44,578 research outputs found
Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications
We provide a theory to establish the existence of nonzero solutions of
perturbed Hammerstein integral equations with deviated arguments, being our
main ingredient the theory of fixed point index. Our approach is fairly general
and covers a variety of cases. We apply our results to a periodic boundary
value problem with reflections and to a thermostat problem. In the case of
reflections we also discuss the optimality of some constants that occur in our
theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page
Coverage, Continuity and Visual Cortical Architecture
The primary visual cortex of many mammals contains a continuous
representation of visual space, with a roughly repetitive aperiodic map of
orientation preferences superimposed. It was recently found that orientation
preference maps (OPMs) obey statistical laws which are apparently invariant
among species widely separated in eutherian evolution. Here, we examine whether
one of the most prominent models for the optimization of cortical maps, the
elastic net (EN) model, can reproduce this common design. The EN model
generates representations which optimally trade of stimulus space coverage and
map continuity. While this model has been used in numerous studies, no
analytical results about the precise layout of the predicted OPMs have been
obtained so far. We present a mathematical approach to analytically calculate
the cortical representations predicted by the EN model for the joint mapping of
stimulus position and orientation. We find that in all previously studied
regimes, predicted OPM layouts are perfectly periodic. An unbiased search
through the EN parameter space identifies a novel regime of aperiodic OPMs with
pinwheel densities lower than found in experiments. In an extreme limit,
aperiodic OPMs quantitatively resembling experimental observations emerge.
Stabilization of these layouts results from strong nonlocal interactions rather
than from a coverage-continuity-compromise. Our results demonstrate that
optimization models for stimulus representations dominated by nonlocal
suppressive interactions are in principle capable of correctly predicting the
common OPM design. They question that visual cortical feature representations
can be explained by a coverage-continuity-compromise.Comment: 100 pages, including an Appendix, 21 + 7 figure
Defects in Lamellar Diblock Copolymers: Chevron- and Omega-shaped Tilt Boundaries
We investigate symmetric grain boundaries in a lamellar diblock copolymer
system. The form of the interface between two grains strongly depends on the
angle , between the normals of the grains. When this angle is small,
the lamellae transform smoothly from one orientation to the other, creating the
chevron morphology. As increases, a gradual transition is observed to
an omega morphology characterized by a protrusion of the lamellae along the
interface between the two phases. We present a theoretical approach to find
these tilt boundaries in two-dimensional systems, based on a Ginzburg-Landau
expansion of the free energy. Calculated order parameter profiles and energies
agree well with transmission electron microscope experiments, and with full
numerical solution of the same problem.Comment: 11 pages, 10 gzipped postscript and jpeg figures, published in PRE
Vol. 61, March 200
Continuation for thin film hydrodynamics and related scalar problems
This chapter illustrates how to apply continuation techniques in the analysis
of a particular class of nonlinear kinetic equations that describe the time
evolution through transport equations for a single scalar field like a
densities or interface profiles of various types. We first systematically
introduce these equations as gradient dynamics combining mass-conserving and
nonmass-conserving fluxes followed by a discussion of nonvariational amendmends
and a brief introduction to their analysis by numerical continuation. The
approach is first applied to a number of common examples of variational
equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including
certain thin-film equations for partially wetting liquids on homogeneous and
heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal
equations. Second we consider nonvariational examples as the
Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard
equations and thin-film equations describing stationary sliding drops and a
transversal front instability in a dip-coating. Through the different examples
we illustrate how to employ the numerical tools provided by the packages
auto07p and pde2path to determine steady, stationary and time-periodic
solutions in one and two dimensions and the resulting bifurcation diagrams. The
incorporation of boundary conditions and integral side conditions is also
discussed as well as problem-specific implementation issues
Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods
The Immersed Boundary method is a simple, efficient, and robust numerical
scheme for solving PDE in general domains, yet it only achieves first-order
spatial accuracy near embedded boundaries. In this paper, we introduce a new
high-order numerical method which we call the Immersed Boundary Smooth
Extension (IBSE) method. The IBSE method achieves high-order accuracy by
smoothly extending the unknown solution of the PDE from a given smooth domain
to a larger computational domain, enabling the use of simple Cartesian-grid
discretizations (e.g. Fourier spectral methods). The method preserves much of
the flexibility and robustness of the original IB method. In particular, it
requires minimal geometric information to describe the boundary and relies only
on convolution with regularized delta-functions to communicate information
between the computational grid and the boundary. We present a fast algorithm
for solving elliptic equations, which forms the basis for simple, high-order
implicit-time methods for parabolic PDE and implicit-explicit methods for
related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat,
Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise
convergence for Dirichlet problems and third-order pointwise convergence for
Neumann problems
Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview
Here, we review the basic concepts and applications of the
phase-field-crystal (PFC) method, which is one of the latest simulation
methodologies in materials science for problems, where atomic- and microscales
are tightly coupled. The PFC method operates on atomic length and diffusive
time scales, and thus constitutes a computationally efficient alternative to
molecular simulation methods. Its intense development in materials science
started fairly recently following the work by Elder et al. [Phys. Rev. Lett. 88
(2002), p. 245701]. Since these initial studies, dynamical density functional
theory and thermodynamic concepts have been linked to the PFC approach to serve
as further theoretical fundaments for the latter. In this review, we summarize
these methodological development steps as well as the most important
applications of the PFC method with a special focus on the interaction of
development steps taken in hard and soft matter physics, respectively. Doing
so, we hope to present today's state of the art in PFC modelling as well as the
potential, which might still arise from this method in physics and materials
science in the nearby future.Comment: 95 pages, 48 figure
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