*Simon Scott*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568360.003.0002
- Subject:
- Mathematics, Analysis

In this Chapter examples of trace structures are given along with some elements of trace theory in general. These facts and examples are called upon and, in the specific case of pseudodifferential ...
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In this Chapter examples of trace structures are given along with some elements of trace theory in general. These facts and examples are called upon and, in the specific case of pseudodifferential operators, developed in more detail in subsequent chapters. The emphasis in this chapter is to provide a broad overview of trace theory and specifically to point out the key ideas on traces of pseudodifferential operators, indicating the relevant sections later in the book for detailed proofs and examples.Less

In this Chapter examples of trace structures are given along with some elements of trace theory in general. These facts and examples are called upon and, in the specific case of pseudodifferential operators, developed in more detail in subsequent chapters. The emphasis in this chapter is to provide a broad overview of trace theory and specifically to point out the key ideas on traces of pseudodifferential operators, indicating the relevant sections later in the book for detailed proofs and examples.

*Simon Scott*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568360.003.0004
- Subject:
- Mathematics, Analysis

In this chapter the constructions of chapters 1 and 2 for are illustrated with explicit computations. The first part of the chapter presents various methods of computation commonly used in the ...
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In this chapter the constructions of chapters 1 and 2 for are illustrated with explicit computations. The first part of the chapter presents various methods of computation commonly used in the literature and which have acted as guide posts to the development of trace and determinant methods in geometric analysis over the past few decades. There are basic equivalences, indispensable in both theoretical developments and exact computations, between coefficients in the asymptotic expansions of zeta traces, heat traces and resolvent traces. The middle part of the chapter presents and proves these identifications, and provides an application to the computation of relative zeta determinants. The final part of the chapter turns to residue trace and residue determinant computations. Based on a residue determinant formula for the spectral zeta function at zero, an elementary proof of the local Atiyah Singer index theorem is given.Less

In this chapter the constructions of chapters 1 and 2 for are illustrated with explicit computations. The first part of the chapter presents various methods of computation commonly used in the literature and which have acted as guide posts to the development of trace and determinant methods in geometric analysis over the past few decades. There are basic equivalences, indispensable in both theoretical developments and exact computations, between coefficients in the asymptotic expansions of zeta traces, heat traces and resolvent traces. The middle part of the chapter presents and proves these identifications, and provides an application to the computation of relative zeta determinants. The final part of the chapter turns to residue trace and residue determinant computations. Based on a residue determinant formula for the spectral zeta function at zero, an elementary proof of the local Atiyah Singer index theorem is given.

*Orri Vésteinsson*

- Published in print:
- 2000
- Published Online:
- October 2011
- ISBN:
- 9780198207993
- eISBN:
- 9780191677885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198207993.003.0007
- Subject:
- History, World Medieval History, History of Religion

This chapter discusses priesthood as a stepping-stone needed for the development of complex power in Iceland. A great number of chieftains were priests and church owners in the early and mid-twelfth ...
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This chapter discusses priesthood as a stepping-stone needed for the development of complex power in Iceland. A great number of chieftains were priests and church owners in the early and mid-twelfth century. Owning churches and being priests helped chieftains to consolidate their powers. They forged strong ties with the households who lived closest to them by providing services for them and receiving tithes from them.Less

This chapter discusses priesthood as a stepping-stone needed for the development of complex power in Iceland. A great number of chieftains were priests and church owners in the early and mid-twelfth century. Owning churches and being priests helped chieftains to consolidate their powers. They forged strong ties with the households who lived closest to them by providing services for them and receiving tithes from them.