129 research outputs found
Global surfaces of section in the planar restricted 3-body problem
The restricted planar three-body problem has a rich history, yet many
unanswered questions still remain. In the present paper we prove the existence
of a global surface of section near the smaller body in a new range of energies
and mass ratios for which the Hill's region still has three connected
components. The approach relies on recent global methods in symplectic geometry
and contrasts sharply with the perturbative methods used until now.Comment: 11 pages, 1 figur
Closed geodesics in Alexandrov spaces of curvature bounded from above
In this paper, we show a local energy convexity of maps into
spaces. This energy convexity allows us to extend Colding and
Minicozzi's width-sweepout construction to produce closed geodesics in any
closed Alexandrov space of curvature bounded from above, which also provides a
generalized version of the Birkhoff-Lyusternik theorem on the existence of
non-trivial closed geodesics in the Alexandrov setting.Comment: Final version, 22 pages, 2 figures, to appear in the Journal of
Geometric Analysi
Birkhoff Theorem and Matter
Birkhoff's theorem for spherically symmetric vacuum spacetimes is a key
theorem in studying local systems in general relativity theory. However
realistic local systems are only approximately spherically symmetric and only
approximately vacuum. In a previous paper, we showed the theorem remains
approximately true in an approximately spherically symmetric vacuum space time.
In this paper we prove the converse case: the theorem remains approximately
true in a spherically symmetric, approximately vacuum space time.Comment: 7 pages, Revtex
Linear stability analysis of resonant periodic motions in the restricted three-body problem
The equations of the restricted three-body problem describe the motion of a
massless particle under the influence of two primaries of masses and
, , that circle each other with period equal to
. When , the problem admits orbits for the massless particle that
are ellipses of eccentricity with the primary of mass 1 located at one of
the focii. If the period is a rational multiple of , denoted ,
some of these orbits perturb to periodic motions for . For typical
values of and , two resonant periodic motions are obtained for . We show that the characteristic multipliers of both these motions are given
by expressions of the form in the limit . The coefficient is analytic in at and
C(e,p,q)=O(e^{\abs{p-q}}). The coefficients in front of e^{\abs{p-q}},
obtained when is expanded in powers of for the two resonant
periodic motions, sum to zero. Typically, if one of the two resonant periodic
motions is of elliptic type the other is of hyperbolic type. We give similar
results for retrograde periodic motions and discuss periodic motions that
nearly collide with the primary of mass
Measurement in biological systems from the self-organisation point of view
Measurement in biological systems became a subject of concern as a
consequence of numerous reports on limited reproducibility of experimental
results. To reveal origins of this inconsistency, we have examined general
features of biological systems as dynamical systems far from not only their
chemical equilibrium, but, in most cases, also of their Lyapunov stable states.
Thus, in biological experiments, we do not observe states, but distinct
trajectories followed by the examined organism. If one of the possible
sequences is selected, a minute sub-section of the whole problem is obtained,
sometimes in a seemingly highly reproducible manner. But the state of the
organism is known only if a complete set of possible trajectories is known. And
this is often practically impossible. Therefore, we propose a different
framework for reporting and analysis of biological experiments, respecting the
view of non-linear mathematics. This view should be used to avoid
overoptimistic results, which have to be consequently retracted or largely
complemented. An increase of specification of experimental procedures is the
way for better understanding of the scope of paths, which the biological system
may be evolving. And it is hidden in the evolution of experimental protocols.Comment: 13 pages, 5 figure
Derivation of the Rules of Quantum Mechanics from Information-Theoretic Axioms
Conventional quantum mechanics with a complex Hilbert space and the Born Rule
is derived from five axioms describing properties of probability distributions
for the outcome of measurements. Axioms I,II,III are common to quantum
mechanics and hidden variable theories. Axiom IV recognizes a phenomenon, first
noted by Turing and von Neumann, in which the increase in entropy resulting
from a measurement is reduced by a suitable intermediate measurement. This is
shown to be impossible for local hidden variable theories. Axiom IV, together
with the first three, almost suffice to deduce the conventional rules but allow
some exotic, alternatives such as real or quaternionic quantum mechanics. Axiom
V recognizes a property of the distribution of outcomes of random measurements
on qubits which holds only in the complex Hilbert space model. It is then shown
that the five axioms also imply the conventional rules for all dimensions.Comment: 20 pages, 6 figure
Constraining the parameters of binary systems through time-dependent light deflection
A theory is derived relating the configuration of the cores of active
galaxies, specifically candidates for presumed super-massive black hole
binaries (SMBHBs), to time-dependent changes in images of those galaxies. Three
deflection quantities, resulting from the monopole term, mass quadrupole term,
and spin dipole term of the core, are examined. The resulting observational
technique is applied to the galaxy 3C66B. This technique is found to under
idealized circumstances surpass the technique proposed by Jenet et al. in
accuracy for constraining the mass of SMBHB candidates, but is exceeded in
accuracy and precision by Jenet's technique under currently-understood likely
conditions. The technique can also under favorable circumstances produce
results measurable by currently-available astronomical interferometry such as
very-long baseline-interferometry (VLBI).Comment: 15 pages, 2 figures, accepted in General Relativity & Gravitatio
A few things I learnt from Jurgen Moser
A few remarks on integrable dynamical systems inspired by discussions with
Jurgen Moser and by his work.Comment: An article for the special issue of "Regular and Chaotic Dynamics"
dedicated to 80-th anniversary of Jurgen Mose
Central extensions of current groups in two dimensions
In this paper we generalize some of these results for loop algebras and
groups as well as for the Virasoro algebra to the two-dimensional case. We
define and study a class of infinite dimensional complex Lie groups which are
central extensions of the group of smooth maps from a two dimensional
orientable surface without boundary to a simple complex Lie group G. These
extensions naturally correspond to complex curves. The kernel of such an
extension is the Jacobian of the curve. The study of the coadjoint action shows
that its orbits are labelled by moduli of holomorphic principal G-bundles over
the curve and can be described in the language of partial differential
equations. In genus one it is also possible to describe the orbits as conjugacy
classes of the twisted loop group, which leads to consideration of difference
equations for holomorphic functions. This gives rise to a hope that the
described groups should possess a counterpart of the rich representation theory
that has been developed for loop groups. We also define a two-dimensional
analogue of the Virasoro algebra associated with a complex curve. In genus one,
a study of a complex analogue of Hill's operator yields a description of
invariants of the coadjoint action of this Lie algebra. The answer turns out to
be the same as in dimension one: the invariants coincide with those for the
extended algebra of currents in sl(2).Comment: 17 page
On collineation groups of finite projective spaces
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46263/1/209_2005_Article_BF01122320.pd
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