In the realm of mathematics, the evaluation of integrals plays a crucial role in various fields such as physics, engineering, and economics. Integrals represent the accumulation of quantities over a continuous range and are essential for understanding the behavior of functions and solving complex problems. Over the years, mathematicians and researchers have developed numerous techniques to evaluate integrals efficiently and accurately. This thesis consists of two parts. In the first part, we present some binomial identities, special numbers and polynomials as well as basic formulas related to ...
In this paper, we'll examine the effect of Cauchy errors in a linear model on the performance of the least squares and maximum likelihood estimators with the aid of two factors; the sample size and the Cauchy scale parameter. A sampling distribution of one hundred experiments was done to judge the estimation process in the case of least squares method. On the other hand, multivariate Newton Raphson was used to calculate the unique solution of the partial derivatives of the log lokelihood function. The uniqueness of the solution is proved for a known location parameter and unknown scale parameter. ...
The paper presents the study of the iterations of rational fractions, that is, the behavior of z_0,z_1=f(z_0 ),…,z_(n+1)=f(z_n ),… We illustrate the conditions needed for this function to behave as its linear part and prove the existence of Siegel and Cremer points. The study extends to a description of Fatou and Julia sets.
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 ...
The purpose of this dissertation is to study pseudo-Riemannian algebras, which are algebras with pseudo-Riemannian non-degenerate symmetric bilinear forms. The paper([1]), the authors Zhiqi Chen, Ke Liang, and Fuhai Zhu find that pseudo-Riemannian algebras whose left centers are isotropic play a key role and show that the decomposition of pseudo-Riemannian algebras whose left centers are isotropic into indecomposable non-degenerate ideals is unique up to a special automorphism. Furthermore, if the left center equals the center, the orthogonal decomposition of any pseudo-Riemannian algebra into ...
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 ...
We need the following definitions: An integral domain is a commutative unitary ring with no zero divisors. A principal ideal domain (PID) is an integral domain in which every ideal can be generated by one element. A unique factorization domain (UFD) is an integral domain in which factorization of integers into primes is unique. (more details later). An integral domain R is said to be a Euclidean ring if for every a≠0 in R there is a defined integer d(a) such that: For all a,b € R, both non zero, d(a) ≤d(ab). For all a,b € R, both non zero, there exists t,r € R such that a=tb+r where either r=0 ...
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 ...
Al Ghandour, Maria Gerges(Notre Dame University-Louaize, 2021)
In today's world, protecting information has become one of the most difficult tasks. Cyber
security events and data breaches continue to be expensive events that affect people and businesses all around the world. A breach occurs when sensitive information is accessed. Moreover, cyber threats are constantly evolving in order to take advantage of online behavior and trends, especially when teleworking has become a necessity due to the global invasion and prevalence of the Coronavirus disease 2019 during the past two years. Therefore, the necessity for cyber insurance, which covers the liability ...
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 ...
Fuzzy measure theory is a generalization of classical measure theory. It was first introduced by Lotfi Zadeh in 1965 in his famous paper ”Fuzzy Sets”. After more than 50 years of the existence and development of classical measure theory, mathematicians felt that the additivity property is, in some applications, too restrictive. It is also unrealistic under real and physical conditions where measurement errors are unavoidable. According to Sugeno, fuzzy measures are obtained by replacing the additivity condition of classical measures with weaker conditions of monotonicity and continuity.
Chapter ...
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 ...
Bitcoin is the most popular purely digital cryptocurrency nowadays. It started in 2009 as a peer to peer payment system, decentralized, pseudo-anonymous and secure system of money. The main objective of this study is to estimate and predict the weekly close bitcoin price by including other variables like commodities, indexes and demand / supply variables through different kind of machine learning and deep learning such as multiple regression, time series ARIMA model, artificial neural network, combination of multiple regression and ARIMA model, and finally combination of multiple regression, ...
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 ...
In this thesis, we will prove a very important theorem in real analysis called The Area formula for the Hausdorff Measure. This theorem is an extension of the well known theorem the Change of variables formula for the Lebesgue measure. In this thesis, we will define the Hausdorff measure and prove some of its properties. We will also define Lipschitz functions and prove some of its properties also. Then, we continue the thesis by proving all the lemmas needed to finalize the proof of the Area formula for the Hausdorff measure. Finally, we finish this thesis by showing three applications of the ...
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 knowledge of whether a time series contains a unit root or not provides guidance to determine whether the series is stationary or not. This topic is one that covers vast amount of research given to its importance in the analysis of economic and other time series data. To understand the behavior, the properties of the series and the influence of any shock that occur to the series, stationary and unit root tests were constructed. In this thesis, we first present the Box and Jenkins ARMA models, discuss the conditions for station-arity. Then, we display different method to test autocorrelation. ...
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 ...
The main objective of this work is to find a more straightforward method for estimating the parameters of an equally spaced discrete autoregressive process by using maximum likelihood estimation (MLE) considering it is challenging to obtain the parameters of a nonlinear optimization procedure. The resulting estimated values are tested through simulation and then compared with those obtained using the previous MLE and Yule-Walker estimation. The achieved result yields slightly increased accuracy.
Another problem we tackle is the Yule-Walker estimators for the continuous autoregressive models ...
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 ...