CWI Tract 133
Summary
This text, derived from the thesis of the author, is concerned
with solving equations of the form $$x^r+y^s=z^t$$ in coprime
integers $x,y,z$ for fixed $r,s,t$. For specific values of
$r,s,t$, this question is reduced to finding the rational points
on finitely many algebraic curves.
These curves have a map to the projective line that factors
through an elliptic curve, but only over an extension of the base
field. This leads to the question of determining the rational
points on an elliptic curve over a number field, subject to
additional arithmetic restrictions. An adaptation of a method
first used by Chabauty enables the resolution of this problem. In
the text, the coprime integer solutions to the equations $x^2+y^3=z^8$,
$x^2+y^8=z^3$, $x^2+y^4=z^5$ and $x^2+y^5=z^4$ are determined.
The methods developed are applicable to hyperelliptic curves in
general, as well as to some other classes of curves.
2002, ISBN 90 6196 508 X, 77 pages,
A publication of the Societe Mathematique de France.
Description
Toric varieties form a beautiful class of algebraic varieties,
which are often used as a testing ground for verifying general
conjectures in algebraic geometry, for example, in Hilbert
schemes, singularity theory, Mori theory, and so on.
This volume gathers expanded versions of lectures presented
during the summer school of "Geometry of Toric Varieties"
in Grenoble (France). These lectures were given during the second
and third weeks of the school. (The first week was devoted to
introductory material.) The paper by D. Cox is an overview of
recent work in toric varieties and its applications, putting the
other contributions of the volume into perspective.
Contents
D. A. Cox -- Update on toric geometry
W. Bruns and J. Gubeladze -- Semigroup algebras and discrete
geometry
A. Craw and M. Reid -- How to calculate A-Hilb $\mathbb{C}^3$
D. I. Dais -- Resolving 3-dimensional toric singularities
D. I. Dais -- Crepant resolutions of Gorenstein toric
singularities and upper bound theorem
J. Hausen -- Producing good quotients by embedding into toric
varieties
Y. Ito -- Special McKay correspondence
Y. Tschinkel -- Lectures on height zeta functions of toric
varieties
J. A. Wisniewski -- Toric Mori theory and Fano manifolds
Details:
Series: Seminaires et Congres, Number: 6
Publication Year: 2002
ISBN: 2-85629-122-8
Paging: 272 pp.
Binding: Softcover
ISBN 0-262-16216-4
5 3/8 x 8, 216 pp., 46 illus.(CLOTH)
In 1824 a young Norwegian named Niels Henrik Abel proved
conclusively that algebraic equations of the fifth order are not
solvable in radicals. In this book Peter Pesic shows what an
important event this was in the history of thought. He also
presents it as a remarkable human story. Abel was twenty-one when
he self-published his proof, and he died five years later, poor
and depressed, just before it started to receive wide acclaim.
Abel's attempts to reach out to the mathematical elite of the day
had been spurned, and he was unable to find a position that would
allow him to work in peace and marry his fiancee.
But Pesic's story begins long before Abel and continues to the
present day, for Abel's proof changed how we think about
mathematics and its relation to the "real" world.
Starting with the Greeks, who invented the idea of mathematical
proof, Pesic shows how mathematics found its sources in the real
world (the shapes of things, the accounting needs of merchants)
and then reached beyond those sources toward something more
universal. The Pythagoreans' attempts to deal with irrational
numbers foreshadowed the slow emergence of abstract mathematics.
Pesic focuses on the contested development of algebra--which even
Newton resisted--and the gradual acceptance of the usefulness and
perhaps even beauty of abstractions that seemed to invoke
realities with dimensions outside of human experience. Pesic
tells this story as a history of ideas, with mathematical details
incorporated in boxes. The book also includes a new annotated
translation of Abel's original proof.
May 2003
ISBN 0-262-57174-9
5 3/8 x 8, 100 pp., 25 illus.(PAPER)
Janos Bolyai (1802-1860) was a mathematician who changed our
fundamental ideas about space. As a teenager he started to
explore a set of nettlesome geometrical problems, including
Euclid's parallel postulate, and in 1832 he published a brilliant
twenty-four-page paper that eventually shook the foundations of
the 2000-year-old tradition of Euclidean geometry. Bolyai's
"Appendix" (published as just that--an appendix to a
much longer mathematical work by his father) set up a series of
mathematical proposals whose implications would blossom into the
new field of non-Euclidean geometry, providing essential
intellectual background for ideas as varied as the theory of
relativity and the work of Marcel Duchamp. In this short book,
Jeremy Gray explains Bolyai's ideas and the historical context in
which they emerged, were debated, and were eventually recognized
as a central achievement in the Western intellectual tradition.
Intended for nonspecialists, the book includes facsimiles of
Bolyai's original paper and the 1898 English translation by G. B.
Halstead, both reproduced from copies in the Burndy Library at
MIT.
Topics in Discrete Mathematics, 11
Description
One of the most frequently occurring types of optimization
problems involves decision variables which have to take integer
values. From a practical point of view, such problems occur in
countless areas of management, engineering, administration, etc.,
and include such problems as location of plants or warehouses,
scheduling of aircraft, cutting raw materials to prescribed
dimensions, design of computer chips, increasing reliability or
capacity of networks, etc. This is the class of problems known in
the professional literature as "discrete optimization"
problems. While these problems are of enormous applicability,
they present many challenges from a computational point of view.
This volume is an update on the impressive progress achieved by
mathematicians, operations researchers, and computer scientists
in solving discrete optimization problems of very large sizes.
The surveys in this volume present a comprehensive overview of
the state of the art in discrete optimization and are written by
the most prominent researchers from all over the world.
This volume describes the tremendous progress in discrete
optimization achieved in the last 20 years since the publication
of Discrete Optimization '77, Annals of Discrete Mathematics,
volumes 4 and 5, 1979 (Elsevier). It contains surveys of the
state of the art written by the most prominent researchers in the
field from all over the world, and covers topics like
neighborhood search techniques, lift and project for mixed 0-1
programming, pseudo-Boolean optimization, scheduling and
assignment problems, production planning, location, bin packing,
cutting planes, vehicle routing, and applications to graph
theory, mechanics, chip design, etc.
Contents
Preface.
Non-standard approaches to integer programming (K. Aardal, R.
Weismantel, L.A. Wolsey).
A survey of very large-scale neighborhood search techniques (R.K.
Ahuja, O. Ergun, J.B Orlin, A.P. Punnen).
Maximum mean weight cycle in a digraph and minimizing cycle time
of a logic chip (C. Albrecht, B. Korte, J. Schietke, J. Vygen).
Lift-and-project for Mixed 0-1 programming: recent progress (E.
Balas, M. Perregaard).
Pseudo-Boolean optimization (E. Boros, P.L. Hammer).
Scheduling and constraint propagation (P. Brucker).
Selected topics on assignment problems (R.E. Burkard).
Ideal clutters (G. Cornuejols, B. Guenin).
Production planning problems in printed circuit board assembly (Y.
Crama, J. van de Klundert, F.C.R. Spieksma).
Discrete location problems with push-pull objectives (J. Krarup,
D. Pisinger, F. Plastria).
Recent advances on two-dimensional bin packing problems (A. Lodi,
S. Martello, D. Vigo).
Cutting planes in integer and mixed integer programming (H.
Marchand, A. Martin, R. Weismantel, L. Wolsey).
Graph connectivity and its augmentation: applications of MA
orderings (H. Nagamochi, T. Ibaraki).
Applications of combinatorics to statics - rigidity of grids (N.
Radics, A. Recski).
Models, relaxations and exact approaches for the capacitated
vehicle routing problem (P. Toth, D. Vigo).
Semidefinite programming for discrete optimization and matrix
completion problems (H. Wolkowicz, M.F. Anjos).
Author index.
Year 2003
Hardbound
ISBN: 0-444-51295-0
588 pages
Description
This is a sequel to volume 19 of Handbook of Statistics on
Stochastic Processes: Modelling and Simulation. It is concerned
mainly with the theme of reviewing and in some cases, unifying
with new ideas the different lines of research and developments
in stochastic processes of applied flavour. This volume consists
of 23 chapters addressing various topics in stochastic processes.
These include, among others, those on manufacturing systems,
random graphs, reliability, epidemic modelling, self-similar
processes, empirical processes, time series models, extreme value
theory, applications of Markov chains, modelling with Monte carlo
techniques, and stochastic processes in subjects such as
engineering, telecommunications, biology, astronomy and chemistry.
(A complete list of the topics addressed in the volume is
available from the "Contents" of the volume.)
An attempt is made to cover in this volume, as in the case of its
predecessor, as many topics as possible. Among various issues
considered in this volume, there are those concerned in
particular with modelling, simulation techniques and numerical
methods concerned with stochastic processes. The scope of the
project involving this volume as well as volume 19 is already
clarified in the "Preface" of volume 19. The present
volume completes the aim of the project, and it should serve as
an aid to students, teachers, researchers and practitioners
interested in applied stochastic processes.
Year 2003
Hardbound
ISBN: 0-444-50013-8
approx 800 pages