27,911 research outputs found
Correspondence between Thermal and Quantum Vacuum Transitions around Horizons
Recently, there are comparable revised interests in bubble nucleation seeded
by black holes. However, it is debated in the literature that whether one shall
interpret a static bounce solution in the Euclidean Schwarzschild spacetime
(with periodic Euclidean Schwarzschild time) as describing a false vacuum decay
at zero temperature or at finite temperature. In this paper, we show a
correspondence that the static bounce solution describes either a thermal
transition of vacuum in the static region outside of a Schwarzschild black hole
or a quantum transition in a maximally extended Kruskal-Szekeres spacetime,
corresponding to the viewpoint of the external static observers or the freely
falling observers, respectively. The Matsubara modes in the thermal
interpretation can be mapped to the circular harmonic modes from an
symmetry in the tunneling interpretation. The complementary tunneling
interpretation must be given in the Kruskal-Szekeres spacetime because of the
so-called thermofield dynamics. This correspondence is general for bubble
nucleation around horizons. We propose a new paradox related to black holes as
a consequence of this correspondence.Comment: 26 pages; v2: typos corrected; v3: references added, discussion on
AdS black holes added, to match the published version; v4(v5): Ref [37]
updated, footnote [10] added v6: two typos correcte
Separable subgroups have bounded packing
In this note, we prove that separable subgroups have bounded packing in
ambient groups. The notion bounded packing was introduced by Hruska and Wise
and in particular, our result answers positively a question of theirs, asking
whether each subgroup of a virtually polycyclic group has the bounded packing
property.Comment: 2 pages, to appear Proc. Amer. Math. So
Stochastic Combinatorial Optimization via Poisson Approximation
We study several stochastic combinatorial problems, including the expected
utility maximization problem, the stochastic knapsack problem and the
stochastic bin packing problem. A common technical challenge in these problems
is to optimize some function of the sum of a set of random variables. The
difficulty is mainly due to the fact that the probability distribution of the
sum is the convolution of a set of distributions, which is not an easy
objective function to work with. To tackle this difficulty, we introduce the
Poisson approximation technique. The technique is based on the Poisson
approximation theorem discovered by Le Cam, which enables us to approximate the
distribution of the sum of a set of random variables using a compound Poisson
distribution.
We first study the expected utility maximization problem introduced recently
[Li and Despande, FOCS11]. For monotone and Lipschitz utility functions, we
obtain an additive PTAS if there is a multidimensional PTAS for the
multi-objective version of the problem, strictly generalizing the previous
result.
For the stochastic bin packing problem (introduced in [Kleinberg, Rabani and
Tardos, STOC97]), we show there is a polynomial time algorithm which uses at
most the optimal number of bins, if we relax the size of each bin and the
overflow probability by eps.
For stochastic knapsack, we show a 1+eps-approximation using eps extra
capacity, even when the size and reward of each item may be correlated and
cancelations of items are allowed. This generalizes the previous work [Balghat,
Goel and Khanna, SODA11] for the case without correlation and cancelation. Our
algorithm is also simpler. We also present a factor 2+eps approximation
algorithm for stochastic knapsack with cancelations. the current known
approximation factor of 8 [Gupta, Krishnaswamy, Molinaro and Ravi, FOCS11].Comment: 42 pages, 1 figure, Preliminary version appears in the Proceeding of
the 45th ACM Symposium on the Theory of Computing (STOC13
Comment on "Next-to-leading order forward hadron production in the small-x regime: rapidity factorization" arXiv:1403.5221 by Kang et al
In a recent paper (arXiv:1403.5221), Kang et al.proposed the so-called
"rapidity factorization" for the single inclusive forward hadron production in
pA collisions. We point out that the leading small-x logarithm was
mis-identified in this paper, and hence the newly added next-to-leading order
correction term is unjustified and should be absent in view of the small-x
factorization.Comment: 3 page
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