PhD

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Generalizations of Cyclic Refinements of Jensen’s Inequalities
(Library Information Services COMSATS University Islamabad Lahore Campus, 2020) Nasir Mehmood; FA14-PMATH-005; LHR TP 7451; Dr. Saad Ihsan Butt
In recent years, the concept of convex functions has been generalized extensively. Applications of convex functions are widely seen in many areas of modern analysis. Convex functions also have significant relation with the theory of inequalities and many useful inequalities are the result of the applications of convex functions. The Jensen's inequality has tremendous implications in many fields of modern analysis. It helps computing useful upper bounds for several entropic measures used in information theory. We consider discrete and continuous cyclic refinements of Jensen's inequality and extend them from convex to higher order convex function by using new Green functions introduced by us and employing different interpolating polynomials and identities. We formulate monotonicity of the linear functionals for nconvex functions at a point. We calculate some new Grüss and Ostrowski type bounds. As an application of our obtained results we give new bounds for Shannon, Relative and Zipf-Mandelbrot entropies.
Predictive Modeling of Diabetes Classification using Artificial Neural Networks
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Amin; FA23-RMT-017; Dr. Muhammad Rafiullah; LHR TP 9764
Diabetes mellitus is a chronic disease that has become a major global public health challenge. Timely and accurate prediction not only aids in immediate treatment but also plays a crucial role in formulating effective strategies. In this research, we compare the prediction of diabetes using two popular neural network models Multi-Layer Perceptron (MLP) and General Regression Neural Network (GRNN). This analysis is based on the PIMA Indian Diabetes Dataset, which contains medical information of female patients, including glucose levels, BMI, insulin amount, age, etc. In this research, the dataset underwent stages of cleaning, normalization, and division into training and testing sets. Then, the mathematical details of the MLP and GRNN models were described, which included forward propagation, activation functions, and loss formulas. The MLP model used two hidden layers with ReLU and Sigmoid activation functions, while the GRNN model used Gaussian radial basis functions and Euclidean distance. After training, the performance of both models was evaluated using metrics such as accuracy, confusion matrix, and ROC-AUC, revealing that both models proved effective in predicting diabetes. However, GRNN demonstrated better overall performance due to its non-repetitive structure and smooth results, while MLP exhibited more effective adaptation thanks to fast computation and deep network architecture, making it suitable for large and complex data. This research indicates that if ANN models are designed and configured correctly, they can help in the timely and effective diagnosis of diabetes. This analysis provides guidance to healthcare professionals and data scientists in selecting appropriate models.
Mathematical Modeling and Stability Analysis of Monkey Pox Transmission in Rodents and Humans
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Aqsa Shehzaadi; FA23-RMT-006; Dr. Yousaf Bhatti; LHR TP 9754
In this thesis, deterministic and stochastic mathematical models for the dynamics of monkey pox transmission across rodent and human populations are developed and analyzed. The study builds systems of nonlinear differential equations to describe disease progression and control strategies, such as vaccination and treatment, using compartmental modeling techniques like SIR and SEIR. Important epidemiological characteristics like interspecies transmission, disease-induced mortality, and incubation times are included in the models. To evaluate the stability of endemic and disease-free equilibria, analytical techniques such as basic reproduction number (R₀) analysis and Jacobian matrices are used. Moreover, random environmental and demographic perturbations are taken into account via stochastic differential equations, and stochastic Euler, Runge-Kutta, and NSFD methods are employed for numerical simulations. The results demonstrate the importance of intervention tactics in lowering the incidence of monkey pox and the usefulness of stochastic modeling in comprehending intricate epidemic dynamics in the face of uncertainty.

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