378 research outputs found
Combinatorics of linear iterated function systems with overlaps
Let be points in , and let
be a one-parameter family of similitudes of : where
is our parameter. Then, as is well known, there exists a
unique self-similar attractor satisfying
. Each has
at least one address , i.e.,
.
We show that for sufficiently close to 1, each has different
addresses. If is not too close to 1, then we can still have an
overlap, but there exist 's which have a unique address. However, we
prove that almost every has addresses,
provided contains no holes and at least one proper overlap. We
apply these results to the case of expansions with deleted digits.
Furthermore, we give sharp sufficient conditions for the Open Set Condition
to fail and for the attractor to have no holes.
These results are generalisations of the corresponding one-dimensional
results, however most proofs are different.Comment: Accepted for publication in Nonlinearit
Golden gaskets: variations on the Sierpi\'nski sieve
We consider the iterated function systems (IFSs) that consist of three
general similitudes in the plane with centres at three non-collinear points,
and with a common contraction factor \la\in(0,1).
As is well known, for \la=1/2 the invariant set, \S_\la, is a fractal
called the Sierpi\'nski sieve, and for \la<1/2 it is also a fractal. Our goal
is to study \S_\la for this IFS for 1/2<\la<2/3, i.e., when there are
"overlaps" in \S_\la as well as "holes". In this introductory paper we show
that despite the overlaps (i.e., the Open Set Condition breaking down
completely), the attractor can still be a totally self-similar fractal,
although this happens only for a very special family of algebraic \la's
(so-called "multinacci numbers"). We evaluate \dim_H(\S_\la) for these
special values by showing that \S_\la is essentially the attractor for an
infinite IFS which does satisfy the Open Set Condition. We also show that the
set of points in the attractor with a unique ``address'' is self-similar, and
compute its dimension.
For ``non-multinacci'' values of \la we show that if \la is close to 2/3,
then \S_\la has a nonempty interior and that if \la<1/\sqrt{3} then \S_\la$
has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of
the model in question.Comment: 27 pages, 10 figure
A dynamical approach to the spatiotemporal aspects of the Portevin-Le Chatelier effect: Chaos,turbulence and band propagation
Experimental time series obtained from single and poly-crystals subjected to
a constant strain rate tests report an intriguing dynamical crossover from a
low dimensional chaotic state at medium strain rates to an infinite dimensional
power law state of stress drops at high strain rates. We present results of an
extensive study of all aspects of the PLC effect within the context a model
that reproduces this crossover. A study of the distribution of the Lyapunov
exponents as a function of strain rate shows that it changes from a small set
of positive exponents in the chaotic regime to a dense set of null exponents in
the scaling regime. As the latter feature is similar to the GOY shell model for
turbulence, we compare our results with the GOY model. Interestingly, the null
exponents in our model themselves obey a power law. The configuration of
dislocations is visualized through the slow manifold analysis. This shows that
while a large proportion of dislocations are in the pinned state in the chaotic
regime, most of them are at the threshold of unpinning in the scaling regime.
The model qualitatively reproduces the different types of deformation bands
seen in experiments. At high strain rates where propagating bands are seen, the
model equations are reduced to the Fisher-Kolmogorov equation for propagative
fronts. This shows that the velocity of the bands varies linearly with the
strain rate and inversely with the dislocation density, consistent with the
known experimental results. Thus, this simple dynamical model captures the
complex spatio-temporal features of the PLC effect.Comment: 17 pages, 18 figure
Don't bleach chaotic data
A common first step in time series signal analysis involves digitally
filtering the data to remove linear correlations. The residual data is
spectrally white (it is ``bleached''), but in principle retains the nonlinear
structure of the original time series. It is well known that simple linear
autocorrelation can give rise to spurious results in algorithms for estimating
nonlinear invariants, such as fractal dimension and Lyapunov exponents. In
theory, bleached data avoids these pitfalls. But in practice, bleaching
obscures the underlying deterministic structure of a low-dimensional chaotic
process. This appears to be a property of the chaos itself, since nonchaotic
data are not similarly affected. The adverse effects of bleaching are
demonstrated in a series of numerical experiments on known chaotic data. Some
theoretical aspects are also discussed.Comment: 12 dense pages (82K) of ordinary LaTeX; uses macro psfig.tex for
inclusion of figures in text; figures are uufile'd into a single file of size
306K; the final dvips'd postscript file is about 1.3mb Replaced 9/30/93 to
incorporate final changes in the proofs and to make the LaTeX more portable;
the paper will appear in CHAOS 4 (Dec, 1993
Banking from Leeds, not London: regional strategy and structure at the Yorkshire Bank, 1859–1952
Industrial philanthropist Edward Akroyd created the Yorkshire Penny Savings Bank in 1859. Despite competition from the Post Office Savings Bank after 1861 and a serious reserve problem in 1911, it sustained his overall strategy to become a successful regional bank. Using archival and contemporary sources to build on recent scholarship illustrating how savings banks were integrated into local economies and the complementary roles of philanthropy and paternalism, we analyse an English regional bank's strategy, including an assessment of strategic innovation, ownership changes and management structure. This will demonstrate that the founder's vision continued, even though the 1911 crisis radically altered both strategy and structure
Data driven optimal filtering for phase and frequency of noisy oscillations: application to vortex flowmetering
A new method for extracting the phase of oscillations from noisy time series
is proposed. To obtain the phase, the signal is filtered in such a way that the
filter output has minimal relative variation in the amplitude (MIRVA) over all
filters with complex-valued impulse response. The argument of the filter output
yields the phase. Implementation of the algorithm and interpretation of the
result are discussed. We argue that the phase obtained by the proposed method
has a low susceptibility to measurement noise and a low rate of artificial
phase slips. The method is applied for the detection and classification of mode
locking in vortex flowmeters. A novel measure for the strength of mode locking
is proposed.Comment: 12 pages, 10 figure
Optimal neural network feature selection for spatial-temporal forecasting
In this paper, we show empirical evidence on how to construct the optimal
feature selection or input representation used by the input layer of a
feedforward neural network for the propose of forecasting spatial-temporal
signals. The approach is based on results from dynamical systems theory, namely
the non-linear embedding theorems. We demonstrate it for a variety of
spatial-temporal signals, with one spatial and one temporal dimensions, and
show that the optimal input layer representation consists of a grid, with
spatial/temporal lags determined by the minimum of the mutual information of
the spatial/temporal signals and the number of points taken in space/time
decided by the embedding dimension of the signal. We present evidence of this
proposal by running a Monte Carlo simulation of several combinations of input
layer feature designs and show that the one predicted by the non-linear
embedding theorems seems to be optimal or close of optimal. In total we show
evidence in four unrelated systems: a series of coupled Henon maps; a series of
couple Ordinary Differential Equations (Lorenz-96) phenomenologically modelling
atmospheric dynamics; the Kuramoto-Sivashinsky equation, a partial differential
equation used in studies of instabilities in laminar flame fronts and finally
real physical data from sunspot areas in the Sun (in latitude and time) from
1874 to 2015.Comment: 11 page
M2-Branes and Fano 3-folds
A class of supersymmetric gauge theories arising from M2-branes probing
Calabi-Yau 4-folds which are cones over smooth toric Fano 3-folds is
investigated. For each model, the toric data of the mesonic moduli space is
derived using the forward algorithm. The generators of the mesonic moduli space
are determined using Hilbert series. The spectrum of scaling dimensions for
chiral operators is computed.Comment: 128 pages, 39 figures, 42 table
Phases of M2-brane Theories
We investigate different toric phases of 2+1 dimensional quiver gauge
theories arising from M2-branes probing toric Calabi-Yau 4 folds. A brane
tiling for each toric phase is presented. We apply the 'forward algorithm' to
obtain the toric data of the mesonic moduli space of vacua and exhibit the
equivalence between the vacua of different toric phases of a given singularity.
The structures of the Master space, the mesonic moduli space, and the baryonic
moduli space are examined in detail. We compute the Hilbert series and use them
to verify the toric dualities between different phases. The Hilbert series,
R-charges, and generators of the mesonic moduli space are matched between toric
phases.Comment: 60 pages, 28 figures, 6 tables. v2: minor correction
Structural Determinants and Children's Oral Health: A Cross-National Study
Much research on children's oral health has focused on proximal determinants at the expense of distal (upstream) factors. Yet, such upstream factors-the so-called structural determinants of health-play a crucial role. Children's lives, and in turn their health, are shaped by politics, economic forces, and social and public policies. The aim of this study was to examine the relationship between children's clinical (number of decayed, missing, and filled teeth) and self-reported oral health (oral health-related quality of life) and 4 key structural determinants (governance, macroeconomic policy, public policy, and social policy) as outlined in the World Health Organization's Commission for Social Determinants of Health framework. Secondary data analyses were carried out using subnational epidemiological samples of 8- to 15-y-olds in 11 countries ( N = 6,648): Australia (372), New Zealand (three samples; 352, 202, 429), Brunei (423), Cambodia (423), Hong Kong (542), Malaysia (439), Thailand (261, 506), United Kingdom (88, 374), Germany (1498), Mexico (335), and Brazil (404). The results indicated that the type of political regime, amount of governance (e.g., rule of law, accountability), gross domestic product per capita, employment ratio, income inequality, type of welfare regime, human development index, government expenditure on health, and out-of-pocket (private) health expenditure by citizens were all associated with children's oral health. The structural determinants accounted for between 5% and 21% of the variance in children's oral health quality-of-life scores. These findings bring attention to the upstream or structural determinants as an understudied area but one that could reap huge rewards for public health dentistry research and the oral health inequalities policy agenda
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