Communications in Mathematical Sciences Articles (Project Euclid)
http://projecteuclid.org/euclid.cms
The latest articles from Communications in Mathematical Sciences on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTTue, 10 May 2011 09:34 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
http://projecteuclid.org/
Minmax variational principle for steady balanced solutions of the rotating shallow water equations
http://projecteuclid.org/euclid.cms/1274816884
<strong>Visweswaran Nageswaran</strong>, <strong>Bruce Turkington</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 321--339.</p>projecteuclid.org/euclid.cms/1274816884_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTStudy of noise-induced transitions in the Lorenz system using the minimum action method
http://projecteuclid.org/euclid.cms/1274816885
<strong>Xiang Zhou</strong>, <strong>Weinan E</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 341--355.</p><p><strong>Abstract:</strong><br/>
We investigate noise-induced transitions in non-gradient systems when complex
invariant sets emerge. Our example is the Lorenz system in three representative Rayleigh number
regimes. It is found that before the homoclinic explosion bifurcation, the only transition state is
the saddle point, and the transition is similar to that in gradient systems. However, when the
chaotic invariant set emerges, an unstable limit cycle continues from the homoclinic trajectory. This
orbit, which is embedded in a local tube-like manifold around the initial stable stationary point as a
relative attractor, plays the role of the most probable exit set in the transition process. This example
demonstrates how limit cycles, the next simplest invariant set beyond fixed points, can be involved
in the transition process in smooth dynamical systems.
</p>projecteuclid.org/euclid.cms/1274816885_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTNonlinear inertia-gravity wave-mode interactions in three dimensional rotating stratified flows
http://projecteuclid.org/euclid.cms/1274816886
<strong>Mark Remmel</strong>, <strong>Jai Sukhatme</strong>, <strong>Leslie M. Smith</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 357--376.</p>projecteuclid.org/euclid.cms/1274816886_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTThe multidimensional maximum entropy moment problem: a review of numerical methods
http://projecteuclid.org/euclid.cms/1274816887
<strong>Rafail V. Abramov</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 377--392.</p><p><strong>Abstract:</strong><br/>
Recently the author developed a numerical method for the multidimensional momentconstrained
maximum entropy problem, which is practically capable of solving maximum entropy
problems in the two-dimensional domain with moment constraints of order up to 8, in the threedimensional
domain with moment constraints of order up to 6, and in the four-dimensional domain
with moment constraints of order up to 4, corresponding to the total number of moment constraints of
44, 83 and 69, respectively. In this work, the author brings together key algorithms and observations
from his previous works as well as other literature in an attempt to present a comprehensive exposition
of the current methods and results for the multidimensional maximum entropy moment problem.
</p>projecteuclid.org/euclid.cms/1274816887_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTApplication of the stochastic mode-reduction strategy and a priori prediction of symmetry
breaking in stochastic systems with underlying symmetry
http://projecteuclid.org/euclid.cms/1274816888
<strong>N. Barlas</strong>, <strong>I. Timofeyev</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 393--408.</p><p><strong>Abstract:</strong><br/>
We consider application of the stochastic mode-reduction strategy to a particular
class of coupled models where a part of self-interactions of the slow variables is given by a rotationally
invariant gradient system. The stochastic mode-reduction strategy is utilized to derive stochastic
reduced models which yield a simple description of the phenomena resulting from breaking the
original rotational symmetry. It is demonstrated that the direction of the symmetry breaking can
be predicted a priori without any knowledge of the statistical behavior of the fast modes.
</p>projecteuclid.org/euclid.cms/1274816888_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTHybrid deterministic stochastic systems with microscopic look-ahead dynamics
http://projecteuclid.org/euclid.cms/1274816889
<strong>M. A. Katsoulakis</strong>, <strong>A. J. Majda</strong>, <strong>A. Sopasakis</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 409--437.</p><p><strong>Abstract:</strong><br/>
We study the impact of stochastic mechanisms on a coupled hybrid system consisting of a general
advection-diffusion-reaction partial differential equation and a spatially distributed
stochastic lattice noise model. The stochastic dynamics include both spin-flip and spin-exchange
type interparticle interactions. Furthermore, we consider a new, asymmetric, single exclusion pro-
cess, studied elsewhere in the context of traffic flow modeling, with an one-sided interaction potential
which imposes advective trends on the stochastic dynamics. This look-ahead stochastic mechanism
is responsible for rich nonlinear behavior in solutions. Our approach relies heavily on first deriving
approximate differential mesoscopic equations. These approximations become exact either in the
long range, Kac interaction partial differential equation case, or, given sufficient time separation con-
ditions, between the partial differential equation and the stochastic model giving rise to a stochastic
averaging partial differential equation. Although these approximations can in some cases be crude,
they can still give a first indication, via linearized stability analysis, of the interesting regimes for the
stochastic model. Motivated by this linearized stability analysis we choose particular regimes where
interacting nonlinear stochastic waves are responsible for phenomena such as random switching,
convective instability, and metastability, all driven by stochasticity. Numerical kinetic Monte Carlo
simulations of the coarse grained hybrid system are implemented to assist in producing solutions and
understanding their behavior.
</p>projecteuclid.org/euclid.cms/1274816889_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTReduced dynamics of stochastically perturbed gradient flows
http://projecteuclid.org/euclid.cms/1274816890
<strong>Ibrahim Fatkullin</strong>, <strong>Gregor Kovacic</strong>, <strong>Eric Vanden-Eijnden</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 439--461.</p><p><strong>Abstract:</strong><br/>
We consider stochastically perturbed gradient flows in the limit when the amplitude
of random fluctuations is small relative to the typical energy scale in the system and the minima of the
energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics
may be described in terms of a diffusion process on these manifolds. We derive explicit equations for
this limiting dynamics and illustrate them on a few finite-dimensional examples. Finally, we formally
extrapolate the reduction technique to several infinite-dimensional examples and derive equations of
the stochastic kink motion in Allen-Cahn-type systems.
</p>projecteuclid.org/euclid.cms/1274816890_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTDiffusion limit of the Vlasov-Poisson-Fokker-Planck system
http://projecteuclid.org/euclid.cms/1274816891
<strong>Najoua El Ghani</strong>, <strong>Nader Masmoudi</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 463--479.</p><p><strong>Abstract:</strong><br/>
We study the diffusion limit of the Vlasov-Poisson-Fokker-Planck System. Here, we
generalize the local in time results and the two dimensional results of Poupaud-Soler and of Goudon to the case of several space dimensions.
Renormalization techniques, the method of moments and a velocity averaging lemma are used to
prove the convergence of free energy solutions (renormalized solutions) to the Vlasov-Poisson-Fokker-
Planck system towards a global weak solution of the Drift-Diffusion-Poisson model.
</p>projecteuclid.org/euclid.cms/1274816891_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTTwo coarse-graining studies of stochastic models in molecular biology
http://projecteuclid.org/euclid.cms/1274816892
<strong>Peter R. Kramer</strong>, <strong>Juan C. Latorre</strong>, <strong>Adnan A. Khan</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 481--517.</p><p><strong>Abstract:</strong><br/>
We examine stochastic coarse-graining strategies for two biomolecular systems. First,
we compute the large-scale transport properties of the basic flashing ratchet mathematical model
for (Brownian) molecular motors and consider in this light whether the underlying continuous-space,
continuous-time Markovian model can be coarse-grained as a discrete-state, continuous-time Markovian
random walk model. Through careful computation of associated statistical signatures of Markovianity,
we find that such a discrete coarse-graining is an excellent approximation over much but not
all of the parameter regime. In particular, for the parameter values associated with the fastest transport
by the flashing ratchet, the discretized model displays non-Markovian features such as waiting
times between jumps which are not exponentially distributed. We provide a theoretical framework
for understanding the conditions under which Markovianity is to be expected in the discretized
model and two mechanisms by which the flashing ratchet model coarse-grains to a non-Markovian
discretized model. Next we turn to a basic question of how the dynamics of water molecules near
the surface of a solute can be represented by a simple drift-diffusion stochastic model. This question
is of most interest for the purpose of accelerating molecular dynamics simulations of proteins, but
for simplicity, here we examine the simple case where the solute is a C60 buckyball, which has a
homogenous, roughly isotropic form. We compare the mathematical drift-diffusion framework with
a statistical quantification of water dynamics near a solute discussed in the biophysical literature.
A key concern is the choice of time interval on which to sample the molecular dynamics data to
generate estimators for the drift and diffusivity. We use a simple mathematical toy model to establish
insight and a strategy, but find for the actual molecular dynamics data that the sampling times
which produce the most faithful drift coefficient and the sampling times which produce the most
faithful diffusion coefficient do not overlap, so that sacrifice of quality in one or the other parameter
appears necessary.
</p>projecteuclid.org/euclid.cms/1274816892_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTIntraseasonal multi-scale moist dynamics of the tropical atmosphere
http://projecteuclid.org/euclid.cms/1274816893
<strong>Joseph A. Biello</strong>, <strong>Andrew J. Majda</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 519--540.</p><p><strong>Abstract:</strong><br/>
We derive a multi-scale model of moist tropical dynamics which is valid on horizontal
synoptic scales, zonal planetary scales, and synoptic and intraseasonal time scales. The Intraseasonal
Multi-Scale Moist Dynamics (IMMD) framework builds on the IPESD framework of A.J. Majda and
R. Klein, J. Atmos. Sci. , 60, 393–408, 2003. It generalizes the latter by allowing for strong zonal
winds (the Trade Winds) and the pressure and stratification variations that they generate. The
framework consists of three pieces. The first, called TH, are planetary scale climatology modulation
equations which govern the Trade Winds and Hadley Circulation. Self-consistency of the asymptotic
theory requires that the meridional component of the Hadley Circulation is an order of magnitude
weaker than the zonal component. The second piece, S, is a linear system of equations which
govern synoptic scale velocity, temperature, and pressure fluctuations forced by synoptic scale heating
fluctuations. Unlike the IPESD theory, these fluctuations are advected by part of the planetary scale
climatology from TH. Since the meridional component of TH is an order of magnitude weaker than
the zonal component, the synoptic scale fluctuations are only advected by the latter. The third,
P, govern the planetary scale anomalies which, like IPESD, are driven both by planetary scale
mean heating and by upscale fluxes from the synoptic scales. These planetary scale anomalies are
advected both by the zonal component of the Trade Winds and by the meridional component of the
Hadley Circulation and, furthermore, respond to an in-scale flux from the mean climatology. We
also present an asymptotic analysis of the equations of bulk cloud thermodynamics in order to lay
out a self-contained path for incorporating synoptic scale cloud models into the IMMD framework.
This framework has potentially important implications for the development of models describing the
Madden-Julian Oscillation (MJO) since the MJO manifests itself as planetary scale anomalies from
a mean climatology which it modulates on intraseasonal time scales.
</p>projecteuclid.org/euclid.cms/1274816893_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTDynamics of current-based, Poisson driven, integrate-and-fire neuronal networks
http://projecteuclid.org/euclid.cms/1274816894
<strong>Katherine A. Newhall</strong>, <strong>Gregor Kovacic</strong>, <strong>Peter R. Kramer</strong>, <strong>Douglas Zhou</strong>, <strong>Aaditya V. Rangan</strong>, <strong>David Cai</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 541--600.</p><p><strong>Abstract:</strong><br/>
Synchronous and asynchronous dynamics in all-to-all coupled networks of identical,
excitatory, current-based, integrate-and-fire (I&F) neurons with delta-impulse coupling currents and
Poisson spike-train external drive are studied. Repeating synchronous total firing events, during
which all the neurons fire simultaneously, are observed using numerical simulations and found to
be the attracting state of the network for a large range of parameters. Mechanisms leading to such
events are then described in two regimes of external drive: superthreshold and subthreshold. In
the former, a probabilistic argument similar to the proof of the Central Limit Theorem yields the
oscillation period, while in the latter, this period is analyzed via an exit time calculation utilizing a
diffusion approximation of the Kolmogorov forward equation. Asynchronous dynamics are observed
computationally in networks with random transmission delays. Neuronal voltage probability density
functions (PDFs) and gain curves—graphs depicting the dependence of the network firing rate on
the external drive strength—are analyzed using the steady solutions of the self-consistency problem
for a Kolmogorov forward equation. All the voltage PDFs are obtained analytically, and asymptotic
solutions for the gain curves are obtained in several physiologically relevant limits. The absence of
chaotic dynamics is proved for the type of network under investigation by demonstrating convergence
in time of its trajectories.
</p>projecteuclid.org/euclid.cms/1274816894_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTThe exact evolution of the scalar variance in pipe and channel flow
http://projecteuclid.org/euclid.cms/1274816895
<strong>Roberto Camassa</strong>, <strong>Zhi Lin</strong>, <strong>Richard M. McLaughlin</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 601--626.</p>projecteuclid.org/euclid.cms/1274816895_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTStochastic homogenization of Hamilon-Jacobi and "viscous"-Hamilton-Jacobi
equations with convex nonlinearities -- Revisited
http://projecteuclid.org/euclid.cms/1274816896
<strong>Pierre-Louis Lions</strong>, <strong>Panagiotis E. Souganidis</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 2, 627--637.</p><p><strong>Abstract:</strong><br/>
In this note we revisit the homogenization theory of Hamilton-Jacobi and “viscous”-
Hamilton-Jacobi partial differential equations with convex nonlinearities in stationary ergodic envi-
ronments. We present a new simple proof for the homogenization in probability. The argument uses
some a priori bounds (uniform modulus of continuity) on the solution and the convexity and coer-
civity (growth) of the nonlinearity. It does not rely, however, on the control interpretation formula
of the solution as was the case with all previously known proofs. We also introduce a new formula
for the effective Hamiltonian for Hamilton-Jacobi and “viscous” Hamilton-Jacobi equations.
</p>projecteuclid.org/euclid.cms/1274816896_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTCoarsening in high order, discrete, ill-posed diffusion equationshttp://projecteuclid.org/euclid.cms/1288725259<strong>Catherine Kublik</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 797--834.</p><p><strong>Abstract:</strong><br/>
We study the discrete version of a family of ill-posed, nonlinear diffusion equations
of order $2n$. The fourth order $(n=2)$ version of these equations constitutes our main motivation, as
it appears prominently in image processing and computer vision literature. It was proposed by You
and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The
second order equation $(n=1)$ corresponds to another famous model from image processing, namely
Perona and Malik’s anisotropic diffusion, and was studied in earlier papers. The equations studied in
this paper are high order analogues of the Perona-Malik equation, and like the second order model,
their continuum versions violate parabolicity and hence lack well-posedness theory. We follow a
recent technique from Kohn and Otto, and prove a weak upper bound on the coarsening rate of the
discrete in space version of these high order equations in any space dimension, for a large class of
diffusivities. Numerical experiments indicate that the bounds are close to being optimal, and are
typically observed.
</p>projecteuclid.org/euclid.cms/1288725259_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTSome decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamicshttp://projecteuclid.org/euclid.cms/1288725260<strong>Jianwen Zhang</strong>, <strong>Junning Zhao</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 835--850.</p>projecteuclid.org/euclid.cms/1288725260_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTA finite time result for vanishing viscosity in the plane with nondecaying vorticityhttp://projecteuclid.org/euclid.cms/1288725261<strong>Elaine Cozzi</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 851--862.</p>projecteuclid.org/euclid.cms/1288725261_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTHeterogeneous multiscale finite element method with novel numerical integration schemeshttp://projecteuclid.org/euclid.cms/1288725262<strong>Rui Du</strong>, <strong>Pingbing Ming</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 863--885.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce two novel numerical integration schemes within the
framework of the heterogeneous multiscale method (HMM), when the finite element method is used
as the macroscopic solver, to resolve the elliptic problem with a multiscale coefficient. For nonself-adjoint
elliptic problems, optimal convergence rate is proved for the proposed methods, which
naturally yields a new strategy for refining the macro-micro meshes and a criterion for determining
the size of the microcell. Numerical results following this strategy show that the new methods
significantly reduce the computational cost without loss of accuracy.
</p>projecteuclid.org/euclid.cms/1288725262_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTTailored finite point method for steady-state reaction-diffusion equationshttp://projecteuclid.org/euclid.cms/1288725263<strong>Houde Han</strong>, <strong>Zhongyi Huang</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 887--899.</p>projecteuclid.org/euclid.cms/1288725263_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTOn the energy conservation by weak solutions of the relativistic Vlasov-Maxwell systemhttp://projecteuclid.org/euclid.cms/1288725264<strong>Reinel Sospedra-Alfonso</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 901--908.</p>projecteuclid.org/euclid.cms/1288725264_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTA level set approach to modeling general service rules in supply chainshttp://projecteuclid.org/euclid.cms/1288725265<strong>Christian Ringhofer</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 909--930.</p><p><strong>Abstract:</strong><br/>
The need for service rules, or policies, in supply chains arises if not all the parts
processed in the chain are considered identical, but are distinguished by certain attributes. We
develop and analyze a methodology to model arbitrary service rules in large supply chains based on
a kinetic (traffic flow like) theory and a level set approach. The final result is a system of hyperbolic
conservation laws for the densities of parts, grouped by their attributes. The validity of the model
is verified against discrete event simulations for several test cases.
</p>projecteuclid.org/euclid.cms/1288725265_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTA strategy for non-strictly convex transport costs and the example of $║x-y║^p$ in $R^2$http://projecteuclid.org/euclid.cms/1288725266<strong>Guillaume Carlier</strong>, <strong>Luigi De Pascale</strong>, <strong>Filippo Santambrogio</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 931--941.</p><p><strong>Abstract:</strong><br/>
This paper deals with the existence of optimal transport maps for some optimal
transport problems with a convex but non-strictly convex cost. We give a decomposition strategy
to address this issue. As a consequence of our procedure, we have to treat some transport problems,
of independent interest, with a convex constraint on the displacement. To illustrate possible results
obtained through this general approach, we prove existence of optimal transport maps in the case
where the source measure is absolutely continuous with respect to the Lebesgue measure and the
transportation cost is of the form $h║x-y║$, with h strictly convex increasing and $║.║$ an arbitrary
norm in $R^2$.
</p>projecteuclid.org/euclid.cms/1288725266_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTSmall amplitude oscillatory shear permeation flow of cholesteric liquid crystal polymershttp://projecteuclid.org/euclid.cms/1288725267<strong>Zhenlu Cui</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 943--963.</p>projecteuclid.org/euclid.cms/1288725267_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTBoundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limithttp://projecteuclid.org/euclid.cms/1288725268<strong>Xiao-Ping Wang</strong>, <strong>Ya-Guang Wang</strong>, <strong>Zhouping Xin</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 965--998.</p><p><strong>Abstract:</strong><br/>
</p>projecteuclid.org/euclid.cms/1288725268_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTStrong convergence of principle of averaging for multiscale stochastic dynamical systemshttp://projecteuclid.org/euclid.cms/1288725269<strong>Di Liu</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 999--1020.</p>projecteuclid.org/euclid.cms/1288725269_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTGlobal existence and finite dimensional global attractor for a 3D double viscous MHD-α modelhttp://projecteuclid.org/euclid.cms/1288725270<strong>Davide Catania</strong>, <strong>Paolo Secchi</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 1021--1040.</p><p><strong>Abstract:</strong><br/>
We consider a magnetohydrodynamic-α model with kinematic viscosity and magnetic
diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). Similar models are
useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current
impossibility to handle non-regularized systems neither analytically nor via numerical simulations.
We prove the existence of a global solution and a global attractor. Moreover, we provide an upper
bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in
terms of degrees of freedom of the system. In some sense, this result provides an intermediate bound
between the number of degrees of freedom for the simplified Bardina model and the Navier–Stokes-α equation.
</p>projecteuclid.org/euclid.cms/1288725270_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTGaussian beam approach for the boundary value problem of high frequency Helmholtz equationhttp://projecteuclid.org/euclid.cms/1288725271<strong>Chunxiong Zheng</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 1041--1066.</p><p><strong>Abstract:</strong><br/>
We propose an asymptotic numerical method called the Gaussian beam approach
for the boundary value problem of high frequency Helmholtz equation. The basic idea is to approximate
the traveling waves with a summation of Gaussian beams by the least squares algorithm.
Gaussian beams are asymptotic solutions of linear wave equations in the high frequency regime. We
deduce the ODE systems satisfied by the Gaussian beams up to third order. The key ingredient
of the proposed method is the construction of a finite-dimensional beam space which has a good
approximating property. If the exact solutions of boundary value problems contain some strongly
evanescent wave modes, the Gaussian beam approach might fail. To remedy this problem, we resort
to the domain decomposition technique to separate the domain of definition into a boundary
layer region and its complementary interior region. The former is handled by a domain-based discretization
method, and the latter by the Gaussian beam approach. Schwarz iterations should then
be performed based on suitable transmission boundary conditions at the interface of two regions.
Numerical tests demonstrate that the proposed method is very promising.
</p>projecteuclid.org/euclid.cms/1288725271_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTPeriodic homogenization of the inviscid G-equation for incompressible flowshttp://projecteuclid.org/euclid.cms/1288725272<strong>Jack Xin</strong>, <strong>Yifeng Yu</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 1067--1078.</p><p><strong>Abstract:</strong><br/>
G-equations are popular front propagation models in combustion literature and
describe the front motion law of normal velocity equal to a constant plus the normal projection of fluid
velocity. G-equations are Hamilton-Jacobi equations with convex but non-coercive Hamiltonians. We
prove homogenization of the inviscid G-equation for space periodic incompressible flows. This extends
a two space dimensional result in "Periodic homogenization of G-equations and viscosity effects,"
Nonlinearity , to appear. We construct approximate correctors to bypass the lack of
compactness due to the non-coercive Hamiltonian. The existence of approximate correctors rely on
a local reachability property of the controlled flow trajectory as well as incompressibility of the flow.
Homogenization then follows from the comparison principle and the perturbed test function method.
The effective Hamiltonian is convex and homogeneous of degree one. It is also coercive if we further
assume that the flow is mean zero.
</p>projecteuclid.org/euclid.cms/1288725272_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTOn the Chang and Cooper scheme applied to a linear Fokker-Planck equationhttp://projecteuclid.org/euclid.cms/1288725273<strong>Christophe Buet</strong>, <strong>Stéphane Dellacherie</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 1079--1090.</p><p><strong>Abstract:</strong><br/>
We show that for a particular linear Fokker-Planck operator, the explicit Chang and
Cooper scheme is positive and entropy satisfying under a CFL criterion when the initial condition
is positive. Then, we deduce that the distribution given by the explicit Chang and Cooper scheme
converges toward a discrete Maxwellian equilibrium.
</p>projecteuclid.org/euclid.cms/1288725273_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTAddendum to: A new median formula with applications to PDE based denoisinghttp://projecteuclid.org/euclid.cms/1288725274<strong>Yingying Li</strong>, <strong>Stanley Osher</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 8, Number 4, 1091--.</p>projecteuclid.org/euclid.cms/1288725274_Tue, 02 Nov 2010 15:14 EDTTue, 02 Nov 2010 15:14 EDTKerr-Debye relaxation shock profiles for Kerr equationshttp://projecteuclid.org/euclid.cms/1294170323<strong>Denise Aregba-Driollet</strong>, <strong>Bernard Hanouzet</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 1--31.</p><p><strong>Abstract:</strong><br/>
The electromagnetic wave propagation in a nonlinear medium can be described by a
Kerr model in the case of an instantaneous response of the material, or by a Kerr-Debye model if the
material exhibits a finite response time. Both models are quasilinear hyperbolic, and the Kerr-Debye
model is a physical relaxation approximation of the Kerr model. In this paper we characterize the
shocks in the Kerr model for which there exists a Kerr-Debye profile. First we consider 1D models
for which explicit calculations are performed. Then we determine the plane discontinuities of the full
vector 3D Kerr system and their admissibility in the sense of Liu and in the sense of Lax. Finally
we characterize the large amplitude Kerr shocks giving rise to the existence of Kerr-Debye relaxation
profiles.
</p>projecteuclid.org/euclid.cms/1294170323_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTMulti-scale methods for wave propagation in heterogeneous mediahttp://projecteuclid.org/euclid.cms/1294170324<strong>Björn Engquist</strong>, <strong>Henrik Holst</strong>, <strong>Olof Runborg</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 33--56.</p><p><strong>Abstract:</strong><br/>
Multi-scale wave propagation problems are computationally costly to solve by traditional
techniques because the smallest scales must be represented over a domain determined by the
largest scales of the problem. We have developed and analyzed new numerical methods for multi-scale
wave propagation in the framework of heterogeneous multi-scale method. The numerical methods
couple simulations on macro- and micro-scales for problems with rapidly oscillating coefficients. We
show that the complexity of the new method is significantly lower than that of traditional techniques
with a computational cost that is essentially independent of the micro-scale. A convergence proof is
given and numerical results are presented for periodic problems in one, two, and three dimensions.
The method is also successfully applied to non-periodic problems and for long time integration where
dispersive effects occur.
</p>projecteuclid.org/euclid.cms/1294170324_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTOn the uniqueness for sub-critical quasi-geostrophic equationshttp://projecteuclid.org/euclid.cms/1294170325<strong>Lucas C. F. Ferreira</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 57--62.</p>projecteuclid.org/euclid.cms/1294170325_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTOn the Cauchy problem for the nonlocal derivative nonlinear Schrodinger equationhttp://projecteuclid.org/euclid.cms/1294170326<strong>Roger Peres de Moura</strong>, <strong>Ademir Pastor</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 63--80.</p>projecteuclid.org/euclid.cms/1294170326_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTWave propagation in shallow-water acoustic random waveguideshttp://projecteuclid.org/euclid.cms/1294170327<strong>Christophe Gomez</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 81--125.</p><p><strong>Abstract:</strong><br/>
In shallow-water waveguides a propagating field can be decomposed in three kinds
of modes: the propagating modes, the radiating modes and the evanescent modes. In this paper we
consider the propagation of a wave in a randomly perturbed waveguide and we analyze the coupling
between these three kinds of modes using an asymptotic analysis based on a separation of scales
technique. Then, we derive the asymptotic form of the distribution of the mode amplitudes and the
coupled power equation for propagating modes. From this equation, we show that the total energy
carried by the propagating modes decreases exponentially with the size of the random section and
we give an expression of the decay rate. Moreover, we show that the mean propagating mode powers
converge to the solution of a diffusion equation in the limit of a large number of propagating modes.
</p>projecteuclid.org/euclid.cms/1294170327_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTRelaxation to equilibrium in diffusive-thermal models with a strongly varying diffusion length-scalehttp://projecteuclid.org/euclid.cms/1294170328<strong>Paul Clavin</strong>, <strong>Laurent Masse</strong>, <strong>Jean-Michel Roquejoffre</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 127--141.</p><p><strong>Abstract:</strong><br/>
We consider reaction-diffusion equations with a strongly varying diffusion lengthscale.
We provide a mathematical study of the relaxation towards the steady planar solution, in the
context of infinitesimal disturbances whose wavelength is much shorter than the total thickness of
the wave. The models under study are relevant in the description of ablation fronts encountered in
inertial confinment fusion, when hydrodynamical effects are neglected.
</p>projecteuclid.org/euclid.cms/1294170328_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTOn the Ostwald ripening of thin liquid filmshttp://projecteuclid.org/euclid.cms/1294170329<strong>Shibin Dai</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 143--160.</p>projecteuclid.org/euclid.cms/1294170329_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTOn the uniqueness of entropy solutions to the Riemann problem for 2x2 hyperbolic systems of conservation lawshttp://projecteuclid.org/euclid.cms/1294170330<strong>Hiroki Ohwa</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 161--185.</p><p><strong>Abstract:</strong><br/>
In this paper we revisit the Riemann problem for 2×2 hyperbolic systems of conservation
laws, which satisfy the condition that the product of non-diagonal elements in the Fréchet
derivative (Jacobian) of the flux is positive, the genuine nonlinearity condition, and the Smoller-
Johnson condition in one space variable. The first condition implies that the system is strictly
hyperbolic. By developing the shock curve approach, we give an alternative shock curve approach
and re-prove the uniqueness of self-similar solutions satisfying the Lax entropy condition at discontinuities.
</p>projecteuclid.org/euclid.cms/1294170330_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTHigh-order entropy-based closures for linear transport in slab geometryhttp://projecteuclid.org/euclid.cms/1294170331<strong>Cory D. Hauck</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 187--205.</p><p><strong>Abstract:</strong><br/>
We compute high-order entropy-based $(M_N)$ models for a linear transport equation
on a one-dimensional slab geometry. We simulate two test problems from the literature: the two-
beam instability and the plane-source problem. In the former case, we compute solutions for systems
up to order $N = 6$; in the latter, up to $N = 15$. The most notable outcome of these results is the
existence of shocks in the steady-state profiles of the two-beam instability for all odd values of $N$.
</p>projecteuclid.org/euclid.cms/1294170331_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTAsymptotic stability of rarefaction waves in radiative hydrodynamicshttp://projecteuclid.org/euclid.cms/1294170332<strong>Chunjin Lin</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 207--223.</p>projecteuclid.org/euclid.cms/1294170332_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTA parametrix construction for the wave equation with low regularity coefficients using a frame of Gaussianshttp://projecteuclid.org/euclid.cms/1294170333<strong>Alden Waters</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 225--254.</p>projecteuclid.org/euclid.cms/1294170333_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTFrom Boltzmann equation to spherical harmonics expansion model: diffusion limit and Poisson couplinghttp://projecteuclid.org/euclid.cms/1294170334<strong>Mohamed Lazhar Tayeb</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 255--275.</p><p><strong>Abstract:</strong><br/>
The diffusion approximation of an initial-boundary value problem for a Boltzmann-
Poisson system is studied. An elastic operator modeling electron-impurity collision is considered.
A relative entropy is used to control the terms coming from the boundary and to prove useful
$L^2$−estimates for the renormalized solutions of the scaled Boltzmann equation (coupled to Poisson).
A careful analysis of a relative entropy for high velocity allows us to show uniform bounds for the
total mass and the kinetic energy which gives the compactness of the self-consistent electrostatic
potential. Then, the moment method is used to prove the convergence of the renormalized solutions
to a weak solution of a Spherical Harmonics Expansion (or SHE-) model coupled to the Poisson
equation.
</p>projecteuclid.org/euclid.cms/1294170334_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTAn improvement of the TYT algorithm for GF(2m) based on reusing intermediate computation resultshttp://projecteuclid.org/euclid.cms/1294170335<strong>Yin Li</strong>, <strong>Gong-liang Chen</strong>, <strong>Yi-yang Chen</strong>, <strong>Jian-hua Li</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 277--287.</p><p><strong>Abstract:</strong><br/>
Multiplicative inversion plays an important role to Elliptic Curve Cryptosystems.
This paper presents an efficient inversion algorithm in $GF(2^m)$ using a normal basis which improves
the Itoh-Tsujii (IT) algorithm and the Takagi et al. (TYT) algorithm . The proposed algorithm
reduces the number of required multiplications by decomposing $m−1$ into several factors plus a
remainder and by reusing intermediate computation values. It is proved that the decomposition
of $m−1$ can be made simpler, but requires even fewer multiplications. Furthermore, a practical
algorithm for finding an optimal decomposition of $m−1$ is investigated.
</p>projecteuclid.org/euclid.cms/1294170335_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTA singular 1-D Hamilton-Jacobi equation, with application to large deviation of diffusionshttp://projecteuclid.org/euclid.cms/1294170336<strong>Xiaoxue Deng</strong>, <strong>Jin Feng</strong>, <strong>Yong Liu</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 289--300.</p><p><strong>Abstract:</strong><br/>
The comparison principle (uniqueness) for the Hamilton-Jacobi equation is usually
established through arguments involving a distance function. In this article we illustrate the subtle
nature of choosing such a distance function, using a special example of one dimensional Hamiltonian
with coefficient singularly (non-Lipschitz) depending upon the state variable. The standard method
of using Euclidean distance as a test function fails in such situation. Once the comparison is established,
we apply it to obtain a new result on small noise Freidlin-Wentzell type probabilistic large
deviation theorem for certain singular diffusion processes.
This article serves to explain basic ideas behind an abstract approach to comparison developed in
J. Feng and T.G. Kurtz, American Mathematical Society, Providence, Rhode Island. Mathematical
Surveys and Monographs , 131, 2006, J. Feng and M. Katsoulakis, Arch. Ration. Mech. Anal. ,
192(2), 275-310, 2009 in a simple manner, removing all technicalities due to infinite dimensionality.
</p>projecteuclid.org/euclid.cms/1294170336_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTA weak trapezoidal method for a class of stochastic differential equationshttp://projecteuclid.org/euclid.cms/1294170337<strong>David F. Anderson</strong>, <strong>Jonathan C. Mattingly</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 301--318.</p><p><strong>Abstract:</strong><br/>
We present a numerical method for the approximation of solutions for the class
of stochastic differential equations driven by Brownian motions which induce stochastic variation
in fixed directions. This class of equations arises naturally in the study of population processes
and chemical reaction kinetics. We show that the method constructs paths that are second order
accurate in the weak sense. The method is simpler than many second order methods in that it
neither requires the construction of iterated Itô integrals nor the evaluation of any derivatives. The
method consists of two steps. In the first an explicit Euler step is used to take a fractional step. The
resulting fractional point is then combined with the initial point to obtain a higher order, trapezoidal
like, approximation. The higher order of accuracy stems from the fact that both the drift and the
quadratic variation of the underlying SDE are approximated to second order.
</p>projecteuclid.org/euclid.cms/1294170337_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTCrank-Nicolson finite element methods using symmetric stabilization with an application to optimal
control problems subject to transient advection-diffusion equationshttp://projecteuclid.org/euclid.cms/1294170338<strong>Erik Burman</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 1, 319--329.</p><p><strong>Abstract:</strong><br/>
We consider a finite element method with symmetric stabilization for transient
advection-diffusion-reaction problems. The Crank-Nicolson finite difference scheme is used for discretization
in time. We prove stability of the numerical method both for implicit and explicit treatment
of the stabilization operator. The resulting convergence results are given and the results are
illustrated by a numerical experiment. We then consider a model problem for pde-constrained optimization.
Using discrete adjoint consistency of our stabilized method we show that both the implicit
and semi-implicit methods proposed yield optimal convergence for the control and the state variable.
</p>projecteuclid.org/euclid.cms/1294170338_Tue, 04 Jan 2011 14:45 ESTTue, 04 Jan 2011 14:45 ESTUnique minimizer for a random functional with double-well potential in dimension 1 and 2http://projecteuclid.org/euclid.cms/1305034458<strong>Nicolas Dirr</strong>, <strong>Enza Orlandi</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 331--351.</p><p><strong>Abstract:</strong><br/>
We add a random bulk term, modelling the interaction with the impurities of the
medium, to a standard functional in the gradient theory of phase transitions consisting of a gradient
term with a double well potential. We show that in $d≤2$ exists, for almost all the realizations
of the random bulk term, a unique random macroscopic minimizer. This result is in sharp contrast
to the case when the random bulk term is absent. In the latter case there are two minimizers which
are (in law) invariant under translations in space.
</p>projecteuclid.org/euclid.cms/1305034458_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTEntropies for radially symmetric higher-order nonlinear diffusion equationshttp://projecteuclid.org/euclid.cms/1305034459<strong>Mario Bukal</strong>, <strong>Ansgar Jungel</strong>, <strong>Daniel Matthes</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 353--382.</p><p><strong>Abstract:</strong><br/>
A previously developed algebraic approach to proving entropy production inequalities
is extended to deal with radially symmetric solutions for a class of higher-order diffusion equations
in multiple space dimensions. In application of the method, novel a priori estimates are derived for
the thin-film equation, the fourth-order Derrida-Lebowitz-Speer-Spohn equation, and a sixth-order
quantum diffusion equation.
</p>projecteuclid.org/euclid.cms/1305034459_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTCorrector theory for elliptic equations in random media with singular Green's function. Application to random boundarieshttp://projecteuclid.org/euclid.cms/1305034460<strong>Guillaume Bal</strong>, <strong>Wenjia Jing</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 383--411.</p><p><strong>Abstract:</strong><br/>
We consider the problem of the random fluctuations in the solutions to elliptic
PDEs with highly oscillatory random coefficients. In our setting, as the correlation length of the
fluctuations tends to zero, the heterogeneous solution converges to a deterministic solution obtained
by averaging. When the Green’s function to the unperturbed operator is sufficiently singular (i.e., not
square integrable locally), the leading corrector to the averaged solution may be either deterministic
or random, or both in a sense we shall explain.
Our main application is the solution of an elliptic problem with random Robin boundary condition
that may be used to model diffusion of signaling molecules through a layer of cells into a bulk of
extracellular medium. The problem is then described by an elliptic pseudo-differential operator (a
Dirichlet-to-Neumann operator) on the boundary of the domain with random potential.
In the physical setting of a three dimensional extracellular medium on top of a two-dimensional
surface of cells forming a layer of epithelium, we show that the approximate corrector to averaging
consists of a deterministic correction plus a Gaussian field of amplitude proportional to the correlation
length of the random medium. The result is obtained under some assumptions on the four-point
correlation function in the medium. We provide examples of such random media based on Gaussian
and Poisson statistics.
</p>projecteuclid.org/euclid.cms/1305034460_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTUnconditionally stable schemes for higher order inpaintinghttp://projecteuclid.org/euclid.cms/1305034461<strong>Carola-Bibiane Schönlieb</strong>, <strong>Andrea Bertozzi</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 413--457.</p>projecteuclid.org/euclid.cms/1305034461_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTA mathematical model for the hard sphere repulsion in ionic solutionshttp://projecteuclid.org/euclid.cms/1305034462<strong>YunKyong Hyon</strong>, <strong>Bob Eisenberg</strong>, <strong>Chun Liu</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 459--475.</p><p><strong>Abstract:</strong><br/>
We introduce a mathematical model for the finite size (repulsive) effects in ionic
solutions. We first introduce an appropriate energy term into the total energy that represents the
hard sphere repulsion of ions. The total energy then consists of the entropic energy, electrostatic
potential energy, and the repulsive potential energy. The energetic variational approach derives a
boundary value problem that includes contributions from the repulsive term with a no flux boundary
condition for charge density which is a consequence of the variational approach, and physically implies
charge conservation. The resulting system of partial differential equations is a modification of the
Poisson-Nernst-Planck (PNP) equations widely if not universally used to describe the drift-diffusion
of electrons and holes in semiconductors, and the movement of ions in solutions and protein channels.
The modified PNP equations include the effects of the finite size of ions that are so important in
the concentrated solutions near electrodes, active sites of enzymes, and selectivity filters of proteins.
Finally, we do some numerical experiments using finite element methods, and present their results
as a verification of the utility of the modified system.
</p>projecteuclid.org/euclid.cms/1305034462_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTPricing and hedging contingent claims with regime switching riskhttp://projecteuclid.org/euclid.cms/1305034463<strong>Robert J. Elliott</strong>, <strong>Tak Kuen Siu</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 477--498.</p><p><strong>Abstract:</strong><br/>
We study the pricing and hedging of contingent claims in a Markov regime-switching
market with a money market account, a zero-coupon bond, and an ordinary share. General contingent
claims with payoffs depending on both the share price and the state of a Markov chain describing
regime switching are considered. A general pricing kernel defined by the product of two density
processes is used to explicitly take into account regime switching risk. Under some differentiability
and boundedness conditions, a martingale representation result is established and the integrands
in the representation are explicitly identified with respect to the general pricing kernel. We then
determine a pricing kernel and a hedging strategy by minimizing the residual risk due to incomplete
hedging. Our analysis is also extended to Asian-style and American-style general contingent claims.
</p>projecteuclid.org/euclid.cms/1305034463_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTExponentially-stable steady flow and asymptotic behavior for the magnetohydrodynamic equationshttp://projecteuclid.org/euclid.cms/1305034464<strong>Lucas C. F. Ferreira</strong>, <strong>Elder J. Villamizar-Roa</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 499--516.</p>projecteuclid.org/euclid.cms/1305034464_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTGaussian processes associated to infinite bead-spring networkshttp://projecteuclid.org/euclid.cms/1305034465<strong>Michael Taylor</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 517--534.</p>projecteuclid.org/euclid.cms/1305034465_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTApproximate solutions to several visibility optimization problemshttp://projecteuclid.org/euclid.cms/1305034466<strong>Rostislav Goroshin</strong>, <strong>Quyen Huynh</strong>, <strong>Hao-Min Zhou</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 535--550.</p><p><strong>Abstract:</strong><br/>
The visibility level set function introduced by Tsai et al. allows for gradient based
and variational formulations of many classical visibility optimization problems. In this work we
propose solutions to two such problems. The first asks where to position $n$-observers such that the
area visible to these observers is maximized. The second problem is to determine the shortest route
an observer should take through a map such that every point in the map is visible from at least
one vantage point on the route. These problems are similar to the "art gallery" and "watchman
route" problems, respectively. We propose a greedy iterative algorithm, formulated in the level set
framework as the solution to the art gallery problem. We also propose a variational solution to the
watchman route problem which achieves complete visibility coverage of the domain while attaining
a local minimum of path length.
</p>projecteuclid.org/euclid.cms/1305034466_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTFrame based segmentation for medical imageshttp://projecteuclid.org/euclid.cms/1305034467<strong>Bin Dong</strong>, <strong>Aichi Chien</strong>, <strong>Zuowei Shen</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 551--559.</p><p><strong>Abstract:</strong><br/>
Medical image segmentation is an important but difficult problem that attracts
tremendous attention from researchers in various fields. In this paper, we propose a frame based
model, as well as a fast implementation, for general medical image segmentation problems. Our
model combines ideas of the frame based image restoration model of J. Cai, S. Osher, and Z. Shen,
Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 8(2), 337–369, 2009 with
ideas of the total variation based segmentation model of T. Chan and L. Vese, Scale-Space Theories
in Computer Vision, 141–151, 1999, T. Chan and L. Vese, IEEE Transactions on image processing,
10(2), 266–277, 2001, T. Chan, S. Esedoglu and M. Nikolova, ALGORITHMS, 66(5), 1632–1648,
and X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Journal of Mathematical
Imaging and Vision, 28(2), 151–167, 2007. Numerical experiments show that the proposed frame
based model outperforms the total variation based model in terms of capturing key features of
biological structures. Successful segmentations of blood vessels and aneurysms in 3D CT angiography
images are also presented.
</p>projecteuclid.org/euclid.cms/1305034467_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTHomogenization of the G-equation with incompressible random drift in two dimensionshttp://projecteuclid.org/euclid.cms/1305034468<strong>James Nolen</strong>, <strong>Alexei Novikov</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 561--582.</p><p><strong>Abstract:</strong><br/>
We study the homogenization limit of solutions to the G-equation with random drift.
This Hamilton-Jacobi equation is a model for flame propagation in a turbulent fluid in the regime of
thin flames. For a fluid velocity field that is statistically stationary and ergodic, we prove sufficient
conditions for homogenization to hold with probability one. These conditions are expressed in terms
of travel times for the associated control problem. When the spatial dimension is equal to two and
the fluid velocity is divergence-free, we verify that these conditions hold under suitable assumptions
about the growth of the random stream function.
</p>projecteuclid.org/euclid.cms/1305034468_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTCritical thresholds in multi-dimensional Restricted Euler equationshttp://projecteuclid.org/euclid.cms/1305034469<strong>Dongming Wei</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 583--596.</p><p><strong>Abstract:</strong><br/>
Using the spectral dynamics, we study the critical threshold phenomena in the multidimensional
restricted Euler (RE) equations. We identify sub-critical and sup-critical initial data for
all space dimensions, which extends the previous result for the 3D and 4D restricted Euler equations.
Our result suggests that: if the number of dimensions is odd, the finite time blowup is generic; in
contrast, if the number of dimensions is even, there is a rich set of initial data which yields global
smooth solutions.
</p>projecteuclid.org/euclid.cms/1305034469_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTDerivation of continuum models for the moving contact line problem based on thermodynamic principleshttp://projecteuclid.org/euclid.cms/1305034470<strong>Weiqing Ren</strong>, <strong>Weinan E</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 597--606.</p><p><strong>Abstract:</strong><br/>
Contact lines arise as the boundaries of free boundaries in fluids. This problem
is interesting and important, not only because it arises in many applications, but also because of
the distinct mathematical and physical features it has, such as singularities, hysteresis, instabilities,
competing scaling regimes, etc. For a long time, this area of study was plagued with conflicting
theories and uncertainties regarding how the problem should be modeled. In the present paper we
illustrate how continuum models for the moving contact line problem can be derived using simple
thermodynamic considerations. Both the sharp interface models and diffuse interface models are
derived.
</p>projecteuclid.org/euclid.cms/1305034470_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTA sharp bound on the L2 norm of the solution of a random elliptic difference equationhttp://projecteuclid.org/euclid.cms/1305034471<strong>Tomasz Komorowski</strong>, <strong>Lenya Ryzhik</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 607--622.</p>projecteuclid.org/euclid.cms/1305034471_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDTKramers' formula for chemical reactions in the context of Wasserstein gradient flowshttp://projecteuclid.org/euclid.cms/1305034472<strong>Michael Herrmann</strong>, <strong>Barbara Niethammer</strong><p><strong>Source: </strong>Commun. Math. Sci., Volume 9, Number 2, 623--635.</p><p><strong>Abstract:</strong><br/>
We derive Kramers’ formula as singular limit of the Fokker-Planck equation with
double-well potential. The convergence proof is based on the Rayleigh principle of the underlying
Wasserstein gradient structure and complements a recent result by Peletier, Savaré and Veneroni.
</p>projecteuclid.org/euclid.cms/1305034472_Tue, 10 May 2011 09:34 EDTTue, 10 May 2011 09:34 EDT