Catálogo de publicaciones - libros

In this chapter, we will define the fundamental notions of limits and continuity of functions and study the properties of continuous functions. We will discuss these properties in more general context of a metric space. The concept of compactness will be introduced. Next, we will focus on connectedness and investigate the relationships between continuity and connectedness. Finally, we will introduce concepts of monotone and inverse functions and prove a set of Intermediate Value Theorems.

In physical terms, differentiation expresses the rate at which a quantity, y , changes with respect to the change in another quantity, x , on which it has a functional relationship. This small chapter will start with the discussion of the derivative, which is one of the two central concepts of calculus (the other is the integral). We will discuss the Mean Value Theorem and look at some applications that include the relationship of the derivative of a function with whether the function is increasing or decreasing. We will expose Taylor’s theorem as a generalization of the Mean Value Theorem. In calculus, Taylor’s theorem gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. There are many OR applications of Taylor’s approximation, especially in linear and non-linear optimization.

In mathematics, power series are devices that make it possible to employ much of the analytical machinery in settings that do not have natural notions of “convergence”. They are also useful, especially in combinatorics, for providing compact representations of sequences and for finding closed formulas for recursively defined sequences, known as the method of generating functions. We will discuss first the notion of series, succeeded by operations on series and tests for convergence/divergence. After power series is formally defined, we will generate exponential, logarithmic and trigonometric functions in this chapter. Fourier series, gamma and beta functions will be discussed as well.

In functional analysis, the Laplace transform is a powerful technique for analyzing linear time-invariant systems. In actual, physical systems, the Laplace transform is often interpreted as a transformation from the time-domain point of view, in which inputs and outputs are understood as functions of time, to the frequency-domain point of view, where the same inputs and outputs are seen as functions of complex angular frequency, or radians per unit time. This transformation not only provides a fundamentally different way to understand the behavior of the system, but it also drastically reduces the complexity of the mathematical calculations required to analyze the system. The Laplace transform has many important Operations Research applications as well as applications in control engineering, physics, optics, signal processing and probability theory. The Laplace transform is used to analyze continuous-time systems whereas its discrete-time counterpart is the Z transform. The Z transform among other applications is used frequently in discrete probability theory and stochastic processes, combinatorics and optimization. In this chapter, we will present an overview of these transformations from differential/difference equation systems’ viewpoint.