5,315 research outputs found
Special complex manifolds
We introduce the notion of a special complex manifold: a complex manifold
(M,J) with a flat torsionfree connection \nabla such that (\nabla J) is
symmetric. A special symplectic manifold is then defined as a special complex
manifold together with a \nabla-parallel symplectic form \omega . This
generalises Freed's definition of (affine) special K\"ahler manifolds. We also
define projective versions of all these geometries. Our main result is an
extrinsic realisation of all simply connected (affine or projective) special
complex, symplectic and K\"ahler manifolds. We prove that the above three types
of special geometry are completely solvable, in the sense that they are locally
defined by free holomorphic data. In fact, any special complex manifold is
locally realised as the image of a holomorphic 1-form \alpha : C^n \to T^* C^n.
Such a realisation induces a canonical \nabla-parallel symplectic structure on
M and any special symplectic manifold is locally obtained this way. Special
K\"ahler manifolds are realised as complex Lagrangian submanifolds and
correspond to closed forms \alpha. Finally, we discuss the natural geometric
structures on the cotangent bundle of a special symplectic manifold, which
generalise the hyper-K\"ahler structure on the cotangent bundle of a special
K\"ahler manifold.Comment: 24 pages, latex, section 3 revised (v2), modified Abstract and
Introduction, version to appear in J. Geom. Phy
Some notes to extend the study on random non-autonomous second order linear differential equations appearing in Mathematical Modeling
The objective of this paper is to complete certain issues from our recent
contribution [J. Calatayud, J.-C. Cort\'es, M. Jornet, L. Villafuerte, Random
non-autonomous second order linear differential equations: mean square analytic
solutions and their statistical properties, Advances in Difference Equations,
2018:392, 1--29 (2018)]. We restate the main theorem therein that deals with
the homogeneous case, so that the hypotheses are clearer and also easier to
check in applications. Another novelty is that we tackle the non-homogeneous
equation with a theorem of existence of mean square analytic solution and a
numerical example. We also prove the uniqueness of mean square solution via an
habitual Lipschitz condition that extends the classical Picard Theorem to mean
square calculus. In this manner, the study on general random non-autonomous
second order linear differential equations with analytic data processes is
completely resolved. Finally, we relate our exposition based on random power
series with polynomial chaos expansions and the random differential transform
method, being the latter a reformulation of our random Fr\"obenius method.Comment: 15 pages, 0 figures, 2 table
Mobility of solitons in one-dimensional lattices with the cubic-quintic nonlinearity
We investigate mobility regimes for localized modes in the discrete nonlinear
Schr\"{o}dinger (DNLS) equation with the cubic-quintic onsite terms. Using the
variational approximation (VA), the largest soliton's total power admitting
progressive motion of kicked discrete solitons is predicted, by comparing the
effective kinetic energy with the respective Peierls-Nabarro (PN) potential
barrier. The prediction is novel for the DNLS model with the cubic-only
nonlinearity too, demonstrating a reasonable agreement with numerical findings.
Small self-focusing quintic term quickly suppresses the mobility. In the case
of the competition between the cubic self-focusing and quintic self-defocusing
terms, we identify parameter regions where odd and even fundamental modes
exchange their stability, involving intermediate asymmetric modes. In this
case, stable solitons can be set in motion by kicking, so as to let them pass
the PN barrier. Unstable solitons spontaneously start oscillatory or
progressive motion, if they are located, respectively, below or above a
mobility threshold. Collisions between moving discrete solitons, at the
competing nonlinearities frame, are studied too.Comment: 12 pages, 15 figure
Tachoastrometry: astrometry with radial velocities
Spectra of composite systems (e.g., spectroscopic binaries) contain spatial
information that can be retrieved by measuring the radial velocities (i.e.,
Doppler shifts) of the components in four observations with the slit rotated by
90 degrees in the sky. By using basic concepts of slit spectroscopy we show
that the geometry of composite systems can be reliably retrieved by measuring
only radial velocity differences taken with different slit angles. The spatial
resolution is determined by the precision with which differential radial
velocities can be measured. We use the UVES spectrograph at the VLT to observe
the known spectroscopic binary star HD 188088 (HIP 97944), which has a maximum
expected separation of 23 milli-arcseconds. We measure an astrometric signal in
radial velocity of 276 \ms, which corresponds to a separation between the two
components at the time of the observations of 18 milli-arcseconds. The
stars were aligned east-west. We describe a simple optical device to
simultaneously record pairs of spectra rotated by 180 degrees, thus reducing
systematic effects. We compute and provide the function expressing the shift of
the centroid of a seeing-limited image in the presence of a narrow slit.The
proposed technique is simple to use and our test shows that it is amenable for
deriving astrometry with milli-arcsecond accuracy or better, beyond the
diffraction limit of the telescope. The technique can be further improved by
using simple devices to simultaneously record the spectra with 180 degrees
angles.With tachoastrometry, radial velocities and astrometric positions can be
measured simultaneously for many double line system binaries in an easy way.
The method is not limited to binary stars, but can be applied to any
astrophysical configuration in which spectral lines are generated by separate
(non-rotational symmetric) regions.Comment: Accepted for publication in A&
Numerical solution of random differential models
This paper deals with the construction of a numerical solution of random initial value problems by means of a random improved Euler method. Conditions for the mean square convergence of the proposed method are established. Finally, an illustrative example is included in which the main statistics properties such as the mean and the variance of the stochastic approximation solution process are given. © 2011 Elsevier Ltd.This work has been partially supported by the Spanish M.C.Y.T. grants MTM2009-08587, DPI2010-20891-C02-01, Universidad Politecnica de Valencia grant PAID06-09-2588 and Mexican Conacyt.Cortés López, JC.; Jódar Sánchez, LA.; Villafuerte Altuzar, L.; Company Rossi, R. (2011). Numerical solution of random differential models. Mathematical and Computer Modelling. 54(7):1846-1851. https://doi.org/10.1016/j.mcm.2010.12.037S1846185154
Dissipative vortex solitons in 2D-lattices
We report the existence of stable symmetric vortex-type solutions for
two-dimensional nonlinear discrete dissipative systems governed by a
cubic-quintic complex Ginzburg-Landau equation. We construct a whole family of
vortex solitons with a topological charge S = 1. Surprisingly, the dynamical
evolution of unstable solutions of this family does not alter significantly
their profile, instead their phase distribution completely changes. They
transform into two-charges swirl-vortex solitons. We dynamically excite this
novel structure showing its experimental feasibility.Comment: 4 pages, 20 figure
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