Final Year Projects (FYPs) - Undergraduates
Permanent URI for this collectionhttps://repository.cuilahore.edu.pk/handle/123456789/53
This collection archives the complete set of theses produced by students of the COMSATS University Islamabad, Lahore Campus.
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Item Geometry of Manifolds and the Gauss-Bonnet Theorem(Library Information Services, CUI Lahore, 2024) Ayodeji Simeon Farominiyi; CIIT/FA21-RMT-108/LHR; Dr. Adeel FarooqThis thesis delves into the fascinating realm of smooth manifolds, investigating their intri- cate geometry through a comprehensive exploration of tangent spaces, connections, curva- ture and the Gauss-Bonnet theorem, a crucial result in the theory of surfaces. This theorem establishes a connection between a surface’s geometric and topological properties and acts as a model for similar statements that hold true in higher-dimensional contexts. The study is organized into three chapters, each addressing key aspects of this subject and culminating with an illustrative example featuring the sphere.Chapter 1 serves as an introduction to the theory of differential manifolds. It begins with a comprehensive overview of the underlying concepts, including topological spaces, charts, atlases, and smooth maps. The chapter provides a detailed examination of tangent spaces and tangent bundles, shedding light on the notion of derivatives and directional derivatives on manifolds. Vector fields and vector bundles are then introduced, enabling a deeper understanding of the interplay between smooth functions and smooth vector fields. The chapter, after the introduction of tensor products, also explores the concept of a Rie- mannian metric, highlighting its significance in defining lengths, angles, and inner products on manifolds. Additionally, differentiable forms are discussed, unveiling their role in cap- turing the geometric properties of manifolds.Item Differentiation and the Maximal Operators(Library Information Services, CUI Lahore, 2023) Hamza Sani; FA21-PMT-107; Dr. Adeel FarooqWe present an overview of certain maximal operators: the Hardy-Littlewood maximal op- erator and the dyadic maximal operator. We show that the two maximal operators are essentially the same. We study their weak type and strong type boundedness. We then use the weak type bounds of the Hardy-Littlewood maximal operator as a tool to differentiate integrals on dimensions higher than 1. Finally, we then study some construction of the dyadic systems in RItem Fractal geometry(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Madiha Iftikhar; SP21-BSM-015; Dr. Adeel Farooq; LHR TP 9884Fractal geometry explores the complex, self-replicating structures found in mathematical, natural, and computational systems. Originating from the foundational work of Benoˆıt B. Mandelbrot, fractals represent shapes exhibiting self-similarity across scales and fractional dimensions. This study delves into key mathematical constructs like the Cantor set, Koch curve, Mandelbrot set, and Julia sets, showcasing their recursive nature and infinite com plexity. The research emphasizes the significance of fractals in modeling natural phenom ena, including coastlines, mountain ranges, and biological patterns, and highlights their transformative applications in computer graphics, medical imaging, and economic analy sis. The iterative methodologies and Hausdorff dimension analysis offer a framework to describe and simulate intricate, real-world patterns. By bridging theoretical and practical realms, fractal geometry not only enriches mathematical theory but also provides innova tive tools for scientific and artistic endeavorsItem Machine Learning: Modern Techniques and Mathematical Approach to Neural Networks(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Sardar Abdul Wahab CH (FA20-BSM-012) : Abdul Raheem (SP20-BSM-003); Dr. Adeel Farooq; LHR TP 9911This thesis explores the mathematical foundations of machine learning algorithms, providing a comprehensive exploration of various key techniques and models. It covers fundamental concepts and methodologies in regression, classification, support vector machines (SVM), decision trees, neural networks, and perceptrons. Each of these algorithms is analyzed in terms of their mathematical underpinnings, operational mechanisms, and practical applications. The study aims to elucidate the core principles that drive these algorithms, offering insights into their theoretical and practical aspects. By understanding the mathematics behind these models, this research contributes to a deeper appreciation and effective utilization of machine learning techniques in solving complex real-world problems.