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

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 $\pm2$ 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&

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

Chemical analysis of NGC 6528: one of the most metal-rich bulge globular cluster

The Bulge Globular Clusters (GCs) are key tracers of this central ancient component of our Galaxy. It is essential to understand their formation and evolution to study that of the bulge, as well as their relationship with the other Galactic GC systems (halo and disk GCs). Our main goals are to obtain detailed abundances for a sample of seven red giant members of NGC 6528 in order to characterize its chemical composition and study the relationship of this GC with the bulge, and with other bulge, halo and disk GCs. Moreover, we analyze this cluster$'$s behavior associated with the Multiple Populations (MPs) phenomenon. We obtained the stellar parameters and chemical abundances of light elements (Na, Al), iron-peak elements (V, Cr, Mn, Fe, Co, Ni, Cu), {\alpha}-elements (O, Mg, Si, Ca, Ti) and heavy elements (Zr, Ba, Eu) in seven red giant members of NGC 6528 using high resolution spectroscopy from FLAMES-UVES. We obtained in six stars of our sample a mean iron content of [Fe/H]=-0.14+/-0.03 dex, in good agreement with other studies. We found no significant internal iron spread. We detected one candidate variable star, which was excluded from the mean in iron content, we derived a metallicity in this star of [Fe/H]=-0.55+/-0.04 dex. Moreover, we found no extended O-Na anticorrelation but instead only an intrinsic Na spread. In addition, NGC 6528 does not exhibit a Mg-Al anticorrelation, and no significant spread in either Mg or Al. The {\alpha} and iron-peak elements show good agreement with the bulge field star trend. The heavy elements are slightly dominated by the r-process. The chemical analysis suggests an origin and evolution similar to that of typical old Bulge field stars. Finally, we find remarkable agreement in the chemical patterns of NGC 6528 and another bulge GC, NGC 6553, suggesting a similar origin and evolution.Comment: Accepted for publication in A&A. 12 pages, 13 figures, 4 table

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