Trees form an important class of graphs, in graph theory. Trees hold a special place in graph theory, due to their simplicity, versatility and widespread applications. From representing hierarchical data structures like family trees and organizational charts, their unique properties, such as having a single path between any two vertices, make them indispensable in solving complex problems with efficiency and clarity. In this chapter, we will explore different types of trees such a spanning trees, rooted trees and binary trees.We will also study different algorithms to construct minimum spanning trees.

3.1 Continuous Functions

3.1 Continuous Functions

We are familiar with the notion of continuous function in real analysis, in general, in a metric space. Recall Definition 2.1.7 of continuity of a function in a metric space. We know that a function is continuous on a set if it is continuous at all its points.

Now the question is, how can we define continuity of a function on a set in the absence of a distance function? In Subsection 2.1.2, we discussed that we need to define continuity of a function in terms of open sets in order to define it in a topological space. Let us see how this can be done. We recall the definition in a metric space below and try to see how we can generalize it to a topological space by bringing the open sets into picture.

Let (X, d) and (Y, ρ) be metric spaces and f : X → Y be a function. Then f is continuous if for every a ∈ X and ∈ > 0, there is δ > 0 such that ρ(f(x), f(a)) ∈ whenever d(x, a) δ.

That is,

f(Bd(a, δ)) ⊆ Bρ(f(a), ∈). (3.1)

Equivalently,

Bd(a, δ) ⊆ f−1(Bρ(f(a), ∈ )).

Thus, in order to have continuity of f, we must have Bd(a, δ) ⊆ f−1(Bρ(f(a), ∈ )). In other words, the set f−1(Bρ(f(a), ∈ )) must contain an open ball Bd(a, δ) containing a. That is, a is an interior point of the set f−1(Bρ(f(a), ∈ )). In fact, it can be easily shown that the set f−1(Bρ(f(a), ∈ )) is open in X. Note that the ball Bρ(f(a), ∈ ) is open in Y. Thus, if we have the condition that the “inverse image of every open subset of Y, under f, is open in X”, then we can conclude the continuity of f. This brings the open subsets of X and Y into picture.

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.

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.