806 research outputs found
Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations
Pattern formation often occurs in spatially extended physical, biological and
chemical systems due to an instability of the homogeneous steady state. The
type of the instability usually prescribes the resulting spatio-temporal
patterns and their characteristic length scales. However, patterns resulting
from the simultaneous occurrence of instabilities cannot be expected to be
simple superposition of the patterns associated with the considered
instabilities. To address this issue we design two simple models composed by
two asymmetrically coupled equations of non-conserved (Swift-Hohenberg
equations) or conserved (Cahn-Hilliard equations) order parameters with
different characteristic wave lengths. The patterns arising in these systems
range from coexisting static patterns of different wavelengths to traveling
waves. A linear stability analysis allows to derive a two parameter phase
diagram for the studied models, in particular revealing for the Swift-Hohenberg
equations a co-dimension two bifurcation point of Turing and wave instability
and a region of coexistence of stationary and traveling patterns. The nonlinear
dynamics of the coupled evolution equations is investigated by performing
accurate numerical simulations. These reveal more complex patterns, ranging
from traveling waves with embedded Turing patterns domains to spatio-temporal
chaos, and a wide hysteretic region, where waves or Turing patterns coexist.
For the coupled Cahn-Hilliard equations the presence of an weak coupling is
sufficient to arrest the coarsening process and to lead to the emergence of
purely periodic patterns. The final states are characterized by domains with a
characteristic length, which diverges logarithmically with the coupling
amplitude.Comment: 9 pages, 10 figures, submitted to Chao
The Cauchy problems for Einstein metrics and parallel spinors
We show that in the analytic category, given a Riemannian metric on a
hypersurface and a symmetric tensor on , the metric
can be locally extended to a Riemannian Einstein metric on with second
fundamental form , provided that and satisfy the constraints on
imposed by the contracted Codazzi equations. We use this fact to study the
Cauchy problem for metrics with parallel spinors in the real analytic category
and give an affirmative answer to a question raised in B\"ar, Gauduchon,
Moroianu (2005). We also answer negatively the corresponding questions in the
smooth category.Comment: 28 pages; final versio
Calabi-Yau cones from contact reduction
We consider a generalization of Einstein-Sasaki manifolds, which we
characterize in terms both of spinors and differential forms, that in the real
analytic case corresponds to contact manifolds whose symplectic cone is
Calabi-Yau. We construct solvable examples in seven dimensions. Then, we
consider circle actions that preserve the structure, and determine conditions
for the contact reduction to carry an induced structure of the same type. We
apply this construction to obtain a new hypo-contact structure on S^2\times
T^3.Comment: 30 pages; v2: typos corrected, presentation improved, one reference
added. To appear in Ann. Glob. Analysis and Geometr
Surgery and the Spectrum of the Dirac Operator
We show that for generic Riemannian metrics on a simply-connected closed spin
manifold of dimension at least 5 the dimension of the space of harmonic spinors
is no larger than it must be by the index theorem. The same result holds for
periodic fundamental groups of odd order.
The proof is based on a surgery theorem for the Dirac spectrum which says
that if one performs surgery of codimension at least 3 on a closed Riemannian
spin manifold, then the Dirac spectrum changes arbitrarily little provided the
metric on the manifold after surgery is chosen properly.Comment: 23 pages, 4 figures, to appear in J. Reine Angew. Mat
Alternative mechanisms of structuring biomembranes: Self-assembly vs. self-organization
We study two mechanisms for the formation of protein patterns near membranes
of living cells by mathematical modelling. Self-assembly of protein domains by
electrostatic lipid-protein interactions is contrasted with self-organization
due to a nonequilibrium biochemical reaction cycle of proteins near the
membrane. While both processes lead eventually to quite similar patterns, their
evolution occurs on very different length and time scales. Self-assembly
produces periodic protein patterns on a spatial scale below 0.1 micron in a few
seconds followed by extremely slow coarsening, whereas self-organization
results in a pattern wavelength comparable to the typical cell size of 100
micron within a few minutes suggesting different biological functions for the
two processes.Comment: 4 pages, 5 figure
Local electronic structure of the peptide bond probed by resonant inelastic soft X-ray scattering.
The local valence orbital structure of solid glycine, diglycine, and triglycine is studied using soft X-ray emission spectroscopy (XES), resonant inelastic soft X-ray scattering (RIXS) maps, and spectra calculations based on density-functional theory. Using a building block approach, the contributions of the different functional groups of the peptides are separated. Cuts through the RIXS maps furthermore allow monitoring selective excitations of the amino and peptide functional units, leading to a modification of the currently established assignment of spectral contributions. The results thus paint a new-and-improved picture of the peptide bond, enhance the understanding of larger molecules with peptide bonds, and simplify the investigation of such molecules in aqueous environment
Size-Dependent Transition to High-Dimensional Chaotic Dynamics in a Two-Dimensional Excitable Medium
The spatiotemporal dynamics of an excitable medium with multiple spiral
defects is shown to vary smoothly with system size from short-lived transients
for small systems to extensive chaos for large systems. A comparison of the
Lyapunov dimension density with the average spiral defect density suggests an
average dimension per spiral defect varying between three and seven. We discuss
some implications of these results for experimental studies of excitable media.Comment: 5 pages, Latex, 4 figure
Parallel spinors and holonomy groups
In this paper we complete the classification of spin manifolds admitting
parallel spinors, in terms of the Riemannian holonomy groups. More precisely,
we show that on a given n-dimensional Riemannian manifold, spin structures with
parallel spinors are in one to one correspondence with lifts to Spin_n of the
Riemannian holonomy group, with fixed points on the spin representation space.
In particular, we obtain the first examples of compact manifolds with two
different spin structures carrying parallel spinors.Comment: 10 pages, LaTeX2
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