83,292 research outputs found
Limit complexities revisited [once more]
The main goal of this article is to put some known results in a common
perspective and to simplify their proofs.
We start with a simple proof of a result of Vereshchagin saying that
equals . Then we use the same argument to prove
similar results for prefix complexity, a priori probability on binary tree, to
prove Conidis' theorem about limits of effectively open sets, and also to
improve the results of Muchnik about limit frequencies. As a by-product, we get
a criterion of 2-randomness proved by Miller: a sequence is 2-random if and
only if there exists such that any prefix of is a prefix of some
string such that . (In the 1960ies this property was
suggested in Kolmogorov as one of possible randomness definitions.) We also get
another 2-randomness criterion by Miller and Nies: is 2-random if and only
if for some and infinitely many prefixes of .
This is a modified version of our old paper that contained a weaker (and
cumbersome) version of Conidis' result, and the proof used low basis theorem
(in quite a strange way). The full version was formulated there as a
conjecture. This conjecture was later proved by Conidis. Bruno Bauwens
(personal communication) noted that the proof can be obtained also by a simple
modification of our original argument, and we reproduce Bauwens' argument with
his permission.Comment: See http://arxiv.org/abs/0802.2833 for the old pape
Limit complexities revisited
The main goal of this paper is to put some known results in a common
perspective and to simplify their proofs. We start with a simple proof of a
result from (Vereshchagin, 2002) saying that \limsup_n\KS(x|n) (here
\KS(x|n) is conditional (plain) Kolmogorov complexity of when is
known) equals \KS^{\mathbf{0'}(x), the plain Kolmogorov complexity with
\mathbf{0'-oracle. Then we use the same argument to prove similar results for
prefix complexity (and also improve results of (Muchnik, 1987) about limit
frequencies), a priori probability on binary tree and measure of effectively
open sets. As a by-product, we get a criterion of Martin-L\"of
randomness (called also 2-randomness) proved in (Miller, 2004): a sequence
is 2-random if and only if there exists such that any prefix
of is a prefix of some string such that \KS(y)\ge |y|-c. (In the
1960ies this property was suggested in (Kolmogorov, 1968) as one of possible
randomness definitions; its equivalence to 2-randomness was shown in (Miller,
2004) while proving another 2-randomness criterion (see also (Nies et al.
2005)): is 2-random if and only if \KS(x)\ge |x|-c for some and
infinitely many prefixes of . Finally, we show that the low-basis
theorem can be used to get alternative proofs for these results and to improve
the result about effectively open sets; this stronger version implies the
2-randomness criterion mentioned in the previous sentence
Multi-dimensional Boltzmann Sampling of Languages
This paper addresses the uniform random generation of words from a
context-free language (over an alphabet of size ), while constraining every
letter to a targeted frequency of occurrence. Our approach consists in a
multidimensional extension of Boltzmann samplers \cite{Duchon2004}. We show
that, under mostly \emph{strong-connectivity} hypotheses, our samplers return a
word of size in and exact frequency in
expected time. Moreover, if we accept tolerance
intervals of width in for the number of occurrences of each
letters, our samplers perform an approximate-size generation of words in
expected time. We illustrate these techniques on the
generation of Tetris tessellations with uniform statistics in the different
types of tetraminoes.Comment: 12p
Local connection forms revisited
Local connection forms provide a very useful tool for handling connections on
principal bundles, because they ignore any complexities of the total space and,
essentially, involve only two fundamental features of the structure group,
namely the adjoint representation and the left (logarithmic) differential. The
main results of this note characterize connections related together by bundle
morphisms, while applications (taken from various sources) refer to connections
on (Banach) associated bundles, in particular vector bundles, and connections
on inverse limit bundles (in the Fr\'echet framework). The role of local
connection forms is further illustrated by their sheaf-theoretic globalization,
resulting in a sheaf operator-like approach to principal connections. The
latter point of view is naturally leading to a theory of connections on
abstract principal sheaves.Comment: AMS-LaTex, 19 pages. Corrected typos. Few stylistic change
On the convergence of the Fitness-Complexity Algorithm
We investigate the convergence properties of an algorithm which has been
recently proposed to measure the competitiveness of countries and the quality
of their exported products. These quantities are called respectively Fitness F
and Complexity Q. The algorithm was originally based on the adjacency matrix M
of the bipartite network connecting countries with the products they export,
but can be applied to any bipartite network. The structure of the adjacency
matrix turns to be essential to determine which countries and products converge
to non zero values of F and Q. Also the speed of convergence to zero depends on
the matrix structure. A major role is played by the shape of the ordered matrix
and, in particular, only those matrices whose diagonal does not cross the empty
part are guaranteed to have non zero values as outputs when the algorithm
reaches the fixed point. We prove this result analytically for simplified
structures of the matrix, and numerically for real cases. Finally, we propose
some practical indications to take into account our results when the algorithm
is applied.Comment: 13 pages, 8 figure
Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization'' by V. Koltchinskii
Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and
oracle inequalities in risk minimization'' by V. Koltchinskii [arXiv:0708.0083]Comment: Published at http://dx.doi.org/10.1214/009053606000001037 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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