Fundamentals of Mathematics in Medical Research: Theory and Cases

In this chapter, we will delve into a variety of fundamental subjects related to the study of real functions and the transformations of such functions. To begin, we will investigate the absolute value, as well as the absolute value function, and examine the behavior and attributes of each of these concepts. Next, we will delve deeper into the world of functions, concentrating on the functions that are associated with a single variable and the graphical representations of these functions. As we move further, we will investigate the most important real-valued functions, such as polynomial functions and piecewise-defined functions, amongst others. Because of these functions, we are able to acquire a deeper comprehension of transformations and how they have an effect on function graphs. In this lesson, we will investigate the scaling and translating transformations, focusing on how they affect the features of functions. Examples and activities are presented throughout the chapter to help readers better understand the fundamental ideas behind graphical representation and function analysis.

This chapter reviews the real-valued functions on space, with fully solved examples that show the application and usefulness of their components to deal with practical problems. Then, we see the types of functions that the reader will often find in academic and research activities. Later, another family of functions called Vector-Valued Functions is defined and its usefulness in solving application problems is shown with examples. Finally, both types of functions are generalized for higher-dimension spaces.

In this chapter, we delve into the key concepts of probability theory and the most common statistical distributions. We begin by introducing Bayes’ Theorem and its application in statistical inference. Next, we address the binomial distribution function and its usefulness in analyzing binary events and calculating probabilities. Then, we examine conditional probability and its importance in decision-making under uncertainty. Subsequently, we immerse ourselves in the main probabilistic functions, including the normal distribution function and its role in modeling natural phenomena. Furthermore, we study the Poisson distribution function, which is applied to situations where the probability of rare events occurring needs to be calculated. Finally, we analyze the general concept of probability and its interpretation in the context of statistical theory. Additionally, we present the Total Probability’s Theorem, which allows for the calculation of the probability of an event based on conditional event information. In summary, this chapter provides a solid foundation for the fundamentals of probability and statistics, exploring key topics such as Bayes’ Theorem, the binomial distribution function, conditional probability, main probabilistic functions, the normal distribution function, the Poisson distribution function, probability, and the Total Probability’s Theorem. 

 This chapter covers the basic concepts of inferential statistics and hypothesis testing. The section commences with an explanation of the confidence intervals for the population mean and population standard deviation. Also discussed are the concepts of cut-off, Type I and Type II errors, false positives, and false negatives. We investigate performance metrics including sensitivity, specificity, true positive rate, and true negative rate. In addition, receiver operating characteristic (ROC) curves are introduced and their usefulness in assessing test performance is emphasized. In addition, binomial and normal distribution tests are discussed. This chapter provides a solid foundation for comprehending the principles and practical applications of inferential statistics in decision-making and the interpretation of scientific results. 

 In this chapter, we will explore various methods and statistical tests used in hypothesis validation and sample comparison. We will begin with an introduction to the formulation of scientific and statistical hypotheses and discuss the determination of the appropriate sample size. Next, we will delve into the analysis of independent and related samples, examining techniques, such as the one-sample binomial test, one-sample method, one-sample runs test, two-sample chi-squared test, two-sample Kolmogorov test, two-sample runs test, and two-sample U test. These tools will provide researchers and scientists with the ability to rigorously evaluate their hypotheses and compare samples in a reliable manner. 

In this chapter, we examine the nature of correlations in Euclidean spaces, focusing on the two-dimensional space R 2 and the three-dimensional space R 3 . We begin by exploring linear correlations in R 2 , where we analyze calculation techniques and association measures to quantify the relationship between two continuous variables. Next, we delve into multiple correlations in R 3 , examining how several variables can be related simultaneously and how their strength and direction can be jointly measured. Subsequently, we address non-linear correlations in R 2 , expanding the focus beyond traditional linear relationships. We explore advanced methods and techniques for detecting and measuring non-linear correlations, allowing us to capture complex and non-linear patterns in the data. Furthermore, examples of practical applications are discussed where the presence of non-linear correlations is crucial for analysis and decision-making.

This chapter presents an overview of curve fitting methods in Euclidean spaces, with a particular emphasis on R 2 and R 3 . In order to represent linear and nonlinear interactions between numerous variables, a number of methodologies, including linear and nonlinear least squares methods, are being investigated. The linear relationship that exists between two variables is broken down in great detail, and a broad variety of examples are provided to show how curve fitting methods can be utilized to build models that are an accurate representation of data sets. In addition to this, the linear relationship that exists between the three variables under consideration is dissected, and detailed strategies for dealing with this scenario are discussed. Curve fitting methods are useful for exploring and evaluating data in Euclidean spaces, as shown by the results and examples shown below, which demonstrate the utility and versatility of these methods.

This chapter examines the analysis of variance (ANOVA) and Fisher statistical tests as fundamental research methods. Introduction to ANOVA and its application to the investigation of significant differences between data groups. Subsequently, we examine the application of Fisher tests in various contexts, including one-factor, two-factor, and even three-factor analyses. In addition, we investigate the design and experimental application of Latin squares. We conclude by discussing multivariate analysis of variance (MANOVA) and emphasizing its usefulness for investigating multiple response variables. This chapter concludes with a comprehensive overview of the previously mentioned statistical techniques and their applicability to scientific research.

 In this chapter, we look at how Monte Carlo simulations and the Markov chain theory can be used to analyze urban transportation problems. Vectors describing the starting and final states of a public transportation network are introduced as fundamental notions. The transition probability matrix and the stochastic matrix are investigated as potential tools for modeling the dynamic evolution of urban mobility using discrete-time Markov Chains. The features of Markov Chains, as revealed by their eigenvalues and eigenvectors, are examined. The Markov Chain Monte Carlo technique for statistical sampling and analysis of urban mobility issues is also discussed. The methodologies’ potential utility in urban transportation planning and decision-making is emphasized. Understanding and addressing the difficulties of urban mobility are greatly aided by the theoretical and conceptual groundwork laid out in this chapter.