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Alluding to the myriad historical references that inspire modern lattice structures, William Mulligan narrates an odyssey spanning different cultures, techniques, and styles. Filled with color photos, the book includes designs for 12 original building projects complete with specs and supply lists, as well as a list of the best 100 vines for latticework gardening. ... Read more

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A state-of-the-art survey of both classical and quantum lattice gas models, this two-volume work will cover the rigorous mathematical studies of such models as the Ising and Heisenberg, an area in which scientists have made enormous strides during the past twenty-five years. This first volume addresses, among many topics, the mathematical background on convexity and Choquet theory, and presents an exhaustive study of the pressure including the Onsager solution of the two-dimensional Ising model, a study of the general theory of states in classical and quantum spin systems, and a study of high and low temperature expansions. The second volume will deal with the Peierls construction, infrared bounds, Lee-Yang theorems, and correlation inequality.This comprehensive work will be a useful reference not only to scientists working in mathematical statistical mechanics but also to those in related disciplines such as probability theory, chemical physics, and quantum field theory. It can also serve as a textbook for advanced graduate students. ... Read more

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This book introduces the lattice approach to quantum field theory. The spectacular successes of this technique include compelling evidence that exchange of gauge gluons can confine the quarks within subnuclear matter. The lattice framework enables novel schemes for quantitative calculation and has caused considerable cross-disciplinary activity between elementary particle and solid state physicists. The treatment begins with the lattice definition of a path integral and ends on Monte Carlo simulation methods. Other topics include invariant group integration, duality, mean field theory and renormalization group techniques. The reader is assumed to have a basic background in relativistic quantum mechanics and some exposure to gauge theories. ... Read more

Customer Reviews (2)

Essential text for any student of lattice gauge theory
This is a dated book (look at the copyright date). This is also an action-packed book. A simple comparison of the page count with (e.g.) Rothe's Lattice Gauge Theories will indicate that Cruetz's treatment is terse and dense. Yet it is surprisingly broad. Come prepared with a good understanding of QFT and statistical physics, and a willingness to supplement this text with others.

The key point is that Creutz is still the definitive reference on LGT. If you already know LGT, then presumably you own this book.

If you don't already know LGT and want to learn it, you should own this book, and probably Rothe as well (Rothe's treatment is much more explicit and easier to follow, if also much longer) or some other companion like Montvay/Munster or Smit. If you only want to buy one book on LGT, then (and only then) I would not suggest Creutz--you can find out the basics from hep-lat and (e.g.) Rothe.

Lattice Field theory classic
While relatively hard to read and using
some unique notation,
this is a master work and a reference in
quantum physics using Keneth G. Wilson'sLattice theory
formulationto illuminate.
The references have been very useful to me over the years as well. ... Read more

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This book is intended to be a thorough introduction to the subject of order and lattices and can be used for a course at the graduate or advanced undergraduate level or for independent study. Prerequisites consist mostly of a bit of mathematical maturity, such as that provided by a basic undergraduate course in abstract algebra.

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The importance of equational axioms emerged initially with the axiomatic approach to Boolean algebras, groups, and rings, and later in lattices. This unique research monograph systematically presents minimal equational axiom-systems for various lattice-related algebras, regardless of whether they are given in terms of join and meet or other types of operations such as ternary operations. Each of the axiom-systems is coded in a handy way so that it is easy to follow the natural connection among the various axioms and to understand how to combine them to form new axiom systems.

A new topic in this book is the characterization of Boolean algebras within the class of all uniquely complemented lattices. Here, the celebrated problem of E V Huntington is addressed, which -- according to G Gratzer, a leading expert in modern lattice theory -- is one of the two problems that shaped a century of research in lattice theory. Among other things, it is shown that there are infinitely many non-modular lattice identities that force a uniquely complemented lattice to be Boolean, thus providing several new axiom systems for Boolean algebras within the class of all uniquely complemented lattices. Finally, a few related lines of research are sketched, in the form of appendices, including one by Dr Willian McCune of the University of New Mexico, on applications of modern theorem-proving to the equational theory of lattices. ... Read more

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Numerical simulation of lattice-regulated QCD has become an important source of information about strong interactions. In the last few years there has been an explosion of techniques for performing ever more accurate studies on the properties of strongly interacting particles. Lattice predictions directly impact many areas of particle and nuclear physics theory and phenomenology.This book provides a thorough introduction to the specialized techniques needed to carry out numerical simulations of QCD: a description of lattice discretizations of fermions and gauge fields, methods for actually doing a simulation, descriptions of common strategies to connect simulation results to predictions of physical quantities, and a discussion of uncertainties in lattice simulations. More importantly, while lattice QCD is a well-defined field in its own right, it has many connections to continuum field theory and elementary particle physics phenomenology, which are carefully elucidated in this book. ... Read more

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Classic work, both in West and in China, on the intricate geometric window grids is a rich source of patterns and ideas for wallpaper, stained glass, fabric work, embroidery, etc. Introduction.
... Read more

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The New Book of Chinese Lattice Designs
The second best book on Chinese Lattice Designs, the first also by Daniel Sheets Dye (Chinese Lattice Designs). This is a very worthwhile addition to the first book and contains a wonderful selection of illustrations. It is not a repeat of the first book, and the larger format provides for larger illustrations.If you are into flat pattern design this just brilliant. ... Read more

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From the first edition: "This book combines the techniques of an introductory text with those of a monograph to introduce the general reader to lattice theory and to bring the expert up to date on the most recent developments. The first chapter, along with a selection of topics from later chapters, can serve as an introductory course covering first concepts, distributive, modular, semimodular, and geometric lattices, and so on. About 900 exercises and almost 130 diagrams help the beginner to learn the basic results and important techniques. The latter parts of each chapter give deeper developments of the fields mentioned above and there are chapters on equational classes (varieties) and free products. More advanced readers will find the almost 200 research problems, the extensive bibliography, and the further topics and references at the end of each chapter of special use." In this present edition, the work has been significantly updated and expanded. It contains an extensive new bibliography of 530 items and has been supplemented by eight appendices authored by an exceptional group of experts. The first appendix, written by the author, briefly reviews developments in lattice theory, specifically, the major results of the last 20 years and solutions of the problems proposed in the first edition. The other subjects concern distributive lattices and duality (Brian A. Davey and Hilary A. Priestley), continuous geometries (Friedrich Wehrung), projective lattice geometries (Marcus Greferath and Stefan E. Schmidt), varieties (Peter Jipsen and Henry Rose), free lattices (Ralph Freese), formal concept analysis (Bernhard Ganter and Rudolf Wille), and congruence lattices (Thomas Schmidt in collaboration with the author).

This is an unchanged reprint, in softcover, of the second (hardcover) edition of this classic book on lattice theory by one of the leading figures of the field. Appendices cover the recent developments. ... Read more

Customer Reviews (1)

Outstanding book on an extraordinarly rich and useful topic
This is an outstanding book on a field of mathematics which, although very accessible and widely applicable, is not considered as fundamental as I believe it should be.The author writes clearly, providing concise proofs with sound logic, good motivation for the material, discussion of historical development of the subject, and directions for future research.Diagrams are used very well in this text.The exercises are numerous and very illuminating, and very fun!The author provides an extensive bibliography and references results throughout the text.

There is something very enticing about the way of thinking used in lattice theory, and in particular, the way Gratzer approaches the subject in this book.Like in most areas of mathematics, a given concept can be viewed in many different ways, and one can study these objects at progressively higher levels of abstraction and generality.What is most remarkable about lattice theory, however, is that these higher levels of abstraction and generality do not become overwhelmingly difficult to comprehend as they develop--something that unfortunately happens in many other areas of mathematics.In particular, I think that algebraic concepts such as congruence relations, equational varieties, and "freeness" are much easier to understand in the context of lattices than in other algebraic structures.However, this is also true of more general mathematical concepts such as the effect of weakening conditions on theorems, searching for counterexamples, finding equivalent formulations of a given condition, and studying properties preserved under maps.This is partly due to the fact that in lattices, much of what is going on can be easily drawn or visualized.For this reason, lattices provide an excellent framework for understanding many of the basic concepts that underly all areas of mathematics.

Although the topic of this book is viewed as specialized and esoteric by some mathematicians, I believe that the material it contains is quite universal.Lattices appear in virtually every area of mathematics, and they are especially useful in universal algebra and combinatorics.While this book does not directly talk much about these applications, this book would certainly enrich the knowledge and understanding of people who work in those fields.I would recommend this book to anyone who is serious about algebra or combinatorics as well, as the ways of thinking developed by reading this book and working exercises will prove invaluable in these disciplines.This book also might be useful to beginning graduate students who want to develop their general mathematical maturity in a setting which can be a lot more fun and accessible than other areas of abstract math.

As a final note, the binding on the hardcover edition is excellent.I rarely encounter books this well-bound, in a day an age when sometimes even hardback books start falling apart after moderate use. ... Read more

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This book provides a concrete introduction to quantum fields on a lattice: a precise and non-perturbative definition of quantum field theory obtained by replacing the space-time continuum by a lattice. Topics covered include quark confinement, chiral symmetry breaking in QCD, quantized non-abelian gauge fields, scaling and universality. The author also discusses the results of simulations on computers. ... Read more

Customer Reviews (1)

An excellent book
The book by Jan Smit "INTRODUCTION TO QUANTUM FIELDS ON A LATTICE"
is an excellent book on Lattice Field Theory. There are very few
books on this subject and I think that it is the best because of
clarity, calculation details and the ability of explaining the
physical aspects of the problems in a clear way. I particularly
appreciated the presentation of the symmetries in QCD and the
introduction of fermion fields on the lattice as well as the
question of scalar theories. ... Read more

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Lattice-valued Logic aims at establishing the logical foundation for uncertain information processing routinely performed by humans and artificial intelligence systems. In this textbook for the first time a general introduction on lattice-valued logic is given. It systematically summarizes research from the basic notions up to recent results on lattice implication algebras, lattice-valued logic systems based on lattice implication algebras, as well as the corresponding reasoning theories and methods. The book provides the suitable theoretical logical background of lattice-valued logic systems and supports newly designed intelligent uncertain-information-processing systems and a wide spectrum of intelligent learning tasks. ... Read more

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A comprehensive and timely survey of the application of the methods of quantum field theory to statistical physics, a very active and fruitful area of modern research, is provided in two volumes. The first volume provides a pedagogical introduction to the subject, discussing Brownian motion, its anticommutative counterpart in the guise of Onsager's solution to the two-dimensional Ising model, the mean field or Landau approximation, scaling ideas exemplified by the Kosterlitz-Thouless theory for the XY transition, the continuous renormalization group applied to the standard phi-to-the-fourth theory (the simplest typical case) and lattice gauge theory as a pathway to the understanding of quark confinement in quantum chromodynamics. ... Read more

Customer Reviews (2)

The worst text-book on theoretical physics I ever read
This book is simply awful.
I couldn't understand a lot of derivations.
Before buying this book I studied the standard course on QFT by Peskin and Schroeder.(I solved 95% of its problems, so I'm not stupid!).
The reason I bought the book by Itzykson and Drouffe was to study some interesting advanced topics in QFT.
I began with Ising model and immediately got stuck.
I read previous reviews on this book. The authors praised it.
I wonder if any of them could proceed from formula 65 to 66?
(first volume)
Did any of them understand the derivation of Szego lemma?
What about Kac determinant?
I found that even original article by Itzykson was written better
and with more care than this book.(!)
Still it's worth buying as the source of references.
That's why one *.

first vol only
i think the first chapter is a b---- to get through, but after i did that i found the entire book either particularly useful or insightful. volume two never motivated it's own purchase.
I read it in frech, so i cant talk about how well things are explained in english, just in case anything was lost in translation, but with something like chapter two on grassman variables i dont think it's such a big deal. to put it succintly chapter 2 makes berezin's book obsolete to me and i feel that way about the whole book. ... Read more

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This book provides an introduction to topological quantum field theory as well as discrete gauge theory with quantum groups. In contrast to much of the existing literature, the present approach is at the same time intuitive and mathematically rigorous, making extensive use of suitable diagrammatic methods. It provides a highly unified description of lattice gauge theory, topological quantum field theory and models of quantum (super)gravity. The reader is thus in a unique position to understand the relations between these subjects as well as the underlying groundwork. ... Read more

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This work presents foundational research on two approaches to studying subgroup lattices of finite abelian $p$-groups. The first approach is linear algebraic in nature and generalizes Knuth's study of subspace lattices. This approach yields a combinatorial interpretation of the Betti polynomials of these Cohen-Macaulay posets. The second approach, which employs Hall-Littlewood symmetric functions, exploits properties of Kostka polynomials to obtain enumerative results such as rank-unimodality. Butler completes Lascoux and Schützenberger's proof that Kostka polynomials are nonnegative, then discusses their monotonicity result and a conjecture on Macdonald's two-variable Kostka functions. ... Read more

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This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. Schramm-Löwner evolutions (SLE) arise in various contexts. The choice of topics is strongly motivated by modern applications and focuses on areas that merit further research. Special features include a simple account of Smirnov's proof of Cardy's formula for critical percolation, and a fairly full account of the theory of influence and sharp-thresholds. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises. ... Read more

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This book explores the surprisingly rich and complex structure of free lattices. The first part of the book presents a complete exposition of the basic theory of free lattices, projective lattices, and lattices which are bounded homomorphic images of a free lattice, as well as applications of these results to other areas. This portion of the book is suitable for use in a graduate course in lattice theory or general algebra. The second part of the book contains new results about free lattices and new proofs of known results, providing the reader with a coherent picture of the fine structure of free lattices. The book closes with an analysis of algorithms for free lattices and finite lattices that is accessible to researchers in other areas and depends only on the first chapter and a small part of the second. Several open problems appear throughout the book and, for easy reference, are assembled in a section at the end. Synthesizing seventy years of research, this is the only comprehensive treatment available of the theory of free lattices. ... Read more

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The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate how they form two sides of the same coin. We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps entitled Algebra and Substructural Logics.
As the book progresses the first objective gains predominance over the second. Although the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions. Decidability of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says that a formula is an intuitionistic tautology if and only if its double negation is a classical one. We generalise it to the substructural setting, identifying for each substructural logic its Glivenko equivalence class with smallest and largest element. This is also where we begin investigating lattices of logics and varieties, rather than particular examples. We continue in this vein by presenting a number of results concerning minimal varieties/maximal logics. A typical theorem there says that for some given well-known variety its subvariety lattice has precisely such-and-such number of minimal members (where values for such-and-such include, but are not limited to, continuum, countably many and two). In the last two chapters we focus on the lattice of varieties corresponding to logics without contraction. In one we prove a negative result: that there are no nontrivial splittings in that variety. In the other, we prove a positive one: that semisimple varieties coincide with discriminator ones.

Within the second, more technical part of the book another transition process may be traced. Namely, we begin with logically inclined technicalities and end with algebraically inclined ones. Here, perhaps, algebraic rendering of Glivenko theorems marks the equilibrium point, at least in the sense that finiteness properties, decidability and Glivenko theorems are of clear interest to logicians, whereas semisimplicity and discriminator varieties are universal algebra par exellence. It is for the reader to judge whether we succeeded in weaving these threads into a seamless fabric.

- Considers both the algebraic and logical perspective within a common framework.
- Written by experts in the area.
- Easily accessible to graduate students and researchers from other fields.
- Results summarized in tables and diagrams to provide an overview of the area.
- Useful as a textbook for a course in algebraic logic, with exercises and suggested research directions.
- Provides a concise introduction to the subject and leads directly to research topics.
- The ideas from algebra and logic are developed hand-in-hand and the connections are shown in every level. ... Read more

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An essential reference for researching ways in which lattice problems can be used to build cryptographic systems. Particular emphasis on problems that are related to the construction of cryptographic functions. ... Read more