*Simon Scott*

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780198568360
- eISBN:
- 9780191594748
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568360.001.0001
- Subject:
- Mathematics, Analysis

This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant ...
More

This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant functionals on geometric operator algebras provide a means of constructing refined invariants in analysis, topology, differential geometry, analytic number theory and QFT. The consequent interactions around such invariants have led to significant advances both in pure mathematics and theoretical physics. As the fundamental tools of trace theory have become well understood and clear general structures have emerged, so the need for specialist texts which explain the basic theoretical principles and the computational techniques has become increasingly exigent. This text is the first to deal with the general theory of traces and determinants of operators on manifolds in a broad context, encompassing a number of the principle applications and backed up by specific computations which set out in detail to newcomers the nuts-and-bolts of the basic theory. Both the microanalytic approach to traces and determinants via pseudodifferential operator theory and the more computational approach directed by applications in geometric analysis, are developed in a general framework that will be of interest to mathematicians and physicists in a number of different fields.Less

This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant functionals on geometric operator algebras provide a means of constructing refined invariants in analysis, topology, differential geometry, analytic number theory and QFT. The consequent interactions around such invariants have led to significant advances both in pure mathematics and theoretical physics. As the fundamental tools of trace theory have become well understood and clear general structures have emerged, so the need for specialist texts which explain the basic theoretical principles and the computational techniques has become increasingly exigent. This text is the first to deal with the general theory of traces and determinants of operators on manifolds in a broad context, encompassing a number of the principle applications and backed up by specific computations which set out in detail to newcomers the nuts-and-bolts of the basic theory. Both the microanalytic approach to traces and determinants via pseudodifferential operator theory and the more computational approach directed by applications in geometric analysis, are developed in a general framework that will be of interest to mathematicians and physicists in a number of different fields.

*Peter Davidson*

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780198722588
- eISBN:
- 9780191789298
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198722588.001.0001
- Subject:
- Mathematics, Applied Mathematics, Mathematical Physics

This book presents the subject of turbulence. The aim of the book is to bridge the gap between the elementary, heuristic accounts of turbulence and the more rigorous accounts given. Throughout, the ...
More

This book presents the subject of turbulence. The aim of the book is to bridge the gap between the elementary, heuristic accounts of turbulence and the more rigorous accounts given. Throughout, the book combines the maximum of physical insight with the minimum of mathematical detail. This second edition covers a decade of advancement in the field, streamlining the original content while updating the sections where the subject has moved on. The expanded content includes large-scale dynamics, stratified & rotating turbulence, the increased power of direct numerical simulation, two-dimensional turbulence, Magnetohydrodynamics, and turbulence in the core of the Earth.Less

This book presents the subject of turbulence. The aim of the book is to bridge the gap between the elementary, heuristic accounts of turbulence and the more rigorous accounts given. Throughout, the book combines the maximum of physical insight with the minimum of mathematical detail. This second edition covers a decade of advancement in the field, streamlining the original content while updating the sections where the subject has moved on. The expanded content includes large-scale dynamics, stratified & rotating turbulence, the increased power of direct numerical simulation, two-dimensional turbulence, Magnetohydrodynamics, and turbulence in the core of the Earth.

*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.

*Karsten Urban*

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780198526056
- eISBN:
- 9780191712340
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526056.001.0001
- Subject:
- Mathematics, Applied Mathematics, Mathematical Finance

Wavelets have become a powerful tool in several applications by now. Their use for the numerical solution of operator equations has been investigated more recently. By now the theoretical ...
More

Wavelets have become a powerful tool in several applications by now. Their use for the numerical solution of operator equations has been investigated more recently. By now the theoretical understanding of such methods is quite advanced and has brought up deep results and additional understanding. Moreover, the rigorous theoretical foundation of wavelet bases has also lead to new insights in more classical numerical methods for partial differential equations (pde's) such as Finite Elements. However, sometimes it is believed that understanding and applying the full power of wavelets needs a strong mathematical background in functional analysis and approximation theory. The main idea of this book is to introduce the main concepts and results of wavelet methods for solving linear elliptic partial differential equations in a framework that allows avoiding technicalities to a maximum extend. On the other hand, the book also describes recent research including adaptive methods also for nonlinear problems, wavelets on general domains and applications.Less

Wavelets have become a powerful tool in several applications by now. Their use for the numerical solution of operator equations has been investigated more recently. By now the theoretical understanding of such methods is quite advanced and has brought up deep results and additional understanding. Moreover, the rigorous theoretical foundation of wavelet bases has also lead to new insights in more classical numerical methods for partial differential equations (pde's) such as Finite Elements. However, sometimes it is believed that understanding and applying the full power of wavelets needs a strong mathematical background in functional analysis and approximation theory. The main idea of this book is to introduce the main concepts and results of wavelet methods for solving linear elliptic partial differential equations in a framework that allows avoiding technicalities to a maximum extend. On the other hand, the book also describes recent research including adaptive methods also for nonlinear problems, wavelets on general domains and applications.