192 research outputs found
Factor versus palindromic complexity of uniformly recurrent infinite words
We study the relation between the palindromic and factor complexity of
infinite words. We show that for uniformly recurrent words one has P(n)+P(n+1)
\leq \Delta C(n) + 2, for all n \in N. For a large class of words it is a
better estimate of the palindromic complexity in terms of the factor complexity
then the one presented by Allouche et al. We provide several examples of
infinite words for which our estimate reaches its upper bound. In particular,
we derive an explicit prescription for the palindromic complexity of infinite
words coding r-interval exchange transformations. If the permutation \pi
connected with the transformation is given by \pi(k)=r+1-k for all k, then
there is exactly one palindrome of every even length, and exactly r palindromes
of every odd length.Comment: 16 pages, submitted to Theoretical Computer Scienc
A geometrical characterization of factors of multidimensional Billiard words and some applications
AbstractWe consider Billiard words in alphabets with k>2 letters. Such words are associated with some k-dimensional positive vector α→=(α1,α2,…,αk). The language of these words is already known in the usual case, i.e. when the αj are linearly independent over Q and so for their inverses. Here we study the language of these words when there exist some linear relationships. We give a new geometrical characterization of the factors of Billiard words. As a consequence, we get some results on the associated language, and on the complexity and palindromic complexity of these words. The situation is quite different from the usual case. The languages of two distinct Billiard words with the same direction generally have a finite intersection. As examples, we get some Standard Billiard words of three letters without any palindromic factor of even length, or Billiard words of three letters whose palindromic factors have a bounded length. These results are obtained by geometrical methods
Effective Physical Processes and Active Information in Quantum Computing
The recent debate on hypercomputation has arisen new questions both on the
computational abilities of quantum systems and the Church-Turing Thesis role in
Physics. We propose here the idea of "effective physical process" as the
essentially physical notion of computation. By using the Bohm and Hiley active
information concept we analyze the differences between the standard form
(quantum gates) and the non-standard one (adiabatic and morphogenetic) of
Quantum Computing, and we point out how its Super-Turing potentialities derive
from an incomputable information source in accordance with Bell's constraints.
On condition that we give up the formal concept of "universality", the
possibility to realize quantum oracles is reachable. In this way computation is
led back to the logic of physical world.Comment: 10 pages; Added references for sections 2 and
Quantal-Classical Duality and the Semiclassical Trace Formula
We consider Hamiltonian systems which can be described both classically and
quantum mechanically. Trace formulas establish links between the energy spectra
of the quantum description and the spectrum of actions of periodic orbits in
the classical description. This duality is investigated in the present paper.
The duality holds for chaotic as well as for integrable systems. For billiards
the quantal spectrum (eigenvalues of the Helmholtz equation) and the classical
spectrum (lengths of periodic orbits) are two manifestations of the billiard's
boundary. The trace formula expresses this link as a Fourier transform relation
between the corresponding spectral densities. It follows that the two-point
statistics are also simply related. The universal correlations of the quantal
spectrum are well known, consequently one can deduce the classical universal
correlations. An explicit expression for the scale of the classical
correlations is derived and interpreted. This allows a further extension of the
formalism to the case of complex billiard systems, and in particular to the
most interesting case of diffusive system. The concept of classical
correlations allows a better understanding of the so-called diagonal
approximation and its breakdown. It also paves the way towards a semiclassical
theory that is capable of global description of spectral statistics beyond the
breaktime. An illustrative application is the derivation of the
disorder-limited breaktime in case of a disordered chain, thus obtaining a
semiclassical theory for localization. A numerical study of classical
correlations in the case of the 3D Sinai billiard is presented. We gain a
direct understanding of specific statistical properties of the classical
spectrum, as well as their semiclassical manifestation in the quantal spectrum.Comment: 42 pages, 17 figure
Normal forms and complex periodic orbits in semiclassical expansions of Hamiltonian systems
Bifurcations of periodic orbits as an external parameter is varied are a
characteristic feature of generic Hamiltonian systems. Meyer's classification
of normal forms provides a powerful tool to understand the structure of phase
space dynamics in their neighborhood. We provide a pedestrian presentation of
this classical theory and extend it by including systematically the periodic
orbits lying in the complex plane on each side of the bifurcation. This allows
for a more coherent and unified treatment of contributions of periodic orbits
in semiclassical expansions. The contribution of complex fixed points is find
to be exponentially small only for a particular type of bifurcation (the
extremal one). In all other cases complex orbits give rise to corrections in
powers of and, unlike the former one, their contribution is hidden in
the ``shadow'' of a real periodic orbit.Comment: better ps figures available at http://www.phys.univ-tours.fr/~mouchet
or on request to [email protected]
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