8,828 research outputs found
Measure Upper Bounds for Nodal Sets of Eigenfunctions of the bi-Harmonic Operator
In this article, we consider eigenfunctions of the bi-harmonic operator,
i.e.,
on with some homogeneous linear boundary
conditions. We assume that () is a
bounded domain, is piecewise analytic and
is analytic except a set which
is a finite union of some compact dimensional submanifolds of
. The main result of this paper is that the measure upper
bounds of the nodal sets of the eigenfunctions is controlled by
. We first define a frequency function and a doubling index
related to these eigenfunctions. With the help of establishing the monotonicity
formula, doubling conditions and various a priori estimates, we obtain that the
dimensional Hausdorff measures of nodal sets of these eigenfunctions in
a ball are controlled by the frequency function and . In order
to further control the frequency function with , we first
establish the relationship between the frequency function and the doubling
index, and then separate the domain into two parts: a domain away from
and a domain near , and develop iteration arguments to deal
with the two cases respectively
Nonlinear backbone torsional pair correlations in proteins
Protein allostery requires dynamical structural correlations. Physical origin
of which, however, remain elusive despite intensive studies during last two
decades. Based on analysis of molecular dynamics (MD) simulation trajectories
for ten proteins with different sizes and folds, we found that nonlinear
backbone torsional pair (BTP) correlations, which are spatially more
long-ranged and are mainly executed by loop residues, exist extensively in most
analyzed proteins. Examination of torsional motion for correlated BTPs
suggested that aharmonic torsional state transitions are essential for such
non-linear correlations, which correspondingly occur on widely different and
relatively longer time scales. In contrast, BTP correlations between backbone
torsions in stable helices and strands are mainly linear and
spatially more short-ranged, and are more likely to associate with intra-well
torsional dynamics. Further analysis revealed that the direct cause of
non-linear contributions are heterogeneous, and in extreme cases canceling,
linear correlations associated with different torsional states of participating
torsions. Therefore, torsional state transitions of participating torsions for
a correlated BTP are only necessary but not sufficient condition for
significant non-linear contributions. These findings implicate a general search
strategy for novel allosteric modulation of protein activities. Meanwhile, it
was suggested that ensemble averaged correlation calculation and static contact
network analysis, while insightful, are not sufficient to elucidate mechanisms
underlying allosteric signal transmission in general, dynamical and time scale
resolved analysis are essential.Comment: 25 pages, 8 figure
Shot Range and High Order Correlations in Proteins
The main chain dihedral angles play an important role to decide the protein
conformation. The native states of a protein can be regard as an ensemble of a
lot of similar conformations, in different conformations the main chain
dihedral angles vary in a certain range. Each dihedral angle value can be
described as a distribution, but only using the distribution can't describe the
real conformation space. The reason is that the dihedral angle has correlation
with others, especially the neighbor dihedral angles in primary sequence. In
our study we analysis extensive molecular dynamics (MD) simulation trajectories
of eleven proteins with different sizes and folds, we found that in stable
second structure the correlations only exist between the dihedrals near to each
other in primary sequence, long range correlations are rare. But in unstable
structures (loop) long range correlations exist. Further we observed some
characteristics of the short range correlations in different second structures
({\alpha}-helix, {\beta}-sheet) and we found that we can approximate good high
order dihedral angle distribution good only use three order distribution in
stable second structure which illustrates that high order correlations (over
three order) is small in stable second structure
Critical points of solutions for mean curvature equation in strictly convex and nonconvex domains
In this paper, we mainly investigate the set of critical points associated to
solutions of mean curvature equation with zero Dirichlet boundary condition in
a strictly convex domain and a nonconvex domain respectively. Firstly, we
deduce that mean curvature equation has exactly one nondegenerate critical
point in a smooth, bounded and strictly convex domain of
. Secondly, we study the geometric structure about the
critical set of solutions for the constant mean curvature equation in a
concentric (respectively an eccentric) spherical annulus domain of
, and deduce that exists (respectively does not
exist) a rotationally symmetric critical closed surface . In fact, in an
eccentric spherical annulus domain, is made up of finitely many isolated
critical points () on an axis and finitely many
rotationally symmetric critical Jordan curves () with
respect to an axis.Comment: 13 pages, 5 figure
End-to-end driving simulation via angle branched network
Imitation learning for end-to-end autonomous driving has drawn attention from
academic communities. Current methods either only use images as the input which
is ambiguous when a car approaches an intersection, or use additional command
information to navigate the vehicle but not automated enough. Focusing on
making the vehicle drive along the given path, we propose a new navigation
command that does not require human's participation and a novel model
architecture called angle branched network. Both the new navigation command and
the angle branched network are easy to understand and effective. Besides, we
find that not only segmentation information but also depth information can
boost the performance of the driving model. We conduct experiments in a 3D
urban simulator and both qualitative and quantitative evaluation results show
the effectiveness of our model.Comment: 10 pages,6 figure
Uniqueness of critical points of solutions to the mean curvature equation with Neumann and Robin boundary conditions
In this paper, we investigate the critical points of solutions to the
prescribed constant mean curvature equation with Neumann and Robin boundary
conditions respectively in a bounded smooth convex domain of
. Firstly, we show the non-degeneracy and uniqueness of
the critical points of solutions in a planar domain by using the local Chen &
Huang's comparison technique and the geometric properties of approximate
surfaces at the non-degenerate critical points. Secondly, we deduce the
uniqueness and non-degeneracy of the critical points of solutions in a
rotationally symmetric domain of by the projection of
higher dimensional space onto two dimensional plane.Comment: 15pages, 4figure
Configurational space discretization and free energy calculation in complex molecular systems
Trajectories provide dynamical information that is discarded in free energy
calculations, for which we sought to design a scheme with the hope of saving
cost for generating dynamical information. We first demonstrated that snapshots
in a converged trajectory set are associated with implicit conformers that have
invariant statistical weight distribution (ISWD). Based on the thought that
infinite number of sets of implicit conformers with ISWD may be created through
independent converged trajectory sets, we hypothesized that explicit conformers
with ISWD may be constructed for complex molecular systems through systematic
increase of conformer fineness, and tested the hypothesis in lipid molecule
palmitoyloleoylphosphatidylcholine (POPC). Furthermore, when explicit
conformers with ISWD were utilized as basic states to define conformational
entropy, change of which between two given macrostates was found to be
equivalent to change of free energy except a mere difference of a negative
temperature factor, and change of enthalpy essentially cancels corresponding
change of average intra-conformer entropy. These findings suggest that entropy
enthalpy compensation is inherently a local phenomenon in configurational
space. By implicitly taking advantage of entropy enthalpy compensation and
forgoing all dynamical information, constructing explicit conformers with ISWD
and counting thermally accessible number of which for interested end
macrostates is likely to be an efficient and reliable alternative end point
free energy calculation strategy.Comment: 27 pages, 8 figures, 1 tabl
Stepwise quantum phonon pumping in plasmon-enhanced Raman scattering
Plasmon-enhanced Raman scattering (PERS) becomes nonlinear when phonon
pumping and phonon-stimulated scattering come into play. It is fundamental to
the understanding of PERS and its photobleaching behavior. By quantization of
the molecular vibration coherent state into phonon number states, we
theoretically predict a stepwise dependence of PERS intensity on laser power.
Experimental evidence is presented by measuring a monolayer of malachite green
isothiocyanate molecules sandwiched in individual gold nanosphere-plane
antennas, under radially polarized laser excitation of sub-microWatt powers.Comment: 26 pages, 7 figure
Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions
In this paper, we mainly investigate the critical points associated to
solutions of a quasilinear elliptic equation with nonhomogeneous Dirichlet
boundary conditions in a connected domain in . Based on
the fine analysis about the distribution of connected components of a
super-level set for any , we obtain
the geometric structure of interior critical points of . Precisely, when
is simply connected, we develop a new method to prove , where are the respective multiplicities of
interior critical points of and is the number of
global maximal points of on . When is an annular
domain with the interior boundary and the external boundary
, where and has
local (global) maximal points on . For the case
or or , we show that
(either or ).Comment: 21pages, 13figure
An accelerated sharp-interface method for multiphase flows simulations
In this work, we develop an accelerated sharp-interface method based on (Hu
et al., JCP, 2006) and (Luo et al., JCP, 2015) for multiphase flows
simulations. Traditional multiphase simulation methods use the minimum time
step of all fluids obtained according to CFL conditions to evolve the fluid
states, which limits the computational efficiency, as the sound speed c of one
fluid may be much larger than the others. To address this issue, based on the
original GFM-like sharp interface methods, the present method is developed by
solving the governing equations of each individual fluid with the corresponding
time step. Without violating the numerical stability requirement, the states of
fluid with larger time-scale features will be updated with a larger time step.
The interaction step between two fluids is solved for synchronization, which is
handled by interpolating the intermediate states of fluid with larger
time-scale features. In addition, an interfacial flux correction is implemented
to maintain the conservative property. The present method can be combined with
a wavelet-based adaptive multi-resolution algorithm (Han et al., JCP, 2014) to
achieve additional computational efficiency. A number of numerical tests
indicate that the accuracy of the results obtained by the present method is
comparable to the original costly method, with a significant speedup
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