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Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic. Ravenel's first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.

Homotopie --- Homotopy theory --- Homotopy theory. --- Deformations, Continuous --- Topology --- Abelian category. --- Abelian group. --- Adams spectral sequence. --- Additive category. --- Affine space. --- Algebra homomorphism. --- Algebraic closure. --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Algebraically closed field. --- Atiyah–Hirzebruch spectral sequence. --- Automorphism. --- Boolean algebra (structure). --- CW complex. --- Canonical map. --- Cantor set. --- Category of topological spaces. --- Category theory. --- Classification theorem. --- Classifying space. --- Cohomology operation. --- Cohomology. --- Cokernel. --- Commutative algebra. --- Commutative ring. --- Complex projective space. --- Complex vector bundle. --- Computation. --- Conjecture. --- Conjugacy class. --- Continuous function. --- Contractible space. --- Coproduct. --- Differentiable manifold. --- Disjoint union. --- Division algebra. --- Equation. --- Explicit formulae (L-function). --- Functor. --- G-module. --- Groupoid. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Hurewicz theorem. --- Inclusion map. --- Infinite product. --- Integer. --- Inverse limit. --- Irreducible representation. --- Isomorphism class. --- K-theory. --- Loop space. --- Mapping cone (homological algebra). --- Mathematical induction. --- Modular representation theory. --- Module (mathematics). --- Monomorphism. --- Moore space. --- Morava K-theory. --- Morphism. --- N-sphere. --- Noetherian ring. --- Noetherian. --- Noncommutative ring. --- Number theory. --- P-adic number. --- Piecewise linear manifold. --- Polynomial ring. --- Polynomial. --- Power series. --- Prime number. --- Principal ideal domain. --- Profinite group. --- Reduced homology. --- Ring (mathematics). --- Ring homomorphism. --- Ring spectrum. --- Simplicial complex. --- Simply connected space. --- Smash product. --- Special case. --- Spectral sequence. --- Steenrod algebra. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subring. --- Symmetric group. --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Vector bundle. --- Zariski topology.

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This self-contained introduction to the distributed control of robotic networks offers a distinctive blend of computer science and control theory. The book presents a broad set of tools for understanding coordination algorithms, determining their correctness, and assessing their complexity; and it analyzes various cooperative strategies for tasks such as consensus, rendezvous, connectivity maintenance, deployment, and boundary estimation. The unifying theme is a formal model for robotic networks that explicitly incorporates their communication, sensing, control, and processing capabilities--a model that in turn leads to a common formal language to describe and analyze coordination algorithms. Written for first- and second-year graduate students in control and robotics, the book will also be useful to researchers in control theory, robotics, distributed algorithms, and automata theory. The book provides explanations of the basic concepts and main results, as well as numerous examples and exercises. Self-contained exposition of graph-theoretic concepts, distributed algorithms, and complexity measures for processor networks with fixed interconnection topology and for robotic networks with position-dependent interconnection topology Detailed treatment of averaging and consensus algorithms interpreted as linear iterations on synchronous networks Introduction of geometric notions such as partitions, proximity graphs, and multicenter functions Detailed treatment of motion coordination algorithms for deployment, rendezvous, connectivity maintenance, and boundary estimation

Robotics. --- Computer algorithms. --- Robots --- Automation --- Machine theory --- Robot control --- Robotics --- Algorithms --- Control systems. --- Computer algorithms --- Control systems --- 1-center problem. --- Adjacency matrix. --- Aggregate function. --- Algebraic connectivity. --- Algebraic topology (object). --- Algorithm. --- Analysis of algorithms. --- Approximation algorithm. --- Asynchronous system. --- Bellman–Ford algorithm. --- Bifurcation theory. --- Bounded set (topological vector space). --- Calculation. --- Cartesian product. --- Centroid. --- Chebyshev center. --- Circulant matrix. --- Circumscribed circle. --- Cluster analysis. --- Combinatorial optimization. --- Combinatorics. --- Communication complexity. --- Computation. --- Computational complexity theory. --- Computational geometry. --- Computational model. --- Computer simulation. --- Computer vision. --- Connected component (graph theory). --- Connectivity (graph theory). --- Consensus (computer science). --- Control function (econometrics). --- Differentiable function. --- Dijkstra's algorithm. --- Dimensional analysis. --- Directed acyclic graph. --- Directed graph. --- Discrete time and continuous time. --- Disk (mathematics). --- Distributed algorithm. --- Doubly stochastic matrix. --- Dynamical system. --- Eigenvalues and eigenvectors. --- Estimation. --- Euclidean space. --- Function composition. --- Hybrid system. --- Information theory. --- Initial condition. --- Instance (computer science). --- Invariance principle (linguistics). --- Invertible matrix. --- Iteration. --- Iterative method. --- Kinematics. --- Laplacian matrix. --- Leader election. --- Linear dynamical system. --- Linear interpolation. --- Linear programming. --- Lipschitz continuity. --- Lyapunov function. --- Markov chain. --- Mathematical induction. --- Mathematical optimization. --- Mobile robot. --- Motion planning. --- Multi-agent system. --- Network model. --- Network topology. --- Norm (mathematics). --- Numerical integration. --- Optimal control. --- Optimization problem. --- Parameter (computer programming). --- Partition of a set. --- Percolation theory. --- Permutation matrix. --- Polytope. --- Proportionality (mathematics). --- Quantifier (logic). --- Quantization (signal processing). --- Robustness (computer science). --- Scientific notation. --- Sensor. --- Set (mathematics). --- Simply connected space. --- Simulation. --- Simultaneous equations. --- State space. --- State variable. --- Stochastic matrix. --- Stochastic. --- Strongly connected component. --- Synchronous network. --- Theorem. --- Time complexity. --- Topology. --- Variable (mathematics). --- Vector field.

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This book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by M. Artin and B. Mazur. Eric M. Friedlander presents many of his own applications of this theory to algebraic topology, finite Chevalley groups, and algebraic geometry. Of particular interest are the discussions concerning the Adams Conjecture, K-theories of finite fields, and Poincare duality. Because these applications have required repeated modifications of the original formulation of etale homotopy theory, the author provides a new treatment of the foundations which is more general and more precise than previous versions.One purpose of this book is to offer the basic techniques and results of etale homotopy theory to topologists and algebraic geometers who may then apply the theory in their own work. With a view to such future applications, the author has introduced a number of new constructions (function complexes, relative homology and cohomology, generalized cohomology) which have immediately proved applicable to algebraic K-theory.

Algebraic topology --- 512.73 --- 515.14 --- Homology theory --- Homotopy theory --- Schemes (Algebraic geometry) --- Geometry, Algebraic --- Deformations, Continuous --- Topology --- Cohomology theory --- Contrahomology theory --- Cohomology theory of algebraic varieties and schemes --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- Homotopy theory. --- Homology theory. --- Abelian group. --- Adams operation. --- Adjoint functors. --- Alexander Grothendieck. --- Algebraic K-theory. --- Algebraic closure. --- Algebraic geometry. --- Algebraic group. --- Algebraic number theory. --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Algebraically closed field. --- Automorphism. --- Base change. --- Cap product. --- Cartesian product. --- Closed immersion. --- Codimension. --- Coefficient. --- Cohomology. --- Comparison theorem. --- Complex number. --- Complex vector bundle. --- Connected component (graph theory). --- Connected space. --- Coprime integers. --- Corollary. --- Covering space. --- Derived functor. --- Dimension (vector space). --- Disjoint union. --- Embedding. --- Existence theorem. --- Ext functor. --- Exterior algebra. --- Fiber bundle. --- Fibration. --- Finite field. --- Finite group. --- Free group. --- Functor. --- Fundamental group. --- Galois cohomology. --- Galois extension. --- Geometry. --- Grothendieck topology. --- Homogeneous space. --- Homological algebra. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Integral domain. --- Intersection (set theory). --- Inverse limit. --- Inverse system. --- K-theory. --- Leray spectral sequence. --- Lie group. --- Local ring. --- Mapping cylinder. --- Natural number. --- Natural transformation. --- Neighbourhood (mathematics). --- Newton polynomial. --- Noetherian ring. --- Open set. --- Opposite category. --- Pointed set. --- Presheaf (category theory). --- Reductive group. --- Regular local ring. --- Relative homology. --- Residue field. --- Riemann surface. --- Root of unity. --- Serre spectral sequence. --- Shape theory (mathematics). --- Sheaf (mathematics). --- Sheaf cohomology. --- Sheaf of spectra. --- Simplex. --- Simplicial set. --- Special case. --- Spectral sequence. --- Surjective function. --- Theorem. --- Topological K-theory. --- Topological space. --- Topology. --- Tubular neighborhood. --- Vector bundle. --- Weak equivalence (homotopy theory). --- Weil conjectures. --- Weyl group. --- Witt vector. --- Zariski topology. --- Homologie --- Topologie algebrique --- Geometrie algebrique --- Homotopie