4.1 CONSTITUTIVE EQUATIONS [LO1]

So far, we have discussed the strain and stress analysis in detail. In this chapter, we shall link the stress and strain by considering the material properties in order to completely describe the elastic, plastic, elasto-plastic, visco-elastic, or other such deformation characteristics of solids. These are known as constitutive equations, or in simpler terms the stress–strain relations. There are endless varieties of materials and loading conditions, and therefore development of a general form of constitutive equation may be challenging. Here we mainly consider linear elastic solids along with their mechanical properties and deformation behavior.

Fundamental relation between stress and strain was first given by Robert Hooke in 1676 in the most simplified manner as, “Force varies as the stretch”. This implies a load–deflection relation that was later interpreted as a stress–strain relation. Following this, we can write P = kδ, where P is the force, δ is the stretch or elongation, and k is the spring constant. This can also be written for linear elastic materials as σ = E∈, where σ is the stress, ∈ is the strain, and E is the modulus of elasticity. For nonlinear elasticity, we may write in a simplistic manner σ = E∈n, where n ≠ 1.

Hooke's Law based on this fundamental relation is given as the stress–strain relation, and in its most general form, stresses are functions of all the strain components as shown in equation (4.1.1).

In various applications of computer vision and imageprocessing, it is required to detect points in animage, which characterize the visual content of thescene in its neighborhood and are distinguishableeven in other imaging instances of the same scene.These points are called key points of an image andthey are characterized by the functionaldistributions, such as distribution of brightnessvalues or color values, around its neighborhood foran image. For example, in the monocular and stereocamera geometries, various analyses involvecomputations of transformation matrices such as,homography between two scenes, fundamental matrixbetween two images of the same scene in a stereoimaging setup, etc. These transformation matricesare computed using key points of the same scenepoint of a pair of images. The image points of thesame scene point in different images of the sceneare called points ofcorrespondence or corresponding points. Key points ofimages are good candidates to form such pairs ofcorresponding points between two images of the samescene. Hence detection and matching of key points ina pair of images are fundamental tasks for suchgeometric analysis.

Consider Fig. 4.1, where images of the same scene arecaptured from two different views. Though theregions of structures in the images visuallycorrespond to each other, it is difficult toprecisely define points of correspondences betweenthem. Even an image of a two-dimensional (2-D)scene, such as 2-D objects on a plane, may gothrough various kinds of transformations, likerotation, scale, shear, etc. It may be required tocompute this transformation among such a pair ofimages. This is also a common problem of imageregistration.

Introduction

Statistical mechanics bridges the gaps between the laws of thermodynamics and the internal structure of the matter. Some examples are as follows:

• 1. Assembly of atoms in gaseous or liquid helium.

• 2. Assembly of water molecules in solid, liquid, or vapor state.

• 3. Assembly of free electrons in metal.

The behavior of all these abovementioned assemblies is totally different in different phases. Therefore, it is most significant to relate the macroscopic behavior of the system to its microscopic structure.

In this mechanics, most probable behavior of assembly are studied instead of individual particle interactions or behavior.

The behavior of assembly that is repeated a maximum time is known as most probable behavior.

hase Space

Six coordinates can fully characterize the state of any system:

• 1. Three for describing the position x, y, z and three for momentum Px, Py, Pz.

• 2. The combined position and momentum space (x, y, z, Px, Py, Pz) is called phase space.

• 3. The momentum space represents the energy of state,

For a system of N particles, there exists 3N position coordinates and 3N momentum coordinates. A single particle in phase space is known as a phase point, and the space occupied by it is known as µ-space.

olume Element ofµ-Space

4. Consider a particle having the position and momentum coordinates in the range.

This chapter looks at some of the experiments of international governance by the United Nations typically listed as cases of international administrations (West Irian, Namibia, Cambodia and Eastern Slavonia). These United Nations peace missions have been considered as international administrations by numerous scholars, in the same category with Kosovo and Timor-Leste and sometimes on par with these two experiments in terms of effective authority deployed by international officials. I will be arguing that contra this opinion generally based on the reading of the mandates, international officials have displayed only limited political authority over these territories. Through archival work conducted in the United Nations Archives, revealing in specific instances the hidden transcripts of the time, I will be analysing each of these cases in turn, and adding a few other cases as well including Cyprus, El Salvador, Mozambique, Western Sahara and Somalia.

The word “Integral Domain” can be interpreted in language as the domain of integrity with the idea that zero divisors are like flaws in the ring as divisibility theory is much more complex in the presence of zero divisors. The term integral comes from rings of algebraic integers—the study of which motivated the abstraction of many algebraic structures. The fact that integral domain embeds into its field of fractions as integers embed into rational supports the terminology.

In this chapter, the theory of commutative rings with unity is extended to define integral domain, which is further abstracted to fields. The study is elaborated through various examples and theorems.

Before defining integral domain, we define a special class of elements in a ring known as zero divisor.

In ring theory, factorization of polynomials constitutes a fundamental aspect of algebraic study, revealing a subtle linkage between the arithmetic properties of polynomials and their structural characteristics. At the core of this exploration lies the distinction between reducible and irreducible polynomials. Reducible polynomials can be expressed as products of two or more nonconstant polynomials, while irreducible polynomials resist such factorization, existing as prime elements within the polynomial ring. Understanding the nature of factorization sheds light on diverse algebraic phenomena and serves as a fundamental building block for various applications across mathematics and its applications.

In this chapter, we explore polynomial factorization in ring theory. We shall discuss reducible and irreducible polynomials, explore the significance of primitive polynomials, and explain the tests and algorithms used to identify the factorization properties of polynomials. Thoroughly examining the polynomial factorization, we aim to provide readers with the tools and insights necessary to understand algebraic structures and properties of polynomial rings.

Rings are algebraic structures that originated from the theory of algebraic integers. The concept was introduced by Richard Dedekind, taking inspiration from the algebraic structure of integers over complex numbers. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. The concept of ring is an extension of groups and has a wide range of applications in mathematics and computer science. Rings are applied in the study of geometric objects, topology, cryptography and various other branches of algebra.

In the vast arena of mathematics, the emergence of ring theory as a pivotal branch of algebra owes much to the visionary insights of two more extraordinary minds: David Hilbert and Emmy Noether. At the beginning of the 20th century, Hilbert provided a unifying framework for understanding sets of numbers endowed with specific algebraic properties. Concurrently, Noether's seminal contributions, notably expounded in her 1921 paper “Ideal Theory in Rings”, brought forth profound advancements in the understanding of commutative rings, laying a robust theoretical foundation for subsequent explorations. Despite encountering formidable obstacles, including entrenched gender biases and political upheavals, Hilbert and Noether remained steadfast in their pursuit of mathematical truth.

Introduction

Humans have had a lengthy history of understanding electricity and magnetism. The tangible characteristics of light have also been studied. But in contrast to optics, electricity and magnetism—now known as electromagnetics—have been believed to be governed by different physical laws. This makes sense because optical physics as it was previously understood by humans differs significantly from the physics of electricity and magnetism. For instance, the ancient Greeks and Asians were aware of lode stone between 600 and 400 BC. Since 200 BC, China has been using the compass. The Greeks described static energy as early as 400 BC. But these oddities had no real effect until the invention of telegraphy. The voltaic cell or galvanic cell was created by Luigi Galvani and Alesandro Volta in the late 1700s, which led to the development of telegraphy. It quickly became clear that information could be transmitted using just two wires attached to a voltaic cell. The development of telegraphy was therefore prompted by this potential by the early 1800s. To learn more about the characteristics of electricity and magnetism, Andre-Marie Ampere (1823) and Michael Faraday (1838) conducted tests. Ampere's law and Faraday's law are consequently called after them. In order to comprehend telegraphy better, Kirchhoff voltage and current rules were also established in 1845. The data transmission mechanism was not well comprehended despite these laws. The cause of the data transmission signal's distortion was unknown. The ideal signal would alternate between ones and zeros, but the digital signal quickly lost its shape along a data transmission line.

A1.1 Basic matrix operations

A1.1.1 Definition (matrix)

A matrix is an ordered rectangular array of numbers (or functions). The numbers (or functions) are called the elements or the entries of the matrix.

A1.1.2 Dimension/size of a matrix

A matrix having m rows and a columns is a matrix of dimension m x n.

We are familiar with the concept of connectedness of a graph and how it decides whether we can traverse the graph, starting from any vertex and reaching another vertex. It is possible for a connected graph to lose its connectedness by the removal of a single vertex. There are also other connected graphs in which the removal of several vertices does not disconnect the graph. Hence we have stumbled upon a parameter that can give a “sense” of the connectedness of a graph. It may answer important questions about the nature of its connectedness. In this chapter, we will deal with connectivity, which measures the connectedness of a graph using some specific parameters.

CHANCE PERMEATES OUR physical and mental universe. While the role of chance in human lives has had a longer history, starting with the more authoritative influence of the nobility, the more rationally sound theory of probability and statistics has come into practice in diverse areas of science and engineering starting from the early to mid-twentieth century. Practical applications of statistical theories proliferated to such an extent in the previous century that the American government-sponsored RAND corporation published a 600-page book that wholly consisted of a random number table and a table of standard normal deviates. One of the primary objectives of this book was to enable a computer-simulated approximate solution of an exact but unsolvable problem by a procedure known as the Monte Carlo method devised by Fermi, von Neumann, and Ulam in the 1930s–40s.

Statistical methods are the mainstay of conducting modern scientific experiments. One such experimental paradigm is known as a randomized control trial, which is widely used in a variety of fields such as psychology, drug verification, testing the efficacy of vaccines, agricultural sciences, and demography. These statistical experiments require sophisticated sampling techniques in order to nullify experimental biases. With the explosion of information in the modern era, the need to develop advanced and accurate predictive capabilities has grown manifold. This has led to the emergence of modern artificial intelligence (AI) technologies. Further, climate change has become a reality of modern civilization. Accurate prediction of weather and climatic patterns relies on sophisticated AI and statistical techniques. It is impossible to think of a modern economy and social life without the influence and role of chance, and hence without the influence of technological interventions based on statistical principles. We must begin this journey by learning the foundational tenets of probability and statistics.

EMPIRICAL TECHNIQUES rely on abstracting meaning from observable phenomena by constructing relationships between different observations. This process of abstraction is facilitated by appropriate measurements (experiments), suitable organization of data generated by measurements, and, finally, rigorous analysis of the data. The latter is a functional exercise that synthesizes information (data) and theory (model) and enables prediction of hitherto unobserved phenomena.1 It is important to underscore that a good theory (model) that explains a certain phenomenon well by appealing to a set of laws and conditions is expected to be a good candidate for predicting the same using reliable data. For example, a good model for the weight of a normal human being is w = m * h, where w and h refer to weight and height of the person, and m can be set to unity if appropriate units are chosen. A rational explanation of such a formula for weight based on anatomical considerations is perhaps very reasonable. From an empirical standpoint, if we collect height and weight data of normal humans, we will notice that a linear model of the form w = m * h represents the data reasonably well and may be used to predict the weight of the person based on the height of the person. This fact ascertains a functional symmetry between explanation and prediction. Therefore, a good predictive model must automatically be able to explain the data (and related events) well.

Take a map of your country where the different regions, provinces, or states are clearly shown. How many colors are required to color each region on a map of the country so that the neighboring regions are colored differently? As a fun exercise, let us create a graph by placing a vertex in the middle of each region, and two vertices are adjacent if the states they represent share a border. Now that we have modeled the regions of a map as vertices of a graph, the graph’s vertices can be colored in the same way that the regions are colored on the map. The graph’s vertices cannot be colored the same when they are adjacent as the neighboring regions cannot have the same color. This interesting problem of coloring has been the source of a great amount of research in graph theory. The applications of graph coloring are numerous, since it is a powerful tool that enables the systematic assignment of resources in situations where certain elements must be distinct or non-overlapping. We will also discuss chromatic number of a graph, which is the minimum number of colors required to have a conflict-free coloring. The parallel concept of edge coloring is essential to frequency assignment in networks, where edges need distinct colors to prevent interference. In this chapter, we will also explore the connection between planarity and coloring, through the five color theorem and four color theorem.