The origin of mathematical matrices lies with the study of systems of simultaneous linear equations. Today, they are used not simply for solving systems of simultaneous linear equations, but also for describing the quantum mechanics of atomic structure, and even for designing computer game graphics. Matrices are very useful due to the fact that they can be easily manipulated. We use the notation A-1 to denote the inverse of a matrix A. One of the major uses of inverses is to solve a system of linear equations. You can write a system in matrix form as AX=B, then X=A-1B. Inverses are also used in ...

In this work we discuss a special kind of stochastic processesX=〖{X_t}〗_(〖t∈R〗_(≥0) ) that are of exceptional interest from both the theoretical ad the applied points of view. These processes are called Affine Processes, and are characterized by the fact that their characteristic function has the form of an exponential of an affine function, i.e. by 〖∅^x〗_X (u) |_t=E^x [e^(<X_(t ),u>) ]=e^(Φ(t,u)+<x,Ψ(t,u)>), Where the exponent Φ(t,u)+ <x,Ψ(t,u)> Is an affine function of its initial state x in the state-space E=〖R^m〗_(≥0)×R^n. The above expectation E^x is the expectation which respect to the ...

In this thesis, we present a new proof of the fundamental theorem of calculus of the Lebesgue Integral [1] and we explore the properties of a new and interesting Lebesgue integrable function [2]. In this thesis, we reveal the main connection between “absolute continuity” and “Lebesgue Integration” that is established from the fundamental theorem of calculus for the Lebesgue integral. And a function which is Lebesgue integrable but not Riemann integrable. In chapter 1, an elementary and simple proof of the fundamental theorem calculus for the Lebesgue integral is given. The proof is based on the ...

The Lagrange’s Mean Value Theorem is a very important result in Analysis. It originated from Rolle’s theorem, which was proved by the French mathematician Michel Rolle (1652-1719) for polynomials in 1691. This theorem appeared for the first time in the book “M´ethode pour r´esoudre les ´egalit´es” without a proof and without any special emphasis. Rolle’s Theorem got its recogni- tion when Joseph Lagrange (1736-1813) presented his mean value theorem in his book “Th´eorie des functions analytiques” in 1797. It received further recognition when Augustin Louis Cauchy (1789- 1857) proved his mean ...

The term operational risk became widespread in the late 1990s when central bank representatives of twelve countries formed a working committee; the Basel Committee on Banking Supervision (BCBS). The BCBS defines operational risk as the risk of loss resulting from inadequate or failed internal processes, people and systems or from external events. This research aims to model operational risk data using the Loss Distribution Approach under BCBS requirements. Simulated data was used consisting of 3,192 operational loss events between the years 2009 and 2018. The implementation of the LDA was ...

The prediction of crystal size distribution from a continuous crystallizer at steady state is important for the simulation, operation and design of crystallizers. In this research, we consider integrodifferential population balance equations (PBE) describing the crystal size distribution for a crystallizer with random growth dispersion and particle agglomeration. We first develop numerical schemes to solve the initial value problem after we establish the well-posedeness of this problem. We then test the performance of these schemes on examples with known solutions. The numerical results from ...

This thesis is an introduction to some of the classical theory and results of Differential Geometry: The geometry of curves and surfaces lying (mostly) in 3-dimensional space. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is adapted to the curve near that point. Given a curve, one can define two quantities: its curvature and torsion. Both quantities are scalar fields and depend on some parameter which parametrizes the curve that is in general the arc length of the curve. The Fundamental ...

The aim of this thesis is to study the qualitative behavior of a specific non-linear Volterra integro-differential equation with finite delays by using Lyapunov's second method. The non-linear Volterra integro-differential equation is: $x'(t)=b(t)x(t-r_1)-\int_{t-r_2}^{t}a(t,s)g(x(s))ds,$ where $r_1$, $r_2$ are positive constants representing 2 finite delays, $t \geq 0$ and $a : \ [0,\infty) \times [-\tau, \infty) \rightarrow \R, \qquad \text{and} \qquad b : \ [0, \infty ) \rightarrow \R$ are two continuous functions. In the first part, we study the qualitative behavior of the constant ...

From ages to ages there had been expectation of individuals on a specific predictions and future occurrences. So also in a game, different participant that involves in those specified game have their various expectations of the results or the output of the game they are involved in. That is why we need a mathematical theory that helps in prediction of the future expectations in our day to day activities. Therefore the Martingale Theory is a very good theory that explains and dissects the expectation of a gamer in a given game of chance. So in this thesis, we shall talk about the Martin-gale ...

Geometric measure theory could be described as differential geometry, generalized through measure theory to deal with maps and surfaces that are not necessarily smooth, and applied to the calculus of variations. Geometric measure theory is important because it studies sets, their variation and their boundaries (from the measure theoretic sense). In particular, a very interesting branch in Geometric measure theory, is the sets of locally finite perimeter. Just as their name actually shows, these sets are essentially sets whose perimeter is (locally) finite. In this thesis we start by giving a ...