Markus Melenk's publications
Publications
Technical reports submitted for publication
(see also
Preprints of the Institute for Analysis and Scientific Computing )
 [1]
Hmatrix approximability of inverses of discretizations of
the fractional Laplacian

M. Karkulik and J.M. Melenk
or at
arXiv:1808.04274[math.NA]
 [1]
Tensor FEM for spectral fractional diffusion

L. Banjai, J.M. Melenk, R.H. Nochetto, E. Otarola, A. Salgado, C. Schwab
ASC Report 13/2017
or at
arXiv:1707.07367[math.NA]
 [2]
On commuting pversion projectionbased interpolation on tetrahedra

J.M. Melenk, C. Rojik
extended version available at
ASC Report 05/2018
or at
arXiv:1802.00197[math.NA]
 [3]
Wavenumberexplicit hpFEM analysis for Maxwell's equations with transparent boundary conditions

J.M. Melenk, S.A. Sauter
ASC Report 09/2018
or at
arXiv:1803.01619[math.NA]
Book
hp finite element methods for singular perturbations
Springer LNM 1796 (2002)
© SpringerVerlag , 318pp.
Errata
Edited Book
Direct and Inverse Problems in Wave Propagation and Applications
(I.G. Graham, U. Langer, J.M. Melenk, M. Sini editors)
Radon Series on Computational and Applied Mathematics , Vol. 14
DeGruyter (2013), ISBN: 9783110282283
Book Chapters

Interpolation and quasiinterpolation in h and hpversion finite element spaces
T. Apel, J.M. Melenk
in: Encyclopedia of Computational Mechanics, second edition, E. Stein, R. de Borst, T. Hughes (eds.), 2018, ISBN: 9781119003793, pp. 133
extended preprint version available as
ASC Report 39/2015

hpVersion of Finite Element Method
in: Encyclopedia of Applied and Computational Mathematics, B. Engquist (ed.), 2016, pp. 656659, ISBN 9783540705284 (Print) 9783540705291 (Online)

On stability of discretizations of the Helmholtz equation
S. Esterhazy, J.M. Melenk
in:
Numerical Analysis of Multiscale Problems,
I.G. Graham, T.Y. Hou, O. Lakkis, R. Scheichl (Eds.),
Springer LNCSE 83 (2012),
pp. 285324
extended version available as
ASC Report 01/2011 or at
arXiv:1105.2112v2 [math.NA]

On Approximation in Meshless Methods
in: Frontiers in Numerical Analysis, Durham 2004, J. Blowey, A. Craig (eds.),
Springer Verlag 2005, pp. 65141
available as
preprint 11/2004, department of mathematics, University of Reading
( alternative link )
(Note: usually the copyright of the papers below is with the publisherthe links
using the DOI point to the published versions. Links to the earlier
technical reports may differ from the final published versions.)
Refereed articles
 [75]
Local convergence of the BEM on polyhedral domains

M. Faustmann, J.M. Melenk
Numer. Math.
available as
ASC Report 03/2017
or at
arXiv:1702.04224[math.NA]
 [74]
On thin plate spline interpolation

M. Löhndorf, J.M. Melenk
ICOSAHOM 2016, Springer LNCSE 119, M. Bittencourt, N. Dumont, J. Hesthaven (eds), pp. 451466
available as
ASC Report 09/2017
or at
arXiv:1705.05178[math.NA]
 [73]
An analysis of a butterfly algorithm

S. Börm, C. Börst, J.M. Melenk
Comp. Math. Appl. 74 (2017), p. 21252143
available at arXiv:1703.01941
or as ASC report 12/2017
 [72]
Robust exponential convergence of hpFEM in balanced norms for singularly perturbed reactiondiffusion problems:
corner domains

M. Faustmann, J.M. Melenk
Comp. Math. Appl. 74 (2017) pp. 15761589
ASC Report 25/2016
or at
arXiv:1610.09211[math.NA]
 [71]
Approximation of the highfrequency Helmholtz kernel by nested directional interpolation

S. Börm, J.M. Melenk
Numer. Math.
137 (2017), pp. 134
ASC Report 33/2015
or at arXiv:1510.07189[math.NA]
 [70]
RungeKutta convolution quadrature and FEMBEM coupling for the timedependent linear Schrödinger equation

J.M. Melenk, A. Rieder
J. Integral equations and applications 29 (2017), pp. 189250
ASC Report 13/2016
or at arXiv:1605.07340[math.NA]
 [69]
Simultaneous quasioptimal convergence rates in FEMBEM coupling

J.M. Melenk, D. Praetorius, B. Wohlmuth
Math. Methods. Appl. Sci. 40 (2017), pp. 463485
also available as
ASC Report 13/2014
or at arXiv:1404.2744 [math.NA]
 [68]
An hpAdaptive NewtonGalerkin Finite Element Procedure for Semilinear Boundary Value Problems

M. Amrein, J.M. Melenk, T. Wihler
Math. Methods Appl. Sci. 40 (2017), pp 19731985
available at
ASC Report 06/2016 or at
arXiv:1602.05354[math.NA]
 [67]
Local inverse estimates for nonlocal boundary integral operators

M. Aurada, M. Feischl, T. Führer, M. Karkulik, J.M. Melenk,
D. Praetorius
Math. Comp.
86 (2017), pp. 26512686
available as
ASC Report 12/2015
or at arXiv:1504.04394 [math.NA]
 [66]
Existence of Hmatrix approximants to the inverse of BEM matrices: the hypersingular integral operator

M. Faustmann, J.M. Melenk, D. Praetorius
IMA J. Numer. Anal. 37 (2017) pp. 12111244
available as
ASC Report 08/2015
or at arXiv:1503.01943[math.NA]
 [65]
Existence of Hmatrix approximants to the inverses of BEM matrices: the simplelayer operator

M. Faustmann, J.M. Melenk, D. Praetorius
Math. Comp. 85 (2016), pp 119152
also available as
ASC Report 37/2013
and arXiv:1311.5028 [math.NA]
 [64]
Robust exponential convergence of hpFEM in balanced norms for singularly perturbed
reactiondiffusion equations

J.M. Melenk, C. Xenophontos
Calcolo 53 (2016), pp. 105132
available as
ASC Report 24/2014
or at arXiv:1408.3328[math.NA]
 [63]
Local highorder regularization and applications to hpmethods

M. Karkulik, J.M. Melenk
Comp. Math. Appl. 70 (2015), pp. 16061639
preprint version available as
ASC Report 38/2014
or at arXiv:1411.5209[math.NA]
 [62]
Optimal additive Schwarz methods for the hpBEM: the hypersingular integral operator in 3D on locally refined meshes

T. Führer, J.M. Melenk, D. Praetorius, A. Rieder
Comp. Math. Appl. 70 (2015), pp. 15831605
available as
ASC Report 41/2014
or at arXiv:1412.2024[math.NA]
 [61]
On optimal L^2 and surface flux convergence in FEM

T. Horger, J.M. Melenk, B. Wohlmuth
CVS 16 (2013), pp. 231246
ASC Report 02/2015
or at arXiv:1501.03003 [math.NA]
 [60]
When is the error in the hBEM for solving the
Helmholtz equation bounded independently of k?

I.G. Graham, M. Löhndorf, J.M. Melenk, E.A. Spence
BIT 55 (2015), pp. 171214
ASC Report 28/2013
 [59]
Hmatrix approximability of the inverse of FEM matrices

M. Faustmann, J.M. Melenk, D. Praetorius
Numer. Math.
131 (2015), pp. 615642
available as
ASC Report 20/2013
and arXiv:1308.0499 [math.NA]
 [58]
FEMBEM coupling for the largebody limit in micromagnetics
 M. Aurada, J.M. Melenk, D. Praetorius
J. Comp. Math. 281 (2015), pp. 1031
also available as
ASC Report 4/2013
 [57]
Quasioptimal convergence rates for adaptive BEMs with data approximation.
Part II: hypersingular integral equation
 M. Feischl, T. Führer, M. Karkulik, J.M. Melenk, D. Praetorius
Electron. Trans. Numer. Anal 44 (2015), pp. 153176
also available as
ASC Report 30/2013
 [56]
A scalable numerical approach for the SteadyState AbInitio Laser Theory

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K.G. Makris, A.D. Stone, J.M. Melenk, S.G. Johnson, S. Rotter
Phys. Rev. A 90 023816 (2014)
also available as
ASC Report 40/2013
and arXiv:1312.2488 [physics.optics]
 [55]
On the stability of the boundary trace of the polynomial L^2 projection on triangles and tetrahedra

J.M. Melenk, T. Wurzer
Comp. Math. Appl. 67 (2014), pp. 944965
also available as extended versions (with slightly different title) at
ASC Report 25/2012
or arXiv:1302.7189 [math.NA]
 [54]
An analysis of discretizations of the Helmholtz equation in L^2 and in negative norms

S. Esterhazy, J.M. Melenk
Comp. Math. Appl. 67 (2014), pp. 830853
extended version available as
ASC Report 31/2012
or as corrected version
 [53]
Quasioptimal convergence rates for adaptive BEMs with data approximation.
Part I: weakly singular integral equations
 M. Feischl, T. Führer, M. Karkulik, J.M. Melenk, D. Praetorius
Calcolo
51 (2014), pp. 531562
available as
ASC Report 24/2013
 [52]
mixed conforming elements for the largebody limit in micromagnetics
 M. Aurada, J.M. Melenk, D. Praetorius
M3AS 24 (2014), pp. 113144
available (with a different title) as
ASC Report 42/2011
 [51]
Quasioptimal a priori estimates for fluxes in
mixed finite element methods and applications
to the StokesDarcy coupling
 J.M. Melenk, H. Rezaijafari, B. Wohlmuth
IMA J. Numer. Anal. 34 (2014), pp. 127
available as
ASC Report 05/2012
 [50]
General DGMethods for Highly Indefinite Helmholtz Problems
 J.M. Melenk, A. Parsania, S. Sauter
J. Sci. Comp. 57 (2013), pp. 536581
available as
corrected version
of
ASC Report 06/2012 (with slightly different title)
 [49]
Symmetryfree, probust equilibrated error indication for the hpversion of the FEM
in almost incompressible linear elasticity

P. Dörsek, J.M. Melenk
CMAM 13 (2013), pp 291304
available as
ASC Report 46/2012
 [48]
A numerical study of averaging error indicators in pFEM

P. Dörsek, J.M. Melenk
in:
Spectral and High Order Methods for Partial Differential Equations
 ICOSAHOM 2012,
Azaiez, M., El Fekih, H., Hesthaven, J. (eds),
LCSE 95, Springer, 2014, pp. 227236
available as
ASC Report 45/2012
 [47]
Quasioptimal convergence rate for an adaptive boundary element method
 M. Feischl, M. Karkulik, J.M. Melenk, D. Praetorius
SIAM J. Numer. Anal. 51 (2013), pp. 13271348
available as
ASC Report 28/2011
 [46]
A high frequency hpboundary element method for scattering by convex polygons
 D.P. Hewett, S. Langdon, J.M. Melenk
SIAM J. Numer. Anal. 51 (2013), pp. 629653
available as
ASC Report 40/2011
 [45]
Analytic regularity for a singularly perturbed system of reactiondiffusion equations with multiple scales: a roadmap
 J.M. Melenk, C. Xenophontos, L. Oberbroeckling
Adv. Comp. Math.
39 (2013), pp. 367394
available as
extended version (with slightly different title): as
ASC Report 29/2011 and at
arXiv:1108.2002 [math.NA]
 [44]
robust exponential convergence of hpFEM for singularly perturbed
reactiondiffusion systems with multiple scales
 J.M. Melenk, C. Xenophontos, L. Oberbroeckling
IMA J. Numer. Anal. 33 (2013), pp. 609628
available as revised version of
ASC Report 31/2011
 [43]
Classical FEMBEM coupling methods:
nonlinearities, wellposedness, and adaptivity

M. Aurada, M. Feischl, T. Führer, M. Karkulik, J.M. Melenk,
D. Praetorius
Comp. Mech. 51 (2013), pp. 399419
available as
ASC Report 08/2012
or at arXiv:1211.4225 [math.NA]
 [42]
Quasioptimal approximation of surface based Lagrange multipliers in finite element methods
 J.M. Melenk, B. Wohlmuth
SIAM J. Numer. Anal. 50 (2012), pp. 20642087
available as
ASC Report 13/2011 and as
corrected version
 [41]
mapping properties of combined field Helmholtz boundary integral operators

SIAM J. Math. Anal. 44 (2012), pp. 25992636
corrected version of
ASC Report 01/2010
 [40]
Numerical quadratic energy minimization bound to convex constraints
in thinfilm micromagnetics
 S. FerrazLeite, J.M. Melenk, D. Praetorius
Numer. Math. 122 (2012), pp. 101131
available as
ASC Report 32/2011
 [39]
Wavenumberexplicit hpBEM for high frequency scattering
 M. Löhndorf, J.M. Melenk
SIAM J. Numer. Anal. 49 (2011), pp. 23402363
available as
corrected version of
ASC Report 02/2010
 [38]
Wavenumber explicit convergence analysis for Galerkin
discretizations of the Helmholtz equation

J.M. Melenk, S. Sauter
SIAM J. Numer. Anal. 49 (2011), pp. 12101243
extended version available as
corrected version of
ASC Report 31/2009
 [37]
RungeKutta convolution quadrature for
operators arising in wave propagation

L. Banjai, C. Lubich, J.M. Melenk
Numer. Math., 119 (2011), pp. 120
available as
ASC Report 24/2010
 [36]
Adaptive hpFEM for the contact problem with Tresca friction in linear elasticity:
the primaldual formulation and a posteriori error estimation

P. Dörsek, J.M. Melenk
Appl. Numer. Math.
60 (2010), pp. 689704
extended version available as
ASC Report 37/2009
 [35]
hpFEM for the contact problem with Tresca friction in linear elasticity:
the primal formulation

P. Dörsek, J.M. Melenk
in:
Spectral and high order methods for PDEs,
J. Hesthaven, E. Rønquist eds,
pp. 117, LNCSE 76, Springer Verlag (2011)
extended version available as
ASC Report 36/2009
 [34]
Convergence Analysis for Finite Element Discretizations of the Helmholtz equation
with DirichlettoNeumann boundary conditions

J.M. Melenk, S. Sauter
Math. Comp.
79 (2010), pp. 18711914
available as
revised version of
ASC Report 15/2008
 [33]
Optimal Convergence of Higher Order Finite Element Methods for Elliptic Interface Problems

Jingzhi Li, J.M. Melenk, Barbara Wohlmuth, Jun Zou
Appl. Numer. Math. , 60 (2010), pp. 1937
available as
revised version of
ASC Report 13/2008
 [32]
On the suboptimality of the pversion interior penalty discontinuous Galerkin method

E. Georgoulis, E. Hall, J.M. Melenk
J. Sci. Comp.
42 (2010), pp. 5467
available as ASC Report 03/2009
 [31]
On the convergence of Filon quadrature

J. Comput. Appl. Math. , 234 (2010), pp. 16921701

extended version available as
ASC Report 08/2008

 [30]
pFEM quadrature error analysis on tetrahedra
 T. Eibner, J.M. Melenk,
in: Proceedings of DD17, Langer, Discacciati, Keyes, Widlund, Zulehner (eds.),
Springer LNSCE vol. 60 (2008), pp. 493500 (ISSN 14397358)

available as
ASC Report 23/2007
 [29]
Multilevel preconditioning for the boundary concentrated hpFEM
 T. Eibner, J.M. Melenk
Comp. Meth. Mech. Eng.
196 (2007), pp. 37133725

available as
Preprint 0513 of the SFB 393
 [28]
Additive Schwarz preconditioning for pversion triangular and tetrahedral finite elements
 J. Schöberl, J.M. Melenk, C. Pechstein, S. Zaglmayr,

IMA J. Numer. Anal. 28 (2008), pp. 124
 this article was among the
top 5 downloads in 2008

available as
Preprint 11/2005 of the Radon institute
 [27]
Fast algorithms for setting up the stiffness matrix in hpFEM: a comparison
 T. Eibner, J.M. Melenk

preprint version
in: Computer Mathematics and its ApplicationsAdvances and Developments (19942005),
Elias A. Lipitakis (Ed.), LEA Publ., Athens, Greece, pp. 575596, ISBN: 9608727596
 [26]
An adaptive strategy for hpFEM based on testing for analyticity
 T. Eibner and J.M. Melenk

Comp. Mech.
39 (2007), pp. 575595
preprint version
 [25]
a local error analysis of the boundary concentrated FEM
 T. Eibner and J.M. Melenk

IMA J. Numer. Anal.
26 (2006), pp.752778

preprint version
 [24]
hpinterpolation of nonsmooth functions

SIAM J. Numer. Anal.
43 (2005), pp. 127155
extended version is available as
Preprint NI03050 of the
Isaac Newton Institute for Mathematical Sciences
 [23]
Approximation of integral operators by variableorder interpolation
 S. Börm, M. Löhndorf, J.M. Melenk

Numer. Math.
99 (2005), pp. 605643

preprint version
 [22]
mortar methods with curved interfaces

B. Flemisch, J.M. Melenk, B. Wohlmuth,
Appl. Numer. Math. 54 (2005), pp. 339361
preprint version

[21]
Twoscale regularity for homogenization problems with nonsmooth fine scale geometry
 A.M. Matache & J.M. Melenk
M3AS, 13 (2003), pp. 10531080

available as
Preprint 63
of the DFG priority program
"multiscale problems"

[20] Boundary concentrated FEM
 B.I. Khoromskij & J.M. Melenk,
SIAM J. Numer. Anal. 41 (2003), pp. 136

preprint version

[19]
An efficient direct solver for the boundary concentrated FEM in 2D
 B.I. Khoromskij & J.M. Melenk,
Computing
69 (2002), pp. 91117

preprint version

[18]
On condition numbers in hpFEM with GaußLobatto based shape functions

J. Comput. Appl. Math. , 139 (2002), pp. 2148

preprint version (220K)

[17]
On residualbased aposteriori error estimation in hp FEM

J.M. Melenk & B. Wohlmuth,
Advances in Comput. Math. , 15 (2001),
pp. 311331

preprint version (275K)

[16]
Approximation orders for natural splines in arbitrary dimensions

J.M. Melenk & T. Gutzmer,
Math. Comput.
70 (2001), pp. 699703

preprint version (73K)

[15]
Fully discrete hp finite elements: fast quadrature

J.M. Melenk, K. Gerdes, & C. Schwab,
Comput. Meths. Appl. Mech. Engrg. 190 (2001), pp. 43394364

preprint version

[14]
The hpversion of the Streamline Diffusion Finite Element Method in two dimensions,
 K. Gerdes, J.M. Melenk, D. Schötzau, & C. Schwab,
M3AS 11 (2001), pp. 301337

abstract and paper

[13]
Spectral Galerkin discretizations for hydrodynamic stability problems
 J.M. Melenk, N.P. Kirchner, & C. Schwab,
Computing
65 (2000), p. 97118

abstract and paper

[12]
On nwidths for elliptic problems

J. Math. Anal. Appl. 247 (2000), p. 272289

preprint version (128K)

[11]
The hpStreamline Diffusion Method for Convection Dominated Problems
in One Space Dimension
 J.M. Melenk & C. Schwab,
EastWest J. Numer. Math.
7 (1999), 3160

abstract and paper

[10]
Operator adapted spectral element methods. I: harmonic and generalized harmonic polynomials

Numer. Math. 84 (1999), p. 3569

preprint version

[9]
An hp FEM for convectiondiffusion problems in
one dimension

J.M. Melenk & C. Schwab,
IMA J. Numer. Anal. 19 (1999), 425453

abstract and paper

[8]
Analytic regularity for a singularly perturbed problem

J.M. Melenk & C. Schwab,
SIAM J. Math. Anal., 30 (1999), 379400

preprint version

[7]
hpFEM for ReactionDiffusion Equations. I: Robust Exponential Convergence

J.M. Melenk & C. Schwab,
SIAM J. Numer. Anal., 35 (1998), 15201557

preprint version

[6]
On the Robust Exponential Convergence of hpFEMs for Problems with Boundary Layers

IMA J. Numer. Anal., 17 (1997), 577601

abstract and paper

[5]
Approximation with harmonic and generalized harmonic polynomials in the partition of unity method
 J.M. Melenk & I. Babuska
Comput. Assist. Mech. Eng. Sci.
4 (1997), 607632

Load Postscript (162K)

[4]
The Partition of Unity Method
 I. Babuska & J.M. Melenk,
Int. J. Numer. Meths. Eng., 40 (1997), 727758
 preprint version (158K)

[3]
The Partition of Unity Finite Element Method: Basic Theory and Applications

J.M. Melenk & I. Babuska,
Comput. Meths. Appl. Mech. Engrg.,139 (1996), 289314

abstract and paper

[2]
Finite Element Method for Solving Problems with Singular Solutions

I. Babuska, B. Andersson, B. Guo, J.M. Melenk, & H.S. Oh,
J. Comput. Appl. Math. , 74 (1996),5170

preprint version (85K)

[1]
Functions with time and frequencygaps

J.M. Melenk & G. Zimmermann,
J. Fourier Anal. Appl. , 2 (6), 1996

preprint version (44K)
Theses

[1]
hpFinite Element Methods for Singular Perturbations
 Habilitation Thesis 2000
 Load Postscript (1050K)

[2]
On Generalized Finite Element Methods
 PhD Thesis 1995
 Load Postscript (608K)

[3] Finite element methods with harmonic shape functions for solving Laplace's equation
 MA Thesis 1992
 Load Postscript (220K)
Selected Proceedings Contributions
 [8]
hpFEM and hpDGFEM for Helmholtz problems

S. Esterhazy, J.M. Melenk, A. Parsania, S. Sauter
Oberwolfach Report , 2013/03
 [7]
Hmatrix approximability of the inverses of first kind BEM matrices

M. Faustmann, J.M. Melenk, and D. Praetorius
Oberwolfach Report , 2012/55
 [6]
A new proof for existence of Hmatrix approximants to the inverse
of FEM matrices: the Dirichlet problem for the Laplacian.

M. Faustmann, J.M. Melenk, and D. Praetorius
in:
Spectral and High Order Methods for Partial Differential Equations
 ICOSAHOM 2012,
Azaiez, M., El Fekih, H., Hesthaven, J. (eds),
LCSE 95, Springer, 2014, pp. 249259
ASC Report 51/2012
 [5]
novel inverse estimates for nonlocal operators
 M. Feischl, Tührer, M. Karkulik, J. Melenk, D. Praetorius:
extended abstract for IABEM 2012
ASC Report 49/2012
 [4]
FEMBEM Couplings without Stabilization
 M. Feischl, T. Führer, M. Karkulik, J. Melenk, D. Praetorius:
extended abstract for IABEM 2012
ASC Report 47/2012
 [3]
Mixed Conforming Elements for the LargeBody Limit in Micromagnetics

M. Aurada, J.M. Melenk, D. Praetorius
proceedings of MathMod 2009
ASC Report 49/2009
 [2]
Reduced Model in ThinFilm Micromagnetics

S. FerrazLeite, J.M. Melenk, D. Praetorius
proceedings of MathMod 2009
ASC Report 02/2009
 [1]
Residual aposteriori error estimates in BEM: Convergence of
hadaptive algorithms
 M. Feischl, M. Karkulik, J. Melenk, D. Praetorius:
extended abstract for IABEM 2009
Unpublished Technical Reports and Selected BA and MA Theses
 [1]
mapping properties of Helmholtz boundary integral operators and their application
to the hpBEM

M. Löhndorf and J.M. Melenk
ASC Report 34/2009
 [2]
stability of the trace of the polynomial L2 projection on triangles

Bachelor thesis of
T. Wurzer
ASC Report 36/2010
an extended version that includes the 3D case is at
arXiv:1302.7189 [math.NA]
 [3]
Analytic regularity for a singularly perturbed system of reactiondiffusion equations with multiple scales: proofs
 J.M. Melenk, C. Xenophontos, L. Oberbroeckling
ASC Report 29/2011 and at
arXiv:1108.2002 [math.NA]
 [4]
Inverse estimates for elliptic integral
operators and their application to the adaptive
coupling of FEM and BEM

M. Aurada, M. Feischl, T. Führer, M. Karkulik, J.M. Melenk,
D. Praetorius
ASC Report 07/2012
or at arXiv:1211.4360 [math.NA]
 [5]
A Posteriori Error Analysis of hpFEM for singularly perturbed problems
reactiondiffusion equations

J.M. Melenk, T. Wihler
ASC Report 25/2014
or at arXiv:1408.6037[math.NA]
Class Notes
[1]
A first course in Numerical Analysis (250pp, in German)
please ask for a copy
[2]
FEM (100pp, in German)
please ask for a copy
[3]
Numerical methods for ODEs (100pp, in German)
please ask for a copy
Last modified: March, 2013