Lattices are geometric objects that can be pictorially described as the set of intersection points of an infinite, regular n-dimensional grid. De spite their apparent simplicity, lattices hide a rich combinatorial struc ture, which has attracted the attention of great mathematicians over the last two centuries. Not surprisingly, lattices have found numerous ap plications in mathematics and computer science, ranging from number theory and Diophantine approximation, to combinatorial optimization and cryptography. The study of lattices, specifically from a computational point of view, was marked by two major breakthroughs: the development of the LLL lattice reduction algorithm by Lenstra, Lenstra and Lovasz in the early 80's, and Ajtai's discovery of a connection between the worst-case and average-case hardness of certain lattice problems in the late 90's. The LLL algorithm, despite the relatively poor quality of the solution it gives in the worst case, allowed to devise polynomial time solutions to many classical problems in computer science. These include, solving integer programs in a fixed number of variables, factoring polynomials over the rationals, breaking knapsack based cryptosystems, and finding solutions to many other Diophantine and cryptanalysis problems.

Publisher Description

The SEWM2002 workshop, like the ones before, brought together theoretical physicists working on thermal field theory and, more generally, on (resummation) techniques for deriving effective actions based on QCD and the electroweak standard model of elementary particle physics, but describing nonstandard situations. The focus was on the temperature/chemical potential phase diagram of QCD, considered both analytically and with lattice gauge theory, equilibrium and nonequilibrium thermo field theory, and on heavy ion physics. Other related topics were “small x physics” in QCD, electroweak baryogenesis, inflation, and dark energy in the early universe.

Particle and nuclear physicists frequently take results from Lattice QCD at their face value without probing into their reliability or sophistication. This attitude usually stems from a lack of knowledge of the field. The aim of the present volume is to rectify this by introducing in an elementary way several topics, which we believe are appropriate for, and of possible interest to, both particle and nuclear physicists who are non-experts in the field.

This volume contains the Proceedings of ICFCA 2004, the 2nd International Conference on Formal Concept Analysis. The ICFCA conference series aims to be the premier forum for the publication of advances in applied lattice and order theory and in particular scienti?c advances related to formal concept analysis. Formal concept analysis emerged in the 1980s from e?orts to restructure lattice theory to promote better communication between lattice theorists and potentialusersoflatticetheory.Sincethen,the?eldhasdevelopedintoagrowing research area in its own right with a thriving theoretical community and an increasing number of applications in data and knowledge processing including data visualization, information retrieval, machine learning, data analysis and knowledge management. In terms of theory, formal concept analysis has been extended into attribute exploration, Boolean judgment, contextual logic and so on to create a powerful general framework for knowledge representation and reasoning. This conference aims to unify theoretical and applied practitioners who use formal concept an- ysis, drawing on the ?elds of mathematics, computer and library sciences and software engineering. The theme of the 2004 conference was ‘Concept Lattices” to acknowledge the colloquial term used for the line diagrams that appear in almost every paper in this volume. ICFCA 2004 included tutorial sessions, demonstrating the practical bene?ts of formal concept analysis, and highlighted developments in the foundational theory and standards. The conference showcased the increasing variety of formal concept analysis software and included eight invited lectures from distinguished speakersinthe?eld.Sevenoftheeightinvitedspeakerssubmittedaccompanying papers and these were reviewed and appear in this volume.

Spatial dynamics, landscape, population.

This is the second volume of the third edition of a successful text, now substantially enlarged and updated to reflect developments over the last decade in the curricula of university courses and in particle physics research. Volume I covered relativistic quantum mechanics, electromagnetism as a gauge theory, and introductory quantum field theory, and ended with the formulation and application of quantum electrodynamics (QED), including renormalization. Building on these foundations, this second volume provides a complete, accessible, and self-contained introduction to the remaining two gauge theories of the standard model of particle physics: quantum chromodynamics (QCD) and the electroweak theory. The treatment significantly extends that of the second edition in several important respects. Simple ideas of group theory are now incorporated into the discussion of non-Abelian symmetries. Two new chapters have been added on QCD, one devoted to the renormalization group and scaling violations in deep inelastic scattering and the other to non-perturbative aspects of QCD using the lattice (path-integral) formulation of quantum field theory; the latter is also used to illuminate various aspects of renormalization theory, via analogies with condensed matter systems. Three chapters treat the fundamental topic of spontaneous symmetry breaking: the (Bogoliubov) superfluid and the (BCS) superconductor are studied in some detail; one chapter is devoted to the implications of global chiral symmetry breaking in QCD; and one to the breaking of local SU(2)xU(1) symmetry in the electroweak theory. Weak interaction phenomenology is extended to include discussion of discrete symmetries and of the possibility that neutrinos are Majorana (rather than Dirac) particles. Most of these topics are normally found only in more advanced texts, and this is the first book to treat them in a manner accessible to the wide readership that the previous editions have attracted.