Polynomials and freeness are discussed in Section 2, while congruence relations form the topic of section 3. A theorem due to N. Funayama & T. Nakayama (of 1942) states that Con L = the lattice of all congruence relations on L is distributive. Boolean algebras R-generated by distributive lattices are presented in Section 4 that includes several results like two lemmas due to J. von Neumann (of 1936) and V. Glivenko (of 1929). Section 5 deals with the topological representation in terms of S(L) = the topological space defined on P(L) = the partially ordered set (poset) of prime ideals. The Stone space S(L) determines L up to an isomorphism. Distributive lattices and pseudocomplementation form the topic of Section 6 in this second chapter. It includes Theorem 12 - Birkhoff's Subdirect Representation Theorem which states that every algebra A in a variety K of algebras can be embedded in a direct product of subdirectly irreducible algebras in K.
Chapter III - Congruences and Ideals - starts with a first section on the weak projevctivity and congruences. It is followed by Section 2 on distributive, standard and neutral elements. A theorem due to G. Grätzer & E.T. Schmidt (1961) characterizes the standard elements in a lattice L. Section 3 deals with distributive, standard and neutral ideals of a lattice. Structure theorems are presented in Section 4. For instance, Theorem 1 in this section states that the direct decompositions of a lattice L into two factors are (up to isomorphism) in one-to-one correspondence with the complemented neutral elements of L. A corollary (to Theorem 6), known as the Birkhoff-Menger Theorem, states that a complemented modular lattice L of finite length is isomorphic to a direct product of simple lattices. The next corollary (R.P. Dilworth - 1950) states the same property for relatively complemented lattices.
Chapter IV - Modular and Semimodular Lattices - is opened with a first section on modular lattices in which the author examines the most important consequences of modularity. A theorem due to G. Birkhoff (of 1940) states that if C and C' are chains in a modular lattice L then the sublattice generated by is distributive. Semimodular lattices are investigated in Section 2. A lattice L is semimodular if it satisfies the upper covering condition: for two elements or The semimodular lattices of finite length are charac-terized in Theorem 2, while Theorem 9 states that a lattice L of finite length is semimodular L is M-symmetric. Geometric lattices are also investigated in this section. A lattice L is said to be geometric if it is semimodular, it is algebraic, and the compact elements of L are exactly the finite joins of atoms of L. Theorem 5 states that every geometric lattice is isomorphic to a direct product of directly indecomposable geometric lattices. Partition lattices are also studied in Section 4. Part A = the set of (partially ordered) partitions of a set A is a complete lattice and it is also a simple geometric lattice (by a theorem due to O. Ore of 1942). Another remarkable result is due to P.M. Whitman (1946), stating that every lattice can be embedded in the lattice of all subgroups of some group. Theorem 5 states that the linear subspaces of a projective space form a modular geometric lattice.
Chapter V - Varieties of Lattices, starts with a first section presenting characterizations of varieties. Let us quote Theorem 3: A class K of lattices is a variety K is closed under the formation of homomorphic images, sublattices, and direct products. Theorem 6 (R. Ville - 1972) states that for any set of finite posets P, Var(P) is a variety of lattices ; Var(K) is the smallest variety containing K. The problem of finding equational bases is approached in Section 3. Theorem 5 (R.N. McKenzie - 1970) shows that any finite lattice has a finite equational basis. Amalgamation properties are studied in Section 4.
Chapter VI - Free Products, is opened by Section 1 that introduces free products of lattices and includes some of their properties. The structure of free lattices is studied in Section 2. Theorem 6 (B. Jónsson - 1961) states that a sublattice A of finite length of a free lattice (of any lattice satisfying = the meet-semidistributive law) is finite. Theorem 9 asserts that there exists a one-to-one correspondence between automorphisms of a free lattice and permutations of its free generating set, while Theorem 10 shows that every chain of a free lattice is countable. Reduced free products are studied in Section 3, and Hopfian lattices are the topic of Section 4. A lattice L is Hopfian if every onto endomorphism is an automorphism. Theorem 5 (G. Grätzer & J. Sichler - 1974) shows that the free product of Hopfian lattices is not necessarily Hopfian.
A short section of Concluding Remarks follows, including some historical references and a list of 66 specific problems with their solvers and corresponding references. A very rich Bibliography is given on 59 pages. A Table of Notation follows it. Not less than eight Appendices complete this volume. We can only mention their titles and authors here. App. A - Retrospective, by G. Grätzer. App. B - Distributive Lattices and Duality, by B.A. Davey & H.A. Priestley. App. C - Congruence Lattices, by G. Grätzer & E.T. Schmidt. App. D - Continuous Geometry, by F. Wehrung. App. E - Projective Lattice Geometries, by M. Greferath & S.S. Schmidt. App. F - Varieties of Lattices, by P. Jipsen & H. Rose. App. G - Free Lattices, by R. Freese. App. H - Applied Lattice Theory: Formal Concept Analysis, by B. Ganter & R. Wille.
A New Bibliography of 530 titles follows, and also a quite rich Index on 23 pages that closes this exceptional book.
Although this a common feature of the volumes edited by Birkhäuser Verlag, we cannot avoid to mention the excellent printing conditions, the clarity and high quality of the numerous figures that are so useful to those ones interested in Lattice Theory.

Prof. Alexandru CARAUSU,Ph.D.

This volume highlights the analysis on noncompact and singular manifolds within the framework of the cone calculus with asymptotics.
The three papers at the beginning deal with parabolic equations, a topic relevant for many applications. The first article presents a calculus for pseudodifferential operators with an anisotropic analytic parameter. The subsequent paper develops an algebra of Mellin operators on the infinite space-time cylinder. It is shown how timelike infinity can be treated as a conical singularity. In the third text - the central article of this volume - the authors use these results to obtain precise information on the long-time asymptotics of solutions to parabolic equations and to construct inverses within the calculus.

There follows a factorization theorem for meromorphic symbols: It is proved that each of these can be decomposed into a holomorphic invertible part and a smoothing part containing all the meromorphic information. It is expected that this result will be important for applications in the analysis of nonlinear hyperbolic equations. The final article addresses the question of the coordinate invariance of the Mellin calculus with asymptotics.

Fundamentals of Error Correcting Codes are an in-depth introduction to coding theory from both an engineering and mathematical viewpoint. It reviews classical topics, and gives much coverage of recent techniques that could previously only be found in specialist publications. Numerous exercises and examples and an accessible writing style make this a lucid and effective introduction to coding theory for advanced undergraduate and graduate students, researchers and engineers - whether approaching the subject from a mathematical, engineering or computer science background.

Contents
Preface, 1. Basic concepts of linear codes, 2. Bounds on size of codes, 3. Finite fields,
4. Cyclic codes, 5. BCH and Reed-Soloman codes, 6. Duadic codes, 7. Weight distributions,
8. Designs, 9. Self-dual codes, 10. Some favourite self-dual codes, 11. Covering radius and cosets, 12. Codes over Z4, 13. Codes from algebraic geometry, 14. Convolutional codes, 15. Soft decision and iterative decoding, Bibliography, Index.

This book covers statics and dynamics to provide the student with everything needed to complete typical first-year undergraduate courses. Although students often find it difficult to visualize problems and grasp the mathematics, Adrian Roberts' approach tackles concepts with many examples, exercises and helpful diagrams. A key section on background mathematics allows students to review the prerequisite mathematics needed to progress through both topics.
Contents
Preface

Part I. Statics: 1. Forces, 2. Moments, 3. Centre of gravity, 4. Distributed forces, 5. Trusses, 6. Beams, 7. Friction, 8. Non-coplanar forces and couples, 9. Virtual work,
Part II. Dynamics: 10. Kinematics of a point, 11. Kinetics of a particle, 12. Plane motion of a rigid body, 13. Impulse and momentum, 14. Work, power and energy,
Part III. Problems: 15. Statics, 16. Dynamics,
Part IV. Background Mathematics: 17.Algebra, 18. Trigonometry, 19. Calculus, 20. Co-ordinate geometry, 21. Vector algebra, 22. Two more topics.

This Book Contains papers presented at the conference "Applied Geometric Algebra in Computer Science and Engineering" (AGACSE 2001) held in the Engineering Department at Cambridge University from July 9th to 13th, 2001. The goal was to demonstrate how the framework of geometric algebra (Clifford algebra) could unify and illuminate diverse fields of science and engineering.

The articles reveal the range fields: from quantum physics to robotics, from crystallographic groups to image understanding, from relativistic mechanics to signal processing. Despite this diversity, the combination of these subjects was not felt to be artificial.
This book should be also useful to mathematicians as to physicians, to mechanics and computers engineers.

Assist. Prof. Ariadna Lucia PLETEA, Ph. D.

This book contains 102 highly selected combinatorial problems used in the training and testing of the USA International Mathematical Olympiad team. Half of the problems are introductory, while the rest are more difficult. All problems have complete solutions. The problems come mostly from American Mathematical Contests, International Mathematical Olympiads, Russian Mathematical Olympiads etc. A glossary and suggestions for further reading are also included.
It is not a collection of very difficult, impenetrable questions. Instead, the book gradually builds students' combinatorial skills and techniques. It aims to broaden a student's view of mathematics in preparation for possible participation in mathematical competitions.

Problem-solving tactics and strategies further stimulate interest and confidence in combinatorics and other areas of mathematics.
Aside from its practical use in training teachers and students engaged in mathematical competitions, it is a source of enrichment that is bound to stimulate interest in a variety of mathematical areas.

Prof. Liliana POPA, Ph.D.