6,950 research outputs found
Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse
We study billiard dynamics inside an ellipse for which the axes lengths are
changed periodically in time and an -small quartic polynomial
deformation is added to the boundary. In this situation the energy of the
particle in the billiard is no longer conserved. We show a Fermi acceleration
in such system: there exists a billiard trajectory on which the energy tends to
infinity. The construction is based on the analysis of dynamics in the phase
space near a homoclinic intersection of the stable and unstable manifolds of
the normally hyperbolic invariant cylinder , parameterised by the
energy and time, that corresponds to the motion along the major axis of the
ellipse. The proof depends on the reduction of the billiard map near the
homoclinic channel to an iterated function system comprised by the shifts along
two Hamiltonian flows defined on . The two flows approximate the
so-called inner and scattering maps, which are basic tools that arise in the
studies of the Arnold diffusion; the scattering maps defined by the projection
along the strong stable and strong unstable foliations of the
stable and unstable invariant manifolds at the homoclinic
points. Melnikov type calculations imply that the behaviour of the scattering
map in this problem is quite unusual: it is only defined on a small subset of
that shrinks, in the large energy limit, to a set of parallel lines
as .Comment: 25 page
Off-Shell Hodge Dualities in Linearised Gravity and E11
In a spacetime of dimension n, the dual graviton is characterised by a Young
diagram with two columns, the first of length n-3 and the second of length one.
In this paper we perform the off-shell dualisation relating the dual graviton
to the double-dual graviton, displaying the precise off-shell field content and
gauge invariances. We then show that one can further perform infinitely many
off-shell dualities, reformulating linearised gravity in an infinite number of
equivalent actions. The actions require supplementary mixed-symmetry fields
which are contained within the generalised Kac-Moody algebra E11 and are
associated with null and imaginary roots.Comment: 33 pages, 2 figures, nomenclature changed and comments added to the
conclusion
Portfolio optimization in the case of an asset with a given liquidation time distribution
Management of the portfolios containing low liquidity assets is a tedious
problem. The buyer proposes the price that can differ greatly from the paper
value estimated by the seller, the seller, on the other hand, can not liquidate
his portfolio instantly and waits for a more favorable offer. To minimize
losses in this case we need to develop new methods. One of the steps moving the
theory towards practical needs is to take into account the time lag of the
liquidation of an illiquid asset. This task became especially significant for
the practitioners in the time of the global financial crises. Working in the
Merton's optimal consumption framework with continuous time we consider an
optimization problem for a portfolio with an illiquid, a risky and a risk-free
asset. While a standard Black-Scholes market describes the liquid part of the
investment the illiquid asset is sold at a random moment with prescribed
liquidation time distribution. In the moment of liquidation it generates
additional liquid wealth dependent on illiquid assets paper value. The investor
has the logarithmic utility function as a limit case of a HARA-type utility.
Different distributions of the liquidation time of the illiquid asset are under
consideration - a classical exponential distribution and Weibull distribution
that is more practically relevant. Under certain conditions we show the
existence of the viscosity solution in both cases. Applying numerical methods
we compare classical Merton's strategies and the optimal consumption-allocation
strategies for portfolios with different liquidation-time distributions of an
illiquid asset.Comment: 30 pages, 1 figur
From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation
We start from known solutions of the Yang-Baxter equation with a spectral
parameter defined on the tensor product of two infinite-dimensional principal
series representations of the group or Faddeev's
modular double. Then we describe its restriction to an irreducible
finite-dimensional representation in one or both spaces. In this way we obtain
very simple explicit formulas embracing rational and trigonometric
finite-dimensional solutions of the Yang-Baxter equation. Finally, we construct
these finite-dimensional solutions by means of the fusion procedure and find a
nice agreement between two approaches
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