Mathematical Inequalities for Optimization and Decision-Making in Engineering and Physical Sciences
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Inequalities, Cauchy--Bunyakovsky--Schwarz inequality,, Bernoulli's inequality, Optimization, Number Theory and Prime Gaps, Inequalities from Physics.Abstract
The primary objective of this article is to provide an accessible exposition of the dynamic and evolving field of mathematical inequalities, with a particular emphasis on their role in optimization and decision-making within engineering and the physical sciences. We begin with an elementary introduction to fundamental inequalities, followed by a range of illustrative examples that span from classical applications to contemporary research challenges. To demonstrate the breadth and utility of inequalities, we explore examples from diverse areas of mathematics, including the ubiquitous triangle inequality, which arises in contexts ranging from Euclidean geometry to matrix norms. We highlight key results, such as the interlacing of roots of orthogonal polynomials, which are elegantly formulated through inequality frameworks. In number theory, we present select theorems and conjectures—particularly from the active domain of prime gaps—that are naturally expressed using inequalities. The article also examines inequalities in physics, such as the Clausius inequality in thermodynamics, constraints on electron localization in atomic structures, and bounds related to the cardinality of resistor networks. Overall, this paper aims to introduce readers to the techniques and significance of mathematical inequalities, while showcasing their applications in optimization, theoretical analysis, and practical decision-making across multiple scientific and engineering domains.
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