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.

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.

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.

Preventing climate change and ensuring sustainable development is one of the most talked about concepts in recent years. The issue of climate change and global warming is closely related to carbon dioxide (CO2) emissions. The rising level of CO2 in the environment is a major concern because of the resulting global warming and its associated adverse effects. In 2015 the United Nations executed the Paris Agreement. The agreement aims to limit the global temperature rise to 2°C, and to make best efforts to keep it to 1.5°C.

This objective has an important connection with energy and, particularly, electrical energy. The most common conventional method for producing electricity is by burning coal, which leads to CO2 emissions. Electricity produced from solar energy and wind energy is considered green electricity because it does not contribute to CO2 emissions. An important component of the CO2 emission reduction initiative of the UN, therefore, is the installation of green energy sources on a large scale throughout the world.

In line with this vision, the most important part of a carbon emission reduction plan worldwide is installation of these sources on a large scale. India, for example, has committed to having 50% of the total installed capacity by 2030 from green sources of electricity generation. This push towards a shift to solar, wind, and hydro-based energy sources is happening at a very fast rate. Under this changed scenario, these technologies are set to become the main technologies in the electric power network. In the Indian grid, for example, the share of solar energy has grown from about 2.6 GW in 2014 to more than 100 GW in 2025. This shift is set to change the fundamental way in which electricity has been generated, transmitted, and utilized.

2.1 Groundwork and Motivation for the Definition

• To begin with, consider the following questions.

  (1) Are the intervals [0, 1] and [0, 5] the same?

  (2) Are the intervals [0, 1] and (0, 1) the same?

  (3) In the plane R2, is a circle the same as a square? Or is a circle of radius 1 the same as the circle of radius 2?

  (4) In R2, is the unit circle the same as the unit disk?

  (5) Is R the same as R2?

  (6) Is the interval (0, 1) the same as R?

  (7) Is the interval (0, 1) the same as the union of intervals (0,#½) ∪ ( #½ , 1)?

  (8) Is the interval [0, 1) the same as the unit circle in R2? Or is (0, 1) or [0, 1] the same as the unit circle in R2?

Of course, to answer this, we must first specify what we mean by the word “same”. Geometrically speaking, clearly, none of the pairs in each of the above questions are the same. For instance, in (1), geometrically the intervals [0, 1] and [0, 5] are of different lengths and as sets [0, 1] ⊆ [0, 5]. However, if we think of the interval [0, 1] as a rubber band of length 1, then we can stretch it from one end to transform it into the interval [0, 5]. Mathematically, we can do this by defining a function f : [0, 1] → [0, 5] by f(x) = 5x. Then we may identify the intervals [0, 1] and [0, 5]. Similarly, in (3), it may not be easy to come up with a function, but if we think of the circle made of a rubber or elastic rope, then we can reshape it into a square to call the circle and a square as “same”. Similarly, we can stretch the circle of radius 1 to form a circle of radius 2. On the other hand, in (7), the interval (0, 1) is a one-piece connected set, but the union (0, ,# ½) ∪ ( #½ , 1) is disconnected as it is missing the point 1 2 . So they do not seem to be the “same” from what we discussed above. In (8), it is difficult to say anything as the interval [0, 1) is a subset of R while the unit circle is in R2.

Medical imaging mostly deals with the visualization ofinternal organs, tissues, etc., using noninvasive orsemi-invasive methods. The primary motive is tounderstand any anomalies in the anatomies and theirfunctions. The signals are acquired inone-dimensional (1-D), two-dimensional (2-D),three-dimensional (3-D), or as videos, depending onthe purpose and mode of imaging. There are variousmodalities in medical imaging. The major modalitiesare as follows.

• • Projection X-ray (radiography)

• • X-ray computed tomography (CT scan)

• • Nuclear medicine images (emissiontomographies like, single photon emission computedtomography (SPECT) and positron emissiontomography (PET))

• • Magnetic resonance imaging (MRI)

• • Ultrasound

There are also several other modalities of medicalimaging. In this chapter, only the above mentionedtechniques are discussed.

14.1 | Projection X-ray imaging

The X-ray imaging technique was discovered by WilhelmConrad Rontgen (inaugural Nobel Prize, 1901) in 1895in Wurzburg, Germany. In its basic form, themechanism for generation of X-ray consists of avacuum tube with a cathode and an anodeappropriately placed as illustrated in Fig. 14.1(a). A beam of electron that emanates from thecathode hits the anode at a very high speed.Depending upon the material of the surface in theanode, there are sub-atomic interactions due to thestriking of sub-atomic particles, which releaseenergy in the form of electromagnetic waves of acertain wavelength band, called X-rays.

QUEUES FEATURE IN our daily lives like never before. From the checkout counter in the community grocery store to customer support over the phone, queues are theatres of great social and engineering drama. Entire business operations of many leading companies are geared towards providing hassle-free customer support and experience – timely and effective resolution of client queries about services on a regular basis. Alternatively, it could be effective traffic management and resource optimization for a multiplex cinema operator involved in ticket sales. Sometimes it may not involve humans at all, like in the case of a database query to a computer server for specific information that may be routed through a job queue. How a queue moves in time and how services are offered over epochs determine how businesses will be able to make profit or how efficiently computer servers will execute tasks. All these have a huge technological and economical impact. No wonder we have seen huge investments by concerned stakeholders to upgrade and upscale hardware and software infrastructure to re-engineer queues towards greater system efficiency and profitability. The mathematical technology of queues is crafted out of models that investigate and replicate stochastic behavior of engineering systems. This is the subject of our study in this chapter.

Definition 1.1 (Binary Operation) Let 𝐺 be a non-empty set. A binary operation on 𝐺 is a function that assigns each ordered pair of elements of 𝐺 an element of 𝐺.

Definition 1.2 (Group) Let 𝐺 be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair (𝑎, 𝑏) of elements of 𝐺 an element in 𝐺 denoted by 𝑎𝑏. We say 𝐺 is a group under this operation if the following three properties are satisfied.