Appendix A

Clustering is a task oforganizing objects into groups whose members aresimilar in some way. A cluster is a collection of objects thatare similar to each other, but dissimilar to theobjects belonging to other clusters. In other words,a cluster is a group of objects with loosely definedsimilarity among them, which may have the potentialto form a class. A class is a known group of objects thatare described by similar characteristics, andclassification isthe task of assigning a defined class to an object.Image segmentationis also a problem that is similar to clustering,where the clusters are formed by groups of pixelsthat are similar in some context. In imagesegmentation, homogeneous regions in an image may beclustered to derive segments in the image. Thesesegments represent clusters. An example of imagesegmentation is shown in Fig. 6.1, where theforeground is represented by mushroom and thebackground is represented by humus substance aroundit. In this case, the image is primarily clusteredinto two regions, which are shown by white solidcontour (foreground) and white dashed contour(background).

The main motivations of clustering techniques are asfollows.

• • To find representative samples of homogeneousgroups in the given data, which would reduce thedata transmission and storage requirements incertain applications. Here, the data isrepresented by a smaller set of representativesamples that capture the characteristics of totaldata.

• • To discover natural groups or categories inthe data, which may be used to describe the datasamples by their unknown properties.

• • To find relevant groups in the data, whichfacilitates to draw attention toward major groupsof the data in the distribution. These groups formthe major clusters in a given context, likesegments in an image.

• • To detect unusual data objects, which are theoutliers in the data, that deviate from thecollective characteristics of groups of data in agiven context.

In the previous chapter, we saw that compactness is a topological property. In this chapter, we shall discuss another important topological property, called connectedness. Vaguely speaking, a connected topological space means a space that is in one single piece. In other words, there is no separation possible for a connected space. The formal definition is given below.

9.1 Connected Spaces

As indicated by the definition, connectedness is a topological property since it is defined in terms of the open sets of X. Therefore, if X is a connected space, then any space that is homeomorphic to X is also connected (see Theorem 9.1.34).

Observe that a separation of a topological space X is a pair of open subsets U and V and U ∩ V = ∅, U ∪ V = X. Thus, it follows that U and V are complements of each other. Since U is open, V = X ∖ U is closed, and since V is open, its complement U = X ∖ V is closed. Alternately, a separation of X can be defined with respect to closed sets as follows.

Primary archaeological knowledge is produced through intensive regional specialization – the antithesis of broad comparative analysis, which demands critical and consistent expert evaluation of information across multiple areas. The availability and quality of data from different regions are spotty, requiring new and more robust analytical approaches for complete reanalysis of primary data for comparative purposes.

INTRODUCTION

Classical mechanics is mainly based on Newton's laws of motion and gravitation. Initially, it was thought that Newton's second law of motion was valid and applicable at all speeds. But new experimental evidence showed that Newton's second law of motion is valid and applicable at low speeds and invalid when the object is moving at high speeds comparable to the velocity of light. This failure of classical mechanics led to the development of the special theory of relativity by young physicist Albert Einstein in 1905, which showed everything in the universe is relative and nothing is absolute. Relativity connects space and time, matter and energy, electricity and magnetism, which are useful and remarkable to our understanding of the physical universe.

The special theory of relativity is applicable to all branches of modern physics, high-energy physics, optics, quantum mechanics, semiconductor devices, atomic theory, nanotechnology, and many other branches of science and technology.

The theory of relativity has two parts: the special theory of relativity and the general theory of relativity. The special theory of relativity deals with the inertial frame of references, while the general theory of relativity deals with the accelerated frame of references. Some common technical terms that are frequently used in relativistic mechanics are as follows:

• 1. Particle:A particle is a tiny bit of matter with almost no linear dimensions and is considered to be located at a single place. Its mass and charge define it. Examples include the electron, proton, and photon, among others.

Remote sensing involves measurements on a targetwithout getting in contact with it and it comprisestechniques for collecting, storing, and processinggeoreferenced and geospatial data to extractvaluable information. In this context, data refer torepresentations stored in computer memory, which canbe manipulated using computers to derive meaningfulinsights. Remote sensing imaging systems primarilywork with georeferenced images, capturing Earth'ssurfaces, environment, atmosphere, etc. Theseimaging systems may be carried by satellites orairborne platforms like airplanes or drones. Forsatellite-based imaging, revolution of the satellitearound continuously rotating Earth allows periodiccapture of images over the same area. There are twomain types of imaging systems: passive and active.In passive systems, sensors detect reflected andemitted electromagnetic (EM) waves from Earth'ssurface, from mainly two types of energy sources,namely sunlight during the day and terrestrial heatat night. These sensors operate within specificspectral bands, converting energy into electricalsignals stored as two-dimension (2-D) images. Theprinciple is similar to optical cameras.Additionally, energy from Earth's thermal emissioncontributes to night time imaging, particularly inthe thermal infrared (IR) or far IR bands. Passiveremote sensing involves capturing images acrossvarious spectral bands, resulting in multispectraland hyperspectral images of specific regions onEarth.

In active imaging systems, microwave radar (RAdioDetection And Ranging) technology is utilized. Aradar transmitter emits a pulse of an EM wave with aspecific wavelength (in the microwave band). Whenthis pulse strikes a target, some of its energyreflects back to the radar antenna to which theradar receiver is connected. The receiver capturesinformation about the location and geometry of thetarget by recording the phase and amplitude of thereturned signal. By scanning the radar beam over anarea, an image of that region is formed. One of thelimitations of radar imaging is that, the size ofthe transmitting antenna is required to be large forobtaining images of high spatial resolution.

An artificial neural network (ANN) is a network ofneural nodes or perceptual nodes.1 In a feed forwardneural network, each node is fed with a weightedinput vector and the net sum of the weighted vectorsfrom several such nodes is passed through anonlinear function, whose response is the output ofthat node. The layers formed by the input nodes andthe output nodes are known as the input layer andthe output layer, respectively. The layers formed byother nodes are known as hidden layers. A network ofseveral such layers (along with input and outputlayers) forms an ANN, as shown in Fig. 9.1.2 Theconventional ANNs have very few hidden layers,usually not more than three. Using only one or twohidden layers is also common in many applications.In contrast, deep neuralarchitectures have relatively more numberof layers. Even more than 100 hidden layers is notuncommon. This is one of the distinguishedcharacteristics of deep neural networks (DNN) froman ordinary ANN.

The concepts used in deep neural computations aredecades older. In fact, they involve the same neuralnetwork computations as in a simple ANN model. Theseconcepts were introduced in 1980's and their basicprinciples still remain the same. However, a boom inusing deep architectures after almost three decadesof their proposition is mainly attributed to theadvancement in technology and science.

Key Concepts

• • Electricity scenario in India and the need for transition to green energy in the country

• • Indian pledge at CoP-26 at Glasgow and the targets of 2030

• • National solar mission and major initiatives that led to exponential growth in solar energy installation in India

• • Net-zero target of India and road map for achieving it

• • Major solar power projects in India

• • Various policies and government organizations involved in achieving the target of solar PV deployment in India

• • Changes required in the grid in view of massive deployment of variable and uncertain sources of electricity

Introduction

India is now the most populous country in the world, with almost 18% of the global population. Being a developing country and one of the major evolving economies, the electricity demand in the country is also growing. India ranked fifth in terms of installed capacity and third in terms of electricity produced, in 2018, in the world. India's annual per capita electricity consumption, although, is about 1122 kWh, which is much lower than the world average of 2674 kWh per year, but this number too is one of the fastest-growing. In the last decade the installed capacity of the Indian grid has increased by more than 200 GW. At this rate India is set to become the biggest electric load centre in the world by 2030 with about 1.5 billion people.

Being the most populous developing country, Indian response to the climate crisis is key to the success of sustainable development and the climate protection mission. India historically accounted for less than 5% of the global emissions.

Directed graphs, also known as digraphs are a fundamental structure in graph theory, where the relations between two vertices are allowed to be asymmetrical. An edge as we have known so far, has always signified a two-way relationship between two vertices. But in directed graphs, a directed edge has a designated direction in which you may go from say a to b but not always from b to a. This directionality introduces the concept of ordered pairs where each edge has a distinct starting point (source) and an ending point (target). Understanding their properties and behavior is crucial for solving problems related to reachability, hierarchy and optimization. This chapter explores the core concepts of directed graphs including their representation, traversal techniques and properties while also providing a succinct introduction to the world of networks.

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.