Course Details:
An understanding of inequalities can be extremely useful to the practicing engineer. In reality inequalities are central to the definitions of all limiting processes, including differentiation and integration. The cases exact solutions are unavailable, inconvenient, or unnecessary, inequalities can be used to obtain error bounds for numerical approximation. Inequalities can lead to an understanding of the qualitative behavior. In this course inequalities will be discussed specifically with engineers and other applied scientists in mind.
Module 1: Review of inequalities
Preliminaries, elementary properties, survival rules, bounded set terminology, quadratic inequalities, absolute value, triangle inequality, examples.
Module 2: Methods from the calculus
Introduction, function terminology, basic results for integrals, results from differential calculus, examples.
Module 3: Standard Inequalities
Introduction, Bernoulli’s inequality, Young’s inequality, inequality of the means, Holder’s inequality, Minkowski’s inequality, Cauchy-Schwarz’s inequality, Chebyshev’s inequality, Jensen’s inequality, examples.
Module 4: Inequalities in abstract spaces
Introduction, Metric Spaces, iterations in a Metric Space, Linear Spaces, Orthogonal Projection and Expansion, examples.
Module 5: Some Applications
Introduction, estimation of integrals, series expansions, Simpson’s rule, Taylor’s method , Special functions of mathematical physics, projectile problem, geometric shapes, electrostatic fields & capacitance, applications to matrices, topics in signal analysis, dynamical system, stability & control, some inequalities of probability, applications in communication systems, existence of solutions, duality theorem & cost minimization.
Text Book:
- Inequalities: with applications to engineering – M. J. Cloud & B. C. Drachman.
References:
- Analytic Inequalities - Nicholas D. Kazarinoff.
- Inequalities - G. H. Hardy, J. E. Littlewood & G. Pólya.