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

Influence de la farine des feuilles de Leucaena leucocephala sur les performances de croissance des lapereaux

Quarante (40) lapereaux locale âgés de 8 semaines environ et pesant en moyenne 1000±200 g ont été utilisés pour évaluer l’effet de la substitution de 0, 5, 10, 15 et 20% de tourteau de coton par la farine des feuilles de Leucaena leucocephala dans la ration, suivant un dispositif complètement randomisé. Les animaux étaient logés individuellement et nourris ad libitum. Il ressort des résultats de cet essai que la consommation alimentaire a augmenté significativement (P.0,05) avec le niveau  d’incorporation de la farine de L.leucocephala. Cette consommation a été de 90,98 ; 86,08 ; 93,00 ; 107,27 et 116,64 g (MS) respectivement pour les rations à 0, 5, 10, 15 et 20% de farine des feuilles de Leucaena leucocephala. Le niveau d’incorporation des feuilles de Leucaena leucocephala n’a eu aucun effet significatif ni sur l’évolution du poids vif, ni sur le gain moyen quotidien et les indices de consommation pendant toute la durée de l’essai. Cependant, la ration contenant 20% de farine des feuilles de L. leucocephala, s’est avérée plus économique. D’une manière générale, les résultats obtenus laissent penser que le taux d’incorporation de 20% de L leucocephala donne de meilleures performances et pourrait être recommandé. Toutefois, d’autres investigations sont nécessaires pour déterminer le niveau optimal d’incorporation de L. leucocephala, en évitant les effets néfastes de la mimosine que contiendraient ces feuilles.Mots clés : Croissance, Leucaena leucocephala, lapereaux

Performance and carcass characteristics of broilers fed boiled cowpea (Vigna unguiculata L Walp) and or black common bean (Phaseolus vulgaris) meal diets

A total of 256 broiler chicks were used to evaluate the effect of replacing meat meal with boiled cowpea and/or black common bean on the performance and carcass characteristics of broilers. At the starterphase, boiled cowpea and or black common bean meal constituted 11% of S1, S2 and a 1:1 ratio of boiled cowpea and black common (S3) diets replacing 100% of meat meal and 25% of fish meal. For thefinisher phase, 14% boiled cowpea (F1), 14% boiled black common bean (F2) and an equal proportion (1:1) of boiled cowpea and black common bean meals (F3) were used to replace 100% meat meal and25% fish meal. During the starter period, feed intake and weight gain were significantly (P 0.05)difference was observed between treatment groups for feed conversion ratio and feed cost for the production of 1 kg live body weight. At finisher, broilers started with the control starter diet and finishedwith any of the test diets significantly (P < 0.05) consumed more feed and acquired heavier weights compared to the other dietary combinations. Feed cost was significantly (P < 0.05) lower for birdsstarted the control starter diet and finished with either the control finisher diet or boiled cowpea diets. Carcass yield was significantly (P < 0.05) higher for birds finished with boiled cowpea diet compared tothe other treatments. Cowpea and black common bean boiled for 30 min under uncontrolled temperature and pressure could not be recommended for broiler's diet formulation

Effect of dietary energy level on growth performance and morphometric parameters of local barred chickens at the starter phase

An experiment was conducted to determine the influence of dietary energy level on the growth performance and morphometric characteristics of local barred chicks. The experiment lasted for 12 weeks. Three hundred and twenty day-old barred chicks were assigned to four different dietary treatments, having four replicates of 20 chicks each. Treatments consisted of D0, D1, D2 and D3 containing 2700, 2800, 2900 and 3000 kcal/kg of metabolisable energy respectively. The highest BW (868 g) and BWG (830 g) were recorded with treatment D1 and D3 (844 g and 804 g respectively). These values were significantly (P&lt;0.05) higher compared to D0 and D2. The feed conversion ratio were significantly (P &lt; 0.05) higher with  treatments D0 and D2 as compared to treatments D1 and D3. Feed cost per kg of body weight was significantly (P&lt;0.05) lower for treatment D1 as compared to other treatments. Thus, metabolisable energy requirement of local barred chicks between 1 to 12 weeks is 2800 kcal/kg.Keywords: Barred chickens, dietary energy, starter phase

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

Effets de l’amprolin-300 Ws sur quelques paramètres de croissance et de reproduction chez le lapin mâle (Oryctolagus cuniculus)

Objectif : L’objectif de la présente étude a été de rechercher les effets de l’Amprolin-300, un anticoccidien largement utilisé en médecine vétérinaire, sur quelques paramètres de croissance et de reproduction chez le lapin mâle de race locale. Méthodologie et résultats : Vingt-quatre (24) lapins âgés de 5 mois environ et pesant entre 2,01 et 2,78 kg ont été répartis en 4 lots comparables de 6 lapins chacun. Ils ont été traités pendant 60 jours avec différentes doses d’Amprolin-300 ws via l’eau de boisson : 0 ; 30 ; 60 et 110 mg/kg de poids corporel. Les résultats obtenus ont révélé que la consommation d’eau a significativement (p&lt;0,05) diminué chez les lapins traités aux doses 60 et 110 mg/kg de poids corporel par rapport aux témoins. Le gain de poids et le gain moyen quotidien ont diminué de manière significative (p&lt;0,05) chez les animaux traités aux doses 30 et 60 mg/kg de poids corporel par rapport aux animaux du lot témoin. Par contre, l’indice de consommation a diminué de façon significative (p&lt;0,05) chez les animaux témoins et chez ceux qui ont ingéré la plus forte dose d’Amprolin-300 ws (110 mg/kg de poids corporel) comparé aux lapins traités avec les doses 30 et 60 mg/kg respectivement. Le pH de la semence a significativement (p&lt;0,05) augmenté chez les mâles traités aux doses 30 et 60 mg/kg de poids corporel par rapport Aux mâles du lot témoin. La motilité individuelle et la concentration des spermatozoïdes ont diminué de manière significative (p&lt;0,05) chez les lapins qui ont ingéré la plus forte dose d’anticoccidien par rapport aux lapins témoins. Conclusion et application des résultats : Ainsi, à la dose de 110 mg/kg de poids corporel, l’Amprolin-300 ws a réduit la consommation alimentaire, le gain de poids, le gain moyen quotidien, le poids spécifique et le pH de la semence. Cette dose a également inhibé la spermatogenèse et réduit le pourcentage de spermatozoïdes mobiles. Son usage devrait donc être règlementé.Mots clés : Amprolin-300 ws, lapin mâle, production et fertilité.Objective: In this study, the effects of Amprolin-300 ws, an anticoccidiostat abundantly used in veterinary medicine, on some growth and reproductive parameters in male rabbits were investigated. Methodology and results: Twenty four (24) rabbits aged 5 months and weighing between 2.01 and 2.78 kg were divided into 4 comparable groups of six rabbits each and treated for 60 days with different doses of Amprolin-300 ws from drinking water: 0, 30, 60 and 110 g/kg of body weight. The results showed that water consumption was significantly (p&lt;0.05) decreased in rabbits treated with 60 and 110 mg/kg of body weight of Amprolin-300 compared to control rabbits. Body weight gain and average daily gain decreased significantly (p&lt;0.05) in animals treated with 30 and 60 mg/kg of body weight compared to control rabbits. On the other hand, feed efficiency decreased significantly (p&lt;0.05) in control animals and those who ingested the highest dose (110 mg/kg of body weight) compared to rabbits treated with 30 and 60 mg/kg of body weight respectively. The pH of semen increased significantly (p&lt;0.05) in animals treated with 30 and 60 mg/kg of body weight compared to control animals. Individual motility and sperm concentration decreased significantly (p&lt;0.05) in rabbits who received the highest dose of Amprolin-300 compared to control rabbits. Conclusion and application of results: Thus, at the dose of 110 mg/kg body weight, Amprolinws-300 reduced in male rabbit, food consumption, body and average daily weight gain, specific weight and pH of semen. It inhibited spermatogenesis and reduced sperm mobility. Therefore its use should be regulated

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

Performance Of Finisher Broiler Chickens As Affected By Graded Levels Of Cooked Cowpeas (Vigna unguiculata) In The Grower-Finisher Diet

Cent soixante (160) poulets de chair mâles âgés de 21 jours et pesant en moyenne 416,32 g ont été répartis au hasard dans 40 unités expérimentales de 4 poulets chacune. Chacune des cinq rations expérimentales contenant respectivement 0% de niébé (F0, ration témoin) ; 15% (F1) ; 20% (F2) ; 25% (F3) et 30% (F4) de niébé cuit a été servie à 8 unités expérimentales pendant 28 jours selon un dispositif expérimental complètement randomisé pour évaluer la performance de production des poulets de chair en finition. Il n’y a pas eu de différence significative (P>0,05) entre les différents lots pour la consommation alimentaire totale. Le poids vif, le gain de poids et l’indice de consommation enregistrés pendant toute la période de l’essai dans les lots F3 (contenant 25% de niébé cuit) et F4 (contenant 30% de niébé cuit) ont été statistiquement faibles (P0.05) between treatment groups for total feed consumption. Live weight, weight gain and feed conversion ratio were statistically poorer and cost of production highest (P0.05) for these parameters. All the treatment groups were not statistically different (P>0.05) for carcass yield, proportionsof organs and serum creatinine level. It was concluded that up to 20% of cooked cowpeas could be used in the finisher diet without negatively affecting the performance of broilers

Performance of finisher broiler chickens as affected by different proportions of cooked cowpeas (Vigna unguiculata) in the grower-finisher diet.

This study was carried out for 4 weeks at the Research Experimental Farm (FAR) of the University of Dschang-Cameroon, on the feeding of broilers with cowpea (Vigna unguiculata) as a source of protein, which replaced animal meal (CMAV 10%) in the finisher diet. The objective was to evaluate the effect of the incorporation level of cooked cowpea on the production performances of broilers in the grower-finisher diet, so cooking could be an easy solution to local farmers if proven efficient. A total of 160 male broiler chicks, 21days old and weighing 416.32g on average, were randomly distributed into 32 experimental units of 4 birds each. Each of the five experimental diets: F0, F1, F2, F3 and F4 containing 0% (control), 15%, 20%, 25%, and 30% of cooked cowpea was respectively allocated to 8 experimental units in a completely randomised design comprising 5 treatments with 4 replicates each. Cowpea grains bought from the local market were cooked for 3 minutes in a pressure cooker at the temperature of about 115oC under a pressure of 155Pa. Cooked grains were sun-dried for one week (under a temperature of about 28-32oC) to a humidity level of about 11%. Parameters measured were feed consumption, live weight and weight gain, feed conversion ratio, feeding cost for the production of one kg live weight, carcass yields, proportions of different parts analysed and the creatinine level in the chickens’ serum. The results showed that there was no significant difference (P>0.05) between treatment groups for total feed consumption. Live weight (1941.93±77.74g and 1804.21±271.73g), weight gain (1207.61±71.50g and 1094.93±177.16g), and feed conversion ratio (2.79 ±0.15 and 3.02 ±0.46) recorded for the F3 group (25% cooked cowpea) and F4 (30% cooked cowpea) respectively were statistically significant (P0.01) during the whole experimental period. F0, F1 and F2 broilers were comparable for these parameters. All the treatment groups were not statistically different (P>0.05) for carcass yield, proportions of organs and serum creatinine level. It was concluded that up to 20% of cooked cowpea could be used in the finisher diet without negatively affecting feed consumption, live weight, weight gain, feed conversion ratio, feeding cost for the production of one Kg live weight and carcass quality of broilers.Key words: Cowpea, Diets, Production Performances, Broiler