4,599 research outputs found
A multifractal zeta function for cookie cutter sets
Starting with the work of Lapidus and van Frankenhuysen a number of papers
have introduced zeta functions as a way of capturing multifractal information.
In this paper we propose a new multifractal zeta function and show that under
certain conditions the abscissa of convergence yields the Hausdorff
multifractal spectrum for a class of measures
Generalised dimensions of measures on almost self-affine sets
We establish a generic formula for the generalised q-dimensions of measures
supported by almost self-affine sets, for all q>1. These q-dimensions may
exhibit phase transitions as q varies. We first consider general measures and
then specialise to Bernoulli and Gibbs measures. Our method involves estimating
expectations of moment expressions in terms of `multienergy' integrals which we
then bound using induction on families of trees
Membrane-type matrix metalloproteinases: expression, roles in metastatic prostate cancer progression and opportunities for drug targeting
YesThe membrane-type matrix metalloproteinases (MT-MMPs), an important subgroup of
the wider MMP family, demonstrate widespread expression in multiple tumor types, and
play key roles in cancer growth, migration, invasion and metastasis. Despite a large body
of published research, relatively little information exists regarding evidence for MT-MMP
expression and function in metastatic prostate cancer. This review provides an appraisal of
the literature describing gene and protein expression in prostate cancer cells and clinical
tissue, summarises the evidence for roles in prostate cancer progression, and examines
the data relating to MT-MMP function in the development of bone metastases. Finally, the
therapeutic potential of targeting MT-MMPs is considered. While MT-MMP inhibition
presents a significant challenge, utilisation of MT-MMP expression and proteolytic capacity
in prostate tumors is an attractive drug development opportunity
Persistent organic pollutants in the Atlantic and southern oceans and oceanic atmosphere
Persistent organic pollutants (POPs) continue to cycle through the atmosphere and hydrosphere despite banned or severely restricted usages. Global scale analyses of POPs are challenging, but knowledge of the current distribution of these compounds is needed to understand the movement and long-term consequences of their global use. In the current study, air and seawater samples were collected Oct. 2007–Jan. 2008 aboard the Icebreaker Oden en route from Göteborg, Sweden to McMurdo Station, Antarctica. Both air and surface seawater samples consistently contained α-hexachlorocyclohexane (α-HCH), γ-HCH, hexachlorobenzene (HCB), α-Endosulfan, and polychlorinated biphenyls (PCBs). Sample concentrations for most POPs in air were higher in the northern hemisphere with the exception of HCB, which had high gas phase concentrations in the northern and southern latitudes and low concentrations near the equator. South Atlantic and Southern Ocean seawater had a high ratio of α-HCH to γ-HCH, indicating persisting levels from technical grade sources. The Atlantic and Southern Ocean continue to be net sinks for atmospheric α-, γ-HCH, and Endosulfan despite declining usage
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
Differentiability of fractal curves
While self-similar sets have no tangents at any single point, self-affine
curves can be smooth. We consider plane self-affine curves without double
points and with two pieces. There is an open subset of parameter space for
which the curve is differentiable at all points except for a countable set. For
a parameter set of codimension one, the curve is continuously differentiable.
However, there are no twice differentiable self-affine curves in the plane,
except for parabolic arcs
Contour lines of the discrete scale invariant rough surfaces
We study the fractal properties of the 2d discrete scale invariant (DSI)
rough surfaces. The contour lines of these rough surfaces show clear DSI. In
the appropriate limit the DSI surfaces converge to the scale invariant rough
surfaces. The fractal properties of the 2d DSI rough surfaces apart from
possessing the discrete scale invariance property follow the properties of the
contour lines of the corresponding scale invariant rough surfaces. We check
this hypothesis by calculating numerous fractal exponents of the contour lines
by using numerical calculations. Apart from calculating the known scaling
exponents some other new fractal exponents are also calculated.Comment: 9 Pages, 12 figure
Sixty Years of Fractal Projections
Sixty years ago, John Marstrand published a paper which, among other things,
relates the Hausdorff dimension of a plane set to the dimensions of its
orthogonal projections onto lines. For many years, the paper attracted very
little attention. However, over the past 30 years, Marstrand's projection
theorems have become the prototype for many results in fractal geometry with
numerous variants and applications and they continue to motivate leading
research.Comment: Submitted to proceedings of Fractals and Stochastics
The fractal structure of elliptical polynomial spirals
We investigate fractal aspects of elliptical polynomial spirals; that is,
planar spirals with differing polynomial rates of decay in the two axis
directions. We give a full dimensional analysis of these spirals, computing
explicitly their intermediate, box-counting and Assouad-type dimensions. An
exciting feature is that these spirals exhibit two phase transitions within the
Assouad spectrum, the first natural class of fractals known to have this
property. We go on to use this dimensional information to obtain bounds for the
H\"older regularity of maps that can deform one spiral into another,
generalising the 'winding problem' of when spirals are bi-Lipschitz equivalent
to a line segment. A novel feature is the use of fractional Brownian motion and
dimension profiles to bound the H\"older exponents.Comment: 19 pages, 7 figure
Adaptive evolution of molecular phenotypes
Molecular phenotypes link genomic information with organismic functions,
fitness, and evolution. Quantitative traits are complex phenotypes that depend
on multiple genomic loci. In this paper, we study the adaptive evolution of a
quantitative trait under time-dependent selection, which arises from
environmental changes or through fitness interactions with other co-evolving
phenotypes. We analyze a model of trait evolution under mutations and genetic
drift in a single-peak fitness seascape. The fitness peak performs a
constrained random walk in the trait amplitude, which determines the
time-dependent trait optimum in a given population. We derive analytical
expressions for the distribution of the time-dependent trait divergence between
populations and of the trait diversity within populations. Based on this
solution, we develop a method to infer adaptive evolution of quantitative
traits. Specifically, we show that the ratio of the average trait divergence
and the diversity is a universal function of evolutionary time, which predicts
the stabilizing strength and the driving rate of the fitness seascape. From an
information-theoretic point of view, this function measures the
macro-evolutionary entropy in a population ensemble, which determines the
predictability of the evolutionary process. Our solution also quantifies two
key characteristics of adapting populations: the cumulative fitness flux, which
measures the total amount of adaptation, and the adaptive load, which is the
fitness cost due to a population's lag behind the fitness peak.Comment: Figures are not optimally displayed in Firefo
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