Cloth | February 2004 | ISBN: 0-691-05853-9
256 pp. | 6 x 9
Lowenheim's theorem reflects a critical point
in the history of
mathematical logic, for it marks the birth
of model theory--that
is, the part of logic that concerns the relationship
between
formal theories and their models. However,
while the original
proofs of other, comparably significant theorems
are well
understood, this is not the case with Lowenheim's
theorem. For
example, the very result that scholars attribute
to Lowenheim
today is not the one that Skolem--a logician
raised in the
algebraic tradition, like Lowenheim--appears
to have attributed
to him. In The Birth of Model Theory, Calixto
Badesa provides
both the first sustained, book-length analysis
of Lowenheim's
proof and a detailed description of the theoretical
framework--and,
in particular, of the algebraic tradition--that
made the theorem
possible.
Badesa's three main conclusions amount to
a completely new
interpretation of the proof, one that sharply
contradicts the
core of modern scholarship on the topic.
First, Lowenheim did not
use an infinitary language to prove his theorem;
second, the
functional interpretation of Lowenheim's
normal form is
anachronistic, and inappropriate for reconstructing
the proof;
and third, Lowenheim did not aim to prove
the theorem's weakest
version but the stronger version Skolem attributed
to him. This
book will be of considerable interest to
historians of logic,
logicians, philosophers of logic, and philosophers
of mathematics.
Calixto Badesa is Associate Professor of
Logic and History of
Logic at the University of Barcelona.
Endorsements:
"A first-rate contribution to the history
and philosophy of
logic, this is scholarship at its best. It
is, to my knowledge,
the first book in the history of logic that
focuses completely on
a single result. Very original in approach
and conception, it
goes against the grain of much recent scholarship.
Given the
complexity of the subject, Badesa could not
have done a better
job of being clear and making the presentation
accessible."--Paolo
Mancosu, University of California, Berkeley
"The Birth of Model Theory represents
a long overdue, in-depth
analysis and exposition of one of the most
important results in
mathematical logic. There are hardly any
informed, sustained
treatments of Lowenheim's work to be found
in the literature.
This well-written book should fill this gap."--Richard
Zach,
University of Calgary
"This book will be extremely useful
to those seeking to make
sense of Lowenheim's work and those seeking
to put it into its
historical context. Calixto Badesa draws
well-supported
conclusions that contradict the entire modern
body of scholarship
on the topic."--Shaughan Lavine, University
of Arizona
Paper | February 2004 | ISBN: 0-691-11899-X
Cloth | February 2004 | ISBN: 0-691-11898-1
376 pp. | 6 x 9
This book provides the first unified examination
of the
relationship between Radon transforms on
symmetric spaces of
compact type and the infinitesimal versions
of two fundamental
rigidity problems in Riemannian geometry.
Its primary focus is
the spectral rigidity problem: Can the metric
of a given
Riemannian symmetric space of compact type
be characterized by
means of the spectrum of its Laplacian? It
also addresses a
question rooted in the Blaschke problem:
Is a Riemannian metric
on a projective space whose geodesics are
all closed and of the
same length isometric to the canonical metric?
The authors comprehensively treat the results
concerning Radon
transforms and the infinitesimal versions
of these two problems.
Their main result implies that most Grassmannians
are spectrally
rigid to the first order. This is particularly
important, for
there are still few isospectrality results
for positively curved
spaces and these are the first such results
for symmetric spaces
of compact type of rank >1. The authors
exploit the theory of
overdetermined partial differential equations
and harmonic
analysis on symmetric spaces to provide criteria
for
infinitesimal rigidity that apply to a large
class of spaces.
A substantial amount of basic material about
Riemannian geometry,
symmetric spaces, and Radon transforms is
included in a clear and
elegant presentation that will be useful
to researchers and
advanced students in differential geometry.
Jacques Gasqui is Professor of Mathematics
at Institut Fourier,
Universite de Grenoble I. Hubert Goldschmidt
is Visiting
Professor of Mathematics at Columbia University
and Professeur
des Universites in France.
Series:
Annals of Mathematics Studies
*
Cloth | February 2004 | ISBN: 0-691-11733-0
160 pp. | 6 x 9 | 16 line illus.
This book provides a clear and authoritative
introduction to the
theory of buildings, a topic of central importance
to
mathematicians interested in the geometric
aspects of group
theory. Its detailed presentation makes it
suitable for graduate
students as well as specialists. Richard
Weiss begins with an
introduction to Coxeter groups and goes on
to present basic
properties of arbitrary buildings before
specializing to the
spherical case. Buildings are described throughout
in the
language of graph theory.
The Structure of Spherical Buildings includes
a reworking of the
proof of Jacques Tits's Theorem 4.1.2. upon
which Tits's
classification of thick irreducible spherical
buildings of rank
at least three is based. In fact, this is
the first book to
include a proof of this famous result since
its original
publication. Theorem 4.1.2 is followed by
a systematic study of
the structure of spherical buildings and
their automorphism
groups based on the Moufang property. Moufang
buildings of rank
two were recently classified by Tits and
Weiss. The last chapter
provides an overview of the classification
of spherical
buildings, one that reflects these and other
important
developments.
Richard M. Weiss is William Walker Professor
at Tufts University.
He is the coauthor, with Jacques Tits, of
Moufang Polygons.
Endorsements:
"This is the best currently available
introduction to the
theory of buildings. And it brings the reader
to a very important
theorem in the theory of spherical buildings.
Moreover, it is
very carefully written: obviously the author
spent quite some
time arranging the different results in the
right order, which
isn't a straightforward task. As for explaining
the really hard
part of the classification of spherical buildings,
this book is a
perfect complement to the existing literature."--Hendrik
Van
Maldeghem, Ghent University
"This book makes an important contribution
to the theory of
buildings, developing some material in ways
that were unavailable
until recently. Weiss is an expert in the
field who has produced
a book that will be useful to anyone learning
the subject."--Mark
A. Ronan, University of Illinois at Chicago
and University
College London
Cloth | April 2004 | ISBN: 0-691-09987-1
208 pp. | 6 x 9 | 70 line illus.
Many industries, such as transportation and
manufacturing, use
control systems to insure that parameters
such as temperature or
altitude behave in a desirable way over time.
For example, pilots
need assurance that the plane they are flying
will maintain a
particular heading. An integral part of control
systems is a
mechanism for failure detection to insure
safety and reliability.
This book offers an alternative failure detection
approach that
addresses two of the fundamental problems
in the safe and
efficient operation of modern control systems:
failure detection--deciding
when a failure has occurred--and model identification--deciding
which kind of failure has occurred. Much
of the work in both
categories has been based on statistical
methods and under the
assumption that a given system was monitored
passively.
Campbell and Nikoukhah's book proposes an
"active"
multimodel approach. It calls for applying
an auxiliary signal
that will affect the output so that it can
be used to easily
determine if there has been a failure and
what type of failure it
is. This auxiliary signal must be kept small,
and often brief in
duration, in order not to interfere with
system performance and
to ensure timely detection of the failure.
The approach is robust
and uses tools from robust control theory.
Unlike some
approaches, it is applicable to complex systems.
The authors
present the theory in a rigorous and intuitive
manner and provide
practical algorithms for implementation of
the procedures.
Stephen L.Campbell is Professor of Mathematics
at North Carolina
State University. Ramine Nikoukhah is Senior
Scientist (Directeur
de Recherche) at Institut National de Recherche
en Informatique
et en Automatique (INRIA) in France.
Endorsements:
"This book describes the first comprehensive
methodology for
active failure detection over finite and
infinite intervals of
observation. The authors are the top researchers
in this field,
and I anticipate their book will prompt other
significant
contributions."--Bernard Levy, University
of California,
Davis
"This is the first book I have seen
that thoroughly and
rigorously addresses an important niche in
failure detection."--Frank
Lewis, University of Texas, Arlington
Series: Princeton Series in Applied Mathematics