1,478,407 research outputs found

    On the resonance eigenstates of an open quantum baker map

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    We study the resonance eigenstates of a particular quantization of the open baker map. For any admissible value of Planck's constant, the corresponding quantum map is a subunitary matrix, and the nonzero component of its spectrum is contained inside an annulus in the complex plane, zminzzmax|z_{min}|\leq |z|\leq |z_{max}|. We consider semiclassical sequences of eigenstates, such that the moduli of their eigenvalues converge to a fixed radius rr. We prove that, if the moduli converge to r=zmaxr=|z_{max}|, then the sequence of eigenstates converges to a fixed phase space measure ρmax\rho_{max}. The same holds for sequences with eigenvalue moduli converging to zmin|z_{min}|, with a different limit measure ρmin\rho_{min}. Both these limiting measures are supported on fractal sets, which are trapped sets of the classical dynamics. For a general radius zmin<r<zmax|z_{min}|< r < |z_{max}|, we identify families of eigenstates with precise self-similar properties.Comment: 32 pages, 2 figure

    On the Min-Max-Delay Problem: NP-completeness, Algorithm, and Integrality Gap

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    We study a delay-sensitive information flow problem where a source streams information to a sink over a directed graph G(V,E) at a fixed rate R possibly using multiple paths to minimize the maximum end-to-end delay, denoted as the Min-Max-Delay problem. Transmission over an edge incurs a constant delay within the capacity. We prove that Min-Max-Delay is weakly NP-complete, and demonstrate that it becomes strongly NP-complete if we require integer flow solution. We propose an optimal pseudo-polynomial time algorithm for Min-Max-Delay, with time complexity O(\log (Nd_{\max}) (N^5d_{\max}^{2.5})(\log R+N^2d_{\max}\log(N^2d_{\max}))), where N = \max\{|V|,|E|\} and d_{\max} is the maximum edge delay. Besides, we show that the integrality gap, which is defined as the ratio of the maximum delay of an optimal integer flow to the maximum delay of an optimal fractional flow, could be arbitrarily large

    Trivial Meet and Join within the Lattice of Monotone Triangles

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    The lattice of monotone triangles (Mn,)(\mathfrak{M}_n,\le) ordered by entry-wise comparisons is studied. Let τmin\tau_{\min} denote the unique minimal element in this lattice, and τmax\tau_{\max} the unique maximum. The number of rr-tuples of monotone triangles (τ1,,τr)(\tau_1,\ldots,\tau_r) with minimal infimum τmin\tau_{\min} (maximal supremum τmax\tau_{\max}, resp.) is shown to asymptotically approach rMnr1r|\mathfrak{M}_n|^{r-1} as nn \to \infty. Thus, with high probability this event implies that one of the τi\tau_i is τmin\tau_{\min} (τmax\tau_{\max}, resp.). Higher-order error terms are also discussed.Comment: 15 page

    On the approximability of robust spanning tree problems

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    In this paper the minimum spanning tree problem with uncertain edge costs is discussed. In order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution. The min-max, min-max regret and 2-stage min-max versions of the problem are discussed. The complexity and approximability of all these problems are explored. It is proved that the min-max and min-max regret versions with nonnegative edge costs are hard to approximate within O(log1ϵn)O(\log^{1-\epsilon} n) for any ϵ>0\epsilon>0 unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within O(logn)O(\log n) unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approximation algorithms with performance ratio of O(log2n)O(\log^2 n) for min-max and 2-stage min-max problems are also proposed
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