496 research outputs found
Formal matched asymptotics for degenerate Ricci flow neckpinches
Gu and Zhu have shown that Type-II Ricci flow singularities develop from
nongeneric rotationally symmetric Riemannian metrics on , for all . In this paper, we describe and provide plausibility arguments for a
detailed asymptotic profile and rate of curvature blow-up that we predict such
solutions exhibit
Complete Embedded Self-Translating Surfaces under Mean Curvature Flow
We describe a construction of complete embedded self-translating surfaces
under mean curvature flow by desingularizing the intersection of a finite
family of grim reapers in general position.Comment: 42 pages, 8 figures. v2: typos correcte
In planta localisation patterns of MADS domain proteins during floral development in Arabidopsis thaliana
Background: MADS domain transcription factors play important roles in various developmental processes in flowering plants. Members of this family play a prominent role in the transition to flowering and the specification of floral organ identity. Several studies reported mRNA expression patterns of the genes encoding these MADS domain proteins, however, these studies do not provide the necessary information on the temporal and spatial localisation of the proteins. We have made GREEN FLUORESCENT PROTEIN (GFP) translational fusions with the four MADS domain proteins SEPALLATA3, AGAMOUS, FRUITFULL and APETALA1 from the model plant Arabidopsis thaliana and analysed the protein localisation patterns in living plant tissues by confocal laser scanning microscopy (CLSM). Results: We unravelled the protein localisation patterns of the four MADS domain proteins at a cellular and subcellular level in inflorescence and floral meristems, during development of the early flower bud stages, and during further differentiation of the floral organs. The protein localisation patterns revealed a few deviations from known mRNA expression patterns, suggesting a non-cell autonomous action of these factors or alternative control mechanisms. In addition, we observed a change in the subcellular localisation of SEPALLATA3 from a predominantly nuclear localisation to a more cytoplasmic localisation, occurring specifically during petal and stamen development. Furthermore, we show that the down-regulation of the homeodomain transcription factor WUSCHEL in ovular tissues is preceded by the occurrence of both AGAMOUS and SEPALLATA3 proteins, supporting the hypothesis that both proteins together suppress WUSCHEL expression in the ovule. Conclusion: This approach provides a highly detailed in situ map of MADS domain protein presence during early and later stages of floral development. The subcellular localisation of the transcription factors in the cytoplasm, as observed at certain stages during development, points to mechanisms other than transcriptional control. Together this information is essential to understand the role of these proteins in the regulatory processes that drive floral development and leads to new hypotheses
Application of the level-set method to the implicit solvation of nonpolar molecules
A level-set method is developed for numerically capturing the equilibrium
solute-solvent interface that is defined by the recently proposed variational
implicit solvent model (Dzubiella, Swanson, and McCammon, Phys. Rev. Lett. {\bf
104}, 527 (2006) and J. Chem.\Phys. {\bf 124}, 084905 (2006)). In the level-set
method, a possible solute-solvent interface is represented by the zero
level-set (i.e., the zero level surface) of a level-set function and is
eventually evolved into the equilibrium solute-solvent interface. The evolution
law is determined by minimization of a solvation free energy {\it functional}
that couples both the interfacial energy and the van der Waals type
solute-solvent interaction energy. The surface evolution is thus an energy
minimizing process, and the equilibrium solute-solvent interface is an output
of this process. The method is implemented and applied to the solvation of
nonpolar molecules such as two xenon atoms, two parallel paraffin plates,
helical alkane chains, and a single fullerene . The level-set solutions
show good agreement for the solvation energies when compared to available
molecular dynamics simulations. In particular, the method captures solvent
dewetting (nanobubble formation) and quantitatively describes the interaction
in the strongly hydrophobic plate system
Critical behavior of collapsing surfaces
We consider the mean curvature evolution of rotationally symmetric surfaces.
Using numerical methods, we detect critical behavior at the threshold of
singularity formation resembling the one of gravitational collapse. In
particular, the mean curvature simulation of a one-parameter family of initial
data reveals the existence of a critical initial surface that develops a
degenerate neckpinch. The limiting flow of the Type II singularity is
accurately modeled by the rotationally symmetric translating soliton.Comment: 23 pages, 10 figure
Simulation of organ patterning on the floral meristem using a polar auxin transport model
An intriguing phenomenon in plant development is the timing and positioning of lateral organ initiation,
which is a fundamental aspect of plant architecture. Although important progress has been made in
elucidating the role of auxin transport in the vegetative shoot to explain the phyllotaxis of leaf formation
in a spiral fashion, a model that explains the whorled organ patterning in the expanding floral meristem
is not available yet. We present an initial simulation approach to study the mechanisms that are expected
to play an important role. Starting point is a confocal imaging study of Arabidopsis floral meristems
at consecutive time points during flower development. These images reveal auxin accumulation patterns
at the positions of the organs, which strongly suggests that the role of auxin in the floral meristem is
similar to the role it plays in the shoot apical meristem. This is the basis for a simulation study of auxin
transport through a growing floral meristem, which may answer the question whether auxin transport can
in itself be responsible for the typical whorled floral pattern. We combined a cellular growth model for
the meristem with a polar auxin transport model. The model predicts that sepals are initiated by auxin maxima arising early during meristem outgrowth. These form a pre-pattern relative to which a series of
smaller auxin maxima are positioned, which partially overlap with the anlagen of petals, stamens, and
carpels. We adjusted the model parameters corresponding to properties of floral mutants and found that
the model predictions agree with the observed mutant patterns. The predicted timing of the primordia
outgrowth and the timing and positioning of the sepal primordia show remarkable similarities with a
developing flower in nature
Dirichlet sigma models and mean curvature flow
The mean curvature flow describes the parabolic deformation of embedded
branes in Riemannian geometry driven by their extrinsic mean curvature vector,
which is typically associated to surface tension forces. It is the gradient
flow of the area functional, and, as such, it is naturally identified with the
boundary renormalization group equation of Dirichlet sigma models away from
conformality, to lowest order in perturbation theory. D-branes appear as fixed
points of this flow having conformally invariant boundary conditions. Simple
running solutions include the paper-clip and the hair-pin (or grim-reaper)
models on the plane, as well as scaling solutions associated to rational (p, q)
closed curves and the decay of two intersecting lines. Stability analysis is
performed in several cases while searching for transitions among different
brane configurations. The combination of Ricci with the mean curvature flow is
examined in detail together with several explicit examples of deforming curves
on curved backgrounds. Some general aspects of the mean curvature flow in
higher dimensional ambient spaces are also discussed and obtain consistent
truncations to lower dimensional systems. Selected physical applications are
mentioned in the text, including tachyon condensation in open string theory and
the resistive diffusion of force-free fields in magneto-hydrodynamics.Comment: 77 pages, 21 figure
A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations on and those
of the parabolic equations on a bounded
domain . We give details on the similarities of these dynamics in the
cases , and and in the corresponding cases ,
and dim() respectively. In addition to
the beauty of such a correspondence, this could serve as a guideline for future
research on the dynamics of parabolic equations
Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres
Using mixed analytical and numerical methods we investigate the development
of singularities in the heat flow for corotational harmonic maps from the
-dimensional sphere to itself for . By gluing together
shrinking and expanding asymptotically self-similar solutions we construct
global weak solutions which are smooth everywhere except for a sequence of
times at which there occurs the type I blow-up at one
of the poles of the sphere. We show that in the generic case the continuation
beyond blow-up is unique, the topological degree of the map changes by one at
each blow-up time , and eventually the solution comes to rest at the zero
energy constant map.Comment: 24 pages, 8 figures, minor corrections, matches published versio
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