627 research outputs found
The Statistics of the Points Where Nodal Lines Intersect a Reference Curve
We study the intersection points of a fixed planar curve with the
nodal set of a translationally invariant and isotropic Gaussian random field
\Psi(\bi{r}) and the zeros of its normal derivative across the curve. The
intersection points form a discrete random process which is the object of this
study. The field probability distribution function is completely specified by
the correlation G(|\bi{r}-\bi{r}'|) = .
Given an arbitrary G(|\bi{r}-\bi{r}'|), we compute the two point
correlation function of the point process on the line, and derive other
statistical measures (repulsion, rigidity) which characterize the short and
long range correlations of the intersection points. We use these statistical
measures to quantitatively characterize the complex patterns displayed by
various kinds of nodal networks. We apply these statistics in particular to
nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of
special interest is the observation that for monochromatic random waves, the
number variance of the intersections with long straight segments grows like , as opposed to the linear growth predicted by the percolation model,
which was successfully used to predict other long range nodal properties of
that field.Comment: 33 pages, 13 figures, 1 tabl
Genetic Classification of Populations using Supervised Learning
There are many instances in genetics in which we wish to determine whether
two candidate populations are distinguishable on the basis of their genetic
structure. Examples include populations which are geographically separated,
case--control studies and quality control (when participants in a study have
been genotyped at different laboratories). This latter application is of
particular importance in the era of large scale genome wide association
studies, when collections of individuals genotyped at different locations are
being merged to provide increased power. The traditional method for detecting
structure within a population is some form of exploratory technique such as
principal components analysis. Such methods, which do not utilise our prior
knowledge of the membership of the candidate populations. are termed
\emph{unsupervised}. Supervised methods, on the other hand are able to utilise
this prior knowledge when it is available.
In this paper we demonstrate that in such cases modern supervised approaches
are a more appropriate tool for detecting genetic differences between
populations. We apply two such methods, (neural networks and support vector
machines) to the classification of three populations (two from Scotland and one
from Bulgaria). The sensitivity exhibited by both these methods is considerably
higher than that attained by principal components analysis and in fact
comfortably exceeds a recently conjectured theoretical limit on the sensitivity
of unsupervised methods. In particular, our methods can distinguish between the
two Scottish populations, where principal components analysis cannot. We
suggest, on the basis of our results that a supervised learning approach should
be the method of choice when classifying individuals into pre-defined
populations, particularly in quality control for large scale genome wide
association studies.Comment: Accepted PLOS On
Last passage percolation and traveling fronts
We consider a system of N particles with a stochastic dynamics introduced by
Brunet and Derrida. The particles can be interpreted as last passage times in
directed percolation on {1,...,N} of mean-field type. The particles remain
grouped and move like a traveling wave, subject to discretization and driven by
a random noise. As N increases, we obtain estimates for the speed of the front
and its profile, for different laws of the driving noise. The Gumbel
distribution plays a central role for the particle jumps, and we show that the
scaling limit is a L\'evy process in this case. The case of bounded jumps
yields a completely different behavior
Tensile Overload and Stress Intensity Shielding Investigations by Ultrasound
Growth of a fatigue crack is modified according to the development of contacts between the crack faces [1,2] creating shielding, thus canceling a portion of the crack driving force. These contacts develop through a number of mechanisms, including plastic deformation, sliding of the faces with respect to each other and the collection of debris such as oxide particles [3]. Compressive stresses are created on either side of the partially contacting crack faces resulting in opening loads that must be overcome in order to apply a driving force at the crack tip. In this way, the crack tip is shielded from a portion of the applied load, thus creating the need for modification [1] of the applied stress intensity range from ΔK = KImax − KImin to ΔKeff = KImax − KIsh. Determination of the contact size and density in the region of closure from ultrasonic transmission and diffraction experiments [4] has allowed estimation of the magnitude of Kish on a crack grown under constant ΔK conditions. The calculation has since [5] been extended to fatigue cracks grown with a tensile overload block. The calculation was also successful in predicting the growth rate of the crack after reinitiation had occurred. This paper reports the further extension to the effects of a variable ΔK on fatigue crack growth. In addition, this paper presents preliminary results on detection of the tightly closed crack extension present during the growth retardation period after application of a tensile overload as well as an observation of the crack surface during reinitiation of growth that presents some interesting questions
Growing interfaces uncover universal fluctuations behind scale invariance
Stochastic motion of a point -- known as Brownian motion -- has many
successful applications in science, thanks to its scale invariance and
consequent universal features such as Gaussian fluctuations. In contrast, the
stochastic motion of a line, though it is also scale-invariant and arises in
nature as various types of interface growth, is far less understood. The two
major missing ingredients are: an experiment that allows a quantitative
comparison with theory and an analytic solution of the Kardar-Parisi-Zhang
(KPZ) equation, a prototypical equation for describing growing interfaces. Here
we solve both problems, showing unprecedented universality beyond the scaling
laws. We investigate growing interfaces of liquid-crystal turbulence and find
not only universal scaling, but universal distributions of interface positions.
They obey the largest-eigenvalue distributions of random matrices and depend on
whether the interface is curved or flat, albeit universal in each case. Our
exact solution of the KPZ equation provides theoretical explanations.Comment: 5 pages, 3 figures, supplementary information available on Journal
pag
Learning a Factor Model via Regularized PCA
We consider the problem of learning a linear factor model. We propose a
regularized form of principal component analysis (PCA) and demonstrate through
experiments with synthetic and real data the superiority of resulting estimates
to those produced by pre-existing factor analysis approaches. We also establish
theoretical results that explain how our algorithm corrects the biases induced
by conventional approaches. An important feature of our algorithm is that its
computational requirements are similar to those of PCA, which enjoys wide use
in large part due to its efficiency
A pedestrian's view on interacting particle systems, KPZ universality, and random matrices
These notes are based on lectures delivered by the authors at a Langeoog
seminar of SFB/TR12 "Symmetries and universality in mesoscopic systems" to a
mixed audience of mathematicians and theoretical physicists. After a brief
outline of the basic physical concepts of equilibrium and nonequilibrium
states, the one-dimensional simple exclusion process is introduced as a
paradigmatic nonequilibrium interacting particle system. The stationary measure
on the ring is derived and the idea of the hydrodynamic limit is sketched. We
then introduce the phenomenological Kardar-Parisi-Zhang (KPZ) equation and
explain the associated universality conjecture for surface fluctuations in
growth models. This is followed by a detailed exposition of a seminal paper of
Johansson that relates the current fluctuations of the totally asymmetric
simple exclusion process (TASEP) to the Tracy-Widom distribution of random
matrix theory. The implications of this result are discussed within the
framework of the KPZ conjecture.Comment: 52 pages, 4 figures; to appear in J. Phys. A: Math. Theo
Characterization of Microstructural Effects on Fatigue Crack Closure
The growth of a fatigue crack is modified by the development of contacts between the crack faces1,2creating shielding and thus canceling a portion of the applied load. These contacts develop through a number of mechanisms, including plastic deformation, sliding of the faces with respect to each other and the creation and collection of debris such as oxide particles3. Compressive stresses are created on either side of the partially contacting crack faces resulting in opening loads that must be overcome in order to apply a driving force to the crack tip for growth. In this way, the crack tip is shielded from a portion of the applied load, thus creating the need for modification1 of the applied stress intensity range from ΔK = KImax — KImin to ΔK = KImax — KIsh. Determination of the contact size and density in the region of closure from ultrasonic transmission and diffraction experiments4has allowed estimation of the magnitude of KIsh on a crack grown under constant ΔK conditions. The calculation has since5 been extended to fatigue cracks grown with a tensile overload block. The calculation was also successful in predicting the growth rate of the crack after reinitiation had occurred. This paper reports the results of attempts to define the amount of retardation remaining before reinitiation of crack growth in terms of the parameters used by the distributed spring model
Functional Renormalization Group and the Field Theory of Disordered Elastic Systems
We study elastic systems such as interfaces or lattices, pinned by quenched
disorder. To escape triviality as a result of ``dimensional reduction'', we use
the functional renormalization group. Difficulties arise in the calculation of
the renormalization group functions beyond 1-loop order. Even worse,
observables such as the 2-point correlation function exhibit the same problem
already at 1-loop order. These difficulties are due to the non-analyticity of
the renormalized disorder correlator at zero temperature, which is inherent to
the physics beyond the Larkin length, characterized by many metastable states.
As a result, 2-loop diagrams, which involve derivatives of the disorder
correlator at the non-analytic point, are naively "ambiguous''. We examine
several routes out of this dilemma, which lead to a unique renormalizable
field-theory at 2-loop order. It is also the only theory consistent with the
potentiality of the problem. The beta-function differs from previous work and
the one at depinning by novel "anomalous terms''. For interfaces and random
bond disorder we find a roughness exponent zeta = 0.20829804 epsilon + 0.006858
epsilon^2, epsilon = 4-d. For random field disorder we find zeta = epsilon/3
and compute universal amplitudes to order epsilon^2. For periodic systems we
evaluate the universal amplitude of the 2-point function. We also clarify the
dependence of universal amplitudes on the boundary conditions at large scale.
All predictions are in good agreement with numerical and exact results, and an
improvement over one loop. Finally we calculate higher correlation functions,
which turn out to be equivalent to those at depinning to leading order in
epsilon.Comment: 42 pages, 41 figure
- …