885 research outputs found
Topology-guided sampling of nonhomogeneous random processes
Topological measurements are increasingly being accepted as an important tool
for quantifying complex structures. In many applications, these structures can
be expressed as nodal domains of real-valued functions and are obtained only
through experimental observation or numerical simulations. In both cases, the
data on which the topological measurements are based are derived via some form
of finite sampling or discretization. In this paper, we present a probabilistic
approach to quantifying the number of components of generalized nodal domains
of nonhomogeneous random processes on the real line via finite discretizations,
that is, we consider excursion sets of a random process relative to a
nonconstant deterministic threshold function. Our results furnish explicit
probabilistic a priori bounds for the suitability of certain discretization
sizes and also provide information for the choice of location of the sampling
points in order to minimize the error probability. We illustrate our results
for a variety of random processes, demonstrate how they can be used to sample
the classical nodal domains of deterministic functions perturbed by additive
noise and discuss their relation to the density of zeros.Comment: Published in at http://dx.doi.org/10.1214/09-AAP652 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
We present a computer-assisted proof of heteroclinic connections in the
one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a
fourth-order parabolic partial differential equation subject to homogeneous
Neumann boundary conditions, which contains as a special case the celebrated
Cahn-Hilliard equation. While the attractor structure of the latter model is
completely understood for one-dimensional domains, the diblock copolymer
extension exhibits considerably richer long-term dynamical behavior, which
includes a high level of multistability. In this paper, we establish the
existence of certain heteroclinic connections between the homogeneous
equilibrium state, which represents a perfect copolymer mixture, and all local
and global energy minimizers. In this way, we show that not every solution
originating near the homogeneous state will converge to the global energy
minimizer, but rather is trapped by a stable state with higher energy. This
phenomenon can not be observed in the one-dimensional Cahn-Hillard equation,
where generic solutions are attracted by a global minimizer
Probabilistic validation of homology computations for nodal domains
Homology has long been accepted as an important computable tool for
quantifying complex structures. In many applications, these structures arise as
nodal domains of real-valued functions and are therefore amenable only to a
numerical study based on suitable discretizations. Such an approach immediately
raises the question of how accurate the resulting homology computations are. In
this paper, we present a probabilistic approach to quantifying the validity of
homology computations for nodal domains of random fields in one and two space
dimensions, which furnishes explicit probabilistic a priori bounds for the
suitability of certain discretization sizes. We illustrate our results for the
special cases of random periodic fields and random trigonometric polynomials.Comment: Published at http://dx.doi.org/10.1214/105051607000000050 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Dissipative Quasigeostrophic Dynamics under Random Forcing
The quasigeostrophic model is a simplified geophysical fluid model at
asymptotically high rotation rate or at small Rossby number. We consider the
quasigeostrophic equation with dissipation under random forcing in bounded
domains. We show that global unique solutions exist for appropriate initial
data. Unlike the deterministic quasigeostrophic equation whose well-posedness
is well-known, there seems no rigorous result on global existence and
uniqueness of the randomly forced quasigeostrophic equation. Our work provides
such a rigorous result on global existence and uniqueness, under very mild
conditions.Comment: LaTeX, 15 page
Connection matrices in combinatorial topological dynamics
Connection matrices are one of the central tools in Conley's approach to the
study of dynamical systems, as they provide information on the existence of
connecting orbits in Morse decompositions. They may be considered a
generalisation of the boundary operator in the Morse complex in Morse theory.
Their computability has recently been addressed by Harker, Mischaikow, and
Spendlove in the context of lattice filtered chain complexes. In the current
paper, we extend the recently introduced Conley theory for combinatorial vector
and multivector fields on Lefschetz complexes by transferring the concept of
connection matrix to this setting. This is accomplished by the notion of
connection matrix for arbitrary poset filtered chain complexes, as well as an
associated equivalence, which allows for changes in the underlying posets. We
show that for the special case of gradient combinatorial vector fields in the
sense of Forman \cite{Fo98a}, connection matrices are necessarily unique. Thus,
the classical results of Reineck have a natural analogue in the combinatorial
setting
Creating Semiflows on Simplicial Complexes from Combinatorial Vector Fields
Combinatorial vector fields on simplicial complexes as introduced by Robin
Forman have found numerous and varied applications in recent years. Yet, their
relationship to classical dynamical systems has been less clear. In recent work
it was shown that for every combinatorial vector field on a finite simplicial
complex one can construct a multivalued discrete-time dynamical system on the
underlying polytope X which exhibits the same dynamics as the combinatorial
flow in the sense of Conley index theory. However, Forman's original
description of combinatorial flows appears to have been motivated more directly
by the concept of flows, i.e., continuous-time dynamical systems. In this
paper, it is shown that one can construct a semiflow on X which exhibits the
same dynamics as the underlying combinatorial vector field. The equivalence of
the dynamical behavior is established in the sense of Conley-Morse graphs and
uses a tiling of the topological space X which makes it possible to directly
construct isolating blocks for all involved isolated invariant sets based
purely on the combinatorial information.Comment: 57 pages, 12 figure
Functional analysis of the magnetosome island in Magnetospirillum gryphiswaldense: the mamAB operon is sufficient for magnetite biomineralization
Bacterial magnetosomes are membrane-enveloped, nanometer-sized crystals of magnetite, which serve for magnetotactic navigation. All genes implicated in the synthesis of these organelles are located in a conserved genomic magnetosome island (MAI). We performed a comprehensive bioinformatic, proteomic and genetic analysis of the MAI in Magnetospirillum gryphiswaldense. By the construction of large deletion mutants we demonstrate that the entire region is dispensable for growth, and the majority of MAI genes have no detectable function in magnetosome formation and could be eliminated without any effect. Only <25% of the region comprising four major operons could be associated with magnetite biomineralization, which correlated with high expression of these genes and their conservation among magnetotactic bacteria. Whereas only deletion of the mamAB operon resulted in the complete loss of magnetic particles, deletion of the conserved mms6, mamGFDC, and mamXY operons led to severe defects in morphology, size and organization of magnetite crystals. However, strains in which these operons were eliminated together retained the ability to synthesize small irregular crystallites, and weakly aligned in magnetic fields. This demonstrates that whereas the mamGFDC, mms6 and mamXY operons have crucial and partially overlapping functions for the formation of functional magnetosomes, the mamAB operon is the only region of the MAI, which is necessary and sufficient for magnetite biomineralization. Our data further reduce the known minimal gene set required for magnetosome formation and will be useful for future genome engineering approaches
- …