We consider the problem of selfish routing in a congested network shared by several users, where each user wishes to minimize the cost of its own flow. Users are atomic, in the sense that each has a nonnegligible amount of flow demand, and flows may be split over different routes. The total cost for each user is the sum of its link costs, which, in turn, may depend on the user’s own flow as well as the total flow on that link. Our main interest here is network topologies that ensure uniqueness of the Nash equilibrium for any set of users and link cost functions that satisfy some mild convexity conditions. We characterize the class of two-terminal network topologies for which this uniqueness property holds, and show that it coincides with the class of nearly parallel networks that was recently shown by Milchtaich [Milchtaich, I. 2005. Topological conditions for uniqueness of equilibrium in networks. Math. Oper. Res. 30 225–244] to ensure uniqueness in nonatomic (or Wardrop) routing games. We further show that uniqueness of the link flows holds under somewhat weaker convexity conditions, which apply to the mixed Nash-Wardrop equilibrium problem. We finally propose a generalized continuum-game formulation of the routing problem that allows for a unified treatment of atomic and nonatomic users.

Abstract. The spectral statistics of the circular billiard with a point-scatterer is investigated. In the semiclassical limit, the spectrum is demonstrated to be composed of two uncorrelated level sequences. The first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point-scatterer. The nearest neighbor spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found. Classical dynamics may be illuminating for the understanding of the corresponding quantum mechanical systems. One of the most studied aspects of the relation between classical and quantum mechanics is the connection between the spectral statistics of the quantum system and the dynamical properties of its classical counterpart. Classically integrable systems typically exhibit Poisson-like spectral statistics [1] while classically chaotic systems exhibit spectral statistics of random matrix ensembles [2,3,4,5]. The spectral statistics of integrable and chaotic systems are universal, that is, they do not depend on specific details of the system but rather on the type of motion and its symmetries. There are systems which are intermediate between integrable and chaotic ones and their spectral properties are not known to be universal. Such systems are of experimental relevance. The spectral statistics of mixed systems, for which the phase space is composed of both integrable and chaotic regions, were studied by Berry and Robnik [6]. The spectrum can be viewed as a superposition of uncorrelated level sequences, corresponding to the various regions, which are either chaotic or integrable. The nearest neighbor spacing distribution (NNSD) of such a superposition of sequences was calculated in [6]

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We study classification problems where features are corrupted by noise and where the magnitude of the noise in each feature is influenced by the resources allocated to its acquisition. This is the case, for example, when multiple sensors share a common resource (power, bandwidth, attention, etc.). We develop a method for computing the optimal resource allocation for a variety of scenarios and derive theoretical bounds concerning the benefit that may arise by non-uniform allocation. We further demonstrate the effectiveness of the developed method in simulations.