In this chapter, we take a step back to discuss, more generally, the language of mathematics and some proof techniques and strategies. In the previous chapters, we have seen numerous mathematical notions, theorems, proofs, and examples. As you have probably noticed, communicating mathematical arguments and ideas in a coherent and precise way is at the core of the subject.

Qualitative research provides an excellent opportunity to study digitalization. The purpose of this chapter is to explore the digitalization of government services by studying the longitudinal development data-sharing practices across different parts of government in the United Kingdom. This chapter reports on a unique, qualitative, interpretive field study based on the author’s role as a participant observer and his analysis of the discourse and contents of the various documents presented in relation to both the creation and running of data-sharing practices in the United Kingdom. The chapter finds that despite government addressing many of the concerns identified in the literature on data sharing, practical and perceptual issues remain – issues that tell us much about the state of digitalization of government services.

Big data has been proclaimed as revolutionary and transformative, and will alter businesses and society in fundamental ways. Such dramatic claims sound suspiciously like prior technological waves which were also promoted as metamorphic. Most of these past ‘revolutionary’ technologies have proved to be more evolutionary than revolutionary, but is the same likely to be true with big data? This chapter examines a number of the underlying assumptions associated with the supposed big data revolution. It highlights some of the fallacies and misconceptions that lie behind big data, and how these assumptions can lead to unintended and dysfunctional consequences. In particular, it explores how these dysfunctions might manifest themselves when it comes to organizational knowledge and practice. The analysis in the chapter leads to the conclusion that while big data offers promise, the hype surrounding it has obscured the potential dangers in its use. The chapter explains that it is important to take a more nuanced view of this new technology, putting safeguards in place to ensure that big data use (including big data analytics, machine learning, artificial intelligence and the algorithms they develop) does not lead to dysfunction.

Today’s information technology is becoming ever-more complex, distributed and pervasive. Therefore, problematizing what we observe as Information Systems (IS) researchers is becoming ever-more difficult. This chapter offers a new perspective for qualitative empirical research in the IS field. It looks at how we can possibly study dynamically changing, evolving, spatially and temporally distributed phenomena that evade our accustomed concepts and assumptions about the locus of agency. Or asked differently: How can we formally approach phenomena evading our concept of ‘identity’?

Using the mathematical-logical framework of the Laws-of-Form, formulated in 1969 by George Spencer-Brown, the chapter introduces the notion of distinction to capture the manifestation of concepts. It provides a short overview and illustrates how it can be used on sample concepts drawn from IS sociomateriality research.

The chapter advances qualitative methodology by suggesting a formal notation to communication analysis that is reflective of technologies’ complex nature. Applying the framework not only alters the epistemological boundaries for how to experience and study the ‘digital’, but also helps to build a bridge between deep technological insights, our immediate, unbiased and mundane experience of technologies, and how we speak about them.

This chapter is devoted to studying, in more depth, the set of integers Z, its structure, and properties. The integers play a fundamental role in many areas of mathematics, science, and beyond. The integers are closely related to the set of natural numbers and thus are often used in problems involving counting, sequences, and structures with finitely many elements (such as finite fields).

This chapter tackles the simple problem of intersecting two (sorted) lists of increasing integers, which constitutes the backbone of every query resolver in databases and (Web) search engines. In dealing with this problem, the chapter describes several approaches of increasing sophistication and elegance, which eventually turn out to be efficient/optimal in terms of time and I/O complexities. A final solution will deploy a proper compression of the input integers and a two-level scheme aimed at reducing the final space occupancy and working efficiently over hierarchical memories.

This chapter describes a data compression technique devised by Mike Burrows and David Wheeler in 1994 at DEC Systems Research Center. This technique is known as the Burrows–Wheeler Transform (or BWT) and offers a revolutionary alternative to dictionary-based and statistical compressors. It is the algorithmic core of a new class of data compressors (such as bzip2), as well as of new powerful compressed indexes (such as the FM-index). The chapter describes the algorithmic details of the BWT and of two other simple compressors, Move-to-Front and Run-Length Encoding, whose combination constitutes the design core of bzip-based compressors. This description is accompanied by the theoretical analysis of the impact of BWT on data compression, in terms of the k-th order empirical entropy of the input data, and by a sketch of the main algorithmic issues that underlie the design of the first provably compressed suffix array to date, namely the FM-index. Given the technicalities involved in the description of the BWT and the FM-index, this chapter offers several running examples and illustrative figures which should ease their understanding.