Browsing by Author "Prof. Dr. Sarfraz Ahmad"

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An Introduction to Projective Geometry and the Klein Quadric
(Library Information Services, CUI Lahore, 2023) Ammara Rashid; CIIT/FA21-RMT-101/LHR; Prof. Dr. Sarfraz Ahmad
In this thesis, the main topic is the Klein quadric and its properties. In order to study this important geometric object, we will need to discuss the basics of projective geometry. We will see how the Klein quadric is defined by a quadratic equation in a projective space which has dimension 5. The main focus of the thesis will be to show that the Klein quadric is an important example of parameter space and in particular, it parameterizes all the lines in the projective space of dimension 3.
Computing Degree Based Topological Indices of Beryllonitrene Network
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Ayesha Abdul Khaliq; FA23-RMT-007; Prof. Dr. Sarfraz Ahmad; LHR TP 9755
The use of graph theory to many fields of study has increased significantly, especially in the field of chemical graph theory. Many scholars have been able to explore a range of new directions in the last few years. A chemical graph is a named graph in which the atoms of a compound are represented at the vertices, and the edges represent chemical bonds between atoms. Determining various physical and chemical properties of the molecular structures under study requires an examination of topological indices. Firstly, we focused on degree based topological indices, namely Randic index, atom bond connectivity index, geomet ric arithmetic index, the first and the second Zagreb indices and co-indices, the first and the second multiple Zagreb indices and co-indices, hyper Zagreb index, forgotten index, augmented Zagreb index, Balaban index, redefined Zagreb type indices, of Beryllonitrene Network BeN4(t). Afterwards, we found the physical measures like heat of formation of Beryllonitrene Network BeN4(t). Then, we fitted curves between different indices and corresponding heat of formation. Curve fitting was done in MATLAB through different methods based on linearity and non-linearity. The performance of the method was tested using mean squared error (MSE), the sum of squared errors (SSE) or R2. Further, we gave graphical representations of these indices. These mathematical frameworks might provide a way to study the thermo dynamical properties of the chemical structure of Beryllonitrene Network BeN4(t).
Novel Study of NLSEs with the Assistance of Different Methods
(Library Information Services COMSATS University Lahore Campus, 2024-03-14) Nuzhat Hanif; FA22-RMT-040; Prof. Dr. Sarfraz Ahmad; LHR TP 9355
It is noteworthy that the significance of nonlinear wave profile in nonlinear sciences and information technology has been increased rapidly for last two decades. For this purpose, many researchers studied nonlinear Schrodinger equations (NLSE) and vari- ¨ ous types of nonlinear evolution equations (NLEEs) appear in nonlinear optics, system of fiber communications, plasma physics, fluid dynamics, quantum physics, high en ergy physics, acoustic gravity waves and optical fibers etc [1]. Many integrable NLEEs possess the soliton solutions which retains the speed as well as the shape, even after col lision with each other. Several efficient techniques have been introduced in literature for obtaining soliton solutions of NLSEs and NLEEs for last two decades, e.g F-expansion approach and its modification, HBM, Lie symmetry technique, first integral method, in verse scattering scheme, Sine-Gordon architectonic, and other approaches. In this thesis firstly, we will examine the optical solitons, the Schrodinger-Hirota equa- ¨ tion in the presence of chromatic dispersion. For this intent, by obtaining the multiple soliton types using the Hirota Bilinear Method (HBM), the Sine-Gordan method (SGE) and the Kudryashov method, opticall soliton solutions were obtained. We obtain some parabolic, anti-parabolic, M-shaped, W-shaped, butterflies, bright, anti-dark, V-shaped, S-shaped, and other solutions for our model. This detailed exploration provides per ception into the dynamic behavior and interactions of nonlinear solutions in NLSHE systems. The nonlinear Schrodinger equation with quintic non-Kerr nonlinear term (NLSE-QNKNT) ¨ will be studied. This equation describes the nonlinear wave state of optical solitons, which is a noteworthy and important model in optical fiber communication. By using the HBM, SGE and the GK approach, the various optical solutions of the nonlinear Schrodinger wquation are obtained. Finally, the graphs of some obtained solutions are ¨ drawn by setting values to tha parameters. We will obtain butterfly, S and W-shaped, parabolic, dark, kink and other solitary wave solutions (SWS) for our governing model.
On Computation of Eigenvectors using Principal Component Analysis (PCA)
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Usama Yaseen FA23-RMT-042; Prof. Dr. Sarfraz Ahmad; LHR TP 9786
This study focuses on the detailed explanation of principle component analysis that is a key technique used in data science for dimensionality reduction. Principle component analysis greatly helps in handling the curse of dimensionality. This research includes detailed description of computation of eigen vectors that includes eigen vectors decomposition, SVD etc. Some ways to apply the computational techniques in PCA have also been discussed in detail such as the implementation of PCA using Numpy and Scipy. PCA has also been implemented using sklearn. Some key applications of Principal Component Analysis in Machine Learning, Image processing, Finance and Bio informatics has also been discussed. The pipeline starting from standardizing the data, computing the covariance matrix, finding eigen values and extracting Principle Components have also been mentioned. In the last part of thesis, World Happiness data set has been used in which principal component analysis has been done in order to reduce the dimensionality. After that a supervised machine learning algorithm Linear Regression has been used to predict the Ladder score.
On Investigating Solutions and Properties of Some Types of Non-Linear Models
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Iqra Yasin; FA23-PMT-016; Prof. Dr. Sarfraz Ahmad; LHR TP 9763
In this thesis, Alfven waves in plasmas with perturbation terms are described using a mod ified form of the nonlinear Schr¨ odinger equation. Dissipative and higher order nonlinearity terms are the perturbation terms taken into account. The integration is performed using the solitary wave ansatz, generating precise topological and non-topological soliton solutions. The existence of the exact solutions is subject to parametric constraints.
t-fold Edgewise Subdivision of a Simplicial Complex
(Library Information Services, CUI Lahore, 2023) Muhammad Asif; CIIT/FA21-RMT-021/LHR; Prof. Dr. Sarfraz Ahmad
In current thesis, the flow of non-Newtonian fluid over variable thicker surface is investi- gated. For the non-Newtonian fluid, five parameters contained Carreau Model is used. To achieve the goal, thesis is divided into three chapters. One chapter contains the basic def- initions and some terminologies. Chapter two contains basic governing equations which are used for flow problem. In the last chapter, the mathematical model is developed by using equations of chapter two. The solution of this model is obtained by RK-method. The results are displayed in form of velocity and temperature profile for discussion.