We introduce modified (p, q)-Bernstein-Durrmeyer operators. We discuss approximation properties for these operators based on Korovkin type approximation theorem and compute the order of convergence using usual modulus of continuity. We also study the local approximation property of the sequence of positive linear operators D∗ n,p,q and compute the rate of convergence for the function f belonging to the class LipM(γ).
Field trials under rain-fed conditions at the International Center for Tropical Agriculture (CIAT) in Colombia were conducted to study the comparative leaf photosynthesis, growth, yield, and nutrient use efficiency in two groups of cassava cultivars representing tall (large leaf canopy and shoot biomass) and short (small leaf canopy and shoot biomass) plant types. Using the standard plant density (10,000 plants ha-1), tall cultivars produced higher shoot biomass, larger seasonal leaf area indices (LAIs) and greater final storage root yields than the short cultivars. At six months after planting, yields were similar in both plant types with the short ones tending to form and fill storage roots at a much earlier time in their growth stage. Root yield, shoot and total biomass in all cultivars were significantly correlated with seasonal average LAI. Short cultivars maintained lower than optimal LAI for yield. Seasonal PN, across cultivars, was 12% greater in short types, with maximum values obtained in Brazilian genotypes. This difference in PN was attributed to nonstomatal factors (i.e., anatomical/biochemical mesophyll characteristics). Compared with tall cultivars, short ones had 14 to 24 % greater nutrient use efficiency (NUE) in terms of storage root production. The lesser NUE in tall plants was attributed mainly to more total nutrient uptake than in short cultivars. It was concluded that short-stemmed cultivars are superior in producing dry matter in their storage roots per unit nutrient absorbed, making them advantageous for soil fertility conservation while their yields approach those in tall types. It was recommended that breeding programs should focus on selection for more efficient short- to medium-stemmed genotypes since resource-limited cassava farmers rarely apply agrochemicals nor recycle residual parts of the crop back to the soil. Such improved short types were expected to surpass tall types in yields when grown at higher than standard plant population densities (>10,000 plants ha-1) in order to maximize irradiance interception. Below a certain population density (<10,000 plants ha-1), tall cultivars should be planted. Findings were discussed in relation to cultivation and cropping systems strategies for water and nutrient conservation and use efficiencies under stressful environments as well as under predicted water deficits in the tropics caused by trends in global climate change. Cassava is expected to play a major role in food and biofuel production due to its high photosynthetic capacity and its ability to conserve water as compared to major cereal grain crops. The interdisciplinary/interinstitutions research reported here, including, an associated release of a drought-tolerant, short-stem cultivar that was eagerly accepted by cassava farmers, reflects well on the productivity of the CIAT international research in Cali, Colombia., and M. A. El-Sharkawy, S. M. de Tafur