The world of graph theory owes its birth to Leonhard Euler (1707–1782) who employed a new strategy to settle a then-unsolved problem called the K¨onigsberg Bridge problem. There were two islands in the middle of the Pregel river, which were connected to each other and also to the mainland by means of seven bridges. The structure of K¨onigsberg and the bridges are described in Figure 1.1.

The question was, “Can a person start at any one of the land masses, walk across each bridge exactly once, touch all land masses and return to the land mass where the person started?” In 1735, Euler correctly identified that there were 4 landmasses and each land mass was connected to the other landmass by means of seven bridges. He intuitively decided that he would model the land masses as “vertices” or “dots” and the seven bridges as “edges” or “lines” connecting the vertices.

A topological space, in general, can have a complex structure. To study such topological spaces, one of the main techniques is to identify such a space, by a homeomorphism, to a space defined in a better way or whose properties are known. We asked relevant questions in the beginning of Chapter 2 of sets (now topological spaces for us) being “same” (now homeomorphic for us), and we also remarked that topology is called “rubber-sheet geometry”. Thus, a topological space need not have a definite geometrical shape or a specific structure, but rather, it can be given by a complex structure. If two topological spaces are homeomorphic, they share certain common properties, which we call “topological properties” or “topological invariants”, which are preserved under a homeomorphism. Therefore, rather than identifying spaces, we can easily distinguish them if such a property is found in one space but not the other. In what follows, till the end of the course of this book, we are going to see many such topological properties and their applications in classifying many topological spaces and answer the questions that we started with in Chapter 2. For instance, we know that ℝ with the usual topology has a countable basis (see Example 2.3.14). This is a topological property called the second countability. That is, a space that does not have a countable basis cannot be homeomorphic to a space with a countable basis. We begin our discussion with the first countability below.

6.1 Local Base and First Countable Spaces

6.1.1 Local base

We have defined neighborhood of a point and neighborhood system at a point in Section 2.6 of Chapter 2. Before we state the first countability axiom, we define local base at a point.

Solid mechanics, compared to mechanics of materials or strength of materials, is generally considered to be a higher level course. It is usually offered in higher semester to senior students. There are many textbooks available on solid mechanics, but they generally include a large part of theory of elasticity with in depth mathematical formulations. The usual prerequisites are one or two semester course on elementary strength of materials and a thorough mathematical background, including scalar, vector, and tensor field theory and cartesian and curvilinear index notation. The difference in levels between these books and elementary texts on strength of materials is generally formidable. However, in our experience of teaching this course for many years at premier institutes like IIT Kharagpur and Jadavpur University, despite its complexity, senior students generally cope well with the course using the readily available textbooks.

However, there is a vast student population pursuing mechanical, civil, or allied engineering disciplines across the country in colleges where AICTE curriculum is followed. Through several years of interaction with this group of students, we have found that there is no suitable textbook that suits their requirements. The book is primarily aimed at this group of students, attempting to bridge the gap between complex formulations in the theory of elasticity and elementary strength of materials in a simplified manner for better understanding. Index notations have been avoided, and the mathematical derivations are restricted to second-order differential equations, their solution methodologies, and only a few special functions, such as stress function and Laplacian operators.

The text follows more or less the AICTE guidelines and consists of twelve chapters. The first five chapters introduce the engineering aspects of solid mechanics and establish the basic theorems of elasticity, governing equations, and their solution methodologies. The next four chapters discuss thick cylinders, rotating disks, torsion of members with both circular and noncircular cross-sections, and stress concentration in some depth using the elasticity approaches. Thermoelasticity is an important issue in the design of high-speed machinery and many other engineering applications. This is dealt with in some detail in the tenth chapter. Problems on contact between curved bodies in two-dimensional and three-dimensional situations can be challenging, and they have wide applications in mechanical engineering such as in bearing and gear technology.

Ideals, in modern algebra, are subrings of a mathematical ring with certain absorption properties. The concept of an ideal was first defined and developed by German mathematician Richard Dedekind in 1871. In particular, he used ideals to translate ordinary properties of arithmetic into properties of sets.

The origin of the notion of ideals in a ring lies in the idea of “ideal numbers”, numbers which are missing but are really ought to be there. Ernst Kummer invented the concept of ideal numbers to serve as the “missing” factors in number rings in which unique factorization fails; here the word “ideal” is in the sense of existing in imagination only.

In this chapter, the abstraction of ideals is explored through various examples. The study is examined through various problems to enable students to apprehend the notion of the ideals.

Key Concepts

• • Steps involved for developing sustainable organizations

• • Case study on a university campus

• • Integration of green sources of energy

• • Implementation of energy efficiency measures

• • Ensuring participation of stakeholders for energy conservation

Introduction

The achievement of SDGs defined under the Paris Agreement requires concerted efforts at the international, national, state, organization, and individual levels. The organizations which follow the principles of sustainable development can serve as a role model for others to follow.

Colleges for higher education and the universities also have an important role to play in achieving the SDGs in general and in the adoption and promotion of green sources of electricity in particular. Goal 4 of SDGs, although, is specific to the availability of quality education to all, but these institutions can play a much broader role in realizing the wide-ranging SDGs. For example, Goal 9: Industry, infrastructure and innovation; Goal 12: Responsible production and consumption; and Goal 13: Climate Action cannot possibly be achieved without the mindful and positive influence of higher education institutions.

More importantly, these institutes need to work on the creation of awareness about the need for sustainable development and SDGs, a crucial requirement for their achievement. The institutes should also make sustainable development an integral part of their future plans. Green and renewable sources of energy like solar PV should be adopted for existing buildings, and these should be made mandatory for the new buildings. The academic institutes, more importantly, should practice on their campuses what they are preaching in the class.

Key Concepts

• " Working of solar PV power plants and their benefits

• " Different configurations of solar PV systems, such as grid-connected, stand-alone, and hybrid solar PV plants

• " Metering mechanisms, such as net metring and gross metring

• " Working and classification of different types of inverters used in solar energy generation

• " Different performance evaluation parameters for solar PV power plants and effect of environmental conditions

• " Components used in solar PV power plants

• " Challenges related to the large-scale integration of solar PV plants with the power grid

Introduction

Solar energy is a renewable source of energy, and when electricity is produced from solar, it does not lead to any CO2 emissions. Apart from being a green and renewable source of energy, solar is the simplest system of electricity generation. As described by Professor Martin Green, ‘The whole photovoltaic technology itself is a bit magical. Sunlight just falls on this inert material and you get electricity straight out of it.’ This technology has emerged as the most powerful solution for decarbonizing the energy system.

The solar PV plants can be installed in two modes: grid-connected and off-grid system. At present, grid-connected solar PV (GCSPV) plants are the most commonly used systems. Although solar PV cells, were discovered in the year 1953, solar PV plants for generating electricity did not gain widespread acceptance primarily because of the panel cost as well as the issues with the batteries involved. GCSPV technology has removed the weak link, the battery from the system, making it an efficient, economical, and durable system with minimum maintenance requirements. These benefits have made the solar PV the fastest rising system in the world.

3.1 STATE OF STRESS AT A POINT [LO1]

When a body is subjected to external forces, its behavior depends on the magnitude and distribution of forces and properties of the body material. Depending on these factors, the body may deform elastically or plastically, or it may fracture. The body may also fail by fatigue when subjected to repetitive loading. Here we are primarily interested in elastic deformation of materials.

In order to establish the concept of stress and stress at a point, let us consider a straight bar of uniform cross-section of area A and subjected to uniaxial force F as shown in Figure 3.1. Stress at a typical section A - A′ is normally given as σ = F/A. This is true only if the force is uniformly distributed over the area A, but this is rarely true. Therefore, definition of stress must be considered by progressively reducing the area until it is small enough such that the force may be considered to be uniformly distributed.

To understand this, consider a body subjected to external forces P1, P2, P3, and P4 as shown in Figure 3.2. If we now cut the body in two pieces,

Internal forces f1, f2, f3, etc. are developed to keep the pieces in equilibrium. Now consider an infinitesimal element of area ΔA Dat the cut section and let the resultant force on the element be Δf.

Introduction

A current mirror is a transistor-based circuit that the current level is controlled in an adjacent transistor, and the adjacent transistor essentially acts as a current source. Such circuits are now considered a commonly used building block in a number of analog integrated circuits (IC). Operational amplifiers, operational transconductance amplifiers, and biasing networks are examples of such circuits that essentially use current mirrors. Analog IC implementation techniques such as current-mode and switched-current circuits use current mirrors as basic circuit elements.

A significant advantage associated with the current mirrors is that they act as a near-ideal current source while fabricated using transistors and can replace large-value passive resistances in analog circuits, saving large chip area.

The later part of the chapter discusses another important analog circuit, namely, differential amplifier. As the name suggests, differential amplifiers amplify the difference between two signals that are applied to their two inputs. In addition to the differential amplification, it is also required that differential amplifiers suppress unwanted signal, which is present on the two input signals in the form of a common-mode signal. A differential amplifier is a particularly very useful and essential part of operational amplifiers. A differential pair is the basic building block of a differential amplifier that comprises of two transistors in a special form of connection.

MARKOV CHAINS WERE first formulated as a stochastic model1 by Russian mathematician Andrei Andreevich Markov. Markov spent most of his professional career at St. Petersburg University and the Imperial Academy of Science. During this time, he specialized in the theory of numbers, mathematical analysis, and probability theory. His work on Markov chains utilized finite square matrices (stochastic matrices) to show that the two classical results of probability theory, namely, the weak law of large numbers and the central limit theorem, can be extended to the case of sums of dependent random variables. Markov chains have wide scientific and engineering applications in statistical mechanics, financial engineering, weather modeling, artificial intelligence, and so on. In this chapter, we will look at a few applications as we build the concepts of Markov chains. Additionally, we will also implement a technique (using Markov chains) to solve a simple and practical engineering problem related to aircraft control and automation.

3.1 Chapter objectives

The chapter objectives are listed as follows.

• 1. Students will learn the definition and applications of Markov processes.

• 2. Students will learn the definition of the stochastic matrix (also known as the probability transition matrix) and perform simple matrix calculations to compute conditional probabilities.

• 3. Students will learn to solve engineering and scientific problems based on discrete time Markov chains (DTMCs) using multi-step transition probabilities.

• 4. Students will learn to compute return times and hitting times to Markov states.

• 5. Students will learn to classify different Markov states.

• 6. Students will learn to use the techniques of DTMCs introduced in this chapter to solve a complex engineering problem related to flight control operations.

This opening chapter introduces readers to the historical backdrop that is crucial to the analysis of contemporary Japan. It positions present-day circumstances in historical perspective, bringing them into relief against the past. The chapter also traces the historical transformations in the patterns of land ownership and tax collection, which conditioned class formation and disintegration at different times.

Japan has long been portrayed as a distinctively uniform society both racially and culturally, despite the firm reality that it has many groups that are subjected to discrimination and prejudice in ethnic and quasi-ethnic terms. This chapter first examines a few aspects of Japan’s ethnocentrism. It then addresses the fallacy of the homogeneity thesis by delineating five minority groups in Japan: indigenous Ainu, burakumin, Zainichi Koreans, Okinawans, and foreign residents. Based on the analysis of minority issues, the latter part of the chapter calls into question the monocultural definition of ‘Japaneseness’ and explores multiple ways of defining ‘the Japanese’.

This chapter endeavors to make sense of Japanese religiosity and to unravel how it has formed an undercurrent in Japanese society. First, it focuses on the characteristics of traditional religions that took hold in premodern Japan: Shinto, Japan’s native religion, and two imported religions, Buddhism and Christianity. The chapter then analyzes newer religions founded in the twentieth century and scrutinizes the more recent emergence of a cultural trend in which individuals seek spirituality outside established religious spheres. It looks at this-worldly financial and political activities of these old and new religions, then sketches how the general trend of secularization faces the revitalization of religious practices.

Contemporary Japan faces a serious dual demographic crisis, with a fast-aging population and a birth rate in rapid decline. As the active labor force diminishes in comparison with the swelling number of retirees, Japan is at the forefront of many advanced economies in confronting this problem, raising the fundamental issue of redistributing economic and social resources across different generations. More broadly, age-based variations are stark, with the young, the middle aged, and the elderly exhibiting dissimilar attitudes and behaviors. The country is also divided geographically: the Japanese have different lifestyles depending on their place of residence. Their eating habits, type of housing, language, style of thinking, and many other aspects of everyday life hinge upon where they live.

This chapter first focuses on the dynamics of an aging society, presenting the most crucial issues in Japan’s demographic change, then examines both generational and geographical variations with a view to assessing the ways in which these primary demographic characteristics condition the options and preferences of various Japanese persons.

The public discourse on class and stratification in Japan experienced a dramatic paradigm shift towards the end of the twentieth century. Although widely portrayed as an egalitarian and predominantly middle-class society during the period of high economic growth until the early 1990s, Japan was suddenly deemed a society divided along class lines under the prolonged stagnation that characterized the Japanese economy for three decades, from the 1990s to the 2020s.

Based on macro-sociological data, this chapter delineates the focal points of debate on the analysis of class and stratification in Japan as a general prelude to specific spheres covered later in the book – cultural diversity and class competition in relation to generation, region, labor, education, gender, ethnicity, and so on.

The ideology of educational credentialism pervades Japanese society and spreads an examination culture across considerable sections of Japan’s schools. In the distribution of occupational positions in Japan, it is widely believed that educational background plays a central role. In higher education, universities are ranked by prestige and reputation, so degrees from top-ranking universities are considered essential qualifications for high positions in the workforce hierarchy. A significant majority of career officials in the nation’s bureaucracy are graduates of the University of Tokyo, particularly its Faculty of Law, even though their share is declining. Large corporations are also believed to promote employees based on the university from which they graduated.

This ideology shapes the normative framework of Japanese education, positioning its top layer as the model to emulate. It compels many university examination candidates to prioritize prestigious universities over disciplines in which they are genuinely interested. Consequently, those aspiring to tertiary education engage in intense competition while preparing for the entrance examinations of reputable universities.

The international evaluation of Japan’s work practice has shifted since the beginning of the twenty-first century from role model to problem case. Once heralded as a model from which every country must learn, the nation’s business world now appears to be seen as a framework to be avoided. Yet, in the middle of the 2020s, Japan remains the fourth-largest economy in the world, even if its potential and challenges are being overshadowed by the struggle for international hegemony between the United States and China. Japan’s world of labor shows a complex mix of continuity and change in the face of globalization, a delicate combination that requires nuanced analysis.

This chapter illustrates such plurality by first highlighting the continuation of the old patterns: the prevailing culture of small business and the perpetuation of the ‘Japanese-style’ management model. The second half focuses on the emerging changes by examining the spread of a new form of capitalism based on the knowledge industry and the production and consumption of symbols, images, and representations. The impact of the new megatrend on employment practices and the power of labor unions is also analyzed.