Productivity and optimum plant density of pigeonpea in different environments in Tanzania

The objective of the present study was to determine the optimum plant density of four pigeonpea genotypes, representing early, medium and late maturing types, grown in five contrasting environments in Tanzania. ICPL 86005 (early), Kat 50/3 and QP 37 (medium) and Local (late) were grown at four plant densities (40 000-320 000 plants/ha) in irrigated and rainfed conditions at Ilonga and under rainfed conditions at Kibaha, Selian and Ismani. At maturity, total above-ground biomass and seed yield (SY) were measured. The highest yields were obtained in the irrigated experiment at Ilonga, where the medium/late genotypes produced 25 t biomass/ha and 5 center dot 6 t seed/ha. The lowest SY were at Kibaha, 0 58 to 1 center dot 76 t/ha, where a severe drought occurred. In nearly all cases the response to density was linear or asymptotic. The response of ICPL 86005 was significantly different from the other three genotypes. The optimum density for SY varied from 37 000 to 227 000 plants/ha in ICPL 86005, compared with 3000 to 101000 plants/ha in the medium/late genotypes. The highest optimum density was at Selian and Ismani and the lowest at Ilonga and Kibaha, where drought occurred. Optimum densities therefore varied greatly with genotype (duration) and environment, and this variation needs to be considered when planning trials.

On asymptotic behavior at infinity and the finite section method for integral equations on the half-line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

Shadow lines in the arithmetic of elliptic curves

Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\otimes Qp. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.