Department of Mathematics
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Item Secure Resolving Dimension of Zero Divisor Graphs Associated to Commutative Rings(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Muhammad Muzammil; CIIT/SP20-RMT-012/LHR; Dr. Hafiz Muhammad Afzal Siddiqui; LHR TP 7626This Thesis contains the study of secure resolving dimensions of certain zero divisor graphs of commutative ring with unity. A subset 𝑇 of 𝐺 is a secure resolving set for 𝐺 if 𝑇 is resolving and for any 𝑥 ∈ 𝑉 − 𝑇, there exists 𝑦 ∈ 𝑇 such that (𝑇 − {𝑦}) ∪ {𝑥} is a resolving set for 𝐺. The minimum cardinality of a secure resolving set for 𝐺 is known as the secure resolving dimension for 𝐺. We determine the secure resolving dimension of various zero divisor graphs derived from commutative ring with unity and prove that all these graphs have constant secure resolving dimensionItem Zero Forcing Number of F-Sum of Some Families of Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Aleeha Shahid; CIIT/SP20-RMT-020/LHR; Dr. Hafiz Muhammad Afzal Siddiqui; LHR TP 7651This thesis deals with zero forcing number of 𝐹-sum of graphs derived from certain families of graph. A zero forcing set of graph 𝑆 is a set 𝑆 ⊆ 𝑉 such that there are no white vertices in the derived coloring of 𝑆. The zero forcing number of 𝐺 is 𝑍(𝐺) = min{|𝑆|: 𝑆is zero forcing set of𝐺}. We study zero forcing number of 𝐹-sum of two path graph, two cycle graph and then path and cycle graph. It has been proved that it is constant in case of 𝑆 operation of 𝐹-sum and unbounded otherwise.Item Extraction of Solitons Emerging in Nonlinear Dynamics(Library Information Services COMSATS University Lahore Campus, 2024-03-18) Bismah Yousaf; SP23-RMT-006; Dr. Hafiz Muhammad Afzal Siddiqui; LHR TP 9574The extraction of solitons that emerge in nonlinear dynamics is examined in this thesis, with particular attention to two significant models: the Perturbed Chen-Lee-Liu and the Biswas Milovic models. These models, which are renowned for their complexity in nonlinear wave propagation, describe a wide range of physical phenomena. The Extended Modified Auxil iary Equation Method (EMAEMM), a powerful yet efficient method for resolving nonlinear partial differential equations (NLPDEs), is used to derive solution solutions to these mod els. The EMAEMM method is a useful tool in nonlinear analysis as it makes it easier to find exact solutions by breaking down the equations into simplified forms. The ability of soliton to hold its shape during propagation makes it useful in a variety of applications, including fluid dynamics, optical fibers, and plasma physics. The stability and sensitivity of the resulting solitons is examined, and a number of graphical representations—all gen erated via Mathematica—are shown, including 2D, 3D, density linear, 1D, slice contour plotting, and stream density plots. The observed solitons exhibit a wide variety of behav iors and include bright, dark, kink, periodic, and optical solitons. Insights into the stability and representation of solitons in mathematical physics are provided by this comprehensive examination of soliton dynamics in nonlinear systems.Item Use of Artificial Intelligence Approach For Visual Prediction & Transfer Learning(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Hafiz Syed Muhammad Hur; FA22-RMT-037; Dr. Hafiz Muhammad Afzal Siddiqui; LHR TP 9381In this work, we introduce the identification of the MNIST database, which will be in handwritten digits that the machine can identify. The human handwriting form may be de tected and converted into computer language. We use several machine learning algorithms, including Convolutional Neural Network (CNN), Recurrent Neural Network (RNN), Long Short-Term Memory (LSTM), and Bidirectional Long Short-Term Memory (BiLSTM). The MNIST database was developed using binary images of handwritten numbers (09) from NIST’s Special Database.Second, introduce the identification of TB photos. Tuber culosis is the biggest cause of mortality worldwide, according to the World Health Organi zation. Inadequate treatment and delayed or incorrect diagnosis have led to several cases of the illness. Accurate and timely diagnosis is critical for successfully managing and preventing tuberculosis. Despite significant progress in deep learning for medical image processing. There are two types of distributions: training and application data.Our findings show that transfer learning from a pre-trained vision transformer outperforms a pre-trained CNN in medical imagingItem Resolving Properties of Graphs Associated to Commutative Rings(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Ammar Mujahid; FA19-RMT-101; LHR TP 7382; Dr. Hafiz Muhammad Afzal SiddiquiLet 𝑆 be a finite commutative unital ring having some non-zero elements 𝑥 and 𝑦 such that 𝑥𝑦=0. The elements of 𝑆 possesses such property are called the zero-divisors, the set of all these elements is denoted by 𝑍(𝑆). We can associate a graph to 𝑆 by means of a zero-divisor set 𝑍(𝑆) denoted by 𝜁(𝑆) (called the zero-divisor graph) to study the algebraic properties of the ring 𝑆. In this research work, we aim to produce some general bounds for edge and mixed version of metric dimension regarding some zero-divisor graphs of some families of rings. Then we prove general result for the upper and lower bounds of edge metric dimension of zero-divisor graphs in terms of maximum degree and diameter of 𝜁(𝑆). Finally, we discuss some relationships between girth, diameter and mixed metric dimension of zero-divisor graphs.Item A Study of Connection-Based Indices of Certain Structures(Library Information Services COMSATS University Lahore Campus, 2023-03-13) Faiza Bibi; SP22-RMT-023; Dr. Hafiz Muhammad Afzal Siddiqui; LHR TP 8729A Study of Connection-Based Indices of Certain Structures