331 research outputs found
A note on entropy estimation
We compare an entropy estimator recently discussed in [10] with two
estimators and introduced in [6][7]. We prove the identity , which has not been taken into account in [10]. Then, we prove that
the statistical bias of is less than the bias of the ordinary likelihood
estimator of entropy. Finally, by numerical simulation we verify that for the
most interesting regime of small sample estimation and large event spaces, the
estimator has a significant smaller statistical error than .Comment: 7 pages, including 4 figures; two references adde
On the uncertainty principle in Rindler and Friedmann spacetimes
We revise the extended uncertainty relations for the Rindler and Friedmann spacetimes recently discussed by Dabrowski and Wagner in [9]. We reveal these results to be coordinate dependent expressions of the invariant uncertainty relations recently derived for general 3-dimensional spaces of constant curvature in [10]. Moreover, we show that the non-zero minimum standard deviations of the momentum in [9] are just artifacts caused by an unfavorable choice of coordinate systems which can be removed by standard arguments of geodesic completion
A reinterpretation of the cosmological vacuum
In this paper, we make a proposal for addressing the cosmological constant problem. Our approach will be based on a reinterpretation of two non-standard de Sitter solutions given by the Einstein vacuum equations with Λ\u3e0. As a first result, we derive an uncertainty principle for both variants of the de Sitter space (Theorem). Subsequently, a decomposition of the cosmological constant in a pair of time-dependent pieces is introduced (Corollary). The time-dependence of the corresponding energy and dark energy density is discussed and especially matched at the Planck scale. Furthermore, we show that for every instant of cosmic time this approach can be revealed in terms of a Schwarzschild-Anti-de Sitter cosmology with Λ\u3e0. The corresponding field equations are provided
The predictability of letters in written english
We show that the predictability of letters in written English texts depends
strongly on their position in the word. The first letters are usually the least
easy to predict. This agrees with the intuitive notion that words are well
defined subunits in written languages, with much weaker correlations across
these units than within them. It implies that the average entropy of a letter
deep inside a word is roughly 4 times smaller than the entropy of the first
letter.Comment: 3 pages, 4 figure
On Lattice-Free Orbit Polytopes
Given a permutation group acting on coordinates of , we
consider lattice-free polytopes that are the convex hull of an orbit of one
integral vector. The vertices of such polytopes are called \emph{core points}
and they play a key role in a recent approach to exploit symmetry in integer
convex optimization problems. Here, naturally the question arises, for which
groups the number of core points is finite up to translations by vectors fixed
by the group. In this paper we consider transitive permutation groups and prove
this type of finiteness for the -homogeneous ones. We provide tools for
practical computations of core points and obtain a complete list of
representatives for all -homogeneous groups up to degree twelve. For
transitive groups that are not -homogeneous we conjecture that there exist
infinitely many core points up to translations by the all-ones-vector. We prove
our conjecture for two large classes of groups: For imprimitive groups and
groups that have an irrational invariant subspace.Comment: 27 pages, 2 figures; with minor adaptions according to referee
comments; to appear in Discrete and Computational Geometr
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