Real analysis is a branch of mathematics focusing on the study of real numbers and related objects. Sets of real numbers, sequences, functions, and series of real numbers are at the core of the subject. The notions of limits and convergence are central in analysis and are used to investigate such objects. Learning real analysis means, in part, deepening our understanding and studying the theoretical foundations of calculus topics. For these reasons, many view real analysis as a rigorous version of calculus. In this chapter, we look at how limits of sequences and function can be formally defined. The precise definitions may require some effort to grasp, but it is absolutely essential for advanced studies in mathematics and related fields. Formal definitions of limits allow us to not only prove various statements (such as the Extreme and the Intermediate Value Theorems, often proved in a Real Analysis course), but also investigate more complicated functions and sequences. Our experience with proof writing and logical statements will be invaluable for our discussion. We also highlight the use of limits to defining continuity and differentiability of functions.

This chapter deals with the design of data structures and algorithms for the substring search problem, which occurs mainly in computational biology and textual database applications to date. Most of the chapter is devoted to describing the two main data-structure champions in this context, the suffix array and the suffix tree. Several pseudocodes and illustrative examples enrich this discussion, which is accompanied by the evaluation of time, space, and I/O complexities incurred by their construction and by the execution of some powerful query operations. In particular, the chapter deals with the efficient/optimal construction of large suffix arrays in external memory, hence describing the DC3 algorithm and the I/O-efficient scan-based algorithm proposed by Gonnet, Baeza-Yates, and Snider, and the efficient direct construction of suffix trees, via McCreight’s algorithm, or via suffix arrays and LCP arrays. It will also detail the elegant construction of this latter array in internal memory, which is fundamental for several text-mining applications, some of which are described at the end of the chapter.

Combinatorics is an area of mathematics that focuses on counting arguments. It is often used to solve problems involving finite or countable sets in statistics, computer science, logic, and other areas of mathematics. Combinatorics is viewed as part of Discrete Mathematics - a broad area in mathematics concerned with the study of finite or countable structures such as logical and algebraic structures, graphs and more. In this chapter, we discuss a few common tools for counting elements in a finite set. The terminology, notation, and proof techniques discussed in earlier chapters, in the context of sets and functions, will prove to be important and useful in addressing counting problems.

An increasingly pervasive digital environment, where technologies mediate social interaction within and outside organizations, creates new rich data sources for IS research. Of primacy to IS scholars, who study phenomena at the intersection of technology, people and organization, is how future research designs can capture such ongoing sociotechnical entanglements occurring in hybrid online and offline spaces. Building on lessons learned from a study of platform workers, this chapter explores three key challenges of a hybrid ethnographic approach to IS research: (1) navigating unbounded fieldsites; (2) managing technological opacity; (3) working with diverse data. The chapter guides researchers by demonstrating how hybrid methods can be used in different configurations across diverse settings. Simultaneously, in the age of web crawlers, data scraping and machine learning, processes that are invaluable in their own rights, this chapter resituates a qualitative ethnographic approach to digital data, introducing participatory digital observation to the rich empirics gathered in face-to-face environments. Rather than reproducing the valuable work done by scholars in digital sociology and ethnography, this chapter brings strands of the conversation together, highlighting the benefits of studying IS phenomena as a hybrid that exists both in physical and virtual spaces.

This chapter discusses the reflexive relationship between qualitative researchers and the process of selecting, forming, processing and interpreting data in algorithmic qualitative research. Drawing on Heidegger’s ideas, it argues that such research is necessarily synthetic – even creative – in that these activities inflect, and are in turn inflected by, the data itself. Thus, methodological transparency is key to understanding how different types of meanings become infused in the process of algorithmic qualitative research. While algorithmic research practices provide multiple opportunities for creating transparent meaning, researchers are urged to consider how such practices can also introduce and reinforce human and algorithmic bias in the form of unacknowledged introduction of perspectives into the data. The chapter demonstrates this reflexive dance of meaning and bias using an illustrative case of topic modelling. It closes by offering some recommendations for engaging actively with the domain, considering a multi-disciplinary approach, and adopting complementary methods that could potentially help researchers in fostering transparency and meaning.

The chapter theorizes power, knowledge and digitalization in the digital era. It theorizes the roles of knowledge and power in the current era and how these are impacted, reinforced, redistributed, challenged and transformed through increased digitalization. The chapter develops a Knowledge-Power-Digitalization framework where the influence of episodic and systemic power on knowledge and the role of Information Systems and digitalization are outlined. The framework outlines the following quadrants: power as possession, power as asymmetries, power as empowerment and power as practice. The role of digitalization is outlined within these quadrants. The Knowledge-Power-Digitalization framework developed outlines avenues for future research in the digital era pertinent to digitalization, knowledge and power dynamics, which are important current and complex phenomena in need of qualitative research understanding and theorization.

Mathematical induction is a proof technique often used to prove that a statement P(n) that depends on a variable n is valid for every natural number. Equivalently, one may think of P(n) as representing an infinite sequence of statements, one for every natural number n: P(1), P(2), P(3), …. When dealing with infinitely many statements, there is no way we can prove them all by proving each statement individually. Induction is a tool we can often use to bypass this difficulty. Mathematical induction is an extremely powerful proof technique. It is not restricted to specific areas of mathematics and thus can be used to prove statements in algebra, geometry, number theory, analysis, etc. Moreover, induction is a useful tool at all levels of mathematics. It is used to prove elementary statements about numbers as well as advanced statements in, say, topology and modern algebra.