6,886 research outputs found
On the Hojman conservation quantities in Cosmology
We discuss the application of the Hojman's Symmetry Approach for the
determination of conservation laws in Cosmology, which has been recently
applied by various authors in different cosmological models. We show that
Hojman's method for regular Hamiltonian systems, where the Hamiltonian function
is one of the involved equations of the system, is equivalent to the
application of Noether's Theorem for generalized transformations. That means
that for minimally-coupled scalar field cosmology or other modified theories
which are conformally related with scalar-field cosmology, like gravity,
the application of Hojman's method provide us with the same results with that
of Noether's theorem. Moreover we study the special Ansatz. , which has been introduced for
a minimally-coupled scalar field, and we study the Lie and Noether point
symmetries for the reduced equation. We show that under this Ansatz, the
unknown function of the model cannot be constrained by the requirement of the
existence of a conservation law and that the Hojman conservation quantity which
arises for the reduced equation is nothing more than the functional form of
Noetherian conservation laws for the free particle. On the other hand, for
teleparallel gravity, it is not the existence of Hojman's conservation
laws which provide us with the special function form of functions, but
the requirement that the reduced second-order differential equation admits a
Jacobi Last multiplier, while the new conservation law is nothing else that the
Hamiltonian function of the reduced equation.Comment: 6 pages; minor corrections; accepted for publication by Physics
Letters B. arXiv admin note: substantial text overlap with arXiv:1503.0846
The interpretation of hard X-ray polarization measurements in solar flares
Observations of polarization of moderately hard X-rays in solar flares are reviewed and compared with the predictions of recent detailed modeling of hard X-ray bremsstrahlung production by non-thermal electrons. The recent advances in the complexity of the modeling lead to substantially lower predicted polarizations than in earlier models and more fully highlight how various parameters play a role in determining the polarization of the radiation field. The new predicted polarizations are comparable to those predicted by thermal modeling of solar flare hard X-ray production, and both are in agreement with the observations. In the light of these results, new polarization observations with current generation instruments are proposed which could be used to discriminate between non-thermal and thermal models of hard X-ray production in solar flares
Lie symmetries of (1+2) nonautonomous evolution equations in Financial Mathematics
We analyse two classes of evolution equations which are of special
interest in Financial Mathematics, namely the Two-dimensional Black-Scholes
Equation and the equation for the Two-factor Commodities Problem. Our approach
is that of Lie Symmetry Analysis. We study these equations for the case in
which they are autonomous and for the case in which the parameters of the
equations are unspecified functions of time. For the autonomous Black-Scholes
Equation we find that the symmetry is maximal and so the equation is reducible
to the Classical Heat Equation. This is not the case for the
nonautonomous equation for which the number of symmetries is submaximal. In the
case of the two-factor equation the number of symmetries is submaximal in both
autonomous and nonautonomous cases. When the solution symmetries are used to
reduce each equation to a equation, the resulting equation is of
maximal symmetry and so equivalent to the Classical Heat Equation.Comment: 15 pages, 1 figure, to be published in Mathematics in the Special
issue "Mathematical Finance
Analytic Behaviour of Competition among Three Species
We analyse the classical model of competition between three species studied
by May and Leonard ({\it SIAM J Appl Math} \textbf{29} (1975) 243-256) with the
approaches of singularity analysis and symmetry analysis to identify values of
the parameters for which the system is integrable. We observe some striking
relations between critical values arising from the approach of dynamical
systems and the singularity and symmetry analyses.Comment: 14 pages, to appear in Journal of Nonlinear Mathematical Physic
Ermakov's Superintegrable Toy and Nonlocal Symmetries
We investigate the symmetry properties of a pair of Ermakov equations. The
system is superintegrable and yet possesses only three Lie point symmetries
with the algebra sl(2,R). The number of point symmetries is insufficient and
the algebra unsuitable for the complete specification of the system. We use the
method of reduction of order to reduce the nonlinear fourth-order system to a
third-order system comprising a linear second-order equation and a conservation
law. We obtain the representation of the complete symmetry group from this
system. Four of the required symmetries are nonlocal and the algebra is the
direct sum of a one-dimensional Abelian algebra with the semidirect sum of a
two-dimensional solvable algebra with a two-dimensional Abelian algebra. The
problem illustrates the difficulties which can arise in very elementary
systems. Our treatment demonstrates the existence of possible routes to
overcome these problems in a systematic fashion.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility
We perform a classification of the Lie point symmetries for the
Black--Scholes--Merton Model for European options with stochastic volatility,
, in which the last is defined by a stochastic differential equation
with an Orstein--Uhlenbeck term. In this model, the value of the option is
given by a linear (1 + 2) evolution partial differential equation in which the
price of the option depends upon two independent variables, the value of the
underlying asset, , and a new variable, . We find that for arbitrary
functional form of the volatility, , the (1 + 2) evolution equation
always admits two Lie point symmetries in addition to the automatic linear
symmetry and the infinite number of solution symmetries. However, when
and as the price of the option depends upon the second
Brownian motion in which the volatility is defined, the (1 + 2) evolution is
not reduced to the Black--Scholes--Merton Equation, the model admits five Lie
point symmetries in addition to the linear symmetry and the infinite number of
solution symmetries. We apply the zeroth-order invariants of the Lie symmetries
and we reduce the (1 + 2) evolution equation to a linear second-order ordinary
differential equation. Finally, we study two models of special interest, the
Heston model and the Stein--Stein model.Comment: Published version, 14pages, 4 figure
WMAP 3yr data with the CCA: anomalous emission and impact of component separation on the CMB power spectrum
The Correlated Component Analysis (CCA) allows us to estimate how the
different diffuse emissions mix in CMB experiments, exploiting also
complementary information from other surveys. It is especially useful to deal
with possible additional components. An application of CCA to WMAP maps
assuming that only the canonical Galactic emissions are present, highlights the
widespread presence of a spectrally flat "synchrotron" component, largely
uncorrelated with the synchrotron template, suggesting that an additional
foreground is indeed required. We have tested various spectral shapes for such
component, namely a power law as expected if it is flat synchrotron, and two
spectral shapes that may fit the spinning dust emission: a parabola in the logS
- log(frequency) plane, and a grey body. Quality tests applied to the
reconstructed CMB maps clearly disfavour two of the models. The CMB power
spectra, estimated from CMB maps reconstructed exploiting the three surviving
foreground models, are generally consistent with the WMAP ones, although at
least one of them gives a significantly higher quadrupole moment than found by
the WMAP team. Taking foreground modeling uncertainties into account, we find
that the mean quadrupole amplitude for the three "good" models is less than 1
sigma below the expectation from the standard LambdaCDM model. Also the other
reported deviations from model predictions are found not to be statistically
significant, except for the excess power at l~40. We confirm the evidence for a
marked North-South asymmetry in the large scale (l < 20) CMB anisotropies. We
also present a first, albeit preliminary, all-sky map of the "anomalous"
component.Comment: 14 pages, 17 figures, submitted to MNRAS, references adde
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
- …