How do you pair individuals or objects in an optimal manner? This question was first framed in the context of marriage and other pairings, but spurred a great deal of research that had combinatorial optimization as its goal. Matchings are a cornerstone concept in graph theory, offering powerful solutions to optimization problems in network design, resource allocaton and scheduling. The concept of matchings play a significant role in finding perfect matches in partnership scenarios, solving complex assignment problems thus providing the framework for making optimal pairings. This chapter also deals with coverings that provide insights into efficient resource allocation. In addition to these concepts, we will explore an algorithm designed to find the maximum matching in bipartite graphs which culminates in the powerful Kuhn–Munkres algorithm for solving the personnel assignment and optimal assignment problems.

In this chapter we explore some powerful tools for efficient traversal across graphs; the Eulerian and the Hamiltonian modes of traversal. These traversals are inspired by the efficiency of traversing an edge or a vertex exactly once. In this chapter, we will discuss two algorithms, namely Fleury’s and Hierholzer’s algorithms to determine an Eulerian circuit in a graph. This chapter also discusses the application of efficient traversals to two well known problems in network optimization: the Chinese postman problem (CPP) and the traveling salesman problem (TSP).

DISTRIBUTIONS ARE GENERALIZATIONS of mathematical functions from a purely technical standpoint. But perhaps it is most pertinent to begin by asking a more utilitarian question. Why should we study distributions? Specifically, why should we study probability distributions? One of the motivations stems from a practical limitation of experimental measurements that is underlined by the uncertainty principle postulated by Werner Heisenberg (see Figure 2.1). The very fabric of reality and the structure of scientific laws that govern our ability to understand physical phenomena demand a probabilistic (statistical) approach. Our inability to make infinite-precision measurements of data necessitates the consideration of averages over many measurements, and under similar conditions, as a more reliable strategy to affix experimental values to unknowns with reasonable accuracy.

We started our discussion in Section 2.1 of Chapter 2 with some motivation from metric spaces to define a topology on a set. Inspired from metric spaces, we defined notions such as open sets and continuous functions in great generality, even in the absence of a metric or a distance function. However, for a metric space, there is a natural way to do this, i.e., we can define a topology on a metric space. The topology so obtained has many desirable properties that are used in analysis. In this chapter, we shall see how a topology is defined on a metric space, and we shall study sequences, continuous functions, etc. on such topological spaces.

5.1 Metric Topology

In this section, we shall see that every metric space is a topological space. Given a metric on a set, we can define topology on it called the metric topology. We begin by giving some examples of metric spaces; their easy verification is left as an exercise as many readers would already be familiar with metric spaces.

This chapter characterizes the very idea of autonomy as a response to two problems: understanding the source of normativity and the reality of freedom. Following debates on normativity (Pufendorf and Leibniz) and freedom (Locke, Hume, Rousseau), Kant introduces the notion of autonomy as a unified response to both problems. On this account, to be positively free and to be normatively bound are one and the same thing: to follow rules one has given to oneself. After discussing the attractiveness of this idea, the chapter elaborates a first fundamental challenge: the so-called paradox of autonomy, suggesting that autonomy at its very foundation reverts to heteronomy or arbitrariness. The chapter shows that Kant’s conception is indeed threatened by this paradox and develops Kant’s ways of avoiding it. It argues that Kant’s most important resource, however, has not yet been fully acknowledged: It consists in his concept of self-organizing beings from the third Critique. To avoid the paradox, we should no longer think of self-determination in terms of self-legislation but rather conceive of it in terms of living self-constitution. The chapter closes by discussing why Kant himself did not fully develop this resource and argues that the main reason resides in his notion of transcendental freedom.

In the arena of abstract algebra, there's a neat idea that helps us understand rings better: subrings. They’re like smaller pieces within bigger algebraic structures. A ring may be characterized by a variety of its subsets, known as subrings. This characterization also gives a geometrical interpretation to a usual abstract ring. The concept of a subring is analogous to the concept of a subgroup in group theory. These concepts came into existence by the efforts of mathematicians like David Hilbert and Emmy Noether, who laid the foundation for modern algebra. Just like how you might break down a big puzzle into smaller parts to understand it better, subrings help us to see the patterns and properties within rings.

In this chapter, the concept of subring is explored through various examples. Then, subring test is stated to check if a given subset of a ring is a subring. The study is examined through various problems to enable readers to apprehend the notion of subring. In this chapter, we’ll take a closer look at these ideas, exploring where they came from, why they’re important, and how they’re used in real-life problems. Let's dive in and uncover the secrets of subrings together!

This book is designed as a textbook for a semester-long introductory course in abstract algebra, with a focus on ring theory. Yes, it’s a “textbook”, but it is not to be conceived as an encyclopedia, nor merely as an additional reference book for your shelf. This is an approachable tool for learners, guiding them through the thought-provoking terrain of ring theory, crafted to make the subject as clear and engaging as possible. We simply aim to provide undergraduate students with a solid understanding of the fundamentals of ring theory while igniting a lifelong curiosity in abstract algebra.

Ring theory, like most areas of modern algebra, has undergone significant changes and developments over the years. This is a subject where math and creativity intersect, and if you’ve believed that math is just about numbers, ring theory will soon convince you otherwise. Our experience tells us that with a clear understanding of algebraic structures, students find it easier to navigate more complex ideas in areas such as coding theory, cryptography, and advanced analysis. Rest assured, we’ve written this book with you, the student, in mind, so expect to find explanations that make sense without needing to pull out your hair or abandon the course.

The advent of the internet and sensor technology has enabled humankind to collect, store, and share data in bulk. In turn, access to a variety of data has amplified a different kind of problem, which is to devise an appropriate strategy to derive meaning from data. Indeed, extracting information from data has acquired the highest priority among tasks performed by engineers and scientists alike. State-ofthe-art machine learning algorithms are used to process and analyze data in order to leverage maximum gains in developing new technology and creating a new body of knowledge.

Further, the data-rich tech-universe has inherent complexity in addition to the vastness in terms of numbers. This complexity arises from the fact that often this data is embedded in a higher-dimensional space. For example, the data acquired by a camera hosted on a robot is in the form of multiple grayscale images (frames); each data-frame is constituted of a sequence of numbers that represents the intensity of grayness of each pixel. If each image has a resolution 100 × 100 (pixel count), then this image data is embedded in a 10000 dimensional space. Additionally, if the camera records 100 frames per second for one minute, then we have 6000 data points in a 10000 dimensional space. This is just an illustrative example of how a high-dimensional large data set may be generated. Quite evidently, not all the 10000 dimensions host most of the information. One of the most important techniques that we will learn in this chapter will allow us to extract a lower dimensional representation of the data set that will retain sufficient information for the robot to navigate and perform its tasks.

In this chapter, we shall study one of the important topological property called compactness, which has roots in analysis. From real analysis, we know the importance of a closed interval [a, b] in ℝ. For instance, if f is a continuous real-valued function on [a, b], then its image f([a, b]) is a closed and bounded subset of R. This is due to the compactness property of [a, b].

Suppose we have a function f : X → R. If X = {x1, . . . , xn} is a finite set, then clearly, f is bounded by M = max{f(x1), . . . , f(xn)}. Suppose X =S α∈Λ Uα and f is bounded on each of the Uα by Mα. Then we do not know that f is bounded because we cannot simply take the maximum of Mα for α ∈ Λ as the indexed set Λ is arbitrary. But suppose there is a finite set {α1, . . . , αn} ⊆ Λ such that X =Sn i=1 Uαi, then f is bounded on X by M = max{Mα1, . . . , Mαn }, i.e., |f(x)| ≤ M for all x ∈ X. In other words, for every collection of subsets of X whose union covers (contains) X, we look for a finite subcollection that covers (contains) X. This motivates the definition of a compact set Y which states that every open cover of Y has a finite subcover. Let us first understand what a cover of a set means.

In contrast to liberal democracy, which translates constituent power into processes and institutions of representation and government, Michael Hardt and Antonio Negri place a premium on constituent power: inspired by social movements, such as Occupy Wall Street, they argue that the constituent power of a multitude should not be translated into the constituted powers of a state. Doing so would deprive the multitude of its revolutionary and radically democratic potential. Pure constituent power seeks to repeat in perpetuity the exception of the revolutionary founding moment of democracy. That Hardt and Negri rely on increasingly theological models in their account of constituent power and revolution highlights the antidemocratic tendency of their conception of “constituent governance.” A political theology that seeks to make the exception permanent is not compatible with democracy. The enactment of pure constituent power in the democracy of the common inevitably leads to what Carl Schmitt described as “sovereign dictatorship.” Hardt’s and Negri’s “constituent governance” is an arbitrary form of governance without any checks and balances.