric object, and highlights numerous metric spaces that groups can act on. It also demonstrates that groups themselves are metric spaces; in other words, groups themselves can be thought of as geometric objects. The chapter concludes by using these ideas to frame the motivating questions of geometric group theory. Exercises relevant to each idea are included.

*David P. Feldman*

- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This book provides an elementary introduction to chaos and fractals. It introduces the key phenomena of chaos — aperiodicity, sensitive dependence on initial conditions, bifurcations — via simple ...
More

This book provides an elementary introduction to chaos and fractals. It introduces the key phenomena of chaos — aperiodicity, sensitive dependence on initial conditions, bifurcations — via simple iterated functions. Fractals are introduced as self-similar geometric objects and analysed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia sets and the Mandelbrot set. The last part of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations.Less

This book provides an elementary introduction to chaos and fractals. It introduces the key phenomena of chaos — aperiodicity, sensitive dependence on initial conditions, bifurcations — via simple iterated functions. Fractals are introduced as self-similar geometric objects and analysed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia sets and the Mandelbrot set. The last part of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations.