1,478,407 research outputs found
On the resonance eigenstates of an open quantum baker map
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, . We consider semiclassical sequences of eigenstates, such that the
moduli of their eigenvalues converge to a fixed radius . We prove that, if
the moduli converge to , then the sequence of eigenstates
converges to a fixed phase space measure . The same holds for
sequences with eigenvalue moduli converging to , with a different
limit measure . Both these limiting measures are supported on
fractal sets, which are trapped sets of the classical dynamics. For a general
radius , 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
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
The lattice of monotone triangles ordered by
entry-wise comparisons is studied. Let denote the unique minimal
element in this lattice, and the unique maximum. The number of
-tuples of monotone triangles with minimal infimum
(maximal supremum , resp.) is shown to
asymptotically approach as . Thus, with
high probability this event implies that one of the is
(, resp.). Higher-order error terms are also discussed.Comment: 15 page
On the approximability of robust spanning tree problems
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 for any unless
the problems in NP have quasi-polynomial time algorithms. Similarly, the
2-stage min-max problem cannot be approximated within unless the
problems in NP have quasi-polynomial time algorithms. In this paper randomized
LP-based approximation algorithms with performance ratio of for
min-max and 2-stage min-max problems are also proposed
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