903 research outputs found
Canonical Representatives of Morphic Permutations
An infinite permutation can be defined as a linear ordering of the set of
natural numbers. In particular, an infinite permutation can be constructed with
an aperiodic infinite word over as the lexicographic order
of the shifts of the word. In this paper, we discuss the question if an
infinite permutation defined this way admits a canonical representative, that
is, can be defined by a sequence of numbers from [0, 1], such that the
frequency of its elements in any interval is equal to the length of that
interval. We show that a canonical representative exists if and only if the
word is uniquely ergodic, and that is why we use the term ergodic permutations.
We also discuss ways to construct the canonical representative of a permutation
defined by a morphic word and generalize the construction of Makarov, 2009, for
the Thue-Morse permutation to a wider class of infinite words.Comment: Springer. WORDS 2015, Sep 2015, Kiel, Germany. Combinatorics on
Words: 10th International Conference. arXiv admin note: text overlap with
arXiv:1503.0618
The Critical Exponent is Computable for Automatic Sequences
The critical exponent of an infinite word is defined to be the supremum of
the exponent of each of its factors. For k-automatic sequences, we show that
this critical exponent is always either a rational number or infinite, and its
value is computable. Our results also apply to variants of the critical
exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni
and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes
or recovers previous results of Krieger and others, and is applicable to other
situations; e.g., the computation of the optimal recurrence constant for a
linearly recurrent k-automatic sequence.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Retention mechanisms and binding states of deuterium implanted into beryllium
doi:10.1088/1367-2630/11/4/043023 Abstract. The retention of 1 keV D+ ions implanted into clean and oxidized single crystalline Be at room and elevated temperatures is investigated by a combination of in situ analytical techniques including temperature programmed desorption (TPD), nuclear reaction analysis, low-energy ion spectroscopy (LEIS) and x-ray photoelectron spectroscopy. For the first time, the whole temperature regime for deuterium release and the influence of thin oxide films on the release processes are clarified. The cleaned and annealed Be sample has residual oxygen concentration equivalent to 0.2monolayer (ML) BeO in the near-surface region as the only contamination. LEIS shows that Be from the volume covers thin BeO surface layers above an annealing temperature of 1000K by segregation, forming a pure Be-terminated surface, which is stable at lower temperatures until again oxidized by residual gas. No deuterium is retained in the sample above 950K. By analyzing TPD spectra, active retention mechanisms and six energetically different binding states are identified. Activation energies (EA
Multifractal eigenstates of quantum chaos and the Thue-Morse sequence
We analyze certain eigenstates of the quantum baker's map and demonstrate,
using the Walsh-Hadamard transform, the emergence of the ubiquitous Thue-Morse
sequence, a simple sequence that is at the border between quasi-periodicity and
chaos, and hence is a good paradigm for quantum chaotic states. We show a
family of states that are also simply related to Thue-Morse sequence, and are
strongly scarred by short periodic orbits and their homoclinic excursions. We
give approximate expressions for these states and provide evidence that these
and other generic states are multifractal.Comment: Substantially modified from the original, worth a second download. To
appear in Phys. Rev. E as a Rapid Communicatio
Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map
We rationalize the somewhat surprising efficacy of the Hadamard transform in
simplifying the eigenstates of the quantum baker's map, a paradigmatic model of
quantum chaos. This allows us to construct closely related, but new, transforms
that do significantly better, thus nearly solving for many states of the
quantum baker's map. These new transforms, which combine the standard Fourier
and Hadamard transforms in an interesting manner, are constructed from
eigenvectors of the shift permutation operator that are also simultaneous
eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal)
symmetry.Comment: Version to appear in J. Phys. A. Added discussions; modified title;
corrected minor error
Shuffling cards, factoring numbers, and the quantum baker's map
It is pointed out that an exactly solvable permutation operator, viewed as
the quantization of cyclic shifts, is useful in constructing a basis in which
to study the quantum baker's map, a paradigm system of quantum chaos. In the
basis of this operator the eigenfunctions of the quantum baker's map are
compressed by factors of around five or more. We show explicitly its connection
to an operator that is closely related to the usual quantum baker's map. This
permutation operator has interesting connections to the art of shuffling cards
as well as to the quantum factoring algorithm of Shor via the quantum order
finding one. Hence we point out that this well-known quantum algorithm makes
crucial use of a quantum chaotic operator, or at least one that is close to the
quantization of the left-shift, a closeness that we also explore
quantitatively.Comment: 12 pgs. Substantially elaborated version, including a new route to
the quantum bakers map. To appear in J. Phys.
Quantum chaos in the spectrum of operators used in Shor's algorithm
We provide compelling evidence for the presence of quantum chaos in the
unitary part of Shor's factoring algorithm. In particular we analyze the
spectrum of this part after proper desymmetrization and show that the
fluctuations of the eigenangles as well as the distribution of the eigenvector
components follow the CUE ensemble of random matrices, of relevance to
quantized chaotic systems that violate time-reversal symmetry. However, as the
algorithm tracks the evolution of a single state, it is possible to employ
other operators, in particular it is possible that the generic quantum chaos
found above becomes of a nongeneric kind such as is found in the quantum cat
maps, and in toy models of the quantum bakers map.Comment: Title and paper modified to include interesting additional
possibilities. Principal results unaffected. Accepted for publication in
Phys. Rev. E as Rapid Com
Palindromic complexity of trees
We consider finite trees with edges labeled by letters on a finite alphabet
. Each pair of nodes defines a unique labeled path whose trace is a
word of the free monoid . The set of all such words defines the
language of the tree. In this paper, we investigate the palindromic complexity
of trees and provide hints for an upper bound on the number of distinct
palindromes in the language of a tree.Comment: Submitted to the conference DLT201
Nivat's conjecture holds for sums of two periodic configurations
Nivat's conjecture is a long-standing open combinatorial problem. It concerns
two-dimensional configurations, that is, maps where is a finite set of symbols. Such configurations are often
understood as colorings of a two-dimensional square grid. Let denote
the number of distinct block patterns occurring in a configuration
. Configurations satisfying for some
are said to have low rectangular complexity. Nivat conjectured that such
configurations are necessarily periodic.
Recently, Kari and the author showed that low complexity configurations can
be decomposed into a sum of periodic configurations. In this paper we show that
if there are at most two components, Nivat's conjecture holds. As a corollary
we obtain an alternative proof of a result of Cyr and Kra: If there exist such that , then is periodic. The
technique used in this paper combines the algebraic approach of Kari and the
author with balanced sets of Cyr and Kra.Comment: Accepted for SOFSEM 2018. This version includes an appendix with
proofs. 12 pages + references + appendi
- …