263,010 research outputs found

    Generalized Toda mechanics associated with classical Lie algebras and their reductions

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    For any classical Lie algebra gg, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers (m,n)(m,n). The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for g=Br,Cr,Dr\mathfrak{g}=B_{r},C_{r},D_{r} with m,n≀3m,n\leq3 are also given. For all m,nm,n, it is shown that the dynamics of the (m,nβˆ’1)(m,n-1)- and the (mβˆ’1,n)(m-1,n)-Toda chains are natural reductions of that of the (m,n)(m,n)-chain, and for m=nm=n, there is also a family of symmetrically reduced Toda systems, the (m,m)Sym(m,m)_{\mathrm{Sym}}-Toda systems, which are also integrable. In the quantum case, all (m,n)(m,n)-Toda systems with m>1m>1 or n>1n>1 describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all (m,n)(m,n)-Toda systems survive after quantization.Comment: 19 pages, bibte

    Distributed Flow Scheduling in an Unknown Environment

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    Flow scheduling tends to be one of the oldest and most stubborn problems in networking. It becomes more crucial in the next generation network, due to fast changing link states and tremendous cost to explore the global structure. In such situation, distributed algorithms often dominate. In this paper, we design a distributed virtual game to solve the flow scheduling problem and then generalize it to situations of unknown environment, where online learning schemes are utilized. In the virtual game, we use incentives to stimulate selfish users to reach a Nash Equilibrium Point which is valid based on the analysis of the `Price of Anarchy'. In the unknown-environment generalization, our ultimate goal is the minimization of cost in the long run. In order to achieve balance between exploration of routing cost and exploitation based on limited information, we model this problem based on Multi-armed Bandit Scenario and combined newly proposed DSEE with the virtual game design. Armed with these powerful tools, we find a totally distributed algorithm to ensure the logarithmic growing of regret with time, which is optimum in classic Multi-armed Bandit Problem. Theoretical proof and simulation results both affirm this claim. To our knowledge, this is the first research to combine multi-armed bandit with distributed flow scheduling.Comment: 10 pages, 3 figures, conferenc

    Distributed Learning in Multi-Armed Bandit with Multiple Players

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    We formulate and study a decentralized multi-armed bandit (MAB) problem. There are M distributed players competing for N independent arms. Each arm, when played, offers i.i.d. reward according to a distribution with an unknown parameter. At each time, each player chooses one arm to play without exchanging observations or any information with other players. Players choosing the same arm collide, and, depending on the collision model, either no one receives reward or the colliding players share the reward in an arbitrary way. We show that the minimum system regret of the decentralized MAB grows with time at the same logarithmic order as in the centralized counterpart where players act collectively as a single entity by exchanging observations and making decisions jointly. A decentralized policy is constructed to achieve this optimal order while ensuring fairness among players and without assuming any pre-agreement or information exchange among players. Based on a Time Division Fair Sharing (TDFS) of the M best arms, the proposed policy is constructed and its order optimality is proven under a general reward model. Furthermore, the basic structure of the TDFS policy can be used with any order-optimal single-player policy to achieve order optimality in the decentralized setting. We also establish a lower bound on the system regret growth rate for a general class of decentralized polices, to which the proposed policy belongs. This problem finds potential applications in cognitive radio networks, multi-channel communication systems, multi-agent systems, web search and advertising, and social networks.Comment: 31 pages, 8 figures, revised paper submitted to IEEE Transactions on Signal Processing, April, 2010, the pre-agreement in the decentralized TDFS policy is eliminated to achieve a complete decentralization among player
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