On the real linear polarization constant problem

The present paper deals with lower bounds for the norm of products of linear forms. It has been proved by J. Arias-de-Reyna [2], that the so-called n(th) linear polarization constant c(n)(C-n) is n(n/2), for arbitrary n is an element of N. The same value for c(n) (R-n) is only conjectured. In a recent work A. Pappas and S. Revesz prove that c(n) (R-n) = n(n/2) for n <= 5. Moreover, they show that if the linear forms are given as f(j)(x) = [x, a(j)),for some unit vectors a(j) (1 <= j <= n), then the product of the f(j)'s attains at least the value n(-n/2) at the normalized signed sum of the vectors having maximal length. Thus they asked whether this phenomenon remains true for arbitrary n is an element of N. We show that for vector systems {a(j)}(j=1)(n) close to an orthonormal system, the Pappas-Revesz estimate does hold true. Furthermore, among these vector systems the only system giving n(-n/2) as the norm of the product is the orthonormal system. On the other hand, for arbitrary vector systems we answer the question of A. Pappas and S. Revesz in the negative when n is an element of N is large enough. We also discuss various further examples and counterexamples that may be instructive for further research towards the determination of c(n)(R-n)

The influence of microlensing on the shape of the AGN Fe K-alpha line

We study the influence of gravitational microlensing on the AGN Fe K-alpha line confirming that unexpected enhancements recently detected in the iron line of some AGNs can be produced by this effect. We use a ray tracing method to study the influence of microlensing in the emission coming from a compact accretion disc considering both geometries, Schwarzschild and Kerr. Thanks to the small dimensions of the region producing the AGN Fe K-alpha line, the Einstein Ring Radii associated to even very small compact objects have size comparable to the accretion disc hence producing noticeable changes in the line profiles. Asymmetrical enhancements contributing differently to the peaks or to the core of the line are produced by a microlens, off-centered with respect to the accretion disc. In the standard configuration of microlensing by a compact object in an intervening galaxy, we found that the effects on the iron line are two orders of magnitude larger than those expected in the optical or UV emission lines. In particular, microlensing can satisfactorily explain the excess in the iron line emission found very recently in two gravitational lens systems, H 1413+117 and MG J0414+0534. Exploring other physical {scenario} for microlensing, we found that compact objects (of the order of one Solar mass) which belong to {the bulge or the halo} of the host galaxy can also produce significant changes in the Fe K$_\alpha$ line profile of an AGN. However, the optical depth estimated for this type of microlensing is {very small, $\tau\sim 0.001$, even in a favorable case.Comment: Astron. Astrophys. accepte

Effect of Dilution on First Order Transitions: The Three Dimensional Three States Potts Model

We have studied numerically the effect of quenched site dilution on a first order phase transition in three dimensions. We have simulated the site diluted three states Potts model studying in detail the second order region of its phase diagram. We have found that the $\nu$ exponent is compatible with the one of the three dimensional diluted Ising model whereas the $\eta$ exponent is definitely different.Comment: RevTex. 6 pages and 6 postscript figure

Latitudinal gradients of cosmic rays and the polarity reversal of the heliospheric magnetic field: A preliminary evaluation

Within the statistical limits imposed by the currently available data and the noise inherent in the determination of the latitudinal gradient, no evidence for the expected change in the latitudinal gradient from pre-1980 to post-1980 epochs can be found. In addition, the rigidity dependence of the gradient appears to be the same in the two epochs. Thus, no evidence is found for a sensitivity of the latitudinal gradient to the polarity of the largescale heliospheric magnetic field such as has been predicted by models incorporating particle drifts

Size effects in multiferroic BiFeO3 nanodots: A first-principles-based study

An effective Hamiltonian scheme is developed to investigate structural and magnetic properties of BiFeO3 nanodots under short-circuit-like electrical boundary conditions. Various striking effects are discovered. Examples include (a) scaling laws involving the inverse of the dots' size for the magnetic and electric transition temperatures; (b) the washing out of some structural phases present in the bulk via size effects; (c) the possibility of tailoring the difference between the Neel and Curie temperatures, by playing with the size and electrical boundary conditions; and (d) an universal critical thickness of the order of 1.6 nm below which the dots do not possess any long-range ordering for the electrical and magnetic dipoles, as well as, for the oxygen octahedral tiltings.Comment: 3 figure