797 research outputs found
Infinite products involving binary digit sums
Let denote the Thue-Morse sequence with values . The
Woods-Robbins identity below and several of its generalisations are well-known
in the literature
\begin{equation*}\label{WR}\prod_{n=0}^\infty\left(\frac{2n+1}{2n+2}\right)^{u_n}=\frac{1}{\sqrt
2}.\end{equation*} No other such product involving a rational function in
and the sequence seems to be known in closed form. To understand these
products in detail we study the function
\begin{equation*}f(b,c)=\prod_{n=1}^\infty\left(\frac{n+b}{n+c}\right)^{u_n}.\end{equation*}
We prove some analytical properties of . We also obtain some new identities
similar to the Woods-Robbins product.Comment: Accepted in Proc. AMMCS 2017, updated according to the referees'
comment
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
Enumeration and Decidable Properties of Automatic Sequences
We show that various aspects of k-automatic sequences -- such as having an
unbordered factor of length n -- are both decidable and effectively enumerable.
As a consequence it follows that many related sequences are either k-automatic
or k-regular. These include many sequences previously studied in the
literature, such as the recurrence function, the appearance function, and the
repetitivity index. We also give some new characterizations of the class of
k-regular sequences. Many results extend to other sequences defined in terms of
Pisot numeration systems
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
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.
An ultrametric state space with a dense discrete overlap distribution: Paperfolding sequences
We compute the Parisi overlap distribution for paperfolding sequences. It
turns out to be discrete, and to live on the dyadic rationals. Hence it is a
pure point measure whose support is the full interval [-1; +1]. The space of
paperfolding sequences has an ultrametric structure. Our example provides an
illustration of some properties which were suggested to occur for pure states
in spin glass models
Role of Ag in textured-annealed Bi2Ca2Co1.7Ox thermoelectric ceramic
Bi2Ca2Co1.7Ox thermoelectric ceramics with small Ag additions (0, 1, 2, 3, 4, and 5 wt.%) have been successfully grown from the melt, using the laser floating zone method and subsequently annealed at 800 °C for 24 h. The microstructure has shown a reduction of the amount of secondary phases for Ag contents up to 4 wt.%. This microstructural evolution leads to a decrease of the electrical resistivity values until an Ag content of 4 wt.%, whereas Seebeck coefficient has been maintained unchanged. This is in agreement with the presence of metallic Ag in all samples, confirmed not only by Energy Dispersive Spectrometry but also by X-ray photoelectron and Auger spectroscopy. These electrical properties lead to maximum power factor values of about 0.30 mW/K2.m at 650 °C for the 4 wt.% Ag containing samples, which is among the best results obtained for this type of materials
The Non-Archimedean Theory of Discrete Systems
In the paper, we study behavior of discrete dynamical systems (automata)
w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be
behavior of the system w.r.t. variety of word transformations performed by the
system: We call a system completely transitive if, given arbitrary pair
of finite words that have equal lengths, the system , while
evolution during (discrete) time, at a certain moment transforms into .
To every system , we put into a correspondence a family of continuous maps of a suitable non-Archimedean metric space
and show that the system is completely transitive if and only if the family
is ergodic w.r.t. the Haar measure; then we find
easy-to-verify conditions the system must satisfy to be completely transitive.
The theory can be applied to analyze behavior of straight-line computer
programs (in particular, pseudo-random number generators that are used in
cryptography and simulations) since basic CPU instructions (both numerical and
logical) can be considered as continuous maps of a (non-Archimedean) metric
space of 2-adic integers.Comment: The extended version of the talk given at MACIS-201
Laser cooling of a diatomic molecule
It has been roughly three decades since laser cooling techniques produced
ultracold atoms, leading to rapid advances in a vast array of fields.
Unfortunately laser cooling has not yet been extended to molecules because of
their complex internal structure. However, this complexity makes molecules
potentially useful for many applications. For example, heteronuclear molecules
possess permanent electric dipole moments which lead to long-range, tunable,
anisotropic dipole-dipole interactions. The combination of the dipole-dipole
interaction and the precise control over molecular degrees of freedom possible
at ultracold temperatures make ultracold molecules attractive candidates for
use in quantum simulation of condensed matter systems and quantum computation.
Also ultracold molecules may provide unique opportunities for studying chemical
dynamics and for tests of fundamental symmetries. Here we experimentally
demonstrate laser cooling of the molecule strontium monofluoride (SrF). Using
an optical cycling scheme requiring only three lasers, we have observed both
Sisyphus and Doppler cooling forces which have substantially reduced the
transverse temperature of a SrF molecular beam. Currently the only technique
for producing ultracold molecules is by binding together ultracold alkali atoms
through Feshbach resonance or photoassociation. By contrast, different proposed
applications for ultracold molecules require a variety of molecular
energy-level structures. Our method provides a new route to ultracold
temperatures for molecules. In particular it bridges the gap between ultracold
temperatures and the ~1 K temperatures attainable with directly cooled
molecules (e.g. cryogenic buffer gas cooling or decelerated supersonic beams).
Ultimately our technique should enable the production of large samples of
molecules at ultracold temperatures for species that are chemically distinct
from bialkalis.Comment: 10 pages, 7 figure
BioMAJ: a flexible framework for databanks synchronization and processing
Large- and medium-scale computational molecular biology projects require accurate bioinformatics software and numerous heterogeneous biological databanks, which are distributed around the world. BioMAJ provides a flexible, robust, fully automated environment for managing such massive amounts of data. The JAVA application enables automation of the data update cycle process and supervision of the locally mirrored data repository. We have developed workflows that handle some of the most commonly used bioinformatics databases. A set of scripts is also available for post-synchronization data treatment consisting of indexation or format conversion (for NCBI blast, SRS, EMBOSS, GCG, etc.). BioMAJ can be easily extended by personal homemade processing scripts. Source history can be kept via html reports containing statements of locally managed databanks
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