This book provides new contributions to the theory of inequalities for integral and sum, and includes four chapters. In the first chapter, linear inequalities via interpolation polynomials and green functions are discussed. New results related to Popoviciu type linear inequalities via extension of the Montgomery identity, the Taylor formula, Abel-Gontscharoff's interpolation polynomials, Hermite interpolation polynomials and the Fink identity with Green’s functions, are presented. The second chapter is dedicated to Ostrowski’s inequality and results with applications to numerical integration and probability theory. The third chapter deals with results involving functions with nondecreasing increments. Real life applications are discussed, as well as and connection of functions with nondecreasing increments together with many important concepts including arithmetic integral mean, wright convex functions, convex functions, nabla-convex functions, Jensen m-convex functions, m-convex functions, m-nabla-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator of order m. The fourth chapter is mainly based on Popoviciu and Cebysev-Popoviciu type identities and inequalities. In this last chapter, the authors present results by using delta and nabla operators of higher order.
Genetically uniform cultivars in many self-pollinated cereal crops dominate commercial production in high-input environments especially due to their high grain yields and wide geographical adaptation. These cultivars generally perform well under favorable and high-input farming systems but their optimal performance cannot be achieved on marginal/organic lands or without the use of external chemical inputs (fertilizers, herbicides and pesticides). Cereal breeding programs aim at evaluating candidate lines/cultivars for agronomic, disease and quality traits in a weed free environment that makes it impossible to identify traits conferring competitive ability against weeds. Moreover, quantification of competitive ability is a complex phenomenon which is affected by range of growth traits. Above (e.g. light) and below (e.g. water and nutrients) ground resources also influence competitiveness to a greater extent. Competitiveness is quantitatively inherited trait which is heavily influenced by many factors including genotype, management, environment and their interaction. Sound plant breeding techniques and good experimental designs are prerequisites for maximizing genetic gains to breed cultivars for organically managed lands. The brief is focused on breeding wheat for enhanced competitive ability along with other agronomic, genetic and molecular studies that have been undertaken to improve weed suppression, disease resistance and quality in organically managed lands. The examples from other cereals have also been highlighted to compare wheat with other cereal crops.
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