Description
Numbers are fascinating. The fascination begins in childhood,
when we first learn to count, and continues as we learn
arithmetic, algebra, geometry, and so on. Eventually, we find
that numbers not only help us to measure the world, but also to
understand it and, to some extent, to control it. In The
Adventure of Numbers, Gilles Godefroy unfolds a great adventure
of the mind by examining our expanding understanding of numbers
throughout history.
The development of mathematics has been punctuated by a need to
reconsider what we mean by "numbers". It is often
during these times that major shifts occur, for example when the
Pythagoreans discovered irrational numbers or when imaginary
numbers were needed to solve the cubic. Each jump takes place in
a context, where mathematics itself is forced to ponder
fundamental questions, many of which led to famous controversies.
Godefroy's adventure starts in the very early days of mathematics
in Mesopotamia and leads to the present day. The adventure does
not end there. Important questions and controversies remain today
that deal with consistency and complexity or with what
constitutes a proof. And the future will hold even more questions.
The author, Gilles Godefroy, is a member of the Institut de
Mathematiques de Jussieu and Directeur de Recherches at the C.N.R.S.
The book is suitable for independent study and supplementary
reading and is recommended for undergraduates, graduate students,
and researchers interested in the history of mathematics.
Contents
Hands, sticks, and stones
By the waters of Babylon
Let none but geometers enter here
Algebra and algorithms
A new world
"Eppur, si muove!"
The century of revolutions
"From the paradise that Cantor has created for us"...
The present perplexity
And now?
Number bases
The Fibonacci sequence
Polynomials
Quaternions
Axioms of set theory and arithmetic
Glossary
Bibliography
Details:
Series: Mathematical World, Volume: 21
Publication Year: 2004
ISBN: 0-8218-3304-9
Paging: 194 pp.
Binding: Softcover
Description
The space of all Riemann surfaces (the so-called moduli space)
plays an important role in algebraic geometry and its
applications to quantum field theory. The present book is devoted
to the study of topological properties of this space and of
similar moduli spaces, such as the space of real algebraic
curves, the space of mappings, and also superanalogs of all these
spaces.
The book can be used by researchers and graduate students working
in algebraic geometry, topology, and mathematical physics.
Contents
Introduction
Moduli of Riemann surfaces, Hurwitz type spaces and their
superanalogs
Moduli of real algebraic curves and their superanalogs.
Differentials, spinors, and Jacobians of real curves
Spaces of meromorphic functions on complex and real algebraic
curves
Bibliography
Index
Details:
Series: Translations of Mathematical Monographs, Volume: 225
Publication Year: 2004
ISBN: 0-8218-3594-7
Paging: 160 pp.
Binding: Hardcover
Description
Stark's conjectures on the behavior of L-functions were
formulated in the 1970s. Since then, these conjectures and their
generalizations have been actively investigated. This has led to
significant progress in algebraic number theory.
The current volume, based on the conference held at Johns Hopkins
University (Baltimore, MD), represents the state-of-the-art
research in this area. The first four survey papers provide an
introduction to a majority of the recent work related to Stark's
conjectures. The remaining six contributions touch on some major
themes currently under exploration in the area, such as non-abelian
and p-adic aspects of the conjectures, abelian refinements, etc.
Among others, some important contributors to the volume include
Harold M. Stark, John Tate, and Barry Mazur.
The book is suitable for graduate students and researchers
interested in number theory.
Contents
C. D. Popescu -- Rubin's integral refinement of the abelian Stark
conjecture
D. S. Dummit -- Computations related to Stark's conjecture
C. Greither -- Arithmetic annihilators and Stark-type conjectures
M. Flach -- The equivariant Tamagawa number conjecture: A survey
J. W. Sands -- Popescu's conjecture in multi-quadratic extensions
D. Solomon -- Abelian conjectures of Stark type in mathbb{Z}_p-extensions
of totally real fields
H. M. Stark -- The derivative of p-adic Dirichlet series at s=0
J. Tate -- Refining Gross's conjecture on the values of abelian L-functions
D. R. Hayes -- Stickleberger functions for non-abelian Galois
extensions of global fields
B. Mazur and K. Rubin -- Introduction to Kolyvagin systems
Details:
Series: Contemporary Mathematics, Volume: 358
Publication Year: 2004
ISBN: 0-8218-3480-0
Paging: 221 pp.
Binding: Softcover
Description
This is the proceedings of the AMS special session on nonstandard
models of arithmetic and set theory held at the Joint Mathematics
Meetings in Baltimore (MD). The volume opens with an essay from
Haim Gaifman that probes the concept of non-standardness in
mathematics and provides a fascinating mix of historical and
philosophical insights into the nature of nonstandard
mathematical structures. In particular, Gaifman compares and
contrasts the discovery of nonstandard models with other key
mathematical innovations, such as the introduction of various
number systems, the modern concept of function, and non-Euclidean
geometries.
Other articles in the book present results related to nonstandard
models in arithmetic and set theory, including a survey of known
results on the Turing upper bounds of arithmetic sets and
functions. The volume is suitable for graduate students and
research mathematicians interested in logic, especially model
theory.
Contents
H. Gaifman -- Non-standard models in a broader perspective
P. D'Aquino and J. F. Knight -- Coding in IDelta_0
A. Enayat -- Automorphisms, Mahlo cardinals, and NFU
T. Forster -- AC fails in the natural analogues of V and L that
model the stratified fragment of ZF
H. M. Friedman -- Working with nonstandard models
K. Hrbacek -- Internally iterated ultrapowers
R. Jin -- On some questions of Hrbacek and Di Nasso
A. M. McAllister -- Turing upper bounds of jump ideals and Scott
sets
J. H. Schmerl -- Diversity in substructures
A. A. Togha -- Automorphisms of countable recursively saturated
models of set theory
Details:
Series: Contemporary Mathematics,Volume: 361
Publication Year: 2004
ISBN: 0-8218-3535-1
Paging: 167 pp.
Binding: Softcover
Description
This proceedings volume is a collection of articles from the Pan-American
Advanced Studies Institute on partial differential equations,
nonlinear analysis and inverse problems held in Santiago (Chile).
Interactions among partial differential equations, nonlinear
analysis, and inverse problems have produced remarkable
developments over the last couple of decades. This volume
contains survey articles reflecting the work of leading experts
who presented minicourses at the event. Contributors include J.
Busca, Y. Capdeboscq, M.S. Vogelius, F. A. Grunbaum, L. F.
Matusevich, M. de Hoop, and P. Kuchment.
The volume is suitable for graduate students and researchers
interested in partial differential equations and their
applications in nonlinear analysis and inverse problems.
Contents
T. Aktosun -- Inverse scattering on the line with incomplete
scattering data
M. S. Ashbaugh -- On universal inequalities for the low
eigenvalues of the buckling problem
J. Baras, C. Berenstein, and F. Gavilanez -- Continuous and
discrete inverse conductivity problems
R. D. Benguria and M. Loss -- Connection between the Lieb-Thirring
conjecture for Schrodinger operators and an isoperimetric problem
for ovals on the plane
J. Busca -- An introduction to PDE methods in finance
Y. Capdeboscq and M. S. Vogelius -- A review of some recent work
on impedance imaging for inhomogeneities of low volume fraction
F. A. C. C. Chalub and J. P. Zubelli -- Matrix bispectrality and
Huygens' principle for Dirac operators
A. J. Corcho and F. Linares -- Well-posedness for the Schrodinger-Debye
equation
A. Dall'Acqua and G. Sweers -- On domains for which the clamped
plate system is positivity preserving
M. Di Francesco and P. A. Markowich -- Entropy dissipation and
Wasserstein metric methods for the viscous Burgers' equation:
convergence to diffusive waves
J. Dolbeault, D. Kinderlehrer, and M. Kowalczyk -- Remarks about
the flashing rachet
N. Ghoussoub and R. J. McCann -- A least action principle for
steepest descent in a non-convex landscape
F. A. Grunbaum and L. F. Matusevich -- A network tomography
problem related to the hypercube
A. Hassell and J. Wunsch -- On the structure of the Schrodinger
propagator
M. V. de Hoop -- The downward continuation approach to modeling
and inverse scattering of seismic data in the Kirchoff
approximation
M. Kowalczyk -- Approximate invariant manifold of the Allen-Cahn
flow in two dimensions
P. Kuchment -- On some spectral problems of mathematical physics
M. Lassas, L. Paivarinta, and E. Saksman -- Inverse problem for a
random potential
R. Mazzeo -- Pseudodifferential analysis for the Laplacian on
noncompact symmetric spaces
G. A. Mendoza -- Boundary structure and cohomology of b-complex
manifolds
G. Nakamura, G. Uhlmann, and J.-N. Wang -- Unique continuation
property for elliptic systems and crack determination in
anisotropic elasticity
M. del Pino, J. Dolbeault, and M. Musso -- Duality in sub-supercritical
bubbling in the Brezis-Nirenberg problem near the critical
exponent
F. Reitich -- High-order domain variations in boundary value and
free boundary problems
A. S. Barreto -- Radiation fields and inverse scattering on
asymptotically Euclidean manifolds
C. Timofte and C. Conca -- Interactive oscillation sources in
Signorini's type problems
R. Weder -- Inverse scattering with time-periodic potentials
R. Weikard -- A local Borg-Marchenko theorem for difference
equations with complex coefficients
Details:
Series: Contemporary Mathematics, Volume: 362
Publication Year: 2004
ISBN: 0-8218-3448-7
Paging: 410 pp.
Binding: Softcover