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 ...

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, ...

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 ...

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 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.

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 ...

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 ...