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.

DIFFRACTION

Diffraction is a phenomenon in which a light beam bends around the corner of an obstacle and spreads into the geometric shadow of that obstacle.

FRESNEL AND FRAUNHOFER DIFFRACTION

Diffraction can be classified into two categories:

• 1. Fresnel diffraction

• 2. Fraunhofer diffraction

The distinction between these two categories is as follows:

• a. In Fresnel diffraction, the screen and source are at a finite distance from an obstacle. The distances are important in this class. In Fraunhofer diffraction, the source and screen are at an infinite distance from an obstacle. Therefore, inclination is important.

• b. The incident wavefront in Fresnel diffraction is either spherical or cylindrical, whereas the incident wavefront in Fraunhofer diffraction is planar.

• c. In Fresnel diffraction, the central point of the screen is either bright or dark depending on the number of zones, whereas in Fraunhofer diffraction, the central point of the screen is always bright.

FRAUNHOFER DIFFRACTION DUE TO SINGLE SLIT

Let us consider a monochromatic light source of wavelength ƛ placed at the focus of convex lens L1. The collimated rays of plane wavefront are incident on a single-slit AB of width “e.” The un-deviated rays from the slit reaches at point O, and the rays diffracted by an angle θ reach at P on the screen, as shown in Figure 12.1.

In graph theory, planar graphs introduce a fascinating area of study that intersects geometry, topology and network analysis. A graph is called planar if it can be drawn on a plane without any edges crossing each other. This property of planarity has significant implications in fields like circuit design, geographic mapping and urban planning where minimizing crossings leads to more efficient and visually accessible structures. In this chapter we will be covering important topics like the Euler’s formula which provides a basis for understanding the structural constraints of a planar graph. We will also be discussing Kuratowski’s and Wagner’s theorems that provide a criteria for non-planarity. As we will see later in Chapter 9, one of the intriguing mathematical characteristics of planar graphs is that their vertices can be colored by at most four colors.

What lurks behind appeals to “community” and a “democracy of the common” as models for the organization of political life is the desire for an existential authenticity that has overcome the contradictions and antagonisms that are part of normal political life under the conditions of democratic pluralism. Placing our hopes in community and the common as alternative, and somehow more authentic, models for the organization of political life always comes at the cost of preparing the ground for abandoning democracy altogether. Real democracy, counterintuitively, does not require community, but it involves distance among those who are represented, those who represent, and those who govern. We might experience this distance as alienating, or as inauthentic, but it allows for what we might call the self-control of self-government. In contrast to appeals to “community” and “the common,” the task of democracy is to negotiate the irreducible pluralism of political life through a normative organization that can be justified to, and is also justifiable by, all those who are subject to such norms.

The subject, Computer Vision, deals with the science ofimparting to a machine or a computer the capabilityof seeing and understanding the environment as wehumans are able to do, and seeks to apply itstheories and models in various applications of ourlife and society. From the late sixties of the lastcentury, there have been efforts in analyzingdigital images captured by a scanner or a camera.Initially, it was the 2-D digital geometry in adiscrete grid of integral coordinate space whichdrew primary attention of the researchers. Inparticular, Prof. Azriel Rosenfeld (1931–2004) ofthe University of Maryland, USA, took a leading andpioneering role in developing theories of digitalpicture processing. Subsequently, the area wasstrengthened by the development and application oftheories of mathematical morphology, textureprocessing, pattern recognition techniques, etc.However, the major development in the theory ofcomputer vision, following the psycho-physiologicalmodels of human vision, happened in the seventies ofthe last century, when Prof. David Marr (1945–1980)of the Massachusetts Institute of Technology (MIT),Cambridge, USA, hypothesized three stages ofprocessing and representation of images by primalsketches consisting of edges, regions, 2.5-Dsketches of the scene, and finally 3-D models.

Over the years, theories of computer vision have beendeveloped from different areas of mathematical andphysical sciences, such as digital geometry,projective geometry, differential geometry, linearand nonlinear systems, human cognition andpsycho-visual perception, color representation andprocessing, computational learning, patternrecognition, etc. As we see, the theoreticalfoundation of the subject has been built fromdifferent domains, and it requires to learn thefundamentals across these disciplines in asystematic and organized manner in the context ofcore agenda of computer vision, which is to solveproblems related to the understanding of a 3-Dscene, static or dynamic, given visual inputs fromimaging systems.

The ways in which human interaction was restructured during complexification in fifty-seven natural experiments from around the world, is characterized in twenty-one variables in the domains of interaction, demography, and political economy. Examination of the data in this way reveals an enormous range of variation among early complex societies on all counts.

In graph theory, independent sets represent collections of vertices that are pairwise nonadjacent, meaning no two vertices within an independent set share an edge. The study of independent sets is often linked to cliques (sets of mutually adjacent vertices) and covering numbers (the smallest set of vertices and edges that cover the entire graph) as they provide contrasting perspectives on how elements within a graph relate to each other.

The world of graph theory is expanding at a pace at which it is hard to keep track of the various disciplines of study that have somehow been irrevocably affected by the techniques owned and created by this versatile subject. However, the beauty of graph theory and its grandeur can also be intimidating for a beginner who wishes to explore its realms. The primary aim of the book is therefore to help a student understand and master the tools and techniques that are inherent to the subject and to serve as a handbook for anyone who wishes to explore the amazing world of graph theory.

We started writing this book with the goal of creating an undergraduate graph theory textbook that would be used by a broad audience, including students and teachers of Mathematics, Computer Science, Economics and perhaps Business Administration. However, mathematical training in proofs and logic is a prerequisite for understanding and following this book.

STATISICAL EXPERIMENTS ENABLE us to make inferences from data about parameters that characterize a population. Generally speaking, inferences may be of two types, namely, deductive inference and inductive inference. Deductive inference pertains to conclusions based on a set of premises (propositions) and their synthesis. Deductive reasoning has a definitive character. For example, all men are mortal (first proposition); Socrates is a man (second proposition); hence, Socrates is mortal (deductive conclusion). On the other hand, inductive inference has a probabilistic character. One conducts an experiment and collects data. Based on this data, certain conclusions are drawn that may have a broader applicability beyond the contours of the particular experiment performed by the researcher. This generalization of the conclusions drawn from the particular experiment constitutes the framework of inductive reasoning. For example, measurement of heights of a small group of people belonging to a certain population is conducted. Based on the calculations of this small sample set, and upon finding that for this small group the average height of men is greater than the average height of women, it is inferred that the men of this population are generally taller than the women.

The formal practice of inductive reasoning dates back to the thesis of Gottfried Wilhelm Leibniz (see Figure 5.1). He was the first to propose that probability is a relation between hypothesis and evidence (data). His thesis was founded on three conceptual pillars: chance (probability), possibilities (realizable random events), and ideas (generalization of inferences by induction). We have encountered the first two concepts in earlier chapters of this textbook. In this chapter, we will delve into the third theme whereby we will discuss methods to draw conclusions from data derived from statistical experiments based on the principles of inductive reasoning.