6,807 research outputs found
Strong convergence rates of probabilistic integrators for ordinary differential equations
Probabilistic integration of a continuous dynamical system is a way of
systematically introducing model error, at scales no larger than errors
introduced by standard numerical discretisation, in order to enable thorough
exploration of possible responses of the system to inputs. It is thus a
potentially useful approach in a number of applications such as forward
uncertainty quantification, inverse problems, and data assimilation. We extend
the convergence analysis of probabilistic integrators for deterministic
ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\
Comput.}, 2017), to establish mean-square convergence in the uniform norm on
discrete- or continuous-time solutions under relaxed regularity assumptions on
the driving vector fields and their induced flows. Specifically, we show that
randomised high-order integrators for globally Lipschitz flows and randomised
Euler integrators for dissipative vector fields with polynomially-bounded local
Lipschitz constants all have the same mean-square convergence rate as their
deterministic counterparts, provided that the variance of the integration noise
is not of higher order than the corresponding deterministic integrator. These
and similar results are proven for probabilistic integrators where the random
perturbations may be state-dependent, non-Gaussian, or non-centred random
variables.Comment: 25 page
Invariants of differential equations defined by vector fields
We determine the most general group of equivalence transformations for a
family of differential equations defined by an arbitrary vector field on a
manifold. We also find all invariants and differential invariants for this
group up to the second order. A result on the characterization of classes of
these equations by the invariant functions is also given.Comment: 13 page
The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures
Newly introduced generalized Poisson structures based on suitable
skew-symmetric contravariant tensors of even order are discussed in terms of
the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are
expressed as conditions on these tensors, the cohomological contents of which
is given. In particular, we determine the linear generalized Poisson structures
which can be constructed on the dual spaces of simple Lie algebras.Comment: 29 pages. Plain TeX. Phyzzx needed. An example and some references
added. To appear in J. Phys.
Symmetry, singularities and integrability in complex dynamics III: approximate symmetries and invariants
The different natures of approximate symmetries and their corresponding first
integrals/invariants are delineated in the contexts of both Lie symmetries of
ordinary differential equations and Noether symmetries of the Action Integral.
Particular note is taken of the effect of taking higher orders of the
perturbation parameter. Approximate symmetries of approximate first
integrals/invariants and the problems of calculating them using the Lie method
are considered
Nuclear symmetry energy and its density slope at normal density extracted from global nucleon optical potentials
Based on the Hugenholtz-Van Hove theorem, it is shown that both the symmetry
energy E and its density slope at normal density
are completely determined by the global nucleon optical potentials
that can be extracted directly from nucleon-nucleus scatterings, (p,n) charge
exchange reactions and single-particle energy levels of bound states. Adopting
a value of for the nucleon effective k-mass in symmetric nuclear
matter at and averaging all phenomenological isovector nucleon
potentials constrained by world data available in the literature since 1969,
the best estimates of MeV and MeV are
simultaneously obtained. Uncertainties involved in the estimates are discussed.Comment: 4 pages including 2 figure
Ordinary differential equations which linearize on differentiation
In this short note we discuss ordinary differential equations which linearize
upon one (or more) differentiations. Although the subject is fairly elementary,
equations of this type arise naturally in the context of integrable systems.Comment: 9 page
Equation of state of the hot dense matter in a multi-phase transport model
Within the framework of a multi-phase transport model, we study the equation
of state and pressure anisotropy of the hot dense matter produced in central
relativistic heavy ion collisions. Both are found to depend on the
hadronization scheme and scattering cross sections used in the model.
Furthermore, only partial thermalization is achieved in the produced matter as
a result of its fast expansion
Nucleon-nucleon cross sections in neutron-rich matter and isospin transport in heavy-ion reactions at intermediate energies
Nucleon-nucleon (NN) cross sections are evaluated in neutron-rich matter
using a scaling model according to nucleon effective masses. It is found that
the in-medium NN cross sections are not only reduced but also have a different
isospin dependence compared with the free-space ones. Because of the
neutron-proton effective mass splitting the difference between nn and pp
scattering cross sections increases with the increasing isospin asymmetry of
the medium. Within the transport model IBUU04, the in-medium NN cross sections
are found to influence significantly the isospin transport in heavy-ion
reactions. With the in-medium NN cross sections, a symmetry energy of
was found most acceptable
compared with both the MSU isospin diffusion data and the presently acceptable
neutron-skin thickness in Pb. The isospin dependent part of isobaric nuclear incompressibility was further narrowed down to
MeV. The possibility of determining simultaneously the in-medium
NN cross sections and the symmetry energy was also studied. The proton
transverse flow, or even better the combined transverse flow of neutrons and
protons, can be used as a probe of the in-medium NN cross sections without much
hindrance from the uncertainties of the symmetry energy.Comment: 32 pages including 14 figures. Submitted to Phys. Rev.
Integrability of Lie systems and some of its applications in physics
The geometric theory of Lie systems will be used to establish integrability
conditions for several systems of differential equations, in particular Riccati
equations and Ermakov systems. Many different integrability criteria in the
literature will be analyzed from this new perspective and some applications in
physics will be given.Comment: 16 page
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