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## Publications of David G. Schaeffer    :chronological  combined listing:

%% Books
@book{fds10093,
Author = {Golubitsky, Martin and Schaeffer, David G.},
Title = {Singularities and groups in bifurcation theory. Vol.
I},
Journal = {pp. xvii+463, 1985, Springer-Verlag, New
York},
MRNUMBER = {86e:58014},
url = {http://www.ams.org/mathscinet-getitem?mr=86e:58014},
Key = {fds10093}
}

@book{fds10084,
Author = {Golubitsky, Martin and Stewart, Ian and Schaeffer, David
G.},
Title = {Singularities and groups in bifurcation theory. Vol.
II},
Journal = {pp. xvi+533, 1988, Springer-Verlag, New York},
MRNUMBER = {89m:58038},
url = {http://www.ams.org/mathscinet-getitem?mr=89m:58038},
Key = {fds10084}
}

@book{fds10078,
Title = {Two phase flows and waves},
Journal = {edited by Joseph, Daniel D. and Schaeffer, David G., pp.
xii+164, 1990, Springer-Verlag, New York},
MRNUMBER = {91e:76008},
url = {http://www.ams.org/mathscinet-getitem?mr=91e:76008},
Key = {fds10078}
}

%% Papers Published
@article{fds10259,
Author = {Coburn, L. A. and Douglas, R. G. and Schaeffer, D. G. and Singer, I. M.},
Title = {$C\sp{\ast}$-algebras of operators on a half-space. II.
Index theory},
Journal = {Inst. Hautes \'Etudes Sci. Publ. Math., no. 40, pp. 69--79,
1971},
MRNUMBER = {50:10884},
url = {http://www.ams.org/mathscinet-getitem?mr=50:10884},
Key = {fds10259}
}

@article{fds10068,
Author = {Shearer, Michael and Schaeffer, David G.},
Title = {A class of fully nonlinear $2\times 2$ systems of partial
differential equations},
Journal = {Comm. Partial Differential Equations, vol. 20, no. 7-8, pp.
1105--1131, 1995},
MRNUMBER = {96b:35133},
url = {http://www.ams.org/mathscinet-getitem?mr=96b:35133},
Key = {fds10068}
}

@article{fds10419,
Author = {Witelski, Thomas P. and Schaeffer, David G. and Shearer,
Michael},
Title = {A discrete model for an ill-posed nonlinear parabolic
PDE},
Journal = {Phys. D, vol. 160, no. 3-4, pp. 189--221,
2001},
MRNUMBER = {1872040},
url = {http://www.ams.org/mathscinet-getitem?mr=1872040},
Key = {fds10419}
}

@article{fds10097,
Author = {Golubitsky, Martin and Schaeffer, David},
Title = {A discussion of symmetry and symmetry breaking},
Journal = {Singularities, Part 1 (Arcata, Calif., 1981), pp. 499--515,
1983, Amer. Math. Soc., Providence, RI},
MRNUMBER = {85b:58018},
url = {http://www.ams.org/mathscinet-getitem?mr=85b:58018},
Key = {fds10097}
}

@article{fds10072,
Author = {Schaeffer, David G.},
Title = {A mathematical model for localization in granular
flow},
Journal = {Proc. Roy. Soc. London Ser. A, vol. 436, no. 1897, pp.
217--250, 1992},
MRNUMBER = {93g:73061},
url = {http://www.ams.org/mathscinet-getitem?mr=93g:73061},
Key = {fds10072}
}

@article{fds10254,
Author = {Schaeffer, David G.},
Title = {A new proof of the infinite differentiability of the free
boundary in the Stefan problem},
Journal = {J. Differential Equations, vol. 20, no. 1, pp. 266--269,
1976},
MRNUMBER = {52:11325},
url = {http://www.ams.org/mathscinet-getitem?mr=52:11325},
Key = {fds10254}
}

@article{fds10267,
Author = {Schaeffer, David G.},
Title = {A note on the representation of a solution of an elliptic
differential equation near an isolated singularity},
Journal = {Proc. amer. Math. Soc., vol. 23, pp. 450--454,
1969},
MRNUMBER = {39:7262},
url = {http://www.ams.org/mathscinet-getitem?mr=39:7262},
Key = {fds10267}
}

@article{fds10105,
Author = {Golubitsky, Martin and Schaeffer, David},
Title = {A qualitative approach to steady-state bifurcation
theory},
Journal = {New approaches to nonlinear problems in dynamics (Proc.
Conf., Pacific Grove, Calif., 1979), pp. 43--51, 1980, SIAM,
MRNUMBER = {81k:58026},
url = {http://www.ams.org/mathscinet-getitem?mr=81k:58026},
Key = {fds10105}
}

@article{fds10261,
Author = {Schaeffer, David G.},
Title = {A regularity theorem for conservation laws},
Journal = {Advances in Math., vol. 11, pp. 368--386,
1973},
MRNUMBER = {48:4523},
url = {http://www.ams.org/mathscinet-getitem?mr=48:4523},
Key = {fds10261}
}

@article{fds8985,
Author = {David G Schaeffer and M. Shearer},
Title = {A Simple Model for Stress Fluctuations in Plasticity, with
Application to Granular Materials},
Journal = {SIAM J. Appl. Math. 58(1998), 1791-1807.},
Key = {fds8985}
}

@article{fds10103,
Author = {Golubitsky, Martin and Keyfitz, Barbara Lee and Schaeffer,
David G.},
Title = {A singularity theory analysis of a thermal-chainbranching
model for the explosion peninsula},
Journal = {Comm. Pure Appl. Math., vol. 34, no. 4, pp. 433--463,
1981},
MRNUMBER = {82h:58010},
url = {http://www.ams.org/mathscinet-getitem?mr=82h:58010},
Key = {fds10103}
}

@article{fds10102,
Author = {Golubitsky, Martin and Keyfitz, Barbara L. and Schaeffer,
David},
Title = {A singularity theory approach to qualitative behavior of
complex chemical systems},
Journal = {New approaches to nonlinear problems in dynamics (Proc.
Conf., Pacific Grove, Calif., 1979), pp. 257--270, 1980,
MRNUMBER = {82i:80011},
url = {http://www.ams.org/mathscinet-getitem?mr=82i:80011},
Key = {fds10102}
}

@article{fds10104,
Author = {Golubitsky, Martin and Schaeffer, David},
Title = {A singularity theory approach to steady-state bifurcation
theory},
Journal = {Nonlinear partial differential equations in engineering and
applied science (Proc. Conf., Univ. Rhode Island, Kingston,
R.I., 1979), pp. 229--254, 1980, Dekker, New
York},
MRNUMBER = {82a:58018},
url = {http://www.ams.org/mathscinet-getitem?mr=82a:58018},
Key = {fds10104}
}

@article{fds10257,
Author = {Schaeffer, David G.},
Title = {A stability theorem for the obstacle problem},
Journal = {Advances in Math., vol. 17, no. 1, pp. 34--47,
1975},
MRNUMBER = {52:994},
url = {http://www.ams.org/mathscinet-getitem?mr=52:994},
Key = {fds10257}
}

@article{fds244126,
Author = {Schaeffer, DG and Matthews, JV},
Title = {A steady-state, hyperbolic free boundary problem for a
granular-flow model},
Journal = {SIAM J. Math Analysis},
Volume = {36},
Pages = {256-271},
Year = {2004},
Key = {fds244126}
}

@article{fds10269,
Author = {Schaeffer, David G.},
Title = {A survey of granular flow},
Journal = {Hyperbolic problems: theory, numerics, applications (Stony
Brook, NY, 1994), pp. 63--80, 1996, World Sci. Publishing,
River Edge, NJ},
MRNUMBER = {1446015},
url = {http://www.ams.org/mathscinet-getitem?mr=1446015},
Key = {fds10269}
}

@article{fds10241,
Author = {Golubitsky, M. and Schaeffer, D.},
Title = {A theory for imperfect bifurcation via singularity
theory},
Journal = {Comm. Pure Appl. Math., vol. 32, no. 1, pp. 21--98,
1979},
MRNUMBER = {80j:58061},
url = {http://www.ams.org/mathscinet-getitem?mr=80j:58061},
Key = {fds10241}
}

@article{fds244129,
Author = {Mitchell, CC and Schaeffer, DG},
Title = {A two-current model for the dynamics of cardiac
membrane.},
Journal = {Bulletin of Mathematical Biology},
Volume = {65},
Number = {5},
Pages = {767-793},
Year = {2003},
Month = {September},
ISSN = {0092-8240},
url = {http://www.ncbi.nlm.nih.gov/pubmed/12909250},
Abstract = {In this paper we introduce and study a model for electrical
activity of cardiac membrane which incorporates only an
inward and an outward current. This model is useful for
three reasons: (1) Its simplicity, comparable to the
FitzHugh-Nagumo model, makes it useful in numerical
simulations, especially in two or three spatial dimensions
where numerical efficiency is so important. (2) It can be
understood analytically without recourse to numerical
simulations. This allows us to determine rather completely
how the parameters in the model affect its behavior which in
turn provides insight into the effects of the many
parameters in more realistic models. (3) It naturally gives
rise to a one-dimensional map which specifies the action
potential duration as a function of the previous diastolic
interval. For certain parameter values, this map exhibits a
new phenomenon--subcritical alternans--that does not occur
for the commonly used exponential map.},
Doi = {10.1016/S0092-8240(03)00041-7},
Key = {fds244129}
}

@article{fds244120,
Author = {Zhao, X and Schaeffer, DG},
Title = {Alternate Pacing of Border-Collision Period-Doubling
Bifurcations.},
Journal = {Nonlinear Dynamics},
Volume = {50},
Number = {3},
Pages = {733-742},
Year = {2007},
Month = {November},
ISSN = {0924-090X},
url = {http://www.ncbi.nlm.nih.gov/pubmed/19132134},
Abstract = {Unlike classical bifurcations, border-collision bifurcations
occur when, for example, a fixed point of a continuous,
piecewise C1 map crosses a boundary in state space. Although
classical bifurcations have been much studied,
border-collision bifurcations are not well understood. This
paper considers a particular class of border-collision
bifurcations, i.e., border-collision period-doubling
bifurcations. We apply a subharmonic perturbation to the
bifurcation parameter, which is also known as alternate
pacing, and we investigate the response under such pacing
near the original bifurcation point. The resulting behavior
is characterized quantitatively by a gain, which is the
ratio of the response amplitude to the applied perturbation
amplitude. The gain in a border-collision period-doubling
bifurcation has a qualitatively different dependence on
parameters from that of a classical period-doubling
bifurcation. Perhaps surprisingly, the differences are more
readily apparent if the gain is plotted vs. the perturbation
amplitude (with the bifurcation parameter fixed) than if
plotted vs. the bifurcation parameter (with the perturbation
amplitude fixed). When this observation is exploited, the
gain under alternate pacing provides a useful experimental
tool to identify a border-collision period-doubling
bifurcation.},
Doi = {10.1007/s11071-006-9174-1},
Key = {fds244120}
}

@article{fds10107,
Author = {Golubitsky, M. and Schaeffer, D.},
Title = {An analysis of imperfect bifurcation},
Journal = {Bifurcation theory and applications in scientific
disciplines (Papers, Conf., New York, 1977), pp. 127--133,
1979, New York Acad. Sci., New York},
MRNUMBER = {81c:58027},
url = {http://www.ams.org/mathscinet-getitem?mr=81c:58027},
Key = {fds10107}
}

@article{fds10243,
Author = {Schaeffer, David G.},
Title = {An application of the Nash-Moser theorem to a free boundary
problem},
Journal = {Nonlinear partial differential equations and applications
(Proc. Special Sem., Indiana Univ., Bloomington, Ind.,
1976-1977), pp. 129--143, 1978, Springer,
Berlin},
MRNUMBER = {80c:35067},
url = {http://www.ams.org/mathscinet-getitem?mr=80c:35067},
Key = {fds10243}
}

@article{fds10260,
Author = {Schaeffer, David G.},
Title = {An application of von Neumann algebras to finite difference
equations},
Journal = {Partial differential equations (Proc. Sympos. Pure Math.,
Vol. XXIII, Univ. California, Berkeley, Calif., 1971), pp.
183--194, 1973, Amer. Math. Soc., Providence,
R.I.},
MRNUMBER = {49:838},
url = {http://www.ams.org/mathscinet-getitem?mr=49:838},
Key = {fds10260}
}

@article{fds10263,
Author = {Schaeffer, David G.},
Title = {An application of von Neumann algebras to finite difference
equations},
Journal = {Ann. of Math. (2), vol. 95, pp. 117--129,
1972},
MRNUMBER = {45:5563},
url = {http://www.ams.org/mathscinet-getitem?mr=45:5563},
Key = {fds10263}
}

@article{fds10255,
Author = {Schaeffer, David G.},
Title = {An example of generic regularity for a non-linear elliptic
equation},
Journal = {Arch. Rational Mech. Anal., vol. 57, pp. 134--141,
1975},
MRNUMBER = {52:8649},
url = {http://www.ams.org/mathscinet-getitem?mr=52:8649},
Key = {fds10255}
}

@article{fds10266,
Author = {Schaeffer, David G.},
Title = {An extension of Hartogs' theorem for domains whose boundary
is not smooth},
Journal = {Proc. Amer. Math. Soc., vol. 25, pp. 714--715,
1970},
MRNUMBER = {41:5650},
url = {http://www.ams.org/mathscinet-getitem?mr=41:5650},
Key = {fds10266}
}

@article{fds10262,
Author = {Schaeffer, David G.},
Title = {An index theorem for systems of difference operators on a
half space},
Journal = {Inst. Hautes \'Etudes Sci. Publ. Math., no. 42, pp.
121--127, 1973},
MRNUMBER = {47:9341},
url = {http://www.ams.org/mathscinet-getitem?mr=47:9341},
Key = {fds10262}
}

@article{fds244121,
Author = {Schaeffer, DG and Cain, JW and Gauthier, DJ and Kalb, SS and Oliver, RA and Tolkacheva, EG and Ying, W and Krassowska, W},
Title = {An ionically based mapping model with memory for cardiac
restitution.},
Journal = {Bulletin of Mathematical Biology},
Volume = {69},
Number = {2},
Pages = {459-482},
Year = {2007},
Month = {February},
ISSN = {0092-8240},
url = {http://www.ncbi.nlm.nih.gov/pubmed/17237915},
Abstract = {Many features of the sequence of action potentials produced
by repeated stimulation of a patch of cardiac muscle can be
modeled by a 1D mapping, but not the full behavior included
in the restitution portrait. Specifically, recent
experiments have found that (i) the dynamic and S1-S2
restitution curves are different (rate dependence) and (ii)
the approach to steady state, which requires many action
potentials (accommodation), occurs along a curve distinct
from either restitution curve. Neither behavior can be
hoc 2D mappings, where the second variable is a "memory"
variable, have been proposed; these models exhibit
qualitative features of the relevant behavior, but a
quantitative fit is not possible. In this paper we introduce
a new 2D mapping and determine a set of parameters for it
that gives a quantitatively accurate description of the full
restitution portrait measured from a bullfrog ventricle. The
mapping can be derived as an asymptotic limit of an
idealized ionic model in which a generalized concentration
acts as a memory variable. This ionic basis clarifies how
the present model differs from previous models. The ionic
basis also provides the foundation for more extensive
cardiac modeling: e.g., constructing a PDE model that may be
used to study the effect of memory on propagation. The
fitting procedure for the mapping is straightforward and can
easily be applied to obtain a mathematical model for data
from other experiments, including experiments on different
species.},
Doi = {10.1007/s11538-006-9116-6},
Key = {fds244121}
}

@article{fds244135,
Author = {Schaeffer, DG and Tolkacheva, E and Mitchell, C},
Title = {Analysis of the Fenton-Karma model through a one-dimensional
map},
Journal = {Chaos},
Volume = {12},
Pages = {1034-1042},
Year = {2002},
Key = {fds244135}
}

@article{fds244131,
Author = {Tolkacheva, EG and Schaeffer, DG and Gauthier, DJ and Mitchell,
CC},
Title = {Analysis of the Fenton-Karma model through an approximation
by a one-dimensional map.},
Journal = {Chaos},
Volume = {12},
Number = {4},
Pages = {1034-1042},
Year = {2002},
Month = {December},
url = {http://www.ncbi.nlm.nih.gov/pubmed/12779627},
Abstract = {The Fenton-Karma model is a simplification of complex ionic
models of cardiac membrane that reproduces quantitatively
many of the characteristics of heart cells; its behavior is
simple enough to be understood analytically. In this paper,
a map is derived that approximates the response of the
Fenton-Karma model to stimulation in zero spatial
dimensions. This map contains some amount of memory,
describing the action potential duration as a function of
the previous diastolic interval and the previous action
potential duration. Results obtained from iteration of the
map and numerical simulations of the Fenton-Karma model are
in good agreement. In particular, the iterated map admits
different types of solutions corresponding to various
dynamical behavior of the cardiac cell, such as 1:1 and 2:1
patterns. (c) 2002 American Institute of
Physics.},
Doi = {10.1063/1.1515170},
Key = {fds244131}
}

@article{fds10252,
Author = {Schaeffer, David G.},
Title = {Approximation of the Dirichlet problem on a half
space},
Journal = {Acta Math., vol. 129, no. 3--4, pp. 281--295,
1972},
MRNUMBER = {52:16058},
url = {http://www.ams.org/mathscinet-getitem?mr=52:16058},
Key = {fds10252}
}

@article{fds244116,
Author = {Schaeffer, DG and Ying, W and Zhao, X},
Title = {Asymptotic approximation of an ionic model for cardiac
restitution.},
Journal = {Nonlinear Dynamics},
Volume = {51},
Number = {1-2},
Pages = {189-198},
Year = {2008},
Month = {January},
ISSN = {0924-090X},
url = {http://www.ncbi.nlm.nih.gov/pubmed/19122809},
Abstract = {Cardiac restitution has been described both in terms of
ionic models-systems of ODE's-and in terms of mapping
models. While the former provide a more fundamental
description, the latter are more flexible in trying to fit
experimental data. Recently we proposed a two-dimensional
mapping that accurately reproduces restitution behavior of a
paced cardiac patch, including rate dependence and
accommodation. By contrast, with previous models only a
qualitative, not a quantitative, fit had been possible. In
this paper, a theoretical foundation for the new mapping is
established by deriving it as an asymptotic limit of an
idealized ionic model.},
Doi = {10.1007/s11071-007-9202-9},
Key = {fds244116}
}

@article{fds10100,
Author = {Schaeffer, David G. and Golubitsky, Martin
A.},
Title = {Bifurcation analysis near a double eigenvalue of a model
chemical reaction},
Journal = {Arch. Rational Mech. Anal., vol. 75, no. 4, pp. 315--347,
1980/81},
MRNUMBER = {83b:80010},
url = {http://www.ams.org/mathscinet-getitem?mr=83b:80010},
Key = {fds10100}
}

@article{fds10098,
Author = {Ball, J. M. and Schaeffer, D. G.},
Title = {Bifurcation and stability of homogeneous equilibrium
tractions},
Journal = {Math. Proc. Cambridge Philos. Soc., vol. 94, no. 2, pp.
315--339, 1983},
MRNUMBER = {84k:73033},
url = {http://www.ams.org/mathscinet-getitem?mr=84k:73033},
Key = {fds10098}
}

@article{fds10094,
Author = {Golubitsky, M. and Marsden, J. and Schaeffer,
D.},
Title = {Bifurcation problems with hidden symmetries},
Journal = {Partial differential equations and dynamical systems, pp.
181--210, 1984, Pitman, Boston, MA},
MRNUMBER = {86a:58020},
url = {http://www.ams.org/mathscinet-getitem?mr=86a:58020},
Key = {fds10094}
}

@article{fds244109,
Author = {Dai, S and Schaeffer, DG},
Title = {Bifurcations in a modulation equation for alternans in a
cardiac fiber},
Journal = {Esaim: Mathematical Modelling and Numerical
Analysis},
Volume = {44},
Number = {6},
Pages = {1225-1238},
Publisher = {E D P SCIENCES},
Year = {2010},
Month = {Winter},
ISSN = {0764-583X},
Abstract = {While alternans in a single cardiac cell appears through a
simple period-doubling bifurcation, in extended tissue the
exact nature of the bifurcation is unclear. In particular,
the phase of alternans can exhibit wave-like spatial
dependence, either stationary or travelling, which is known
as discordant alternans. We study these phenomena in simple
cardiac models through a modulation equation proposed by
Echebarria-Karma. As shown in our previous paper, the zero
solution of their equation may lose stability, as the pacing
rate is increased, through either a Hopf or steady-state
bifurcation. Which bifurcation occurs first depends on
parameters in the equation, and for one critical case both
modes bifurcate together at a degenerate (codimension 2)
bifurcation. For parameters close to the degenerate case, we
investigate the competition between modes, both numerically
and analytically. We find that at sufficiently rapid pacing
(but assuming a 1:1 response is maintained), steady patterns
always emerge as the only stable solution. However, in the
parameter range where Hopf bifurcation occurs first, the
evolution from periodic solution (just after the
bifurcation) to the eventual standing wave solution occurs
through an interesting series of secondary bifurcations. ©
2010 EDP Sciences, SMAI.},
Doi = {10.1051/m2an/2010028},
Key = {fds244109}
}

@article{fds10101,
Author = {Golubitsky, Martin and Schaeffer, David},
Title = {Bifurcations with ${\rm O}(3)$\ symmetry including
applications to the B\'enard problem},
Journal = {Comm. Pure Appl. Math., vol. 35, no. 1, pp. 81--111,
1982},
MRNUMBER = {83b:58026},
url = {http://www.ams.org/mathscinet-getitem?mr=83b:58026},
Key = {fds10101}
}

@article{fds10106,
Author = {Schaeffer, David and Golubitsky, Martin},
Title = {Boundary conditions and mode jumping in the buckling of a
rectangular plate},
Journal = {Comm. Math. Phys., vol. 69, no. 3, pp. 209--236,
1979},
MRNUMBER = {81k:35019},
url = {http://www.ams.org/mathscinet-getitem?mr=81k:35019},
Key = {fds10106}
}

@article{fds10095,
Author = {Holder, E. J. and Schaeffer, D.},
Title = {Boundary conditions and mode jumping in the von K\'arm\'an
equations},
Journal = {SIAM J. Math. Anal., vol. 15, no. 3, pp. 446--458,
1984},
MRNUMBER = {85m:73029},
url = {http://www.ams.org/mathscinet-getitem?mr=85m:73029},
Key = {fds10095}
}

@article{fds244123,
Author = {Schaeffer, DG and Shearer, M and Witelski, T},
Title = {Boundary-value problems for hyperbolic partial differential
equations related to steady granular flow},
Journal = {Math. and Mech. of Solids},
Volume = {12},
Pages = {665-699},
Year = {2007},
Key = {fds244123}
}

@article{fds244113,
Author = {Dai, S and Schaeffer, DG},
Title = {Chaos in a one-dimensional model for cardiac
dynamics},
Journal = {Chaos},
Volume = {20},
Number = {2},
Year = {2010},
Month = {June},
Key = {fds244113}
}

@article{fds244136,
Author = {Tolkacheva, EG and Schaeffer, DG and Gauthier, DJ and Krassowska,
W},
Title = {Condition for alternans and stability of the 1:1 response
pattern in a "memory" model of paced cardiac
dynamics.},
Journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter
Physics},
Volume = {67},
Number = {3 Pt 1},
Pages = {031904},
Year = {2003},
Month = {March},
ISSN = {1539-3755},
url = {http://www.ncbi.nlm.nih.gov/pubmed/12689098},
Abstract = {We analyze a mathematical model of paced cardiac muscle
consisting of a map relating the duration of an action
potential to the preceding diastolic interval as well as the
preceding action potential duration, thereby containing some
degree of "memory." The model displays rate-dependent
restitution so that the dynamic and S1-S2 restitution curves
are different, a manifestation of memory in the model. We
derive a criterion for the stability of the 1:1 response
pattern displayed by this model. It is found that the
stability criterion depends on the slope of both the dynamic
and S1-S2 restitution curves, and that the pattern can be
stable even when the individual slopes are greater or less
than one. We discuss the relation between the stability
criterion and the slope of the constant-BCL restitution
curve. The criterion can also be used to determine the
bifurcation from the 1:1 response pattern to alternans. We
demonstrate that the criterion can be evaluated readily in
experiments using a simple pacing protocol, thus
establishing a method for determining whether actual
myocardium is accurately described by such a mapping model.
We illustrate our results by considering a specific map
recently derived from a three-current membrane model and
find that the stability of the 1:1 pattern is accurately
described by our criterion. In addition, a numerical
experiment is performed using the three-current model to
illustrate the application of the pacing protocol and the
evaluation of the criterion.},
Doi = {10.1103/PhysRevE.67.031904},
Key = {fds244136}
}

@article{fds29019,
Author = {D.G. Schaeffer and E. Tolkacheva and D. Gauthier and W.
Krassowska},
Title = {Condition for alternans and stability of the 1:1 response
pattern in a memory model of paced cardiac
dynamics},
Journal = {Phys Rev E},
Volume = {67},
Pages = {031904},
Year = {2003},
Key = {fds29019}
}

@article{fds244133,
Author = {Socolar, JES and Schaeffer, DG and Claudin, P},
Title = {Directed force chain networks and stress response in static
granular materials},
Journal = {European Physical Journal E},
Volume = {7},
Number = {4},
Pages = {353-370},
Year = {2002},
Abstract = {A theory of stress fields in two-dimensional granular
materials based on directed force chain networks is
presented. A general Boltzmann equation for the densities of
force chains in different directions is proposed and a
complete solution is obtained for a special case in which
chains lie along a discrete set of directions. The analysis
and results demonstrate the necessity of including nonlinear
terms in the Boltzmann equation. A line of nontrivial
fixed-point solutions is shown to govern the properties of
large systems. In the vicinity of a generic fixed point, the
response to a localized load shows a crossover from a
single, centered peak at intermediate depths to two
propagating peaks at large depths that broaden
diffusively.},
Key = {fds244133}
}

@article{fds244110,
Author = {Beck, M and Jones, CKRT and Schaeffer, D and Wechselberger,
M},
Title = {Electrical waves in a one-dimensional model of cardiac
tissue},
Journal = {Siam Journal on Applied Dynamical Systems},
Volume = {7},
Number = {4},
Pages = {1558-1581},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2008},
Month = {December},
url = {http://dx.doi.org/10.1137/070709980},
Abstract = {The electrical dynamics in the heart is modeled by a
two-component PDE. Using geometric singular perturbation
theory, it is shown that a traveling pulse solution, which
corresponds to a single heartbeat, exists. One key aspect of
the proof involves tracking the solution near a point on the
slow manifold that is not normally hyperbolic. This is
achieved by desingularizing the vector field using a blow-up
technique. This feature is relevant because it distinguishes
cardiac impulses from, for example, nerve impulses.
Stability of the pulse is also shown, by computing the zeros
of the Evans function. Although the spectrum of one of the
fast components is only marginally stable, due to essential
spectrum that accumulates at the origin, it is shown that
the spectrum of the full pulse consists of an isolated
eigenvalue at zero and essential spectrum that is bounded
away from the imaginary axis. Thus, this model provides an
example in a biological application reminiscent of a
previously observed mathematical phenomenon: that connecting
an unstable-in this case marginally stable-front and back
can produce a stable pulse. Finally, remarks are made
regarding the existence and stability of spatially periodic
pulses, corresponding to successive heartbeats, and their
relationship with alternans, irregular action potentials
Industrial and Applied Mathematics.},
Doi = {10.1137/070709980},
Key = {fds244110}
}

@article{fds244125,
Author = {Schaeffer, DG and Kalb, S and Tolkacheva, E and Gauthier, D and Krassowska, W},
Title = {Features of the restitution portrait for mapping models with
an arbitrary amount of memory},
Journal = {Chaos},
Volume = {15},
Pages = {023701},
Year = {2005},
Key = {fds244125}
}

@article{fds9766,
Author = {David G Schaeffer and M. Sexton and J. Socolar},
Title = {Force Distribution in a Scalar Model for Non-Cohesive
Granular Material},
Journal = {Phys. Rev. Lett. E 60 (1999), 1999-2008},
Key = {fds9766}
}

@article{fds244122,
Author = {Schaeffer, DG and Tighe, B and Socolar, J and Michener, G and Huber,
M},
Title = {Force distribution in granular media},
Journal = {PRE},
Volume = {72},
Pages = {031306},
Year = {2005},
Key = {fds244122}
}

@article{fds8986,
Author = {G. Tardos and M.I. Khan and David G Schaeffer},
Title = {Forces On a Slowly Rotating, Rough Cylinder in a Couette
Device Containing a Dry, Frictional Powder},
Journal = {Physics of Fluids 10(1998), 335-341.},
Key = {fds8986}
}

@article{fds8981,
Author = {F.X. Garzizar and David G Schaeffer and M. Shearer and J.
Trangenstein},
Title = {Formation and Development of Shear Bands in Granular
Material},
Journal = {Trans. of 11th Army Conf. on Appl. Math. &
Computing.},
Key = {fds8981}
}

@article{fds10256,
Author = {Guillemin, V. and Schaeffer, D.},
Title = {Fourier integral operators from the Radon transform point of
view},
Journal = {Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII,
Stanford Univ., Stanford, Calif., 1973), Part 2, pp.
297--300, 1975, Amer. Math. Soc., Providence,
R.I.},
MRNUMBER = {52:1420},
url = {http://www.ams.org/mathscinet-getitem?mr=52:1420},
Key = {fds10256}
}

@article{fds8980,
Author = {Shearer, Michael and Schaeffer, David G.},
Title = {Fully nonlinear hyperbolic systems of partial differential
equations related to plasticity},
Journal = {Comm. Partial Differential Equations, vol. 20, no. 7-8, pp.
1133--1153, 1995},
MRNUMBER = {96b:35134},
url = {http://www.ams.org/mathscinet-getitem?mr=96b:35134},
Key = {fds8980}
}

@article{fds10099,
Author = {Schaeffer, David},
Title = {General introduction to steady state bifurcation},
Journal = {Dynamical systems and turbulence, Warwick 1980 (Coventry,
1979/1980), pp. 13--47, 1981, Springer, Berlin},
MRNUMBER = {83j:58037},
url = {http://www.ams.org/mathscinet-getitem?mr=83j:58037},
Key = {fds10099}
}

@article{fds244137,
Author = {Gremaud, P and Schaeffer, DG and Shearer, M},
Title = {Granular Flow Past a Binsert},
Journal = {Report to Jenike & Johanson, Inc.},
Year = {1997},
Month = {January},
Key = {fds244137}
}

@article{fds10420,
Author = {G. Metcalfe and L. Kondic and D. Schaeffer and S. Tennakoon and R.
Behringer},
Title = {Granular friction and the fluid-solid transition for shaken
granular materials},
Journal = {Phys. Rev. E 65 (2002)},
Key = {fds10420}
}

@article{fds10083,
Author = {Schaeffer, David G. and Pitman, E. Bruce},
Title = {Ill-posedness in three-dimensional plastic
flow},
Journal = {Comm. Pure Appl. Math., vol. 41, no. 7, pp. 879--890,
1988},
MRNUMBER = {89m:73018},
url = {http://www.ams.org/mathscinet-getitem?mr=89m:73018},
Key = {fds10083}
}

@article{fds10242,
Author = {Golubitsky, M. and Schaeffer, D.},
Title = {Imperfect bifurcation in the presence of
symmetry},
Journal = {Comm. Math. Phys., vol. 67, no. 3, pp. 205--232,
1979},
MRNUMBER = {80j:58017},
url = {http://www.ams.org/mathscinet-getitem?mr=80j:58017},
Key = {fds10242}
}

@article{fds10082,
Author = {Pitman, E. Bruce and Schaeffer, David G.},
Title = {Instability and ill-posedness in granular
flow},
Journal = {Current progress in hyberbolic systems: Riemann problems and
computations (Brunswick, ME, 1988), pp. 241--250, 1989,
Amer. Math. Soc., Providence, RI},
MRNUMBER = {90k:73037},
url = {http://www.ams.org/mathscinet-getitem?mr=90k:73037},
Key = {fds10082}
}

@article{fds10079,
Author = {Schaeffer, David G.},
Title = {Instability and ill-posedness in the deformation of granular
materials},
Journal = {Internat. J. Numer. Anal. Methods Geomech., vol. 14, no. 4,
pp. 253--278, 1990},
MRNUMBER = {91e:73071},
url = {http://www.ams.org/mathscinet-getitem?mr=91e:73071},
Key = {fds10079}
}

@article{fds10081,
Author = {Schaeffer, David G. and Shearer, Michael and Pitman, E.
Bruce},
Title = {Instability in critical state theories of granular
flow},
Journal = {SIAM J. Appl. Math., vol. 50, no. 1, pp. 33--47,
1990},
MRNUMBER = {90k:73044},
url = {http://www.ams.org/mathscinet-getitem?mr=90k:73044},
Key = {fds10081}
}

@article{fds10089,
Author = {Schaeffer, David G.},
Title = {Instability in the evolution equations describing
incompressible granular flow},
Journal = {J. Differential Equations, vol. 66, no. 1, pp. 19--50,
1987},
MRNUMBER = {88i:35169},
url = {http://www.ams.org/mathscinet-getitem?mr=88i:35169},
Key = {fds10089}
}

@article{fds10271,
Author = {Schaeffer, David G.},
Title = {Instability in the flow of granular materials},
Journal = {Mathematics applied to fluid mechanics and stability (Troy,
N.Y., 1985), pp. 274, 1986, SIAM, Philadelphia,
PA},
MRNUMBER = {869642},
url = {http://www.ams.org/mathscinet-getitem?mr=869642},
Key = {fds10271}
}

@article{fds10065,
Author = {Howle, Laurens and Schaeffer, David G. and Shearer, Michael and Zhong, Pei},
Title = {Lithotripsy: the treatment of kidney stones with shock
waves},
Journal = {SIAM Rev., vol. 40, no. 2, pp. 356--371 (electronic),
1998},
MRNUMBER = {99d:92009},
url = {http://www.ams.org/mathscinet-getitem?mr=99d:92009},
Key = {fds10065}
}

@article{fds10076,
Author = {Schaeffer, David G. and Shearer, Michael},
Title = {Loss of hyperbolicity in yield vertex plasticity models
Journal = {Nonlinear evolution equations that change type, pp.
192--217, 1990, Springer, New York},
MRNUMBER = {92f:73022},
url = {http://www.ams.org/mathscinet-getitem?mr=92f:73022},
Key = {fds10076}
}

@article{fds10077,
Author = {Schaeffer, David G.},
Title = {Mathematical issues in the continuum formulation of slow
granular flow},
Journal = {Two phase flows and waves (Minneapolis, MN, 1989), pp.
118--129, 1990, Springer, New York},
MRNUMBER = {91f:73014},
url = {http://www.ams.org/mathscinet-getitem?mr=91f:73014},
Key = {fds10077}
}

@article{fds8983,
Author = {David G Schaeffer},
Title = {Memoirs From a Small-Scale Course On Industrial
Math},
Journal = {Notices AMS, 43(1996), 550-557.},
Key = {fds8983}
}

@article{fds244114,
Author = {Gonzales, K and Kayikci, O and Schaeffer, DG and Magwene,
P},
Title = {Modeling mutant phenotypes and oscillatory dynamics in the
\emph{Saccharomyces cerevisiae} cAMP-PKA
pathway},
Journal = {PLoS Computational Biology},
Year = {2010},
Month = {Winter},
Key = {fds244114}
}

@article{fds8987,
Author = {David G Schaeffer and M. Shearer},
Title = {Models of Stress Fluctuations in Granular Materials, Powders
and Grains},
Journal = {R.P. Behringer and J. Jenkins (eds.), Balkema,
1997.},
Key = {fds8987}
}

@article{fds10071,
Author = {Schaeffer, David G. and Schecter, Stephen and Shearer,
Michael},
Title = {Non-strictly hyperbolic conservation laws with a parabolic
line},
Journal = {J. Differential Equations, vol. 103, no. 1, pp. 94--126,
1993},
MRNUMBER = {94d:35102},
url = {http://www.ams.org/mathscinet-getitem?mr=94d:35102},
Key = {fds10071}
}

@article{fds10244,
Author = {Schaeffer, David G.},
Title = {Non-uniqueness in the equilibrium shape of a confined
plasma},
Journal = {Comm. Partial Differential Equations, vol. 2, no. 6, pp.
587--600, 1977},
MRNUMBER = {58:29210},
url = {http://www.ams.org/mathscinet-getitem?mr=58:29210},
Key = {fds10244}
}

@article{fds10085,
Author = {Beale, J. Thomas and Schaeffer, David G.},
Title = {Nonlinear behavior of model equations which are linearly
ill-posed},
Journal = {Comm. Partial Differential Equations, vol. 13, no. 4, pp.
423--467, 1988},
MRNUMBER = {89h:35329},
url = {http://www.ams.org/mathscinet-getitem?mr=89h:35329},
Key = {fds10085}
}

@article{fds8977,
Author = {Garaizar, F. Xabier and Schaeffer, David
G.},
Title = {Numerical computations for shear bands in an antiplane shear
model},
Journal = {J. Mech. Phys. Solids, vol. 42, no. 1, pp. 21--50,
1994},
MRNUMBER = {94j:73029},
url = {http://www.ams.org/mathscinet-getitem?mr=94j:73029},
Key = {fds8977}
}

@article{fds10063,
Author = {Gremaud, Pierre Alain and Schaeffer, David G. and Shearer,
Michael},
Title = {Numerical determination of flow corrective inserts for
granular materials in conical hoppers},
Journal = {Internat. J. Non-Linear Mech., vol. 35, no. 5, pp. 869--882,
2000},
MRNUMBER = {2001a:76129},
url = {http://www.ams.org/mathscinet-getitem?mr=2001a:76129},
Key = {fds10063}
}

@article{fds8978,
Author = {Gardner, Carl L. and Schaeffer, David G.},
Title = {Numerical simulation of uniaxial compression of a granular
material with wall friction},
Journal = {SIAM J. Appl. Math., vol. 54, no. 6, pp. 1676--1692,
1994},
MRNUMBER = {95g:76010},
url = {http://www.ams.org/mathscinet-getitem?mr=95g:76010},
Key = {fds8978}
}

@article{fds10249,
Author = {Guillemin, Victor and Schaeffer, David},
Title = {On a certain class of Fuchsian partial differential
equations},
Journal = {Duke Math. J., vol. 44, no. 1, pp. 157--199,
1977},
MRNUMBER = {55:3504},
url = {http://www.ams.org/mathscinet-getitem?mr=55:3504},
Key = {fds10249}
}

@article{fds244117,
Author = {Schaeffer, DG and Catlla, A and Witelski, T and Monson, E and Lin,
A},
Title = {On spiking models of synaptic activity and impulsive
differential equations},
Journal = {SIAM Review},
Volume = {50},
Number = {553--569},
Year = {2008},
Key = {fds244117}
}

@article{fds244127,
Author = {Schaeffer, DG and Matthews, M and Gremaud, P},
Title = {On the computation of steady hopper flows III: Comparison of
von Mises and Matsuoka-Nakai materials"},
Journal = {J Comp. Phy.},
Volume = {219},
Pages = {443-454},
Year = {2006},
Key = {fds244127}
}

@article{fds10247,
Author = {Schaeffer, David G.},
Title = {On the existence of discrete frequencies of oscillation in a
rotating fluid},
Journal = {Studies in Appl. Math., vol. 54, no. 3, pp. 269--274,
1975},
MRNUMBER = {56:10385},
url = {http://www.ams.org/mathscinet-getitem?mr=56:10385},
Key = {fds10247}
}

@article{fds244132,
Author = {Schaeffer, DG and Shearer, M and Witelski, T},
Title = {One-dimensional solutions of an elastoplasticity model of
granular material},
Journal = {Math. Models and Methods in Appl. Sciences},
Volume = {13},
Pages = {1629-1671},
Year = {2003},
Key = {fds244132}
}

@article{fds10248,
Author = {Schaeffer, David G.},
Title = {One-sided estimates for the curvature of the free boundary
in the obstacle problem},
Journal = {Advances in Math., vol. 24, no. 1, pp. 78--98,
1977},
MRNUMBER = {56:6506},
url = {http://www.ams.org/mathscinet-getitem?mr=56:6506},
Key = {fds10248}
}

@article{fds244118,
Author = {Berger, CM and Zhao, X and Schaeffer, DG and Dobrovolny, HM and Krassowska, W and Gauthier, DJ},
Title = {Period-doubling bifurcation to alternans in paced cardiac
tissue: crossover from smooth to border-collision
characteristics.},
Journal = {Physical Review Letters},
Volume = {99},
Number = {5},
Pages = {058101},
Year = {2007},
Month = {August},
ISSN = {0031-9007},
url = {http://www.ncbi.nlm.nih.gov/pubmed/17930795},
Abstract = {We investigate, both experimentally and theoretically, the
period-doubling bifurcation to alternans in heart tissue.
Previously, this phenomenon has been modeled with either
smooth or border-collision dynamics. Using a modification of
existing experimental techniques, we find a hybrid behavior:
Very close to the bifurcation point, the dynamics is smooth,
whereas further away it is border-collision-like. The
essence of this behavior is captured by a model that
exhibits what we call an unfolded border-collision
bifurcation. This new model elucidates that, in an
experiment, where only a limited number of data points can
be measured, the smooth behavior of the bifurcation can
easily be missed.},
Doi = {10.1103/physrevlett.99.058101},
Key = {fds244118}
}

@article{fds8982,
Author = {M. K. Gordon and David G Schaeffer and M. Shearer},
Title = {Plane Shear Waves in a Fully Saturated Granular Medium with
Velocity-and Stress-Controlled Boundary Conditions},
Journal = {Int. J. Nonlinear Mechancis 32(1997), 489-503.},
Key = {fds8982}
}

@article{fds10064,
Author = {Hayes, Brian T. and Schaeffer, David G.},
Title = {Plane shear waves under a periodic boundary disturbance in a
saturated granular medium},
Journal = {Phys. D, vol. 121, no. 1-2, pp. 193--212,
1998},
MRNUMBER = {99g:73052},
url = {http://www.ams.org/mathscinet-getitem?mr=99g:73052},
Key = {fds10064}
}

@article{fds10240,
Author = {Schaeffer, David G.},
Title = {Qualitative analysis of a model for boundary effects in the
Taylor problem},
Journal = {Math. Proc. Cambridge Philos. Soc., vol. 87, no. 2, pp.
307--337, 1980},
MRNUMBER = {81c:35007},
url = {http://www.ams.org/mathscinet-getitem?mr=81c:35007},
Key = {fds10240}
}

@article{fds244130,
Author = {Schaeffer, DG and Cain, J and Tolkacheva, E and Gauthier,
D},
Title = {Rate-dependent waveback velocity of cardiac action
potentials in a done-dimensional cable},
Journal = {Phys Rev E},
Volume = {70},
Pages = {061906-?},
Year = {2004},
Key = {fds244130}
}

@article{fds10270,
Author = {Shearer, Michael and Schaeffer, David G.},
Title = {Recent developments in nonstrictly hyperbolic conservation
laws},
Journal = {Transactions of the fourth Army conference on applied
mathematics and computing (Ithaca, N.Y., 1986), pp. 43--52,
1987, U.S. Army Res. Office, Research Triangle Park,
NC},
MRNUMBER = {905075},
url = {http://www.ams.org/mathscinet-getitem?mr=905075},
Key = {fds10270}
}

@article{fds10251,
Author = {Guillemin, V. and Schaeffer, D.},
Title = {Remarks on a paper of D. Ludwig},
Journal = {Bull. Amer. Math. Soc., vol. 79, pp. 382--385,
1973},
MRNUMBER = {53:13800},
url = {http://www.ams.org/mathscinet-getitem?mr=53:13800},
Key = {fds10251}
}

@article{fds244134,
Author = {Schaeffer, DG},
Title = {Review of W. Cheney's "Analysis for applied
mathematics"},
Journal = {Amer. Math Monthly},
Volume = {110},
Pages = {550},
Year = {2003},
Key = {fds244134}
}

@article{fds10067,
Author = {Shearer, Michael and Schaeffer, David G.},
Title = {Riemann problems for $5\times 5$ systems of fully non-linear
equations related to hypoplasticity},
Journal = {Math. Methods Appl. Sci., vol. 19, no. 18, pp. 1433--1444,
1996},
MRNUMBER = {97m:73028},
url = {http://www.ams.org/mathscinet-getitem?mr=97m:73028},
Key = {fds10067}
}

@article{fds10087,
Author = {Schaeffer, David G. and Shearer, Michael},
Title = {Riemann problems for nonstrictly hyperbolic $2\times 2$
systems of conservation laws},
Journal = {Trans. Amer. Math. Soc., vol. 304, no. 1, pp. 267--306,
1987},
MRNUMBER = {88m:35101},
url = {http://www.ams.org/mathscinet-getitem?mr=88m:35101},
Key = {fds10087}
}

@article{fds10073,
Author = {Schaeffer, David G. and Shearer, Michael},
Title = {Scale-invariant initial value problems in one-dimensional
dynamic elastoplasticity, with consequences for
multidimensional nonassociative plasticity},
Journal = {European J. Appl. Math., vol. 3, no. 3, pp. 225--254,
1992},
MRNUMBER = {93g:73057},
url = {http://www.ams.org/mathscinet-getitem?mr=93g:73057},
Key = {fds10073}
}

@article{fds244128,
Author = {Gremaud, PA and Matthews, JV and Schaeffer, DG},
Title = {Secondary circulation in granular flow through
nonaxisymmetric hoppers},
Journal = {Siam Journal on Applied Mathematics},
Volume = {64},
Number = {2},
Pages = {583-600},
Year = {2004},
Month = {June},
ISSN = {0036-1399},
Abstract = {Jenike's radial solution, widely used in the design of
materials-handling equipment, is a similarity solution of
steady-state continuum equations for the flow under gravity
of granular material through an infinite, right-circular
cone. In this paper we study how the geometry of the hopper
influences this solution. Using perturbation theory, we
compute a first-order correction to the (steady-state)
velocity resulting from a small change in hopper geometry,
either distortion of the cross section or tilting away from
vertical. Unlike for the Jenike solution, all three
components of the correction velocity are nonzero; i.e.,
there is secondary circulation in the perturbed
flow.},
Doi = {10.1137/S0036139903415124},
Key = {fds244128}
}

@article{fds244119,
Author = {Schaeffer, DG and Cain, J},
Title = {Shortening of action potential duraction near an insulating
boundary},
Journal = {Math Medicine and Biology},
Volume = {25},
Number = {21--36},
Year = {2008},
Key = {fds244119}
}

@article{fds10246,
Author = {Schaeffer, David G.},
Title = {Singularities and the obstacle problem},
Journal = {Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII,
Stanford Univ., Stanford, Calif., 1973), Part 2, pp.
339--340, 1975, Amer. Math. Soc., Providence,
R.I.},
MRNUMBER = {57:10227},
url = {http://www.ams.org/mathscinet-getitem?mr=57:10227},
Key = {fds10246}
}

@article{fds244124,
Author = {Zhao, X and Schaeffer, DG and Berger, CM and Gauthier,
DJ},
Title = {Small-Signal Amplification of Period-Doubling Bifurcations
in Smooth Iterated Maps.},
Journal = {Nonlinear Dynamics},
Volume = {48},
Number = {4},
Pages = {381-389},
Year = {2007},
Month = {June},
ISSN = {0924-090X},
url = {http://www.ncbi.nlm.nih.gov/pubmed/19112525},
Abstract = {Various authors have shown that, near the onset of a
period-doubling bifurcation, small perturbations in the
control parameter may result in much larger disturbances in
the response of the dynamical system. Such amplification of
small signals can be measured by a gain defined as the
magnitude of the disturbance in the response divided by the
perturbation amplitude. In this paper, the perturbed
response is studied using normal forms based on the most
general assumptions of iterated maps. Such an analysis
provides a theoretical footing for previous experimental and
numerical observations, such as the failure of linear
analysis and the saturation of the gain. Qualitative as well
as quantitative features of the gain are exhibited using
selected models of cardiac dynamics.},
Doi = {10.1007/s11071-006-9092-2},
Key = {fds244124}
}

@article{fds10090,
Author = {Shearer, M. and Schaeffer, D. G. and Marchesin, D. and Paes-Leme, P. L.},
Title = {Solution of the Riemann problem for a prototype $2\times 2$
system of nonstrictly hyperbolic conservation
laws},
Journal = {Arch. Rational Mech. Anal., vol. 97, no. 4, pp. 299--320,
1987},
MRNUMBER = {88a:35156},
url = {http://www.ams.org/mathscinet-getitem?mr=88a:35156},
Key = {fds10090}
}

@article{fds10245,
Author = {Schaeffer, David G.},
Title = {Some examples of singularities in a free
boundary},
Journal = {Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), vol. 4, no. 1, pp.
133--144, 1977},
MRNUMBER = {58:24345},
url = {http://www.ams.org/mathscinet-getitem?mr=58:24345},
Key = {fds10245}
}

@article{fds244112,
Author = {Dai, S and Schaeffer, DG},
Title = {Spectrum of a linearized amplitude equation for alternans in
a cardiac fiber},
Journal = {Siam Journal on Applied Mathematics},
Volume = {69},
Number = {3},
Pages = {704-719},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2008},
Month = {December},
ISSN = {0036-1399},
Abstract = {Under rapid periodic pacing, cardiac cells typically undergo
a period-doubling bifurcation in which action potentials of
short and long duration alternate with one another. If these
action potentials propagate in a fiber, the short-long
alternation may suffer reversals of phase at various points
along the fiber, a phenomenon called (spatially) discordant
alternans. Either stationary or moving patterns are
possible. Using a weak approximation, Echebarria and Karma
proposed an equation to describe the spatiotemporal dynamics
of small-amplitude alternans in a class of simple cardiac
models, and they showed that an instability in this equation
predicts the spontaneous formation of discordant alternans.
To study the bifurcation, they computed the spectrum of the
relevant linearized operator numerically, supplemented with
partial analytical results. In the present paper we
calculate this spectrum with purely analytical methods in
two cases where a small parameter may be exploited: (i)
small dispersion or (ii) a long fiber. From this analysis we
estimate the parameter ranges in which the phase reversals
of discordant alternans are stationary or moving. © 2008
Society for Industrial and Applied Mathematics.},
Doi = {10.1137/070711384},
Key = {fds244112}
}

@article{fds10258,
Author = {Golubitsky, Martin and Schaeffer, David G.},
Title = {Stability of shock waves for a single conservation
law},
Journal = {Advances in Math., vol. 16, pp. 65--71, 1975},
MRNUMBER = {51:10889},
url = {http://www.ams.org/mathscinet-getitem?mr=51:10889},
Key = {fds10258}
}

@article{fds10088,
Author = {Pitman, E. Bruce and Schaeffer, David G.},
Title = {Stability of time dependent compressible granular flow in
two dimensions},
Journal = {Comm. Pure Appl. Math., vol. 40, no. 4, pp. 421--447,
1987},
MRNUMBER = {88i:35170},
url = {http://www.ams.org/mathscinet-getitem?mr=88i:35170},
Key = {fds10088}
}

@article{fds244111,
Author = {Schaeffer, DG and Iverson, RM},
Title = {Steady and intermittent slipping in a model of landslide
motion regulated by pore-pressure feedback},
Journal = {Siam Journal on Applied Mathematics},
Volume = {69},
Number = {3},
Pages = {769-786},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2008},
Month = {December},
ISSN = {0036-1399},
Abstract = {This paper studies a parsimonious model of landslide motion,
which consists of the one-dimensional diffusion equation
(for pore pressure) coupled through a boundary condition to
a first-order ODE (Newton's second law). Velocity weakening
of sliding friction gives rise to nonlinearity in the model.
Analysis shows that solutions of the model equations exhibit
a subcritical Hopf bifurcation in which stable, steady
sliding can transition to cyclical, stick-slip motion.
Numerical computations confirm the analytical predictions of
the parameter values at which bifurcation occurs. The
existence of stick-slip behavior in part of the parameter
space is particularly noteworthy because, unlike stick-slip
behavior in classical models, here it arises in the absence
of a reversible (elastic) driving force. Instead, the
driving force is static (gravitational), mediated by the
effects of pore-pressure diffusion on frictional resistance.
© 2008 Society for Industrial and Applied
Mathematics.},
Doi = {10.1137/07070704X},
Key = {fds244111}
}

@article{fds10075,
Author = {Wang, Feng and Gardner, Carl L. and Schaeffer, David
G.},
Title = {Steady-state computations of granular flow in an
axisymmetric hopper},
Journal = {SIAM J. Appl. Math., vol. 52, no. 4, pp. 1076--1088,
1992},
MRNUMBER = {93c:73040},
url = {http://www.ams.org/mathscinet-getitem?mr=93c:73040},
Key = {fds10075}
}

@article{fds9762,
Author = {Hayes, Brian T. and Schaeffer, David G.},
Title = {Stress-controlled shear waves in a saturated granular
medium},
Journal = {European J. Appl. Math., vol. 11, no. 1, pp. 81--94,
2000},
MRNUMBER = {2000k:74037},
url = {http://www.ams.org/mathscinet-getitem?mr=2000k:74037},
Key = {fds9762}
}

@article{fds10250,
Author = {Schaeffer, David G.},
Title = {Supersonic flow past a nearly straight wedge},
Journal = {Duke Math. J., vol. 43, no. 3, pp. 637--670,
1976},
MRNUMBER = {54:1850},
url = {http://www.ams.org/mathscinet-getitem?mr=54:1850},
Key = {fds10250}
}

@article{fds10253,
Author = {Schaeffer, David G.},
Title = {The capacitor problem},
Journal = {Indiana Univ. Math. J., vol. 24, no. 12, pp. 1143--1167,
1974/75},
MRNUMBER = {52:14607},
url = {http://www.ams.org/mathscinet-getitem?mr=52:14607},
Key = {fds10253}
}

@article{fds10091,
Author = {Schaeffer, David G. and Shearer, Michael},
Title = {The classification of $2\times 2$ systems of nonstrictly
hyperbolic conservation laws, with application to oil
recovery},
Journal = {Comm. Pure Appl. Math., vol. 40, no. 2, pp. 141--178,
1987},
MRNUMBER = {88a:35155},
url = {http://www.ams.org/mathscinet-getitem?mr=88a:35155},
Key = {fds10091}
}

@article{fds10265,
Author = {Schaeffer, David G.},
Title = {The Dirichlet problem with generalized functions as
data},
Journal = {Ann. Mat. Pura Appl. (4), vol. 83, pp. 153--174,
1969},
MRNUMBER = {41:7271},
url = {http://www.ams.org/mathscinet-getitem?mr=41:7271},
Key = {fds10265}
}

@article{fds10074,
Author = {An, Lian Jun and Schaeffer, David G.},
Title = {The flutter instability in granular flow},
Journal = {J. Mech. Phys. Solids, vol. 40, no. 3, pp. 683--698,
1992},
MRNUMBER = {93c:73053},
url = {http://www.ams.org/mathscinet-getitem?mr=93c:73053},
Key = {fds10074}
}

@article{fds244115,
Author = {Farjoun, Y and Schaeffer, DG},
Title = {The hanging thin rod: a singularly perturbed eigenvalue
problem},
Journal = {SIAM Sppl. Math.},
Year = {2010},
Month = {July},
Key = {fds244115}
}

@article{fds10066,
Author = {Schaeffer, David G. and Shearer, Michael},
Title = {The influence of material non-uniformity preceding
shear-band formation in a model for granular
flow},
Journal = {European J. Appl. Math., vol. 8, no. 5, pp. 457--483,
1997},
MRNUMBER = {98g:73016},
url = {http://www.ams.org/mathscinet-getitem?mr=98g:73016},
Key = {fds10066}
}

@article{fds10069,
Author = {Shearer, Michael and Schaeffer, David G.},
Title = {The initial value problem for a system modelling
unidirectional longitudinal elastic-plastic
waves},
Journal = {SIAM J. Math. Anal., vol. 24, no. 5, pp. 1111--1144,
1993},
MRNUMBER = {95f:73038},
url = {http://www.ams.org/mathscinet-getitem?mr=95f:73038},
Key = {fds10069}
}

@article{fds10080,
Author = {Shearer, Michael and Schaeffer, David G.},
Title = {The quasidynamic approximation in critical state
plasticity},
Journal = {Arch. Rational Mech. Anal., vol. 108, no. 3, pp. 267--280,
1989},
MRNUMBER = {91d:73031},
url = {http://www.ams.org/mathscinet-getitem?mr=91d:73031},
Key = {fds10080}
}

@article{fds10086,
Author = {Schaeffer, David G. and Shearer, Michael},
Title = {Three phase flow in a porous medium and the classification
of nonstrictly hyperbolic conservation laws},
Journal = {International workshop on applied differential equations
(Beijing, 1985), pp. 154--162, 1986, World Sci. Publishing,
Singapore},
MRNUMBER = {89c:35100},
url = {http://www.ams.org/mathscinet-getitem?mr=89c:35100},
Key = {fds10086}
}

@article{fds10092,
Author = {Shearer, Michael and Schaeffer, David G.},
Title = {Three-phase flow in a porous medium and the classification
of non-strictly hyperbolic conservation laws},
Journal = {Transactions of the third Army conference on applied
mathematics and computing (Atlanta, Ga., 1985), pp.
509--517, 1986, U.S. Army Res. Office, Research Triangle
Park, NC},
MRNUMBER = {87j:76093},
url = {http://www.ams.org/mathscinet-getitem?mr=87j:76093},
Key = {fds10092}
}

@article{fds10096,
Author = {Schaeffer, David G.},
Title = {Topics in bifurcation theory},
Journal = {Systems of nonlinear partial differential equations (Oxford,
1982), pp. 219--262, 1983, Reidel, Dordrecht},
MRNUMBER = {85e:58107},
url = {http://www.ams.org/mathscinet-getitem?mr=85e:58107},
Key = {fds10096}
}

@article{fds8979,
Author = {Shearer, Michael and Schaeffer, David G.},
Journal = {Quart. Appl. Math., vol. 52, no. 3, pp. 579--600,
1994},
MRNUMBER = {95m:73030},
url = {http://www.ams.org/mathscinet-getitem?mr=95m:73030},
Key = {fds8979}
}

@article{fds10070,
Author = {Schaeffer, David G. and Shearer, Michael},
Title = {Unloading near a shear band: a free boundary problem for the
wave equation},
Journal = {Comm. Partial Differential Equations, vol. 18, no. 7-8, pp.
1271--1298, 1993},
MRNUMBER = {94i:35203},
url = {http://www.ams.org/mathscinet-getitem?mr=94i:35203},
Key = {fds10070}
}

@article{fds10264,
Author = {Schaeffer, David G.},
Title = {Wiener-Hopf factorization of the symbol of an elliptic
difference operator},
Journal = {J. Functional Analysis, vol. 5, pp. 383--394,
1970},
MRNUMBER = {41:7491},
url = {http://www.ams.org/mathscinet-getitem?mr=41:7491},
Key = {fds10264}
}

%% Papers Submitted
@article{fds184863,
Author = {S. Payne and B. Li and H. Song and D.G. Schaeffer and L.
You},
Title = {Self-organized pattern formation by a pseudo-Turing
mechanism},
Year = {2010},
Month = {Winter},
Key = {fds184863}
}

%% Preprints
@article{fds139451,
Author = {D.G. Schaeffer and A. Catlla and T. Witelski and E. Monson and A.
Lin},
Title = {Annular patterns in reaction-diffusion systems and their
implications for neural-glial interactions},
Year = {2008},
Key = {fds139451}
}



dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320