In this chapter, we shall see how new topological spaces may be created from given topological space(s). We already saw one such technique Section 2.4 of Chapter 2, where, given a subset Y of a topological space X, we endow it with the subspace topology, also called the relative topology inherited from X. In the first section, we shall see a possible way to define a topology on a set by considering a function from the set to some topological space satisfying specific properties. On the other hand, the same can be done by means of a function from a topological space to our given set. Furthermore, given topological space, not necessarily distinct, we shall see how topologies can be defined on their Cartesian product and their disjoint union (coproduct). In Chapter 2, we saw many topologies on ℝ, such as the usual topology, the lower limit topology, and the K-topology, which were defined by means of a basis. Most of the topologies that we shall see in this chapter are defined by means of a subbasis. It is advisable to refer Section 2.3.2 once before proceeding with this chapter for a better grasp of these concepts.

As we discuss the topic of traversal within a graph, it becomes necessary to define the concept of distance in a graph. The definition of distance within a graph follows the same style of definition of distance between two cities or from one's location to one's destination. Just as it is logical to seek shortest paths in one's day-to-day life, we discuss the distance between vertices, in terms of the shortest path from vertex u to vertex v. In a connected graph G, the distance between two vertices u and v denoted as d(u,v) is the shortest path between two vertices in a graph. In a disconnected graph, there are no paths between two vertices which lie in different components and so it follows that if u and v lie in different components of a disconnected graph, then d(u,v) = ∞. In this chapter, we will also be introduced to shortest path algorithms and graph traversal algorithms, that help one to navigate a graph while also paying attention to distance and connectivity.

This chapter introduces the fundamental idea of The Life of Freedom in Kant and Hegel: the notion that we can only make sense of autonomy by returning to the concept of life. This return is needed to understand fully the genesis, the form, and the reality of human freedom. Such an account can be developed by means of a systematic reconstruction of Kant’s and Hegel’s philosophies of freedom. As we can learn from Kant’s account, the notion of autonomy is threatened by the paradox of self-legislation and an opposition of freedom and nature that makes the reality of freedom unintelligible. As Kant already indicates and Hegel goes on to develop, we can overcome these problems by reconceiving of autonomy as a form of life. The chapter outlines the reading of Kant and Hegel supporting this view, situates the resulting systematic position in current debates on the sources of normativity and the nature of human freedom, and defines its relation to other approaches norm and nature (ethical naturalism, forms of life, and biopolitics).

Object tracking dealswith the estimation of the trajectory of a movingobject in the image plane. The tracking takes placein a temporal sequence of images or frames in agiven input video. The task of object tracking isperformed in various contexts. Accordingly thecomputational problems are formulated with varyingset of objectives. For example, the tracking may berequired for either a single object or multipleobjects. It can either be a trajectory intwo-dimensional (2-D) space, as in the image space,or it can be a three-dimensional (3-D) trajectory,where the object is moving in a world coordinatesystem. It can either be a tracking of a set ofpoints, or it can be tracking of the entire body ofan object. The motion involved with the object caneither be a rigid body motion, or it can be adeformable body motion. The processing can either beperformed offline on recorded videos, or it may berequired to process the video in real-time or quasireal-time, while capturing the videos. On the otherhand, it could be an online tracking, which meanstracking results are obtained within a very shorttime interval of capturing of video frames. Thechoice and design of a tracking algorithm depends onthe context and application. A few examples ofapplications of tracking are shown in Fig. 8.1. TheFig. 8.1 (a) illustrates tracking of the vehiclesand pedestrians to check if they are on the correctlane. In (b), visitors of a monument in crowd arebeing tracked and in (c), cars are being tracked toensure the speed limit.

Climate change, global warming, and shifting to sustainable ways of growth are major concerns of present times. Although these issues were identified many years ago, for a long time the exact nature of these phenomena and their impact were under debate. However, recent years have seen clearly visible and regularly occurring phenomena that confirm the adverse effects and grave threat of global warming and climate change.

It is noteworthy that 17 November 2023 was an important day for climate change, as on this day the average temperature of the earth exceeded by more than 2°C compared to the pre-industrial age temperature for the first time. The Copernicus Climate Change Service of the European Union (EU) has confirmed that in 2023 the average temperature of the earth's surface was 1.48°C higher than the temperature during the pre-industrialization days. In absolute terms also, the global emissions of CO2 are rising; in 2022, for example, these emissions were at least a billion tonnes higher than in 2019.

The Conference of Parties (CoP) is the supreme decision, making body under the United Nations Framework Convention on Climate Change (UNFCC). CoP is an important annual event initiated about 30 years ago. But the increasing interest of public, over the years, in the event shows that we have come a long way in understanding the adverse impacts of climate change. From sceptics doubting the very idea of climate change to it becoming the most debated topic in the world is a major change.

Sinusoidal Oscillations

The feedback amplifiers having negative feedback are likely to become unstable because of some unavoidable phase lag at the output due to the frequency-dependent nature of the primary amplifier gain or feedback network, or due to both. The feedback becomes positive if the additional phase lag becomes 180o at a certain frequency. If the loop gain becomes unity at any frequency, the amplifier starts oscillating at this frequency. This instability and consequent oscillations are undesirable in circuits used as amplifiers. However, practically, such oscillations are generated intentionally for widespread applications also. Hence, many schemes and approaches are in use to obtain oscillations for different frequency ranges. Such circuits are known as oscillators, or signal generators.

To understand this instability and the development of oscillations, intended or otherwise, consider the basic block diagram of a negative feedback amplifier shown in Figure 6.1. Here, vin is the input signal, the net input to the amplifier is ve, the voltage gain of the amplifier is (Av) , βff is the feedback factor, and vout is the output voltage.

The voltage gain of the feedback amplifier is (vout/vin ) = Av/(1 + βf Av). When the voltage gain Av and feedback factor βf are negative (or positive) and real constants, the feedback becomes negative. However, because of the frequency-dependent nature of the primary (and/or in feedback circuit) amplifier, the loop gain has an additional phase shift of 180o at some critical frequency ωo.

10.1 INTRODUCTION [LO1]

There are many applications where structures or machine parts are subjected to significant changes in temperature, for example, turbine blades, high-speed rotating machinery, and boilers in thermal power plants. Large thermal stresses may be developed in such applications, and sometimes such stresses may exceed the yield limit. It is therefore necessary to make provisions in the design of components to avoid failure due to thermal stresses. If the ends of a rod or any other machine parts are rigidly fixed such that the expansion or compression is prevented and the temperature is changed, tensile or compressive stress would be set up, and in simple terms, these stresses are called thermal stresses. In a long steam pipe, expansion joints are sometimes inserted, and in bridges, one end may be rigidly fastened to the main structure while the other end rests on rollers to avoid thermal stresses. In simple terms, this may be demonstrated considering a rod of length l and cross-sectional area A fixed at both ends (Figure 10.1) and temperature is raised by ΔT. This would produce a thermal strain ∈t in the rod such that

where a is the coefficient of thermal expansion. Since the rod is not free to expand, a compressive stress σc would be developed in the rod, and this is given by

If the rise in temperature is significant, the rod may buckle, which is of serious consideration in the design of machine parts or structures. To avoid this, we need to find the critical force Pcr for buckling. Using the basic buckling criterion, this can be given for a column pin ended at both ends, by

where I is the least moment of inertia of the constant cross-section column (rod) and A is the cross-sectional area of the column.

Challenging typological definitions and unilineal evolution, this chapter advocates for a comparative methodology centered on understanding variation in trajectories of complexification. It underscores the limitations of “dichotomania” and the need for richer conceptual and linguistic tools to characterize societal variation.

rystallography

The science that deals with the geometrical structure and physical properties of crystalline solids is called crystallography.

Solids are classified into two categories:

• 1. Crystalline solids

• 2. Amorphous solids

CRYSTALLINE SOLIDS

Crystalline solids are those that contain the regular repeated pattern of atoms or molecules, as shown in Figure 10.1. The physical properties of crystalline solids are different in different directions. Therefore, crystalline solids are anisotropic. Examples are rock salt, quartz, calcite, sugar, and so on.

AMORPHOUS SOLIDS

Figure 10.2 illustrates an amorphous material, which lacks the regular arrangement of atoms or molecules. The amorphous solid's physical characteristics are uniform throughout. As a result, amorphous solids are isotropic. Examples are glass, rubber, polymers, and so on.

SPACE LATTICES

A crystal is made up of identical structural units (atoms, molecules, or ions) that are infinitely repeated in space; each unit can be replaced by a geometrical point. The outcome is a pattern of dots with crystal-like geometrical characteristics. The crystal lattice or space lattice is this geometric arrangement. Lattice points are the name given to the geometrical points.

The regular pattern of points that describes the three-dimensional arrangement of points (atoms, molecules, or ions) in the crystal structure is called the crystal lattice or space lattice.

BASIS

The unit assembly of atoms, molecules, or ions identical in the composition, arrangement, and orientation is called basis. If we add the basis to every lattice point, then it forms a crystal structure, as shown in Figure 10.3.

Space lattice + Basis = Crystal structure.

Engineering physics plays a crucial role in providing the foundational knowledge necessary for the development of innovative technologies. It is an essential part of the curriculum for students in various streams of science and engineering at the undergraduate level.

The goal of this book is to develop a solid understanding of the basic principles of physics and highlight their relevance to engineering. The content is structured to progressively build the knowledge and skills necessary for further studies in both theoretical and applied sciences. Each chapter begins with the basic concepts and gradually moves to more advanced topics, supported by numerical examples, illustrations, and problem sets that reinforce learning. The problems included are designed to improve the problem-solving skills of students and provide practical insight into the engineering applications of physics.

The manuscript includes 14 chapters that were prepared in accordance with the syllabus taught in various Indian colleges and universities. In addition to core topics, the manuscript also covers advanced topics such as relativistic mechanics, quantum mechanics, optical fiber, lasers, semiconducting materials, superconducting materials, and nanomaterials. Students who want to pursue higher education and a career in research, as well as instructors who instruct postgraduate courses at universities, will find these topics helpful for building a solid foundational understanding and developing problem-solving abilities.