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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,
             Philadelphia, Pa.},
   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,
             SIAM, Philadelphia, Pa.},
   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
             produced by a 1D mapping. To address these shortcomings, ad
             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 (Woodbury, N.Y.)},
   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
             configurations of an elastic body under dead-load
             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},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000284759600003&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   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 = {The European Physical Journal. E, Soft Matter},
   Volume = {7},
   Number = {4},
   Pages = {353-370},
   Year = {2002},
   Month = {April},
   url = {http://dx.doi.org/10.1140/epje/i2002-10011-7},
   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.},
   Doi = {10.1140/epje/i2002-10011-7},
   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
             that have been linked with arrhythmia. © 2008 Society for
             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
             under nonproportional loading},
   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 Kayıkçı, O and Schaeffer, DG and Magwene,
             PM},
   Title = {Modeling mutant phenotypes and oscillatory dynamics in the
             Saccharomyces cerevisiae cAMP-PKA pathway.},
   Journal = {Bmc Systems Biology},
   Volume = {7},
   Pages = {40},
   Year = {2013},
   Month = {Winter},
   url = {http://dx.doi.org/10.1186/1752-0509-7-40},
   Abstract = {The cyclic AMP-Protein Kinase A (cAMP-PKA) pathway is an
             evolutionarily conserved signal transduction mechanism that
             regulates cellular growth and differentiation in animals and
             fungi. We present a mathematical model that recapitulates
             the short-term and long-term dynamics of this pathway in the
             budding yeast, Saccharomyces cerevisiae. Our model is aimed
             at recapitulating the dynamics of cAMP signaling for
             wild-type cells as well as single (pde1Δ and pde2Δ) and
             double (pde1Δpde2Δ) phosphodiesterase mutants.Our model
             focuses on PKA-mediated negative feedback on the activity of
             phosphodiesterases and the Ras branch of the cAMP-PKA
             pathway. We show that both of these types of negative
             feedback are required to reproduce the wild-type signaling
             behavior that occurs on both short and long time scales, as
             well as the the observed responses of phosphodiesterase
             mutants. A novel feature of our model is that, for a wide
             range of parameters, it predicts that intracellular cAMP
             concentrations should exhibit decaying oscillatory dynamics
             in their approach to steady state following glucose
             stimulation. Experimental measurements of cAMP levels in two
             genetic backgrounds of S. cerevisiae confirmed the presence
             of decaying cAMP oscillations as predicted by the model.Our
             model of the cAMP-PKA pathway provides new insights into how
             yeast respond to alterations in their nutrient environment.
             Because the model has both predictive and explanatory power
             it will serve as a foundation for future mathematical and
             experimental studies of this important signaling
             network.},
   Doi = {10.1186/1752-0509-7-40},
   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},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000220192800010&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   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},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000263103100004&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   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},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000263103100008&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   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.},
   Title = {Unloading near a shear band in granular material},
   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
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Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320