Department of Mathematics

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4D Black Holes and Their Parameters
(Library Information Services COMSATS University Lahore Campus, 2024-03-18) Sania sobedar; SP23-RMT-036; Dr. Muhammad Zubair; LHR TP 9576
Black holes are among the most enigmatic objects in the universe, characterized by their immense gravitational pull and profound influence on surrounding matter. In this thesis we considered Dyonic Modmax black hole, which is static and we apply Newman-Janis Algorithm to make the black hole rotating. The rotating metric then explored to study parametric constraints and frequencies of black hole by using the Quasi-Periodic Oscilla tions. Moreover, we investigated the effective potential. We also find the orbital frequency (or Keplerian frequency) of Dyonic ModMax black hole. It is the frequency of a particle which is orbiting around the BH. In general, it is function containing charge and radial distance of a particle from BH.
A Comparative Study of Machine Learning Algorithms For Cancer Detection
(Library Information Services COMSATS University Lahore Campus, 2024-03-14) Rukhseena Ashfaq; FA22-RMT-001; FA22-RMT-001; Dr. Muhammad Yousaf
A Comparative Study of Machine Learning Algorithms For Cancer Detection
A Comparative Study of Topological Descriptors for Benzyl Sulfamoyl Network
(Library Information Services COMSATS University Lahore Campus, 2024-03-14) Ayesha Maqbool; FA22-RMT-043; Dr. Muhammad Kamran Siddiqui; LHR TP 9358
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 Benzyl Sul famoyl Network BSb. Afterwards, we found the physical measures like heat of formation of Benzyl Sulfamoyl Network BSb. 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 R 2 . 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 Benzyl Sulfamoyl Network BSb.
A Composite Modeling Approach for the 3D Simulations and Medical Imaging of Cardiac ATTR Amyloidosis
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Usman Ashiq; CIIT/FA19-RMT-031/LHR; Dr. Ayesha Sohail; LHR TP 7441
Mathematical Modeling and Machine Learning is playing a key role in applied mathematics. Machine learning has witnessed a tremendous amount of attention over the last few years. Deep neural networks are now the state-of-the-art machine learning models across a variety of areas, from image analysis to natural language processing. These developments have a huge potential for medical imaging technology, medical data analysis, medical diagnostics and healthcare in general, slowly being realized. Imaging has played a variety of roles in the study of Cardiac Amyloid (CA) over the past four decades. We provide a short overview of Cardiac Amyloid and Medical imaging techniques used for diagnosis of amyloids in our heart as well as recent advances in techniques. We also working on different neural networks especially on CNN for diagnosis of cardiac amyloid with the aid of Resnet-50 network on MATLAB and tensor-flow on Python. This study aims to assist doctors in choosing an acceptable classification method for each patient's condition. The challenge for the future will be to availability to most efficient imaging data for this disease.
A Computational Investigation of Fluid and Air Bubble Dynamics in Two-Phase Flow
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Shamsul Jahan Shamsa (FA23-RMT-038); Dr. Mohsan Hassan; LHR TP 9783
This work investigates heat transfer in two–phase flow of Newtonian fluid (water bubbles) and power-law non-Newtonian fluid (carboxymethylcellulose) in a L-shaped pipe geometry. The problem is simulated using Computational Fluid Dynamics (CFD) techniques, specifically the Level Set interface tracking method. Focusing on pressure, velocity and temperature distributions under different flow conditions, the research addresses challenges related to computing efficiency, numerical stability, and interface dynamics. These findings lay a foundation for optimizing fluid system by illustrating how flow parameters affect phase interactions and pressure gradients.
A Multi-person Decision Model for Low-Carbon Supplier Selection using Intuitionistic Preference Relations
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Zurwa Abrar; SP20-RMT-041; LHR TP 7909; Dr. Atiq ur Rehman
In this thesis we presented a group decision framework to solve the low carbon supplier issue by employing intuitionistic preference relations, (IOWA) intuitionistic ordered weighted averaging operators and induced intuitionistic ordered weighted averaging operator (IIOWA) We use rectification and obtain weight vector and we then rank them and get the best decision out of it.
A Note on Determinantal Varieties
(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Ayesha Bashir; FA22-RMT-011; Dr. Imran Ahmed; LHR TP 9340
The disappearance of all minors of a specific size in a r ×s matrix indicates determinantal varieties. “Essential isolated determinantal singularities (EIDS)” are a special class of de terminantal variations that were created by Ebeling and Zade in 2009. Pereira demonstrated that, for all 1 ≤ k ≤ min{r,s} the determinantal variety Z k = A−1
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 8729
A Study of Connection-Based Indices of Certain Structures
A Study Of Cosmic Accelerating Behavior with Interacting Dark Sectors
(Library Information Services, COMSATS University, Lahore Campus, 2019) Muhammad Jameel Imtiaz; , FA17-RMT-031; Dr. Rabia Saleem, Assistant Profesor; LHR TP 5659
The main focus of this thesis is to investigate an interacting phenomena be tweendarkmatter(DM)-darkenergy(DE)withinanon-flatFriedmann-Lemaitre Robertson-Walker (FLRW) spacetime geometry boundedbyahorizonwithspe cific Ricci cut-off. The Ricci model of DE has been under taken as a cosmic fluid ingredient along with DM. To check consistency of the considered model with astrophysical data coming from recent observations, we assume a positive interaction term Q between two components of fluid along with “Chevallier Polarsky-Linder type parametrization" of the coincidence parameter r(z). By calculating all the physical quantities like energy density and pressure, we con straint the model parameters and found specific region of validity for these pa rameters. It is observed that our model possess a future singularity of Type III. After finding the singular behavior, we examine the nature of our model via cosmological parameters like q, r, s. It is noted that our model is very close to Λcold dark matter (ΛCDM)
A study of generlazied scound law of Themodaynamic in masssive Gravity
(2024) Muhammad Shoib ( Sp 21- BSM-024), Muhammad Awais (sp21- BSM-002); Dr. Rabia Saleem; LHR TP -9878
A Study of Topological Properties of Certain Chemical Structures
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Rizwan Saleem; CIIT/FA19-RMT-040/LHR; Dr.Hafiz Muhammad Afzal Siddiqui; LHR TP 7448
This study examines the topological properties of selected chemical structures using tools from graph theory and mathematical chemistry. Chemical compounds are modeled as graphs in which atoms correspond to vertices and chemical bonds to edges. The research focuses on analyzing structural characteristics such as connectivity, degree distribution, and symmetry, along with the computation of important topological indices. These properties are used to gain insights into the stability, reactivity, and physicochemical behavior of the compounds. Comparative analysis of different chemical structures highlights the role of topology in predicting molecular properties and supporting the design of new materials.
A Study of Triglyceride by Using Topological Indices
(Library Information Services, CUI Lahore, 2023) Ali Ahmad; FA20-RMT-007; Dr. Hafiz Muhammad Afzal Siddiqui
A topological graph index, also called a molecular descriptor, is a mathematical formula that can be applied to any graph which provides the information about its chemical properties. From this index, it is possible to analyse mathematical values and further investigate some physicochemical properties of a molecule. Therefore, it is an efficient method in avoiding expensive and time-consuming laboratory experiments. In this thesis, we study some degree based topological indices of triglyceride its line and paraline graphs.
A Study on Astragaloside IV in View of Connectivity Indices
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Muhammad Tahir Hussain; CIIT/SP20-RMT-003/LHR; Dr. Sana Javed; LHR TP 7622
Topological indices are frequently used in the analysis of different chemical or molecular compounds including drugs. This thesis mainly deals with the computation of degree-based topological indices of the drug Astrgaloside IV and Benzenoid and the construction of a network of the indices. The main objective is to detect the subnetworks of the network with highly connected links called modules and the master regulators in that module. This approach would help the researchers to investigate the physio-chemical characteristics of a Astrgaloside IV based on the master regulatory index in the module where a master regulator index is an index that is at the top of a regulatory hierarchy in a module and is not influenced by any other index. This thesis is divided into five chapters. Chapter 1 includes the basic definitions , notions , and terminologies related to this thesis. Chapter 2 provides a review of literature about our work. The main results are gives in chapter 3. Chapter 4 contains a bried discussion about the work while chapter 5 consists of the references used in this thesis.
Accelerating Expansion in Curved FLRW Space-Time for Interaction/Non-interacting DE-DM within Rastall Gravity
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Shahnila; CIIT/FA19-RMT-086/LHR; Dr. Rabia Saleem; LHR TP 7446
In this thesis, we explore the phenomenon of cosmic evolution using curved FLRW space-time bounded by apparent horizon with a speciﬁc holographic cut-off. To this end, we use the framework of Rastall gravity and universe is assumed to be consists of interacting/non-interacting dark energy and dark matter. In both scenarios, we evaluate exact solutions of the dynamical equa- tions and constraint the holographic parameter c2(z), assuming a slowly vary- ing function of red-shift during non-interacting model. For interacting model, we consider c2 as a constant function of red-shift. Moreover, we analyze nature of the obtained results via deceleration parameter (q), stateﬁnder pair (j, s) and Om(z)-diagnostic by constraining the involved model parameters using latest observational data. The graphical analysis showed that interacting model is very close to ΛCDM model as compared to non-interacting case. We conclude that this holographic proposal is enough to describe the cosmic evolution at an accelerating rate.
Accelerating Expansion of the Universe within Rastall Gravity Via Cosmography of Some Dark Energy Models
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Shan Ali; CIIT/SP20-RMT-029/LHR; Dr. Rabia Saleem; LHR TP 7634
This thesis is mainly focused to investigate the cosmological implications of Rastall gravity with the help of two dark energy models, i.e., new Tsallis agegraphic dark energy (NTADE) and Sharma-Mittal holographic dark energy (SMHDE) for interacting and non-interacting cases. The background of ﬂat FLRW space-time is being considered to develop the dynamical system of equa- tions. To check the viability of these models and to distinguish between them, we develop important cosmological parameters including equation of state pa- rameter, deceleration parameter, square speed of sound, stateﬁnder and Om(z) diagnostic for both cases. By constraining the involved model parameters, it is observed that equation of state parameter for NTADE model lies in phantom re- gion for both interacting and non-interacting cases. Whereas for non-interacting SMHDE, it shows quintom as well as quintessence regions depending on the values of model parameter δ, and it indicates only the phantom region for inter- acting case. The ωD − ω′ D pair shows freezing and thawing regions for non-interacting NTADE while only freezing region for interacting NTADE and non-interacting SMHDE. For interacting SMHDE case, it shows freezing and thawing regions as well as region transition from freezing to thawing. The stateﬁnder pair (j, s) for NTADE and SMHDE indicates a rich behavior as it shows different DE eras; phantom, quintessence as well as Chaplygin gas for both cases. We conclude that non-interacting SMHDE is more physically viable model than NTADE model because it approaches to ΛCDM limit and also the results for this model lie in stable region as shown by graphical analysis of the square speed of sound
AConsistency-Driven Group Decision Making Approach with Triangular Fuzzy Reciprocal Preference Relations
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Malaika Ali; FA23-RMT-021; Dr. Atiq-ur-Rehman; LHR TP 9768
Triangular fuzzy reciprocal preference relations (TFRPRs) are one of the effective tools for expressing the fuzzy and uncertain preferences of decision makers in group decision making. The consistency of TFRPRs is a key prerequisite for reasonable and reliable de cision making. To this end, this study developed an improved multiplicative acceptable consistency group decision-making method based on TFRPRs. By analyzing the compo sition of TFRPRs, the multiplicative consistency index and the corresponding threshold are defined to measure whether TFRPRs have acceptable consistency. This consistency measure reflects the essential characteristics of triangular fuzzy numbers and fully consid ers the multiplicative consistency of the mode value and the multiplicative consistency of the geometric mean based on the central tendency. Then, three inconsistent TFRPR cases are analyzed, and an algorithm for solving TFRPR inconsistency based on mathematical derivation and linear programming model is proposed. Based on the multiplicative accep tance consistency (MACD) of TFRPRs, a MACD-IOWA is proposed to aggregate individ ual TFRPRs into group TFRPRs. The basic principle is that the higher the MACD value, the higher the decision maker’s weight. Finally, the feasibility and effectiveness of the proposed method are verified through case studies, comparative analysis and discussion.
Advance Numerical Methods for Special Differential Equations with Structure
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Fozia Yasin; CIIT/FA19-RMT-017/LHR; Dr. Yousaf Habib; LHR TP 7422
The current work is concerned with structure-preserving numerical meth- ods for particular diﬀerential equations representing dynamically in classical mechanics from the Hamiltonian perspective. These diﬀerential equations involve quantities that should remain constant throughout the investiga- tion. These methods includes the symplectic Runge-Kutta methods and the G-symplectic general linear methods. They faithfully maintain the un- derline characteristic properties of a Hamiltonian system during numerical discritization of the problem, whereas conventional numerical methods do not account for the preservation of these invariants. The conservation of total energy and the symplecticity of ﬂow are two physically signiﬁcant invariant properties of a Hamiltonian system which we to preserve numeri- cally. This thesis contains four chapters. In chapter 1, the fundamental theory of ordinary diﬀerential equations is presented. The exposition combines deﬁnitions with the development and application of numerical methods for special diﬀerential equations, and it serves as the foundation for all subsequent investigations in this work. In chapter 2, numerical methods for the Hamiltonian system are thoroughly discussed. In chapter 3, the crucial concept of projection is introduced. The basic concept of the standard projection technique is generalised to create the heterogeneous class of numerical methods known as general lin- ear methods. In chapter 4, the symmetric general linear methods are used with the Runge-Kutta methods as starting methods.
AHP Framework for Group Decision-Making under Hesitant Fuzzy Preference Relations
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Rumaisa Ahsan (FA23-RMT-034); Dr. Atiq-ur-Rehman; LHR TP 9780
multi-criteria group decision-making problems with uncertainty based on hesitant fuzzy reference relations by the analytic hierarchy process. Although existing methods are advanced, they basically rely on the traditional preference models, which cannot effectively describe the hesitation and inconsistency that are seen naturally in an expert evaluation due to the limited characteristics of the existing methods. As the existing methods only rely on traditional preference models and cannot effectively describe the hesitation and inconsistency, this study introduces a TL-consistency based normalization strategy to normalize the incomplete hesitant fuzzy preference relations. The handling of group preferences involves a systematic process consisting of normalization, consistency assessment, agreement evaluation, and aggregation methods to ensure a consistent and analytically reliable representation of the group preference. In this phase, priority weights of the decision-makers from the previous two steps. The global alternatives rankings are derived by combining normalized hesitant fuzzy preference relations and forming a consistent evaluation matrix with a combined AHP. A numerical example illustrates the applicability of the approach, demonstrating the clarity and practical interpretability of the derived results and providing a case for generating strong decisions. This framework adds to the existing literature by increasing the possibility of consensus and consistency under conditions of ambiguity, subjectivity, and incomplete preference information in group decision-making settings.
Algebraic Characteristics of Graphs
(Library Information Services COMSATS University Islamabad Lahore Campus, 2020) Muhammad Asim Razzaq; FA13-PMATH-004; LHR TP 7454; Dr. Kashif Ali
Graph theory is a vital and special branch of discrete mathematics. It deals with geometric computational study of various objects. The main component and object of the theory is graph and its generalization approach. Therefore, it has vast application to other mathematical and provides computational assistance to non-mathematical sciences. The most prominent use of graph theory as a significant tool in chemistry, this branch is named as chemical graph theory. It provides helpful assistance for obtaining molecular descriptor to develop essential relationship between molecular graphs and chemical compounds. Graph theory also serves its specialization in algebraic graph theory specifically labellings. In this dissertation, we formulate and discuss all these types of indices. We develop a new approach to calculate the eccentricities of vertices of any graph by using computer aid softwares. This approach help us to find the exact expression of the ECI and ABC5 for the butterfly, bene and torodial grid graphs. We also express a new type of counting polynomial associated with the previous counting polynomials. Moreover, we work some topological indices relative to degree dependent on generalized subdivision of line graphs. We also determine the general results about subdivision of line graphs for neighborhood valency-based indices. We also discuss the stability of line graph of chemical molecules.”Line0graphs furnish0a tool for0studying the0topological properties of0alpha-systems.”In addition to this, we study four important counting polynomials called omega, theta, PI and Sadhana for V-Phenylenic. We also introduced a new polynomial Upsilon on the basis of ”co” relation and find new results relative to new counting polynomial.”We also determine that inertia indices for V-phenylenic nanotube and these indices are not0equal for the line0graph of V-phenylenic0nanotube. We also find that the0nullity for this0nanotube and of line0graph of these0nanotube.”Lastly, we also discuss H-group magic labellings of some graphs. We mainly study about the H-groupmagic0total graph0labelings G(v; e) over finite0abelian0group Zt Zv, where H =K2; Ct and e = (t 􀀀1)v:
Algebraic Characterization of Fuzzy Graphs
(Library Information Services COMSATS University Lahore Campus, 2024-03-18) Mubeen Naz; SP23-RMT-017; Dr. Imran Ahmed; LHR TP 9579
The aim of the thesis is to investigate the algebraic properties of fuzzy graphs with par ticular focus placed on three main topics which are the square of a graph, minimal free resolutions of modules, and fuzzy edge ideals. We will examine the fuzzy graphs relation ships to fundamental algebraic concepts like rings, ideals, and edge ideals by examining their structural and algebraic characteristics. The study focuses on how graph-theoretical concepts and algebraic methods work together. It gives information about minimal free resolutions and the behavior of fuzzy square graph and edge ideals of fuzzy square graph. These results advance the theoretical underpinnings and practical uses of fuzzy commuta tive algebra