Springer Science+Business Media, LLC Graduate Texts in MathematicsTAKEUTIIZARING. Introduction to 37 MONK. MathematicalLogic. Axiomatic Set Theory. 2nd ed. 38 GRAUERTIFRITZSCHE. Several Complex 2 OXTOBY. Measure and Category. 2nd ed. Variables. 3 SCHAEFER. Topological Vector Spaces. 39 ARVESON. An Invitation to C*-Algebras. 2nd ed. 40 KEMENY/SNEUJKNAPP. Denumerable 4 HILTON/STAMMBACH. A Course in Markov Chains. 2nd ed. Homological Algebra. 2nd ed. 41 APOSTOL. Modular Functions and Dirichlet 5 MAc LANE. Categories for the Working Series in Number Theory. Mathematician. 2nd ed. 2nd ed. 6 HUGHESIPIPER. Projective Planes. 42 SERRE. Linear Representations ofFinite 7 SERRE. A Course in Arithmetic. Groups. 8 TAKEUTIIZARING. Axiomatic Set Theory. 43 GILLMAN/JERISON. Rings ofContinuous 9 HUMPHREYS. Introductionto Lie Algebras Functions. and Representation Theory. 44 KENDIG. E1ementary Algebraic Geometry. 10 COHEN. A Course in Simple Homotopy 45 LOEVE. Probability Theory I. 4th ed. Theory. 46 LOEVE. Probability Theory II . 4th ed. II CONWAY. Functions ofOne Complex 47 MOISE. Geometrie Topology in Variable I. 2nd ed. Dimensions 2 and 3. 12 BEALS. Advanced Mathematical Analysis. 48 SACHSlWu. General Relativity for 13 ANDERSONIFULLER. Rings and Categories Mathematicians. ofModules. 2nd ed. 49 GRUENBERGIWEIR. Linear Geometry. 14 GOLUBITSKy/GUILLEMIN. Stable Mappings 2nded.

This paper reverse-engineers backoff-based random-access MAC protocols in ad-hoc networks. We show that the contention resolution algorithm in such protocols is implicitly participating in a non-cooperative game. Each link attempts to maximize a selfish local utility function, whose exact shape is reverse-engineered from the protocol description, through a stochastic subgradient method in which the link updates its persistence probability based on its transmission success or failure. We prove that existence of a Nash equilibrium is guaranteed in general. Then we establish the minimum amount of backoff aggressiveness needed, as a function of density of active users, for uniqueness of Nash equilibrium and convergence of the best response strategy. Convergence properties and connection with the best response strategy are also proved for variants of the stochastic-subgradient-based dynamics of the game. Together with known results in reverse-engineering TCP and BGP, this paper further advances the recent efforts in reverse-engineering layers 2-4 protocols. In contrast to the TCP reverse-engineering results in earlier literature, MAC reverse-engineering highlights the non-cooperative nature of random access.

Because of their significance in physics and chemistry, representation of Lie groups has been an area of intensive study by physicists and chemists, as well as mathematicians. This introduction is designed for graduate students who have some knowledge of finite groups and general topology, but is otherwise self-contained. The author gives direct and concise proofs of all results yet avoids the heavy machinery of functional analysis. Moreover, representative examples are treated in some detail.

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