HomePage of Maciej P. Wojtkowski
My papers:
-
Geometry of Kalman filters
. . .
dvi file: 52676 bytes
Abstract
We present a geometric explanation of Kalman filters
in terms of a symplectic linear space and a special
quadratic form on it. It is an extension of the work
of Bougerol \cite{B1} with application of a different
metric introduced in \cite{L-W}.
The new results are contained in Theorems 1 and 4.
-
Gaussian thermostats as geodesic flows of nonsymmetric linear connections
. . .
dvi file: 52676 bytes
Abstract
We establish that Gaussian thermostats are geodesic flows
of special metric connections. We give sufficient conditions for
hyperbolicity of geodesic flows of metric connections
in terms of their curvature and torsion.
- Design of hyperbolic billiards
. . .
pdf file: 216674 bytes
Abstract
We formulate a general framework for the construction of
hyperbolic billiards.
Spherical symmetry is exploited for a simple treatment
of billiards with spherical caps and soft billiards in
higher dimensions. Other examples include the Papenbrock stadium.
- Hyperbolic billiards
. . .
pdf file: 216674 bytes
Abstract
The article published in the Encyclopedia of Mathematical Physics,
Elsevier 2006.
- Rigidity of some Weyl manifolds with
nonpositive sectional curvature
. . .
PostScript file: 427907 bytes
Abstract
We provide a list of all locally metric
Weyl connections with nonpositive sectional
curvatures on two types of manifolds, n-dimensional tori $\Bbb T^n$
and $\Bbb M^n =\Bbb S^1\times\Bbb S^{n-1}$
with the standard conformal structures.
For $\Bbb M^n$ we prove that it carries no other Weyl
connections with nonpositive sectional curvatures, locally metric or
not. For $\Bbb T^n$ we prove the same in the more narrow class of integrable
connections.
- Rigidity of some Weyl manifolds with
nonpositive sectional curvature
. . .
pdf file: 169915 bytes
- Weyl Manifolds and Gaussian Thermostats
. . .
PostScript file: 460820 bytes
Abstract
A relation betwen Weyl connections and Gaussian
Thermostats is exposed and exploited.
Published in the Proceedings of the International Congress of
Mathematicians, Beijing 2002, Vol III, (2002), 511-523.
- Weyl Manifolds and Gaussian Thermostats
. . .
pdf file: 218122 bytes
- Monotonicity, J-algebra of Potapov
and Lyapunov exponents . . .
PostScript file: 283019 bytes
Abstract We present a new
approach and a generalization of
the estimates of Lyapunov exponents
developed first in \cite{W2} in the symplectic case.
The work of Lewowicz \cite{L}, Markarian \cite{M},
and our \cite{W1}, \cite{W2}, \cite{W5},
are combined with the $\Cal J$--algebra of Potapov,
\cite{P1},\cite{P2},\cite{P3}.
We obtain a general theory which we then specify to the
symplectic case. The appendix contains a simple
application to the gas of hard spheres.
Published in Smooth Ergodic Theory and Its Applications,
Proceedings of Symposia in Pure Mathematics, Vol 69, AMS (2001), 499-521.
- W-flows on Weyl manifolds and Gusssian
Thermostats . . .
PostScript file: 291876 bytes
Abstract We introduce
W-flows, by modifying the geodesic flow on
a Weyl manifold, and show that they coincide with the
isokinetic dynamics. We establish some connections between
negative curvature of the Weyl structure and the
hyperbolicity of W-flows, generalizing in dimension 2
the classical result of
Anosov on Riemannian geodesic flows. In higher dimensions
we establish only weaker hyperbolic properties.
We extend the theory to billiard W-flows and introduce the
Weyl counterparts of Sinai billiards.
We obtain that the isokinetic Lorentz gas with the constant
external field $E$ and scatterers of radius $r$, studied
by Chernov, Eyink, Lebowitz and Sinai in \cite{Ch-E-L-S},
is uniformly hyperbolic, if only $r|E| < 1$, and this condition is
sharp.
Published in
J. Math. Pures Appl. 79,10 (2000) 953-974.
- Magnetic Flows and Gaussian Thermostats . . .
PostScript file: 182546 bytes
Abstract We consider a class of
flows which include both magnetic flows and Gaussian thermostats of
external fields. We give sufficient conditions for such
flows on manifolds of negative sectional curvature to be Anosov.
Published in Fundamenta Mathematicae, Vol. 163, (177 -- 191), 2000.
- Complete Hyperbolicity in
Hamiltonian Systems with Linear Potential
and Elastic Collisions . . .
PostScript file: 166586 bytes
Abstract We present a class of
systems with all
Lyapunov exponents different from zero (completely hyperbolic).
They are obtained by the restriction of the configuration
space of a simple completely
integrable system with linear potential and elastic collisions.
We show that special geometry of the configuration space
is necessary.
We survey concrete realizations of this scheme discussed
previously in \cite{W6} and elsewhere.
Published in Reports on Mathematical Physics, Vol. 44,
(301 -- 312), 1999.
- (with Carlangelo Liverani)
Conformally Symplectic Dynamics and Symmetry
of the Lyapunov Spectrum . . .
PostScript file: 208248 bytes
Abstract A generalization of the
Hamiltonian formalism is studied and the symmetry of
the Lyapunov spectrum established for the resulting systems.
The formalism is applied to the Gausssian isokinetic
dynamics of interacting particles with hard core collisions
and other systems.
Published in Communications in Mathematical Physics, Vol.
194, (47 -- 60), 1998.
- Hamiltonian Systems with Linear Potential
and Elastic Constraints . . .
PostScript file: 356987 bytes
Abstract We consider a class of Hamiltoniansystems with linear
potential, elastic
constraints and arbitrary number of degrees of freedom.
We establish sufficient conditions for complete hyperbolicity of the
system.
Published in Fundamenta Mathematicae, Vol. 157, (305 -- 341), 1998.
- Hamiltonian Systems with Linear Potential
and Elastic Constraints . . .
pdf file: 461552 bytes