10,805 research outputs found
Computational studies of biomembrane systems: Theoretical considerations, simulation models, and applications
This chapter summarizes several approaches combining theory, simulation and
experiment that aim for a better understanding of phenomena in lipid bilayers
and membrane protein systems, covering topics such as lipid rafts, membrane
mediated interactions, attraction between transmembrane proteins, and
aggregation in biomembranes leading to large superstructures such as the light
harvesting complex of green plants. After a general overview of theoretical
considerations and continuum theory of lipid membranes we introduce different
options for simulations of biomembrane systems, addressing questions such as:
What can be learned from generic models? When is it expedient to go beyond
them? And what are the merits and challenges for systematic coarse graining and
quasi-atomistic coarse grained models that ensure a certain chemical
specificity
Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast
We construct finite-dimensional approximations of solution spaces of
divergence form operators with -coefficients. Our method does not
rely on concepts of ergodicity or scale-separation, but on the property that
the solution space of these operators is compactly embedded in if source
terms are in the unit ball of instead of the unit ball of .
Approximation spaces are generated by solving elliptic PDEs on localized
sub-domains with source terms corresponding to approximation bases for .
The -error estimates show that -dimensional spaces
with basis elements localized to sub-domains of diameter (with ) result in an
accuracy for elliptic, parabolic and hyperbolic
problems. For high-contrast media, the accuracy of the method is preserved
provided that localized sub-domains contain buffer zones of width
where the contrast of the medium
remains bounded. The proposed method can naturally be generalized to vectorial
equations (such as elasto-dynamics).Comment: Accepted for publication in SIAM MM
Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems
In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study
Second-order sensitivity relations and regularity of the value function for Mayer's problem in optimal control
This paper investigates the value function, , of a Mayer optimal control
problem with the state equation given by a differential inclusion. First, we
obtain an invariance property for the proximal and Fr\'echet subdifferentials
of along optimal trajectories. Then, we extend the analysis to the
sub/superjets of , obtaining new sensitivity relations of second order. By
applying sensitivity analysis to exclude the presence of conjugate points, we
deduce that the value function is twice differentiable along any optimal
trajectory starting at a point at which is proximally subdifferentiable. We
also provide sufficient conditions for the local regularity of on
tubular neighborhoods of optimal trajectories
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