Vered Rom-Kedar
Department
of computer
science and applied mathematics
Weizmann Institute
Research interests and
publications:
To
chronological publication
list
|
Billiard like and Impact-like systems
|
Transport
theory |
Chaotic fluid mixing |
|
Exponentially
small splitting Abstract
Publications |
Forced NLS
& Parabolic resonances |
Modeling & medicine |
with V. Gelfreich, A. Rapoport, K. Shah and D. Turaev
Main
motivation and
results:
One of the underlying assumptions of statistical mechanics is the Boltzmann ergodic
hypothesis
which roughly states that the dynamics of a sufficiently large number
of small
molecules in a box is ergodic. The
mathematical
investigation of this hypothesis had focused mainly on studying the
dynamics of
hard spheres, and lead to fundamental developments in the theory of
dynamical
systems. Studying the motion of n "real" particles, having smooth
steep repelling potentials, is a mathematical challenge with important
implications to Physics. These are the billiard-like systems.
To
this aim we develop methodologies for analyzing Hamiltonian systems
with steep
repelling potentials by using the singular billiard limit.
Mathematically -
they allow to study far from integrable
systems in
arbitrary dimension. Physically - they allow to
study
when the widely used idealized billiard model works, and, more
surprisingly,
when it fails. These methodologies are being extended to steep impact
systems - here, in the singular limit, particles move under non-trivial
potential in the domain interior and reflect from the domain boundary
as in the billiards.
In
[R15,R17,R21] we establish that in the two
dimensional
case the smooth steep potentials can indeed ruin ergodicity.
These findings lead to an experimental realization of this phenomenon
in an
atom-optic system (Kaplan et al, PRL, 2001). A similar
approach may be utilized to predict and control the form of the chaotic
scattering function of various scattering-by-steep-potentials problems
[R29].
In [R28], the multi-dimensional analog is analyzed: we proved that elliptic orbits appear in systems with steep smooth potentials that limit to Sinai billiards for arbitrarily large dimension (see figure for effective stability islands in a 20-dimensional phase space). Admittedly, the considered special example cannot be directly linked to the smooth many particles case. Nonetheless, this construction, of stabilizing mechanism in multi-dimensional highly unstable systems, suggests that analogous stabilizing mechanisms may exist in the smooth Boltzmann problem.
More recently [R34,R38, R44], we study classical particles accelerators: billiards with oscillating boundaries. While it has been already established that 2D accelerators can lead to unbounded growth of the averaged energy of ensembles of particles (i.e. to heating), we are the first to construct accelerators in which this growth is exponential in time.
We have demonstrated that the exponential in time energy growth mechanism may be induced by a billiard with mixed phase space (the Oscillating Mushroom). Moreover, we derived adiabatic theory for non-ergodic systems from which the exponential growth rate is predicted and is then corroborated numerically [R46].
In [R59] we proved that the N repelling particle system in d>1 dimensions is not ergodic even in the high energy limit: we proved that choreographic KAM stable motion exists whenever the billiard table has a non-degenerate elliptic periodic orbit and also that for N particles in a d-dimensional box the motion of particles moving in parallel lines without colliding is KAM stable for some sets of distances between the particles.
| [R59] | V.Rom-Kedar and D.Turaev | Stable motions of high energy particles interacting via a repelling potential | Communications in Mathematical Physics 405 (6), 150, 2024 | |
[R48] |
K.
Shah, V.Gelfreich,
V.
Rom-Kedar and D. Turaev |
Leaky
Fermi Accelerators |
PRE 91, 062920 (2015) | |
| [R46] | Oscillating mushrooms: adiabatic theory for a non-ergodic system | Journal of Physics A : Mathematical and Theoretical, Volume 47 (Number 39). Article number 395101 . ISSN 1751-8113 | ||
| [R44] | V.Gelfreich, V. Rom-Kedar and D. Turaev | Fermi acceleration and adiabatic invariants for non-autonomous billiards | Chaos 22,
033116 (2012); http://dx.doi.org/10.1063/1.4736542 (21 pages). |
|
| [R43] | V. Rom-Kedar and D.Turaev, | Billiards: a singular perturbation limit of smooth Hamiltonian flows" | Chaos 22,
026102 (2012); http://dx.doi.org/10.1063/1.4722010 (21 pages) |
|
| [R38] | V. Gelfreich, V. Rom-Kedar, K. Shah, D. Turaev, | Robust
exponential
accelerators |
PRL 106, 074101, 2011. | |
| [R34] | K. Shah, D. Turaev and V. Rom-Kedar, | Exponential energy growth in a Fermi accelerator, | Phys. Rev. E 81, 056205, 2010 | |
|
[R29] |
A.
Rapoport and V. Rom-Kedar |
Phys. Rev E., 77, 016207 (2008). |
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|
A. Rapoport, V. Rom-Kedar and D. Turaev |
Comm. Math. Phys, 279, 497-534, 2008. |
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V. Rom-Kedar and D.Turaev |
Physica D, 130:187-210, 1999. |
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A. Rapoport, V. Rom-Kedar and D. Turaev. |
Approximating
multi-dimensional Hamiltonian flows by billiards. |
Comm. Math. Phys., 272(3), 567-600, 2007. |
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|
[RP8]
|
D. Turaev and V. Rom-Kedar |
In
Chaos, kinetics and nonlinear dynamics in fluids and Plasmas (Carry-Le Rouet, 1997), volume 511 of Lecture Notes in
Phys., pages 17-50. Springer, |
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|
[R15] |
D. Turaev and V. Rom-Kedar |
Nonlinearity, 11(3):575-600, 1998. |
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D. Turaev and V. Rom-Kedar |
|
J. Stat. Phys., 112(3-4):765-813, 2003 |
|
|
[R25]
|
A. Rapoport, V. Rom-Kedar |
Non-ergodicity of the motion in three dimensional steep repelling dispersing potentials. |
Chaos 16, 043108, 2006. |
with M. Kloc, L. Lerman, M. Pnueli, L. Becker, S. Elliott, B. Firester, S. Gonen Cohen, K. Fraczek, M. Pnueli, O. Yaniv
Main
motivation and
results: By impact like systems (sometimes called generalized billiards) we mean mechanical Hamiltonian systems which have smooth dynamics inside a domain D whereas near the boundary there is a steep, billiard like potential, so that when the steepness paramter goes to zero the system approaches elastic reflections from the domain boundary.
In [R40] we proposed that such systems provide good models to chemical reactions: the steep potentials appear from the atomic repulsion forces whereas the smooth potential models the attraction forces between the atoms. In particular, we showed that the saddle in the corner system provides insights regarding the validity of transition state theore for co-linear reactions. In [R45] we developed the framework for analyzing general systems of the impact-like type and explained the significance of the Hill's region in their analysis. In [R50,R52,R54] we introduced the Near Integrable Hamiltonian impact-like systems and analyzed their behavior near regular and near tangent tori. In [R57] (submitted in 2020, published in 2024) we introduced the class of pseudointegrable Hamiltonian impact system. In [R53] we analyzed the ergodic properties of a class PIHIS and in [R56] we studied the quantization of such systems.
[R57] L. Becker, S. Elliott, B. Firester, S. Gonen Cohen, M. Pnueli and V. Rom-Kedar, Impact Hamiltonian systems and polygonal billiards, in "Hamiltonian Systems: Dynamics, Analysis, Applications" edited by Albert Fathi, Philip J. Morrison, Tere M-Seara, Sergei Tabachnikov, MSRI publications, Vol 72 ,2024
[R56] O. Yaniv and V. Rom-Kedar Quantum pseudointegrable Hamiltonian impact system , Phys. Rev. E 107, 054221, 2023
[R54] M. Pnueli and V. Rom-Kedar, Near tangent dynamics in a class of Hamiltonian impact systems, SIAM J. Appl. Dyn. Syst., 21 (3), 2000-2046 , 2022
[R53] K. Fraczek and V. Rom-Kedar, Non-uniform ergodic properties of Hamiltonian flows with impacts , Ergodic Theory and Dynamical Systems, First View , pp. 1 - 63, DOI:10.1017/etds.2021.106, 2021
[R52] M. Pnueli and V. Rom-Kedar, On the structure of Hamiltonian impact systems, Nonlinearity 34 (4), 2611, 2021. https://doi.org/10.1088/1361-6544/abb450
[R50] M. Pnueli and V. Rom-Kedar, On Near Integrability
of Some Impact Systems, SIAM J. Appl. Dyn.
Syst., 17(4), 2707–2732. (26 pages), 2018, https://doi.org/10.1137/18M1177937.
[R45] M. Kloc and V. Rom-Kedar "Smooth Hamiltonian systems with soft
impacts" SIAM J. Applied
Dynamical Systems (Vol. 13, Issue 3), 2014.
Multi-dimensional
systems, forced NLS and Parabolic Resonances
with
Main
motivation and
results:
Near-integrable systems had been
extensively studied
since the days of Poincaré, yet,
the structure
of their chaotic solutions is still mostly unknown. A decade ago we
discovered
a new kind of instability in such systems, the parabolic resonance
instability.
It was discovered while studying a geophysical application
[R9,R11,R12], and
then we studied it in the multi-dimensional setting [R19,R20,R22]. More
recently, we proposed that the hierarchy of bifurcation framework
[RP11,R23] is
convenient for identifying the dominant instability mechanisms that
appear in
specific multi-dimensional models.
Notably, applying this framework to the conservatively forced Non-Linear Schrödinger (NLS) equation, conditions under which a truncated form of this equation has a parabolic resonance instability were derived [R23]. It was then demonstrated that such instabilities do appear in the original PDE model - the forced NLS - and correspond to a new type of solutions of this extensively studied equation [R24]. As a blind search for such solutions is a formidable numerical task, our analysis is pivotal for identifying what are the interesting parameter values and initial conditions for which new types of solutions - like the parabolic resonant ones - appear.
NEWS:
1. In [R32], we showed that the parabolic resonance mechanism can even cause spatial decoherence of small amplitude nearly flat initial data.
2. We have classified the structure of the unpertubed energy surfaces of a "generic" class of near-integrable two-degrees of freedom systems [R31] . This classification shows that there are some mechanisms of instability in two-d.o.f. systems that have not been studied yet!
3. In [R36] we developed a framework to study the near integrable PDE structure. This paper was selected to be in the Nonlinearity Highly downloaded Collection for 2010.
4. In [R35] we provide a framework for studyng analytically the parabolic resonance instability by using the adiabatic chaos framework. Most importantly, we show that the volume of the chaotic zone depends on the perturbation parameter as a power law, thus it is numerically observable.
| [R35] | V. Rom-Kedar and D. Turaev, | The symmetric Parabolic Resonance Instability, | Nonlinearity 23 1325-1351, 2010. |
| [R36] | E. Shlizermann and V Rom-Kedar, | Classification of solutions of the forced periodic nonlinear Schr\"{o}dinger equation, | Nonlinearity 23 (9), 2183-2218, 2010. |
| [R32] |
E.
Shlizermann
and V. Rom-Kedar; |
Parabolic
Resonance: |
Physical
Review Letters, 102, 033901, 2009. |
| [R31] | M.
Radnovic and V. Rom-Kedar |
Foliations of Isonergy Surfaces and Singularities of Curves | Regular and Chaotic Dynamics, Vol. 13, No. 6, pp. 645–668, J. Moser-80 memorial issue, 2008. |
| [R24] |
E. Shlizermann
and V Rom-Kedar |
Three types of chaos in the forced nonlinear Schrodinger
equation. |
Physical Review Letters, 96, 024104,
2006. |
| [R23] |
E. Shlizermann
and V. Rom-Kedar |
Hierarchy of bifurcations in the truncated
and forced NLS model |
Chaos 15(1) 013107,
22 pp, 2005. |
|
[R9] |
V. Rom-Kedar |
Chaos, 7(1):148-158, 1997. |
|
|
[R22] |
A. Litvak-Hinenzon
and V. Rom-Kedar |
SIAM J. Appl.
Dyn. Syst. 3(4), 525—573, 2004. |
|
|
[R11] |
V. Rom-Kedar and N.
Paldor |
Bull. Amer. Mete. Soc., 78(12):2779-2784, 1997. |
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|
[R12] |
V.Rom-Kedar, Y. Dvorkin, and N. Paldor;
|
Chaotic
Hamiltonian dynamics of particle's
horizontal motion in the atmosphere. |
Physica D, 106(3-4):389-431, 1997. |
|
[RP5] |
V. Rom-Kedar; Y.
Dvorkin, and N.Paldor |
In M.F. Shlesinger,
Zaslavsky, G.M.thinspace
and U.Frisch, editors, Proc. on Le'vy Flights and Related Phenomena, Lecture
Notes in Physics, pages 72-87. Springer-Verlag,
1995. |
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|
[RP6] |
V. Rom-Kedar |
Some characteristics of
two-degrees-of-freedom Hamiltonian flows. |
In Doveil Benkadda and Elskens,
editors, Transport, Chaos and Plasma Physics, II, Proc., Marseille,
pages 164-178, |
|
[RP7] pdf Abstract |
V. Rom-Kedar |
Transport in a class of n-d.o.f. systems. |
In Hamiltonian systems with three or
more degrees of freedom (S'Agar'o,
1995) , pages 538-543. Kluwer Acad. Publ., |
|
[RP9] pdf Abstract |
A. Litvak-Hinenzon
and V. Rom-Kedar |
Parabolic resonances in near integrable Hamiltonian systems. |
In D.S. Broomhead,
E.A. Luchinskaya, P.V.E. McClintock, and T.Mullin, editors, emStochaos:
Stochastic and Chaotic Dynamics in the Lakes, pages 358-368.
American |
|
[R19] pdf Abstract |
A. Litvak-Hinenzon
and V. Rom-Kedar |
Resonant tori
and instabilities in Hamiltonian systems. |
Nonlinearity, 15(4):1149-1177, 2002. |
|
[R20] pdf Abstract |
A. Litvak-Hinenzon
and V. Rom-Kedar |
Parabolic resonances in 3 degree of freedom
near-integrable Hamiltonian systems |
Physica
D, 164(3-4):213-250, 2002. |
|
[RP12] pdf Abstract |
E. Shlizermann
and V. Rom-Kedar |
Characterization of Orbits in the Truncated
and Forced Nonlinear Shrodinger Model. |
ENOC, 2005 |
|
[RP11] pdf Abstract |
E. Shlizerman
and V. Rom-Kedar |
Energy surfaces and hierarchies of bifurcations - instabilities in the forced truncated NLS. |
In P. Collet
et al. editors, Chaotic Dynamics and Transport in Classical and Quantum
Systems. Kluwer Academic Press in NATO
Science Series C, 2004. |
Modeling
and Medicine: models of the hematopoesis system:
with R. Malka,
E. Shochat
MD/PhD, and Prof. L. Segel (deceased,
2005).
Main
motivation
and results: Why do G-CSF (Granulocyte Colony Stimulating Factor)
injections
help some neutropenic patients (patients
with very
low counts of this kind of white blood cells) and not others?
In [R27] we employ dynamical
systems approach to develop a simple model (not Hamiltonian) that is
adequate
for describing the natural dynamics of the neutrophils
and the G-CSF levels in the blood on a time scale of several days. In
[R30] we
use a variant of this model to study the response of patients that
received
chemotherapy and thus became neutropenic
to various
protocols of G-CSF support.
More precisely, in [R30] we built axiomatically a mathematical model that describes the neutrophils’ dynamics. We showed that patients fall into three groups according to their blood counts, and that these groups respond differently to the growth factor therapy. The first group responds well. The current treatment is ineffective for the second group, but our results imply that a change in the administration protocol may help. In the third group, treatment with growth factors alone is insufficient at any dose, and the patients need an alternative therapy. These conclusions stem directly from the mathematical model, which makes it possible to predict outcomes and improve treatment. While the immune system is exceedingly complex, the model is simple, and that’s precisely its strength. We are proposing to test these recommendations in clinical trials. If these succeed, treatment efficiency would increase, and perhaps some lives could be saved.
NEWS:
We study the development of bacterial infections under neutropenic conditions by studying the neutrophils-bacteria interaction in-vitro [R33] and by combining this study with the blood neutrophils dynamics [R39].
| [R42] | Roy Malka, Baruch Wolach, Ronit Gavrieli, Eliezer Shochat and Vered Rom-Kedar | Evidence for bistable bacteria-neutrophil interaction and its clinical implications | J. Clin Invest. doi:10.1172/JCI59832, 2012. See also commentary . |
| [R39] | R. Malka and V. Rom-Kedar | Bacteria--Phagocytes Dynamics, Axiomatic Modelling and Mass-Action Kinetics, | Mathematical Biosciences and Engineering, 8(2), 475-502, 2011. |
| [R33] | R. Malka R, E. Shochat, V. Rom-Kedar; | Bistability and Bacterial Infections. | PLoS ONE 5(5): e10010. doi:10.1371/journal.pone.0010010, 2010. |
|
[R27] pdf
Abstract |
E. Shochat, V.
Rom-Kedar and L. Segel |
Bull. Math. Biology 69(7), 2299-2338, 2007. |
|
|
[R30] pdf
Abstract |
E. Shochat and
V. Rom-Kedar |
Clinical Cancer Research 14, 6354-6363, October 15, 2008 2008. |
Chaotic
fluid mixing :
with R. Aharon, D. F. Carlson, E. Fredj, H. Gildor, R. Mundel
A. Poje,
A. Leonard and S. Wiggins.
Main
motivation and
results:
The field of chaotic advection is concerned with the study of mixing of
passive
scalars in fluids with specified velocity fields that are
time-dependent yet
non-turbulent. Since the important paper by Aref
in
1984, it attracts a growing community of researchers coming from
diverse areas
of science and engineering and involves both practical problems and
beautiful
mathematical questions.
In my
Ph.D. studies we
considered
time periodic planar flows (the OVP model) and introduced several
dynamical
systems concepts and tools to this field; We identified the unstable
manifold
and KAM tori as the observed objects in
flow
visualizations and as controllers of transport, discovered lobe
dynamics, and
established the direct relation between lobe dynamics and fluid
transport
[R3,RP1]. These ideas influence theoretical and experimental studies of
fluid
mixing problems arising in geophysical applications, traditional lab
scale
experiments and microfluidics (e.g. these
findings
became the main topic of S. Wiggins popular book "Chaotic Transport in
Dynamical systems").
In [R16] we proved that for a large class of two-dimensional time-periodic fluid flows some mixing characteristics have a universal dependence on the flow frequency even in the non-perturbative regime. These results were realized experimentally (see e.g. Horner et al. JFM 02). In [R36] we show that upper bounds on eddy diffusivity may be found by combining LCS computations from RF data and airial photos of separating lines.
In [41] we show that minute 3d convection can untangle the chaotic tangles of the surface mixing.
In [47] we show that extreme values of observables
provide simple, intuitive,
and computationally cheap diagnostics for fluid mixing; Moreover, they
also enable a significant data reduction, since it is possible to
extract from their cumulative distribution functions much of the
relevant information regarding the existence, location, size and motion
of the coherent structures.
| R. Mundel, E. Fredj, H. Gildor and V. Rom-Kedar, | "New Lagrangian diagnostics for characterizing fluid flow mixing" | ||
| [R41] | Rotem Aharon, Vered Rom-Kedar, and Hezi Gildor, | "When complexity leads to simplicity: ocean surface mixing simplified by vertical convection" | Phys.
Fluids 24,
056603 (2012); \http://dx.doi.org/10.1063/1.4719147. ON THE COVER. |
| [R36] | D. F. Carlson, E. Fredj, H. Gildor, V. Rom-Kedar | Deducing an upper bound to the horizontal eddy diffusivity using a stochastic Lagrangian model, | Environmental Fluid Mechanics, 1567-7419 (Print) 1573-1510 (Online), 2010. |
|
[R3] |
V. Rom-Kedar, A. Leonard, and S. Wiggins |
An
analytical study of transport, mixing and
chaos in an unsteady vortical flow |
J. Fluid Mech., 214:347-394, 1990. |
|
A. Leonard, V. Rom-Kedar, and S.Wiggins |
proc. intl. conf. on the physics of chaos and systems far from equilibrium. In Nuclear Physics B (Proc. Suppl.), volume2, pages 179-190, 1987 |
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[R16] |
V. Rom-Kedar and A.C. Poje |
Universal
properties of chaotic transport in
the presence of diffusion |
Phys. of Fluids, 11(8):2044-2057, 1999. |
Transport
theory of periodically forced systems :
with A. Litvak Hinenzon and
G.M. Zaslavsky
Main
motivation
and results: The transport theory of periodically forced
Hamiltonian
systems is concerned with characterizing the motion of ensembles of
initial
conditions from one region in phase space to another.
In
[R2] we observed that the geometry of the unstable and stable manifolds
determines exactly all the transport rates between certain phase space
regions
for a general class of forced planar flows. This discovery of the
popular lobe
dynamics lead to a burst of theoretical and even some experimental
studies in
diverse fields of Physics ([R2] was generalized to various settings and
was
followed by numerous publications on its role in fluid mixing, chemical
reactions,
optics, astrophysics, ship stability, electroconvection
etc.). In subsequent works I developed perturbational
methods for calculating topological characteristics of these manifolds'
complex
geometry (called homoclinic tangles).
These studies
lead to analytical estimates of the transport rates for non-trivial
time
scales, to a lower bound on the topological entropy of homoclinic
tangles in open flows [R1,R7,R8], to analytical methods for detecting
multi-pulse solutions [R8] and to insights regarding the structure of
strange
attractors in slightly dissipative system [R10].
|
[R1] |
V. Rom-Kedar |
Transport rates of a family of
two-dimensional maps and flows |
Physica
D, 43:229-268, 1990. |
|
[R2] |
V. Rom-Kedar and S. Wiggins |
Transport in two-dimensional maps |
Archive for Rational Mech. and Anal.,
109(3):239-298, 1990. |
|
[R7] |
V.Rom-Kedar |
Homoclinic
tangles - classification and applications. |
Nonlinearity, 7:441-473, 1994. |
|
[R8] |
V.Rom-Kedar |
Secondary homoclinic
bifurcation theorems |
Chaos, 5(2):385-401, 1995. |
|
[R10] |
A. Litvak Hinenzon
and V. Rom-Kedar |
Symmetry breaking perturbations and strange
attractors. |
Phys. Rev. E, 55(5):4964-4978, 1997. |
|
[R18] |
V. Rom-Kedar and
G.M. Zaslavsky |
Islands of accelerator modes and homoclinic tangles.
|
Chaos, 9 (3):697-705, 1999. |
|
[RP2] |
V. Rom-Kedar |
Homoclinic
structures in open flows. |
In J.-D. Fournier and P.-L. Sulem, editors, emLarge Scale Structures in Nonlinear Physics, Proc., Lecture Notes in Physics, pages 50-72, Villefranche-sur-Mer, 1991. Springer-Verlag. |
|
[RP3] |
Y. Ponty, A. Pouquet, V. Rom-Kedar, and P.L. Sulem |
Dynamo in a nearly integrable
chaotic flow. |
In M.R.C. Proctor, editor, Proc. NATO
ASI: Theory of Solar and Planetary Dynamos, pages 241-248, |
|
[RP4] |
V. Rom-Kedar |
The topological approximation method. |
In Doveil Benkadda and Elskens,
editors, Transport, Chaos and Plasma Physics, Proc. , pages 39-57,
Marseille, 1994. World Scientific |
|
[RE1] |
V. Rom-Kedar and
G.M. Zaslavsky |
Chaotic kinetics and transport (overview). |
Chaos, 10(1):1-2, 2000. Focus Issue:
Chaotic kinetics and Transport, eds V.
Rom-Kedar and G. Zaslavsky. Including
papers from the Workshop on Nonlinear Dynamics: Chaos, Transport and
Transition to Turbulence held at the New York University, New York,
November 1998. |
|
[RP10] |
V. Rom-Kedar |
Frequency spanning homoclinic
families |
Commun.
Nonlinear Sci. Numer.
Simul., 8(3-4):149-169, 2003.
Chaotic transport and complexity in classical and quantum dynamics. |
Exponentially
small splitting :
|
[R5] |
C.Amick, S.C.E. Ching, L.P. Kadanoff,
and V.Rom-Kedar |
Beyond all orders: Singular perturbations in
a mapping |
J. Nonlinear Sci., 2:9-67, 1992. |
|
[R6] |
V.Rom-Kedar,
L.P. Kadanoff, S.C.E. Ching,
and C.Amick |
The breakup of a heteroclinic
connection in a volume preserving mapping. |
Physica D, 62:51-65, 1993 |
|
[R13] |
S. Pekarsky and V.
Rom-Kedar |
Uniform stochastic web in low-dimensional
Hamiltonian systems. |
Phys. Lett. A, 225:274-286, 1997. |
|
[R14] |
S. Pekarsky and V.
Rom-Kedar |
Degeneracies and
instabilities of the motion of a charged particle - the 4d web-map
model. |
Nonlinearity, 10(4):949-963, 1997. |