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 ...

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 ...

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 ...