537,517 research outputs found
Afterthoughts | Complexities of Potency
Afterthoughts to themed perspectives on Complexities of Potency
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
Configuration Complexities of Hydrogenic Atoms
The Fisher-Shannon and Cramer-Rao information measures, and the LMC-like or
shape complexity (i.e., the disequilibrium times the Shannon entropic power) of
hydrogenic stationary states are investigated in both position and momentum
spaces. First, it is shown that not only the Fisher information and the
variance (then, the Cramer-Rao measure) but also the disequilibrium associated
to the quantum-mechanical probability density can be explicitly expressed in
terms of the three quantum numbers (n, l, m) of the corresponding state.
Second, the three composite measures mentioned above are analytically,
numerically and physically discussed for both ground and excited states. It is
observed, in particular, that these configuration complexities do not depend on
the nuclear charge Z. Moreover, the Fisher-Shannon measure is shown to
quadratically depend on the principal quantum number n. Finally, sharp upper
bounds to the Fisher-Shannon measure and the shape complexity of a general
hydrogenic orbital are given in terms of the quantum numbers.Comment: 22 pages, 7 figures, accepted i
State Complexity of Catenation Combined with Star and Reversal
This paper is a continuation of our research work on state complexity of
combined operations. Motivated by applications, we study the state complexities
of two particular combined operations: catenation combined with star and
catenation combined with reversal. We show that the state complexities of both
of these combined operations are considerably less than the compositions of the
state complexities of their individual participating operations.Comment: In Proceedings DCFS 2010, arXiv:1008.127
- …