Dietmar Cieslik
Ernst-Moritz-Arndt University, Greifswald, Germany

The Steiner Ratio

COMBINATORIAL OPTIMIZATION Volume 10

Steiner's Problem concerns finding a shortest interconnecting network for a finite set of points in a metric space. A solution must be a tree, which is called a Steiner Minimal Tree (SMT), and may contain vertices different from the points which are to be connected. Steiner's Problem is one of the most famous combinatorial鉾eometrical problems, but unfortunately it is very difficult in terms of combinatorial structure as well as computational complexity. However, if only a Minimum Spanning Tree (MST) without additional vertices in the interconnecting network is sought, then it is simple to solve. So it is of interest to know what the error is if an MST is constructed instead of an SMT. The worst case for this ratio running over all finite sets is called the Steiner ratio of the space.

The book concentrates on investigating the Steiner ratio. The goal is to determine, or at least estimate, the Steiner ratio for many different metric spaces. The author shows that the description of the Steiner ratio contains many questions from geometry, optimization, and graph theory.

Audience: Researchers in network design, applied optimization, and design of algorithms.

Contents
Preface. 1. The Historical Genesis. 2. Networks, Spaces and Algorithms. 3. Shortest Trees in Metric Spaces ・A Survey. 4. The Steiner Ratio of Metric Spaces. 5. The Steiner Ratio of Banach-Minkowski Spaces. 6. Euclidean Spaces. 7. The Steiner Ratio of Neighboured Spaces. 8. Banach-Minkowski Planes. 9. The Steiner Ratio and the Embedding of Spaces. 10. The Steiner Ratio and Discrete Geometry. 11. The Dependence of the Steiner Ratio on the Dimension. 12. Related Questions. References. Index.

Kluwer Academic Publishers, Dordrecht
Hardbound, ISBN 0-7923-7015-5
July 2001, 256 pp.

edited by
Alfredo Macias / Universidad Autonoma Metropolitana - Iztapalapa, Mexico City, Mexico
Jorge L. Cervantes-Cota / Instituto Nacional de Investigaciones Nucleares, Mexico City, Mexico
Claus Lammerzahl / Heinrich-Heine-Universitat, Dusseldorf, Germany

Exact Solutions and Scalar Fields in Gravity
Recent Developments

Divided into four parts, this book covers recent developments in topics pertaining to gravity theories, including discussions on the presence of scalar fields.

Part One is devoted to exact solutions in general relativity, and is mainly concerned with the results of rotating null dust beams and fluids. Also included is a panoramic vision of new research directions in this area, which would require revising certain theorems and their possible extensions within gravity theories, new aspects concerning the Ernst potentials, double Kerr spacetimes, and rotating configurations. In particular, there is a detailed discussion of totally symmetric and totally geodesic spaces, in which a method for generating (2+1)-dimensional solutions from (3+1)-dimensional solutions is given.

Part Two deals with alternative theories of gravity, all of which include scalar fields and gauge fields. Here, quantum and cosmological effects, which arise from both gravity theories in four and higher dimensions and from metric-affine theories, are investigated.

Part Three is devoted to cosmological and inflationary scenarios. Local effects, such as the influence of scalar fields in protogalactic interactions, numerical studies of the collapse of molecular cores, as well as the inverse inflationary problem and the blue eigenvalue spectrum of it, are considered. Moreover, the role of scalar fields as dark matter and quantum cosmology in the Bergman-Wagoner and Gowdy theories, together with the relation of the conformal symmetry and deflationary gas universe, are likewise presented.

The last part of the book includes some mixed topics which are still in the experimental stage. Among them are the foundation of the Maxwell theory, a discussion on electromagnetic Thirring problems, a note on the staticity of black holes with non-minimally coupled scalar fields, and a study of the Lorentz force free charged fluids in general relativity.

Thus, this book is the most up-to-date, comprehensive collection of papers on the subject of exact solutions and scalar fields in gravity and is a valuable tool for researchers in the area.

Kluwer Academic/Plenum Publishers
Hardbound, ISBN 0-306-46618-X
August 2001, 352 pp.

P.A. Grillet / Tulane University, New Orleans, LA, USA

Commutative Semigroups

ADVANCES IN MATHEMATICS Volume 2

This is the first book about commutative semigroups in general. Emphasis is on structure but the other parts of the theory are at least surveyed and a full set of about 850 references is included. The book is intended for mathematicians who do research on semigroups or who encounter commutative semigroups in their research.

Contents

Preface. General structure theory I. Elementary properties. Il. Cancellative semigroups. III. Semilattice decompositions. IV. Subdirect decompositions. V. Group coextensions. VI. Finitely generated semigroups. VII. Subcomplete semigroups. VIII. Other results. Congruences IX. Nilsemigroups. X. Group-free semigroups. XI. Subcomplete semigroups. Cohomology XII. Commutative semigroup cohomology. XIII. The overpath method. XIV. Semigroups with zero cohomology. References. Author. Index. Notation. Index.

Kluwer Academic Publishers, Dordrecht
Hardbound, ISBN 0-7923-7067-8
August 2001, 456 pp.

Thomas Fiedler
University of Paul Sabatier, Toulouse, France

Gauss Diagram Invariants for Knots and Links

MATHEMATICS AND ITS APPLICATIONS Volume 532

This book contains new numerical isotopy invariants for knots in the product of a surface (not necessarily orientable) with a line and for links in 3-space. These invariants, called Gauss diagram invariants, are defined in a combinatorial way using knot diagrams. The natural notion of global knots is introduced. Global knots generalize closed braids. If the surface is not the disc or the sphere then there are Gauss diagram invariants which distinguish knots that cannot be distinguished by quantum invariants. There are specific Gauss diagram invariants of finite type for global knots. These invariants, called T-invariants, separate global knots of some classes and it is conjectured that they separate all global knots. T-invariants cannot be obtained from the (generalized) Kontsevich integral.

Audience: The book is designed for research workers in low-dimensional topology.

Contents

Preface. Introduction and announcement. 1. The space of diagrams. 2. Invariants of knots and links by Gauss sums. 3. Applications. 4. Global knot theory in F2 × . 5. Isotopies with restricted cusp crossing for fronts with exactly two cusps of Legendre knots in ST★2. Bibliography. Index.

Kluwer Academic Publishers, Dordrecht
Hardbound, ISBN 0-7923-7112-7
August 2001, 428 pp.

edited by
Derek Holton / University of Otago, Dunedin, New Zealand

The Teaching and Learning of Mathematics at University Level
An ICMI Study

NEW ICMI STUDY SERIES Volume 7

This book arose from the ICMI Study into the teaching and learning of mathematics at university level that began with a conference in Singapore in 1998. The book looks at tertiary mathematics and its teaching from a number of aspects including practice, research, mathematics and other disciplines, technology, assessment, and teacher education. Over 50 authors, all international experts in their field, combined to produce a text that contains the latest in thinking and the best in practice. It therefore provides in one book a state-of-the-art statement on tertiary teaching from a multi-perspective standpoint. No previous book has attempted to take such a wide view of the topic. The book will be of special interest to academic mathematicians, mathematics educators, and educational researchers.

Kluwer Academic Publishers, Dordrecht
Hardbound, ISBN 0-7923-7191-7
October 2001, 568 pp.

Paperback, ISBN 1-4020-0072-3
October 2001, 568 pp.