595 research outputs found
Novel continuum modeling of crystal surface evolution
We propose a novel approach to continuum modeling of the dynamics of crystal
surfaces. Our model follows the evolution of an ensemble of step
configurations, which are consistent with the macroscopic surface profile.
Contrary to the usual approach where the continuum limit is achieved when
typical surface features consist of many steps, our continuum limit is
approached when the number of step configurations of the ensemble is very
large. The model can handle singular surface structures such as corners and
facets. It has a clear computational advantage over discrete models.Comment: 4 pages, 3 postscript figure
Continuum description of profile scaling in nanostructure decay
The relaxation of axisymmetric crystal surfaces with a single facet below the
roughening transition is studied via a continuum approach that accounts for
step energy g_1 and step-step interaction energy g_3>0. For diffusion-limited
kinetics, free-boundary and boundary-layer theories are used for self-similar
shapes close to the growing facet. For long times and g_3/g_1 < 1, (a) a
universal equation is derived for the shape profile, (b) the layer thickness
varies as (g_3/g_1)^{1/3}, (c) distinct solutions are found for different
g_3/_1, and (d) for conical shapes, the profile peak scales as
(g_3/g_1)^{-1/6}. These results compare favorably with kinetic simulations.Comment: 4 pages including 3 figure
Decay of one dimensional surface modulations
The relaxation process of one dimensional surface modulations is re-examined.
Surface evolution is described in terms of a standard step flow model.
Numerical evidence that the surface slope, D(x,t), obeys the scaling ansatz
D(x,t)=alpha(t)F(x) is provided. We use the scaling ansatz to transform the
discrete step model into a continuum model for surface dynamics. The model
consists of differential equations for the functions alpha(t) and F(x). The
solutions of these equations agree with simulation results of the discrete step
model. We identify two types of possible scaling solutions. Solutions of the
first type have facets at the extremum points, while in solutions of the second
type the facets are replaced by cusps. Interactions between steps of opposite
signs determine whether a system is of the first or second type. Finally, we
relate our model to an actual experiment and find good agreement between a
measured AFM snapshot and a solution of our continuum model.Comment: 18 pages, 6 figures in 9 eps file
A Local Computation Approximation Scheme to Maximum Matching
We present a polylogarithmic local computation matching algorithm which
guarantees a (1-\eps)-approximation to the maximum matching in graphs of
bounded degree.Comment: Appears in Approx 201
Origin Gaps and the Eternal Sunshine of the Second-Order Pendulum
The rich experiences of an intentional, goal-oriented life emerge, in an
unpredictable fashion, from the basic laws of physics. Here I argue that this
unpredictability is no mirage: there are true gaps between life and non-life,
mind and mindlessness, and even between functional societies and groups of
Hobbesian individuals. These gaps, I suggest, emerge from the mathematics of
self-reference, and the logical barriers to prediction that self-referring
systems present. Still, a mathematical truth does not imply a physical one: the
universe need not have made self-reference possible. It did, and the question
then is how. In the second half of this essay, I show how a basic move in
physics, known as renormalization, transforms the "forgetful" second-order
equations of fundamental physics into a rich, self-referential world that makes
possible the major transitions we care so much about. While the universe runs
in assembly code, the coarse-grained version runs in LISP, and it is from that
the world of aim and intention grows.Comment: FQXI Prize Essay 2017. 18 pages, including afterword on
Ostrogradsky's Theorem and an exchange with John Bova, Dresden Craig, and
Paul Livingsto
On computational irreducibility and the predictability of complex physical systems
Using elementary cellular automata (CA) as an example, we show how to
coarse-grain CA in all classes of Wolfram's classification. We find that
computationally irreducible (CIR) physical processes can be predictable and
even computationally reducible at a coarse-grained level of description. The
resulting coarse-grained CA which we construct emulate the large-scale behavior
of the original systems without accounting for small-scale details. At least
one of the CA that can be coarse-grained is irreducible and known to be a
universal Turing machine.Comment: 4 pages, 2 figures, to be published in PR
Coarse-graining of cellular automata, emergence, and the predictability of complex systems
We study the predictability of emergent phenomena in complex systems. Using
nearest neighbor, one-dimensional Cellular Automata (CA) as an example, we show
how to construct local coarse-grained descriptions of CA in all classes of
Wolfram's classification. The resulting coarse-grained CA that we construct are
capable of emulating the large-scale behavior of the original systems without
accounting for small-scale details. Several CA that can be coarse-grained by
this construction are known to be universal Turing machines; they can emulate
any CA or other computing devices and are therefore undecidable. We thus show
that because in practice one only seeks coarse-grained information, complex
physical systems can be predictable and even decidable at some level of
description. The renormalization group flows that we construct induce a
hierarchy of CA rules. This hierarchy agrees well with apparent rule complexity
and is therefore a good candidate for a complexity measure and a classification
method. Finally we argue that the large scale dynamics of CA can be very
simple, at least when measured by the Kolmogorov complexity of the large scale
update rule, and moreover exhibits a novel scaling law. We show that because of
this large-scale simplicity, the probability of finding a coarse-grained
description of CA approaches unity as one goes to increasingly coarser scales.
We interpret this large scale simplicity as a pattern formation mechanism in
which large scale patterns are forced upon the system by the simplicity of the
rules that govern the large scale dynamics.Comment: 18 pages, 9 figure
Non-invertible transformations and spatiotemporal randomness
We generalize the exact solution to the Bernoulli shift map. Under certain
conditions, the generalized functions can produce unpredictable dynamics. We
use the properties of the generalized functions to show that certain dynamical
systems can generate random dynamics. For instance, the chaotic Chua's circuit
coupled to a circuit with a non-invertible I-V characteristic can generate
unpredictable dynamics. In general, a nonperiodic time-series with truncated
exponential behavior can be converted into unpredictable dynamics using
non-invertible transformations. Using a new theoretical framework for chaos and
randomness, we investigate some classes of coupled map lattices. We show that,
in some cases, these systems can produce completely unpredictable dynamics. In
a similar fashion, we explain why some wellknown spatiotemporal systems have
been found to produce very complex dynamics in numerical simulations. We
discuss real physical systems that can generate random dynamics.Comment: Accepted in International Journal of Bifurcation and Chao
Slow fluctuations in enhanced Raman scattering and surface roughness relaxation
We propose an explanation for the recently measured slow fluctuations and
``blinking'' in the surface enhanced Raman scattering (SERS) spectrum of single
molecules adsorbed on a silver colloidal particle. We suggest that these
fluctuations may be related to the dynamic relaxation of the surface roughness
on the nanometer scale and show that there are two classes of roughness with
qualitatively different dynamics. The predictions agree with measurements of
surface roughness relaxation. Using a theoretical model for the kinetics of
surface roughness relaxation in the presence of charges and optical electrical
fields, we predict that the high-frequency electromagnetic field increases both
the effective surface tension and the surface diffusion constant and thus
accelerates the surface smoothing kinetics and time scale of the Raman
fluctuations in manner that is linear with the laser power intensity, while the
addition of salt retards the surface relaxation kinetics and increases the time
scale of the fluctuations. These predictions are in qualitative agreement with
the Raman experiments
Genetic analysis of four consanguineous multiplex families with inflammatory bowel disease
Background: Family studies support a genetic predisposition to inflammatory bowel diseases (IBD), but known genetic variants only partially explain the disease heritability. Families with multiple affected individuals potentially harbour rare and high-impact causal variants. Long regions of homozygosity due to recent inbreeding may increase the risk of individuals bearing homozygous loss-of-function variants. This study aimed to identify rare and homozygous genetic variants contributing to IBD. Methods: Four families with known consanguinity and multiple cases of IBD were recruited. In a family-specific analysis, we utilised homozygosity mapping complemented by whole-exome sequencing. Results: We detected a single region of homozygosity shared by Crohn's disease cases from a family of Druze ancestry, spanning 2.6 Mb containing the NOD2 gene. Whole-exome sequencing did not identify any potentially damaging variants within the region, suggesting that non-coding variation may be involved. In addition, affected individuals in the families harboured several rare and potentially damaging homozygous variants in genes with a role in autophagy and innate immunity including LRRK1, WHAMM, DENND3, and C5. Conclusion: This study examined the potential contribution of rare, high-impact homozygous variants in consanguineous families with IBD. While the analysis was not designed to achieve statistical significance, our findings highlight genes or loci that warrant further research. Non-coding variants affecting NOD2 may be of importance in Druze patients with Crohn's disease
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