Transformations: A Mathematical Approach- Fundamental Concepts

Operator algebra is a branch of mathematics that can be applied to several fields of science as operators connecting vector spaces to interpret data from different algebras. Operator algebra can also be used for its axiomized properties. Both approaches are explored in this book.

This chapter focuses on the characterization of a mathematical operator called transformation; it describes the different types of transformations, their basic properties, restrictions, and extensions. The aspects here reviewed will be described in detail in the next chapters.

This chapter is focused on the characterization of “linear space” as a mathematical abstraction. It describes the different types of linear spaces, their basic properties, and main components. It also gives a first classification and explains their restrictions and extensions.

This chapter introduces the definition of nonlinear transformations, their basic properties, and components. It explains their main applications through different examples and gives a first classification, their restrictions, and extensions. It includes special cases showing that it is possible to transform nonlinear equations into linear equations, using nonlinear transformations and it also describes the Least Squares method.

Acoustics is the branch of physics that studies all the physical phenomena associated with the generation, propagation, and detection of mechanical waves that are heard in a band of frequencies called sound waves.

Actuary is a discipline oriented to financial and commercial matters, where a mixture of algorithms of deterministic and stochastic nature comes together not only to predict the future value of specific parameters but also to estimate their past values. In this chapter, we review three issues: annuity, mortality rate, and linear programming.

Bioinformatics is a scientific field that combines mathematics and computer sciences to generate algorithms that model biological and medical scenarios. This chapter describes the basic properties of algorithms, their main classification as well as their restrictions and extensions.

An integral, as a mathematical operator, is substantially simplified by transformations over the integration area. This chapter defines the main types of transformations, surface integrals, line integrals, and the Jacobian determinant.

Cybernetics is especially focused on processing since our telecommunications and computers work with electromagnetic signals processed through electronic circuits that produce specific responses under specific conditions that can encrypt information. This chapter reviews three issues: signal processing, artificial neural network, and natural language processing.

Epidemiological surveillance is a crucial field of Medicine, an early warning of an outbreak depends on the efficiency of a surveillance system. At present, the methodology to give an early warning has not changed, it is based on a minimum number of verified cases tested in a laboratory. However, there are already valuable contributions to assume that in the near future, this methodology will have a stochastic component.

Genetics focuses on the study of the hereditary transmission of the anatomical, cytologic, and functional characters from parents to offspring. This chapter reviews the linear transformations that can be related to that and their extensions.

This scientific discipline focuses on the study of the behavior and properties of light such as the energy of a waveform or an electromagnetic wave. The concept of light is geometrically represented as a set of rays travelling in straight line, bending when it passes through or reflects from a surface. The study of the reflection and refraction of rays in different forms is called Optics [Cornejo-Rodríguez and Urcid-Serrano, 2005] p. 8.

Experimental verification of different scientific hypotheses requires one or more statistical tests. These tests assume that the samples studied behave according to different distributions: Normal, Binomial and Exponential. In this chapter, we will study the linear or nonlinear nature of these distributions.

The Stokes, Gauss, and Green theorems play an important role in Vector Analysis. The transformations involved simplify line and surface integrals. In this chapter, these transformations and integral theorems will be reviewed.