32,408 research outputs found
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
Well-posed two-point initial-boundary value problems with arbitrary boundary conditions
We study initial-boundary value problems for linear evolution equations of
arbitrary spatial order, subject to arbitrary linear boundary conditions and
posed on a rectangular 1-space, 1-time domain. We give a new characterisation
of the boundary conditions that specify well-posed problems using Fokas'
transform method. We also give a sufficient condition guaranteeing that the
solution can be represented using a series.
The relevant condition, the analyticity at infinity of certain meromorphic
functions within particular sectors, is significantly more concrete and easier
to test than the previous criterion, based on the existence of admissible
functions.Comment: 21 page
The chebop system for automatic solution of differential equations
In MATLAB, it would be good to be able to solve a linear differential equation by typing u = L\f, where f, u, and L are representations of the right-hand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to be able to exponentiate an operator with expm(L) or determine eigenvalues and eigenfunctions with eigs(L). A system is described in which such calculations are indeed possible, based on the previously developed chebfun system in object-oriented MATLAB. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution
The asymptotic homogenization elasticity tensor properties for composites with material discontinuities
The classical asymptotic homogenization approach for linear elastic composites with discontinuous material properties is considered as a starting point. The sharp length scale separation between the fine periodic structure and the whole material formally leads to anisotropic elastic-type balance equations on the coarse scale, where the arising fourth rank operator is to be computed solving single periodic cell problems on the fine scale. After revisiting the derivation of the problem, which here explicitly points out how the discontinuity in the individual constituentsâ elastic coefficients translates into stress jump interface conditions for the cell problems, we prove that the gradient of the cell problem solution is minor symmetric and that its cell average is zero. This property holds for perfect interfaces only (i.e., when the elastic displacement is continuous across the compositeâs interface) and can be used to assess the accuracy of the computed numerical solutions. These facts are further exploited, together with the individual constituentsâ elastic coefficients and the specific form of the cell problems, to prove a theorem that characterizes the fourth rank operator appearing in the coarse-scale elastic-type balance equations as a composite material effective elasticity tensor. We both recover known facts, such as minor and major symmetries and positive definiteness, and establish new facts concerning the Voigt and Reuss bounds. The latter are shown for the first time without assuming any equivalence between coarse and fine-scale energies (Hillâs condition), which, in contrast to the case of representative volume elements, does not identically hold in the context of asymptotic homogenization. We conclude with instructive three-dimensional numerical simulations of a soft elastic matrix with an embedded cubic stiffer inclusion to show the profile of the physically relevant elastic moduli (Youngâs and shear moduli) and Poissonâs ratio at increasing (up to 100 %) inclusionâs volume fraction, thus providing a proxy for the design of artificial elastic composites
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A class of cubic and quintic spline modified collocation methods for the solution of two-point boundary value problems
This paper is concerned with the study of a class of methods for solving second and fourth-order two-point boundary-value problems. The methods under
consideration are modifications of the standard cubic and quintic spline
collocation techniques, and are derived by making use of recent results con- cerning the a posteriori correction of cubic and quintic interpolating spline
Spectral methods for CFD
One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched
Relaxed micromorphic model of transient wave propagation in anisotropic band-gap metastructures
In this paper, we show that the transient waveforms arising from several
localised pulses in a micro-structured material can be reproduced by a
corresponding generalised continuum of the relaxed micromorphic type.
Specifically, we compare the dynamic response of a bounded micro-structured
material to that of bounded continua with special kinematic properties: (i) the
relaxed micromorphic continuum and (ii) an equivalent Cauchy linear elastic
continuum. We show that, while the Cauchy theory is able to describe the
overall behaviour of the metastructure only at low frequencies, the relaxed
micromorphic model goes far beyond by giving a correct description of the pulse
propagation in the frequency band-gap and at frequencies intersecting the
optical branches. In addition, we observe a computational time reduction
associated with the use of the relaxed micromorphic continuum, compared to the
sensible computational time needed to perform a transient computation in a
micro-structured domain
Lower Bounds for Ground States of Condensed Matter Systems
Standard variational methods tend to obtain upper bounds on the ground state
energy of quantum many-body systems. Here we study a complementary method that
determines lower bounds on the ground state energy in a systematic fashion,
scales polynomially in the system size and gives direct access to correlation
functions. This is achieved by relaxing the positivity constraint on the
density matrix and replacing it by positivity constraints on moment matrices,
thus yielding a semi-definite programme. Further, the number of free parameters
in the optimization problem can be reduced dramatically under the assumption of
translational invariance. A novel numerical approach, principally a combination
of a projected gradient algorithm with Dykstra's algorithm, for solving the
optimization problem in a memory-efficient manner is presented and a proof of
convergence for this iterative method is given. Numerical experiments that
determine lower bounds on the ground state energies for the Ising and
Heisenberg Hamiltonians confirm that the approach can be applied to large
systems, especially under the assumption of translational invariance.Comment: 16 pages, 4 figures, replaced with published versio
Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
This paper presents two new approaches for finding the homogenized
coefficients of multiscale elliptic PDEs. Standard approaches for computing the
homogenized coefficients suffer from the so-called resonance error, originating
from a mismatch between the true and the computational boundary conditions. Our
new methods, based on solutions of parabolic and elliptic cell-problems, result
in an exponential decay of the resonance error
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