A Mathematical Theory of Stochastic Microlensing II. Random Images, Shear, and the Kac-Rice Formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (p.d.f.) of the random shear tensor at a general point in the lens plane due to point masses in the limit of an infinite number of stars. Up to this order, the p.d.f. depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the stars' masses. As a consequence, the p.d.f.s of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic p.d.f. of the shear magnitude in the limit of an infinite number of stars is also presented. Extending to general random distributions of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of {\it global} expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars.Comment: To appear in JM

A Mathematical Theory of Stochastic Microlensing I. Random Time-Delay Functions and Lensing Maps

Stochastic microlensing is a central tool in probing dark matter on galactic scales. From first principles, we initiate the development of a mathematical theory of stochastic microlensing. Beginning with the random time delay function and associated lensing map, we determine exact expressions for the mean and variance of these transformations. We characterize the exact p.d.f. of a normalized random time delay function at the origin, showing that it is a shifted gamma distribution, which also holds at leading order in the limit of a large number of point masses at a general point of the lens plane. For the large number of point masses limit, we also prove that the asymptotic p.d.f. of the random lensing map under a specified scaling converges to a bivariate normal distribution. We show analytically that the p.d.f. of the random scaled lensing map at leading order depends on the magnitude of the scaled bending angle due purely to point masses as well as demonstrate explicitly how this radial symmetry is broken at the next order. Interestingly, we found at leading order a formula linking the expectation and variance of the normalized random time delay function to the first Betti number of its domain. We also determine an asymptotic p.d.f. for the random bending angle vector and find an integral expression for the probability of a lens plane point being near a fixed point. Lastly, we show explicitly how the results are affected by location in the lens plane. The results of this paper are relevant to the theory of random fields and provide a platform for further generalizations as well as analytical limits for checking astrophysical studies of stochastic microlensing.Comment: New layout, more details and discussion. To appear, Journal of Mathematical Physic

Genetic diversity analysis in the section Caulorrhizae (genus Arachis) using microsatellite markers

Diversity in 26 microsatellite loci from section Caulorrhizae germplasm was evaluated by using 33 accessions of A. pintoi Krapov. & W.C. Gregory and ten accessions of Arachis repens Handro. Twenty loci proved to be polymorphic and a total of 196 alleles were detected with an average of 9.8 alleles per locus. The variability found in those loci was greater than the variability found using morphological characters, seed storage proteins and RAPD markers previously used in this germplasm. The high potential of these markers to detect species-specific alleles and discriminate among accessions was demonstrated. The set of microsatellite primer pairs developed by our group for A. pintoi are useful molecular tools for evaluating Section Caulorrhizae germplasm, as well as that of species belonging to other Arachis sections

Effet du taux d’incorporation de la farine de patate douce crue dans l’aliment sur les performances de croissance du poulet de chair

Objectifs : Cette étude a été menée dans le but d’évaluer l’effet du taux d’incorporation de la farine de patate douce crue sur les performances de croissance du  poulet de chair.Méthodologie et résultats : A cet effet, 192 poussins d’un jour de souche Starbro ont été repartis dans 16 unités expérimentales suivant un dispositif   complètement randomisé comportant 4 traitements et 4 répétitions chacun. Une ration témoin (R0) contenant le maïs comme principale source d’énergie a été comparée à trois autres rations R50, R75 et R100 dans lesquelles 50, 75 et 100% de maïs ont été remplacés avec de la farine de patate douce crue de    granulométrie comprise 1 et 2 mm. En dehors de l’indice de consommation, tous les paramètres de croissance étudiés ont baissé avec le taux  ’incorporation  croissant de la patate dans l’aliment. De même, le rendement carcasse tend à baisser avec l’augmentation du taux de patate dans la ration, la ration R100 sans  maïs ayant induit le rendement carcasse le plus faible de tous les groupes de poulets. Par contre, le poids et la densité de l’intestin ont augmenté de façon  significative (P<0,05) avec la ration R100 contenant uniquement la patate douce comme source d’énergie. Tous les traitements ont été comparables pour le poids du gésier et la longueur de l’intestin.Conclusions et applications des résultats: Dans les conditions de la présente étude, il a été conclu qu’il est possible de substituer le maïs par de la farine de patate de granulométrie comprise entre 1 et 2 mm pourvu que le taux d’incorporation ne dépasse pas 50% dans la ration.Mots clés : Granulométrie, patate crue, performances de croissance, poulets de chair, taux d’incorporation

Effects of heat stress on some reproductive parameters of male cavie (Cavia porcellus) and mitigation strategies using guava (Psidium guajava) leaves essential oil

Climate changes, particularly the increase of temperature are among the main causes behind the decline of fertility in humans as well as animals. In this study, the effects of heat stress on some reproductive parameters of male cavies and mitigation strategies using guava leaves essential oil (GLEO) were studied. For this purpose, 40 male cavies aged 2.5\u20133 months and weighing between 348 and 446 g were divided into 4 groups of 10 animals each and subjected to the following temperatures: Ambient temperature (20\u201325 \ub0C) for the control group, 35 \ub0C for group 1, 45 \ub0C for group 2 and 45 \ub0C+100 \ub5l GLEO/kg body weight, administered by gavage to animals for group 3. Exposure time of heat was 7 h per day for 60 days. Results reveal that the relative weights of testes, epididymis, vas deferens and seminal vesicles were hardly affected by the temperature levels considered (P>0.05). The mass and individual sperm motility was significantly lower (P<0.05) in cavies exposed to the temperature of 35 and 45 \ub0C as compared with those which received GLEO and controls. The percentages of abnormal sperm and altered sperm DNA were higher in animals exposed to temperature of 35 and 45 \ub0C as compared with the controls. The activity of superoxide dismutase significantly increased (P<0.05) in animals exposed to temperature of 45 \ub0C and in those of 45 \ub0C and orally treated with GLEO, compared with cavies exposed to temperature of 45 \ub0C without receiving GLEO. The level of malondialdehyde was significantly increased (P<0.05) in animals exposed to temperature of 35 and 45 \ub0C, whereas the level of nitric oxide was significantly lower (P<0.05) in exposed animals as compared with controls. It was concluded that the exposure of male cavies at 35 and 45 \ub0C for 60 days induce heat stress that causes deterioration of sperm characteristics. These effects that can be mitigated by the administration of guava leaves essential oil