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Browsing by Author "Nakad, Roger, Ph.D."

Browsing by Author "Nakad, Roger, Ph.D."

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  • Fuzzy measure spaces and fuzzy integrals: an overview and an application 
    Mahmoud, Faten (Notre Dame University-Louaize, 2021)
    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 ...
  • Introduction to differential geometry of space curves and surfaces 
    Nicolakis, Tatiana (Notre Dame University-Louaize., 2020)
    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 ...
  • Lebesgue integration : a new proof of the fundamental theorem of calculus and a new lebesgue integrable function 
    Fayad, Rawan (Notre Dame University-Louaize, 2021-12)
    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 ...
  • Mean value theorems : generalizations and associated functional equations 
    Yammine, Rebecca (Notre Dame University-Louaize, 2020-12)
    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 ...