Series: Sources and Studies in the History of Mathematics and
Physical Sciences
2004, Approx. 170 p. 50 illus., Hardcover
ISBN: 0-387-20709-0
About this book
John Wallis was appointed Savilian Professor of Geometry at
Oxford University in 1649. He was then a relative newcomer to
mathematics, and largely self-taught, but in his first few years
at Oxford he produced his two most significant works: De
sectionibus conicis and Arithmetica infinitorum. In both books,
Wallis drew on ideas originally developed in France, Italy, and
the Netherlands: analytic geometry and the method of indivisibles.
He handled them in his own way, and the resulting method of
quadrature, based on the summation of indivisible or
infinitesimal quantities, was a crucial step towards the
development of a fully fledged integral calculus some ten years
later. To the modern reader, the Arithmetica Infinitorum reveals
much that is of historical and mathematical interest, not least
the mid seventeenth-century tension between classical geometry on
the one hand, and arithmetic and algebra on the other. Newton was
to take up Wallis?s work and transform it into mathematics that
has become part of the mainstream, but in Wallis?s text we see
what we think of as modern mathematics still struggling to emerge.
It is this sense of watching new and significant ideas force
their way slowly and sometimes painfully into existence that
makes the Arithmetica Infinitorum such a relevant text even now
for students and historians of mathematics alike. Dr J.A. Stedall
is a Junior Research Fellow at Queen's University. She has
written a number of papers exploring the history of algebra,
particularly the algebra of the sixteenth and seventeenth
centuries. Her two previous books, A Discourse Concerning Algebra:
English Algebra to 1685 (2002) and The Greate Invention of
Algebra: Thomas Harriot's Treatise on Equations (2003), were both
published by Oxford University Press.
Table of contents
* Acknowledgements * Introduction by J. A. Stedall * To the Most
Distinguished and Worthy Gentleman and Most Skilled
Mathematician, Dr. William Oughtre, Rector of the Church of
Aldbury in the County of Surrey * To the Most Respected Gentleman
Doctor William Oughtred,Most Widely Famed Amongst Mathematicians,
by John Wallis, Savilian Professor Of Geometry at Oxford *
Glossary * Bibliography
Series: Universitext
2004, Approx. 300 p., Softcover
ISBN: 3-540-20921-2
Analysis II
Series: Universitext
Godement, Roger
2004, Approx. 300 p., Softcover
ISBN: 3-540-20921-2
About this textbook
Functions in R and C, including the theory of Fourier series,
Fourier integrals and part of that of holomorphic functions, form
the focal topic of these two volumes. Based on a course given by
the author to large audiences at Paris VII University for many
years, the exposition proceeds somewhat nonlinearly, blending
rigorous mathematics skilfully with didactical and historical
considerations. It sets out to illustrate the variety of possible
approaches to the main results, in order to initiate the reader
to methods, the underlying reasoning, and fundamental ideas. It
is suitable for both teaching and self-study. In his familiar,
personal style, the author emphasizes ideas over calculations
and, avoiding the condensed style frequently found in textbooks,
explains these ideas without parsimony of words. The French
edition in four volumes, published from 1998, has met with
resounding success: the first two volumes are now available in
English.
Written for:
Undergraduate students, and lecturers in analysis
Keywords:
Convergence
derivations
integrals
Series: Grundlehren der mathematischen Wissenschaften, Vol.
329
2004, XV, 483 p., Hardcover
ISBN: 3-540-21058-X
About this book
Tauberian theory compares summability methods for series and
integrals, helps to decide when there is convergence, and
provides asymptotic and remainder estimates. The author shows the
development of the theory from the beginning and his expert
commentary evokes the excitement surrounding the early results.
He shows the fascination of the difficult Hardy-Littlewood
theorems and of an unexpected simple proof, and extolls Wiener's
breakthrough based on Fourier theory. There are the spectacular
"high-indices" theorems and Karamata's "regular
variation", which permeates probability theory. The author
presents Gelfand's elegant algebraic treatment of Wiener theory
and his own distributional approach. There is also a new unified
theory for Borel and "circle" methods. The text
describes many Tauberian ways to the prime number theorem. A
large bibliography and a substantial index round out the book.
Table of contents
The Hardy-Littlewood Theorems.- Wiener's Theory.- Complex
Tauberian Theorems.- Karamata's Heritage: Regular Variation.-
Extensions of the Classical Theory.- Borel Summability and
General Circle Methods.- Tauberian Remainder Theory.- References.-
Index.
2004, Approx. 300 pp., Hardcover
ISBN: 3-540-43405-4
About this book
This is the English translation of Bourbaki's text Groupes et
Algebres de Lie, Chapters 7 to 9. It completes the previously
published translations of Chapters 1 to 3 (3-540-64242-0) and 4
to 6 (3-540-42650-7) by covering the structure and representation
theory of semi-simple Lie algebras and compact Lie groups.
Chapter 7 deals with Cartan subalgebras of Lie algebras, regular
elements and conjugacy theorems. Chapter 8 begins with the
structure of split semi-simple Lie algebras and their root
systems. It goes on to describe the finite-dimensional modules
for such algebras, including the character formula of Hermann
Weyl. It concludes with the theory of Chevalley orders. Chapter 9
is devoted to the theory of compact Lie groups, beginning with a
discussion of their maximal tori, root systems and Weyl groups.
It goes on to describe the representation theory of compact Lie
groups, including the application of integration to establish
Weyl's formula in this context. The chapter concludes with a
discussion of the actions of compact Lie groups on manifolds. The
nine chapters together form the most comprehensive text available
on the theory of Lie groups and Lie algebras.
Written for:
Mathematicians and graduate students
Keywords:
Lie algebra
Lie groups
harmonic analysis
representation theory
root system