Weizhu Bao
I have conducted research in BoseEinstein condensation;
computational fluid dynamics; computational quantum physics and chemistry;
hyperbolic conservation
laws; multiscale modeling, simulation and applications;
quantized vortices in superfluidity and superconductivity;
numerical methods for problems in
unbounded domains;
finite element method for some nonlinear problems;
numerical analysis and scientific computing;
computational and applied mathematics in general.
Multiscale Methods and Analysis for
Oscillatory PDEs
For
KleinGordon and Dirac equations
(nonrelativistic limit)

Numerical methods and comparison for the Dirac equation in the
nonrelativistic limit regime (with Y. Cai, X. Jia and Q.
Tang), J. Sci. Comput., to appear (arXiv: 1504.02881).

A uniformly accurate (UA) multiscale time integrator Fourier
pseoduspectral method for the KleinGordonSchrodinger equations in the
nonrelativistic limit regime (with X. Zhao),
Numer. Math., to appear
(arXiv: 1505.00083).

A uniformly accurate (UA) multiscale time integrator pseudospectral
method for the Dirac equation in the nonrelativistic limit
regime (with Y. Cai, X. Jia and Q. Tang),
SIAM J. Numer. Anal.,
Vol. 54 (2016),
pp. 17851812 (arXiv: 1507.04103).

Error estimates of numerical methods for
the nonlinear Dirac equation in the nonrelativistic limit regime
(with Y. Cai, X. Jia and J. Yin),
Sci. China Math.,
Vol. 59 (2016),
pp. 14611494 (arXiv: 1511.01192v3).

A uniformly accurate multiscale time integrator pseudospectral method
for the KleinGordon equation in the nonrelativistic limit
regime (with Y. Cai and X. Zhao), SIAM J. Numer.
Anal., Vol. 52 (2014),
pp. 24882511.

Uniformly accurate multiscale time integrators for highly
oscillatory second order differential equations (with
X. Dong and X. Zhao), J. Math. Study, Vol. 47
(2014), pp. 111150.

Analysis and comparison of numerical methods for
KleinGordon equation in nonrelativistic limit regime
(with X. Dong), Numer. Math.,
Vol. 120 (2012), pp. 189229.

An efficient and stable
numerical method for the MaxwellDirac system
(with X.G. Li), J. Comput. Phys., Vol.
199 (2004), pp. 663687.
For nonlinear Schrodinger
equation (semiclassical limit)

Uniform and optimal error estimates of an exponential wave integrator
sine pseudospectral method for the nonlinear
Schrodinger equation with wave operator
(with Y. Cai), SIAM J. Numer. Anal., Vol. 52
(2014), pp. 11031127.

Dimension reduction of the Schrodinger equation with Coulomb
and anisotropic confining potentials (with
H. Jian, N. J. Mauser and Y. Zhang), SIAM J. Appl.
Math., Vol. 73 (2013), pp. 21002123.

Computational methods for the dynamics of the nonlinear
Schrodinger/GrossPitaevskii equations (with
X. Antoine and C. Besse), Comput. Phys. Commun.,
Vol. 184 (2013), pp. 26212633 (An
Invited Feature Article).

Numerical methods and comparison for computing dark and bright
solitons in the nonlinear Schrodinger equation (with
Q. Tang and Z. Xu), J. Comput. Phys., Vol.
235 (2013), pp. 423445.

Uniform error estimates of finite difference methods for
the nonlinear Schrodinger equation with wave operator
(with Y. Cai), SIAM J Numer. Anal.,
Vol. 50 (2012), pp. 492521.

Numerical methods for computing ground state and dynamics of
nonlinear relativistic Hartree equation for boson stars
(with X. Dong), J. Comput. Phys.,
Vol. 230 (2011),
pp. 54495469.

Comparisons between sineGordon equation and
perturbed nonlinear Schrodinger equations for modeling
light bullets beyond critical collapse
(with X. Dong and J. Xin),
Physica D, Vol. 239 (2010), pp.
11201134.

Continuous configuration timedependent selfconsistent
field method for polyatomic quantum dynamical problems
(with D. H. Zhang, M. H. Yang and S.Y. Lee),
J. Chem. Phys.,
Vol. 122 (2005), pp. 11011104.

Numerical methods
for the nonlinear Schrodinger equation with nonzero farfield
conditions, Methods and Applications of Analysis,
Vol. 11 (2004), pp. 367388.

Effective one particle quantum dynamics of electrons:
a numerical study of the SchrodingerPoissonX_\alpha model
(with N. J. Mauser, H. P. Stimming), Comm. Math. Sci.,
Vol. 1 (2003), pp. 809828.
 An explicit unconditionally stable numerical
method
for solving damped
nonlinear Schrodinger equations with a focusing nonlinearity
(with D. Jaksch), SIAM J. Numer. Anal.,
Vol. 41 (2003), pp. 14061426.

Numerical study of timesplitting spectral discretizations of
nonlinear Schrodinger equations in the semiclassical regimes
(with S. Jin and P. A. Markowich ), SIAM J. Sci. Comput.,
Vol. 25 (2003), pp. 2764.

On timesplitting spectral approximation for the Schrodinger
equation in the semiclassical regime (with J. Shi and
P. A. Markowich), J. Comput. Phys.,
Vol. 175 (2002), pp. 487524.
For Zakharov system
(subsonic limit)
For coupled dispersive PDEs
 A uniformly
accurate multiscale time integrator pseudospectral method
for the KleinGordonZakharov system in the highplasmafrequency
limit regime (with X. Zhao), J. Comput. Phys.,
Vol. 327 (2016),
pp. 270293

An exponential wave integrator pseudospectral method
for the KleinGordonZakharov system
(with X. Dong and X. Zhao),
SIAM J. Sci. Comput., Vol. 35
(2013), pp. A2903A2927.

Singular limits of KleinGordonSchrodinger equations to
SchrodingerYukawa equations
(with X. Dong and S. Wang), Multiscale Modeling and Simulation:
a SIAM Interdisciplinary Journal,
Vol. 8 (2010),
pp. 17421769.

Efficient and accurate numerical methods for
the KleinGordonSchrodinger equations
(with L. Yang), J. Comput. Phys.,
Vol. 225 (2007),
pp. 18631893.

A timesplitting spectral method for
threewave interactions in media with
competing quadratic and cubic nonlinearities
(with C. Zheng), Commun. Comput. Phys.,
Vol. 2 (2007), pp. 123140.
Modeling and Simulation for SolidState
Deweeting Problems
Triple junction drag effects during topological changes in the
evolution of polycrystalline microstructures
(with Q. Zhao, W. Jiang and D. J. Srolovitz), Acta Mater., to appear
(arXiv: 1611.09449).
Stable equilibria of anisotropic particles on substrates: a generalized
Winterbottom construction (with W. Jiang,
D. J. Srolovitz and Y. Wang), arXiv: 1608.08481.
A parametric finite element method for solidstate dewetting problems
with anisotropic surface energies (with W. Jiang, Y. Wang and
Q. Zhao), J. Comput. Phys., Vol. 330 (2017),
pp. 380400 (arXiv: 1601.05877).
Solidstate dewetting and island morphologies in strongly anisotropic
materials (with W. Jiang, Y. Wang, Q. Zhao, D. J. Srolovitz),
Scripta Materialia, Vol. 115 (2016),
pp. 123127 (arXiv: 1510.03303).
Sharp interface
model for solidstate
dewetting problems with weakly anisotropic surface energy
(with W. Jiang, D. J.
Srolovitz, Y. Wang), Phys. Rev. B, Vol. 91
(2015), article 045303.
Phase field approach for simulating solidstate dewetting
problems
(with W. Jiang, C. V. Thompson and D. J. Srolovitz),
Acta Mater., Vol. 60 (2012), pp. 55785592.
BoseEinstein Condensation (BEC)
Review Papers
For singlecomponent BEC

Fundamental gaps of the
GrossPitaevskii equation with
repulsive interaction (with X. Ruan),
arXiv: 1512.07123.

A regularized Newton method for computing ground states of
BoseEinstein condensates (with X. Wu and Z. Wen),
arXiv: 1504.02891.
 Meanfield
regime and ThomasFermi approximations of trapped BoseEinstein
condensates with higher order interactions in one
and two dimensions (with Y. Cai and X. Ruan),
J. Phys. B: At. Mol. Opt. Phys.,
Vol. 49 (2016),
article 125304 (arXiv: 1511.00141).

Dimension reduction for anisotropic BoseEinstein condensates in the
strong interaction regime (with L. Le Treust and F. Mehats),
Nonlinearity, Vol. 28 (2015),
pp. 755772.

Subdiffusive spreading of a BoseEinstein condensate in random
potentials
(with B. Min, T. Li, and M. Rosenkranz),
Phys. Rev. A, Vol. 86 (2012), article 053612.

Breathing oscillations of a trapped impurity in a Bose
gas (with T. J. Johnson, M. Bruderer, Y. Cai,
S. R. Clark and D. Jaksch), EPL  Europhysics Letters,
Vol. 98 (2012), article 26001.

Selftrapping of BoseEinstein condensates expanding in shallow
optical lattices
(with M. Rosenkranz, D. Jaksch and F. Y. Lim),
Phys. Rev. A,
Vol. 77 (2008), article 063607.

Selftrapping of impurities in BoseEinstein condensates: Strong
attractive and repulsive coupling
(with M. Bruderer and D. Jaksch),
EPL  Europhysics Letters ,
Vol. 82 (2008), article 30004
(was chosen to be part of special online
collection of most frequently downloaded articles on
ultracold gases and related areas
published in EPL,
see details ).
 A uniformly
convergent numerical method for singularly
perturbed nonlinear eigenvalue problems (with M.H. Chai),
Commun. Comput. Phys.,
Vol. 4 (2008), pp. 135160.

Convergence rate of dimension reduction
in BoseEinstein condensates
(with Y. Ge, D. Jaksch,
P. A. Markowich and R. M. Weishaeupl),
Comput. Phys. Commun., Vol. 177 (2007),
pp. 832850.

Energy and chemical potential asymptotics
for the ground state of BoseEinstein condensates
in the semiclassical regime
(with F. Y. Lim and Y. Zhang),
Bulletin of the Institute of
Mathematics, Academia Sinica, Vol. 2 (2007),
pp. 495532.

Efficient and spectrally accurate numerical methods
for computing ground and first excited states
in BoseEinstein condensates
(with IL. Chern and F. Y. Lim),
J. Comput. Phys., Vol. 219 (2006), pp.
836854.

Dynamics of the ground state and central vortex states
in BoseEinstein condensation
(with Y. Zhang), Math. Models Meth. Appl. Sci.,
Vol. 15 (2005), pp. 18631896.

A fourthorder timesplitting LaguerreHermite
pseudospectral method
for BoseEinstein condensates
(with J. Shen), SIAM J. Sci. Comput.,
Vol. 26 (2005), pp. 20102028.

On the GrossPitaevskii equation with strongly
anisotropic confinement: formal asymptotics and
numerical experiments
(with P. A. Markowich, C. Schmeiser
and R. M. Weishaupl ), Math. Models Meth. Appl. Sci.,
Vol. 15 (2005), pp. 767782.

Computing the ground state solution of
BoseEinstein condensates by a normalized gradient flow
(with Q. Du), SIAM J. Sci. Comput.,
Vol. 25 (2004), pp. 16741697.

Three dimensional simulation of jet formation in
collapsing condensates
(with D. Jaksch and P. A. Markowich),
J. Phys. B: At. Mol. Opt. Phys.,
Vol. 37 (2004), pp. 329343.

Numerical solution of the GrossPitaevskii equation for
BoseEinstein condensation
(with D. Jaksch and P. A. Markowich), J. Comput. Phys.,
Vol. 187 (2003), pp. 318  342.

Ground state solution of
BoseEinstein condensate by directly minimizing the energy functional
(with W. J. Tang), J. Comput. Phys.,
Vol. 187 (2003), pp. 230  254.
For rotating BEC

Hubbard model for atomic impurities bound by the vortex lattice of a
rotating BEC (with T. H. Johnson, Y. Yuan,
S. R. Clark, C. Foot and D. Jaksch), Phys. Rev. Lett.,
Vol. 116 (2016),
article 240402 (arXiv: 1512.09334) and its
suplemmentary material.

A simple and efficient numerical method for computing
the dynamics of rotating BoseEinstein condensates via a
rotating Lagrangian coordinate (with
D. Marahrens, Q. Tang and Y. Zhang),
SIAM J. Sci. Comput.,
Vol. 35 (2013), pp. A2671A2695.

Optimal error estimates of finite difference methods for
the GrossPitaevskii equation with angular momentum rotation
(with Y. Cai), Math. Comp.,
Vol. 82 (2013), pp. 99128.

A generalizedLaguerreFourierHermite pseudospectral method for
computing the dynamics of rotating BoseEinstein condensates
(with H. Li and J. Shen),
SIAM J. Sci. Comput., Vol. 31 (2009), pp.
36853711.

A generalizedLaguerreHermite pseudospectral
method for computing symmetric and central vortex states
in BoseEinstein condensates
(with J. Shen),
J. Comput. Phys.,
Vol. 227 (2008), pp. 97789793.

Dynamics of the center of mass in rotating BoseEinstein
condensates (with Y. Zhang),
Appl. Numer. Math.,
Vol. 57 (2007), pp. 697709.

An efficient and spectrally accurate numerical method for
computing dynamics of rotating BoseEinstein condensates
(with H. Wang), J. Comput. Phys.,
Vol. 217 (2006), pp. 612626.

Dynamics of rotating BoseEinstein condensates and
their efficient and accurate numerical computation
(with Q. Du and Y. Zhang), SIAM
J. Appl. Math., Vol. 66 (2006), pp. 758786.

Ground, symmetric and central vortex states
in rotating BoseEinstein condensates
(with H. Wang and P. A. Markowich),
Comm. Math. Sci.,
Vol. 3 (2005), pp. 5788.
For dipolar BEC

Dimension reduction for dipolar BoseEinstein condensates in the
strong interaction regime (with L. Le Treust and F. Mehats),
Kinet. Relat. Mod., Vol.
10 (2017), pp. 553571 (arXiv: 1501.02177).

Accurate and efficient numerical methods for computing
ground states and dynamics of dipolar BoseEinstein condensates via the
nonuniform FFT (with Q. Tang and Y. Zhang),
Commun. Comput. Phys., Vol. 19 (2016),
pp. 11411166 (arXiv: 1504.02897).
 Computing
the ground state and dynamics of the nonlinear Schroedinger equation
with nonlocal interactions via the nonuniform FFT
(with S. Jiang, Q. Tang and Y. Zhang),
J. Comput. Phys., Vol. 296 (2015),
pp. 7289 (arXiv: 1410.3584).

Fast and accurate evaluation of nonlocal
Coulomb and dipoledipole interactions via the nonuniform FFT
(with L. Greengard and S. Jiang), SIAM J. Sci. Comput.,
Vol. 36 (2014),
pp. B777B794.
 Effective dipoledipole interactions in multilayered dipolar
BoseEinstein condensates
(with Y. Cai and M. Rosenkranz),
Phys. Rev. A, Vol.
88 (2013), article 013616.

GrossPitaevskiiPoisson equations for dipolar BoseEinstein
condensate with anisotropic confinement
(with N. Ben Abdallah and Y. Cai),
SIAM J. Math. Anal., Vol. 44 (2012),
pp. 17131741.

Scattering and bound states in twodimensional
anisotropic potentials
(with M. Rosenkranz), Phys. Rev. A,
Vol. 84
(2011), article 050701(R).

Meanfield regime of trapped dipolar BoseEinstein
condensates in one and two dimensions (with Y. Cai,
M. Rosenkranz and Z. Lei), Phys. Rev. A,
Vol. 82 (2010), article 043623.

Efficient numerical methods for computing ground states and
dynamics of dipolar BoseEinstein condensates
(with Y. Cai and H. Wang),
J. Comput. Phys., Vol. 229 (2010),
pp. 78747892.

Symmetry breaking and selftrapping of a dipolar BoseEinstein
condensate in a doublewell potential
(with B. Xiang, J. Gong, H. Pu and B. Li),
Phys. Rev. A,
Vol. 79 (2009), article 013626.
For spinor and multicomponent BEC

Ground states and dynamics of spinorbitcoupled BoseEinstein
condensates (with Y. Cai), SIAM J. Appl. Math.,
Vol. 75 (2015),
pp. 492517.

Efficient methods for computing ground states of spin1 BoseEinstein
condensates based on their characterizations (with
IL. Chern and Y. Zhang), J. Comput. Phys.,
Vol. 253 (2013), pp. 189208.

Ground states of twocomponent BoseEinstein condensates with
an internal atomic Josephson junction
(with Y. Cai),
East Asia Journal on Applied Mathematics, Vol. 1
(2011), pp. 4981.

Dynamical
laws of the coupled GrossPitaevskii equations
for spin1 BoseEinstein condensates
(with Y. Zhang), Methods and Applications of Analysis,
Vol. 17 (2010),
pp. 4980.

Numerical methods for computing the ground state of spin1
BoseEinstein condensates in uniform magnetic field
(with F. Y. Lim),
Phys. Rev. E,
Vol. 78 (2008), article 066704.

Computing ground states of spin1 BoseEinstein condensates
by the normalized gradient flow
(with F. Y. Lim),
SIAM J. Sci. Comput.,
Vol. 30 (2008), pp. 19251948.

A mass and magnetization conservative and energy
diminishing numerical method for computing ground state of
spin1 BoseEinstein condensates
(with H. Wang),
SIAM J. Numer. Anal., Vol. 45 (2007),
pp. 21772200.

Dynamics of rotating twocomponent BoseEinstein condensates
and its efficient computation
(with H. Li and Y. Zhang),
Physica D, Vol. 234 (2007),
pp. 4969.

Ground states and dynamics of multicomponent
BoseEinstein condensates
, Multiscale Modeling and Simulation: a SIAM
Interdisciplinary Journal,
Vol. 2 (2004), pp. 210236.
For polariton condensate
Quantized Vortex in Superfluidity and
Superconductivity
The kinematic effects of the defects
in liquid crystal dynamics (with X. Zhao),
Commun. Comput. Phys.,
Vol. 20 (2016),
pp. 234249.
Numerical study of quantized vortex interaction
in nonlinear Schroedinger equation on bounded domains (with
Q. Tang), Multiscale Modeling and Simulation:
a SIAM Interdisciplinary Journal, Vol. 12
(2014), pp. 411439.
Numerical study of quantized vortex interaction
in the GinzburgLandau equation on bounded domains
(with Q. Tang), Commun. Comput. Phys., Vol.
14 (2013), pp. 819850.
Quantized vortex stability and interaction
in the nonlinear wave equation
(with R. Zeng and Y. Zhang),
Physica D, Vol. 237 (2008),
pp. 23912410.
The dynamics and interaction of quantized vortices in
GinzburgLandauSchrodinger equations
(with Q. Du and Y. Zhang),
SIAM J. Appl. Math., Vol. 67 (2007),
pp. 17401775.
Dynamics of vortices in weakly interacting BoseEinstein condensates
(with A. Klein, D. Jaksch and Y. Zhang),
Phys. Rev. A, Vol. 76 (2007), article 043602.
Numerical simulation of vortex dynamics in
GinzburgLandauSchrodinger equation
(with Q. Du and Y. Zhang),
Eur. J. Appl. Math.,
Vol. 18 (2007),
pp. 607630.
Artificial Boundary Conditions and Error
Estimates
For incompressible flow

Artificial boundary conditions for incompressible NavierStokes
equations: A
wellposed result, Comput. Methods Appl. Mech. Engrg.,
Vol. 188 (2000), pp. 595611.

The artificial boundary conditions for computing the flow around a
submerged
body (with. X. Wen ),
Comput. Methods Appl. Mech. Engrg., Vol. 188
(2000), pp. 473482.
 The approximations of the exact boundary
condition at an artificial boundary for linearized incompressible viscous
flow , J. Comput. Math., Vol.
16 (1998),
pp. 239256.

Artificial boundary conditions for twodimensional incompressible
viscous flows around an obstacle,
Comput. Methods Appl. Mech. Engrg., Vol.
147
(1997), pp. 263273.
 Local artificial boundary conditions for
the incompressible viscous flow in a slip channel
(with H. Han), J. Comput. Math., Vol.
15 (1997),
pp. 335344.

Nonlocal artificial boundary conditions for the incompressible
viscous flow in a channel using spectral techniques
(with H. Han), J. Comput. Phys., Vol. 126
(1996), pp. 5263.

An artificial boundary condition for the incompressible viscous
flows using the method of lines (with H. Han),
Int. J. Numer. Methods Fluids,
Vol. 22 (1996), pp. 483493.

An artificial boundary condition for the incompressible viscous
flows in a
noslip channel (with H. Han), J. Comput. Math.,
Vol. 13 (1995), pp. 5165.

A discrete artificial boundary condition for steady incompressible
viscous flows in a noslip channel using a fast iterative method
(with H. Han and J. Lu), J. Comput. Phys., Vol.
114 (1994), pp. 201208.
For linear elasticity and
incompressile materials

Error bounds for the finite element approximation of
the exterior Stokes equations in two dimensions
, IMA J. Numer. Anal.,
Vol. 23 (2003), pp. 125148.

Error bounds for the finite element approximation of an
incompressible material in an unbounded domain (with H. Han),
Numer. Math., Vol. 93 (2003), pp. 415444.
 Error estimates for the
finite element approximation of linear elastic
equations in an unbounded domain (with H. Han),
Math. Comput.,
Vol. 70 (2001), pp. 14371459.

The direct method of lines for the problem of infinite elastic
foundation (with H. Han), Comput. Methods
Appl. Mech. Engrg., Vol. 175 (1999),
pp. 157173.

The artificial boundary conditions for incompressible materials on
an unbounded domain (with H. Han), Numer. Math.,
Vol. 77 (1997), pp. 347363.

Numerical simulation for the problem of infinite elastic
foundation (with H. Han and T. Wang),
Comput. Methods Appl. Mech. Engrg., Vol. 147
(1997), pp. 369385.
For elliptic problems

Numerical simulations of fracture problems by coupling the
FEM and the direct
method of lines (with H. Han and Z. Huang),
Comput. Methods
Appl. Mech. Engrg., Vol. 190 (2001),
pp. 48314846.

Highorder local artificial boundary conditions for problems in
unbounded domains (with H. Han),
Comput. Methods Appl. Mech. Engrg., Vol. 188
(2000), pp. 455471.

Error estimates for the finite element approximation of problems
in unbounded
domains (with H. Han),
SIAM J. Numer. Anal., Vol. 37 (2000),
pp. 11011119.

The discrete artificial boundary condition on a polygonal artificial
boundary for the exterior problem of Poisson equation by using the
direct method
of lines (with H. Han),
Comput. Methods Appl. Mech. Engrg., Vol. 179
(1999), pp. 345360.
Computational Fluid Dynamics
Highorder Istable central difference schemes for
viscous compressible flows (with J. Shi),
J. Comput. Math., Vol.
21 (2003), pp. 101112.
Error estimates on the random projection methods for
hyperbolic conservation laws with stiff reaction terms
(with S. Jin), Appl. Numer. Math.,
Vol. 43 (2002), pp. 315333.
An economical finite element approximation of a generalized
Newtonian flow
, Comput. Methods Appl. Mech. Engrg.,
Vol. 191 (2002), pp. 36373648.
The random projection method for stiff multispecies detonation
capturing (with S. Jin), J. Comput. Phys.,
Vol. 178 (2002), pp. 3757.
The random projection method for stiff detonation capturing
(with S. Jin), SIAM J. Sci. Comput.,
Vol. 23 (2001), pp. 10001026.
On infsup conditions of mixed finite element
formulations for acoustic
fluids (with X. Wang and K. J. Bathe), Math. Mod.
Meth. Appl. Sci., Vol. 11 (2001),
pp. 883901.
Weakly compressible highorder Istable central difference schemes
for incompressible viscous flows (with S. Jin),
Comput. Methods Appl.
Mech. Engrg., Vol. 190 (2001),
pp. 50095026.
The random projection method
for hyperbolic systems with stiff reaction terms
(with S. Jin), J. Comput. Phys., Vol.
163 (2000), pp. 216248.
A priori and posteriori error bounds for
nonconforming linear finite element
approximation of a nonNewtonian flow (with J. W. Barrett),
M^2AN Math. Model. Numer. Anal., Vol. 32
(1998), pp. 843858.
Miscelansous
Singlepolymer `flyfishing' effect for nanoscale motors
and machines: an exact wormlikechain model study
(with N. Wang, R. Hou and Z. Wang),
arXiv: .
A variationaldifference numerical method for designing
progressiveaddition lenses (with
W. Jiang, Q. Tang and H. Wang),
ComputerAided Design, Vol. 48
(2014), pp. 1727.
Approximation and comparison for motion by mean
curvature with intersection points ,
Computers & Math. Appl. , Vol.
46 (2003), pp. 12111228.