Department of Mathematics

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    Cosmological Consequences of Equation of State Parameterizations in Mimetic Gravity
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Muhammad Rizwan Azhar; CIIT/FA20-RMT-032/LHR; Dr. Shamaila Rani; LHR TP 7935
    This thesis is devoted to study the cosmic aspects of the evolving uni- verse by taking into account mimetic gravity. Mimetic gravity is a Weyl- symmetric expansion of general relativity linked by a single disformal trans- formation, in which the presence of a dust-like perfect fluid may be used to imitate cold dark matter on a cosmic scale. We consider three well-knows equation of state parameterizations in terms of redshift function. Using mimetic gravity energy density, we calculate the potential, coincidence pa- rameter and deceleration parameter for all parametrization models with the help of energy conservation equation, Lagrange multiplier and normal- ized Hubble parameter. Considering the observational values of parameters in parametrization models, we get some useful results for the underlying scenario.
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    Newton and Simpson Formula Type Inequalities Via Generalized Convexity in Quantum Calculus
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Qurat UL Ain Tariq; FA20-RMT-038; LHR TP 7939
    This study establishes Newton- and Simpson-type integral inequalities within the framework of generalized convexity in quantum calculus. By extending classical convexity concepts to the quantum calculus setting, new bounds are derived for differentiable functions defined on quantum time scales. The work develops refined integral inequalities using generalized convex functions, providing sharper estimates than their classical counterparts. Newton and Simpson-type formulas are adapted to incorporate q-calculus operators, and the resulting inequalities are analyzed for accuracy and convergence behavior. Applications of these results are discussed in numerical integration and approximation theory. The findings demonstrate that generalized convexity in quantum calculus offers powerful tools for improving classical numerical inequality estimates.
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    Computing Degree Based Topological Indices of Calcium Hydroxide
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Hammad Ali; CIIT/FA20-RMT-036LHR; Dr. Muhammad Kamran Siddiqui; LHR TP 7938
    ‘Graph Theory is playing a key role in applied and pure Mathematics. The scope of Graph Theory is expanding day by day. ‘ In this thesis, we computed some topological indices, Co-indices and reverse degree- based indices such as ffirst‘ and ‘second Zagreb‘co-index, first and‘second multiple ‘Zagreb co-index‘forgotton‘topological co-index, atom bond‘connectivity co- index,‘geometric arithmetic co-index , Balaban index and general‘Randic index ‘for‘∝ = 1, −1, 1 2 , − 1 2 ‘for Calcium Hydroxide. My thesis comprising of five chapters. In the first chapter, we will define some basic definitions. In the 2nd chapter, we will find topological indices of Calcium Hydroxide structure. In the 3rd chapter, we will find topological Co-Indices of Calcium Hydroxide structure. In the 4th chapter, we will find reverse degree based topological indices of Calcium Hydroxide structure. In the 5th chapter, we will present the conclusion of our thesis.
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    Interior Solutions of Charged Anisotropic Gravastar in f (R, T ) Gravity
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Muhammad Taimoor Murtaza; CIIT/FA20-RMT-028/LHR; Dr. Rabia Saleem; LHR TP 7932
    In 1998, the Nobel Prize-winning discovery of the accelerating cosmic expan- sion demands physically consistent models to describe the origin of this current acceleration. As a result, during the last two decades, there have been several theoretical and experimental efforts into improving our understanding of the dynamics of the Universe. To support the lines, we consider the imperfect fluid distribution with in gravastar in the framework of f(R, T ) gravity and evaluate the major matter contents in the quintessence regime via conformal symmetry. After considering the well-known matching conditions with the exterior one, we investigate the various realistic features of thin shell gravastars with radial perturbation along with an electromagnetic field through a particular equation of state (EoS). The matter configuration around the black hole (BH) geometry is examined through various physical properties and checks the stability of gravastar for various values of involved model parameters. It is worth men- tioning that the realistic features of gravastars such as Proper Length, Entropy, and Thin Shell Energy satisfy the linear relationship between the thickness of the shell. Moreover, the speed of sound parameter η(a), is utilized to examine the stable configuration of the thin shell which supports the acceptability of our considered model.
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    Study of Cosmological Aspects in Chern - Simons Modified Gravity
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Waheed Ahmad; FA20-RMT-006; LHR TP 7914
    We offer a cosmological model where the cosmos recovers in an non-unique recovery. Following the bounce, it slows down in some cases, the main evolu tion is similar to matter The era of deceleration, and finally the current era, is evolving through a phase of acceleration. Our goal is to investigate this behav ior using the modified F(R) gravity, Chern-Simons theory, and to compare the concept to other empirical evidence. We calculate the shape of the entire F(R) conceivable cosmic time horizon using enabled the development and the rele vant process parameters. The quasi jump appears to be associated with periods of DE, particularly from quasi bounces to slowdowns as well as from the re tardation era to the late acceleration era, according to the structure of F(R). It is vital to recognize that the script’s rebound in this situation is an uneven bounce, particularly because the Hubble radius is monotonic. It gradually increases and gradually differentiates in the latter part of ca, while it seems to decrease grad ually over time now. Decreasing behavior of Hubble beam ensures delayed acceleration cosmic age. Furthermore, due to the aforementioned the Hubble radius’s development, prototype disturbance patterns occur during deep con tractions awayfromrebounds,whenalltheperturbationfunctioniscoming. We calculate gradient and tensile strength separately spectral, so we evaluate raw observations such as spectral metrics gradient curvature misalignment, stretch ing the gradient ratio, therefore, in I agree with the most recent Planck 2018 limitations. The term CS¢aappears to, in this meaning, it has a significant effect on the development of tensor barriers, but keeps the gradient part constant, the perturbation in the case of the vacuum modelF(R), therefore, CS the term seems to play an vital role in reconciling observable with Planck result.
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    Effects of Spatial Parts of the External Input on Bumps in the Two-Population Neural Field Model
    (library Information Services COMSATS University Islamabad Lahore.Campus, 2022-03-12) Sadia Jamil; FA20-RMT-011; LHR TP 7918; Dr. M. Yousaf Bhatti
    The Two-Population Neural Field Model (TPNF) of Wilson-Cowan type represents excitatory and inhibitory activity levels in the brain. The stationary and symmetric solution of this model is Bump pair which serves as working memory/self-sustained activity in the brain. Therefore, it is very important to investigate the appearance and disappearance of these activities. There exist two Bumps Pair solution (i.e. one narrow and other named as broad bump pair) for the case of without external input. The narrow bump pair is not stable for all values of τ (Relative inhibition time constant). The Broad Bump pair is stable for the suitable value (small and moderate value) of the τ. The impact of spatio-temporal external input on appearance of bumps for different temporal part has already been investigated in detail but the spatial part of the external input is equally important and has not been investigated that much. We have investigated the impacts of spatiotemporal external input on appearance bump solutions of the TPNF Model with a focus on spatial parts of the external input. We considered four possible functions for this purpose and found that the broad bump choice as a spatial function in the external input is the most suitable to evoke the activity in the network.
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    Application of Ranking Heptagonal Neutrosophic Fuzzy Numbers to a Tansportation Problem
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Mudasir Nazar, Umer Farooq; CIIT/FA22-BSM-031/LHR, CIIT/FA22-BSM-59/LHR; Dr. Madiha Khalid; LHR TP 9924
    This thesis introduces the concept of neutrosophic heptagonal numbers and presents a novel approach for their ranking. Neutrosophic heptagonal numbers extend the realm of neutrosophic numbers to a seven-dimensional space, offering a more comprehensive representation of uncertainty in various real-world appli- cations. The developed ranking method provides a structured framework to assess and order these complex numbers, facilitating decision-making processes in uncertain environments. To demonstrate the practical utility of this approach, we apply it to solve a transportation problem framed within the context of a cost matrix. By lever- aging the ranking methodology on the cost matrix, we effectively address the uncertainties inherent in transportation planning, optimizing routes, and mini- mizing costs. Through computational experiments, we validate the effectiveness and efficiency of the proposed method in handling real-world scenarios charac- terized by intricate uncertainties. This research contributes to advancing the understanding and application of neutrosophic heptagonal numbers in decision sciences, offering a valuable tool for analyzing complex systems where uncertainty plays a significant role
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    Computational aspects of SSC of Wheel Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Rimsha Amber; CIIT/FA20-BSM-058/LHR; Dr. Imran Ahmed; LHR TP 9923
    Simplicial complexes represent crucial mathematical structures with broad applications. Stanley-Reisner ideals encode combinatorial information about these complexes, bridg- ing combinatorial geometry and commutative algebra. Simplicial complexes, comprised of simplices, capture complex topological and combinatorial relationships, serving as founda- tional tools in numerous mathematical disciplines. Shellable complexes, a significant sub- class, allow systematic decomposition into ”shells,” revealing insights into their topological and homological properties. Studying the spanning simplicial complexes of wheel graphs elucidates the connection between graph theory and algebraic topology. Investigation into the shellability of line simplicial complexes sheds light on their combinatorial intricacies, contributing to a deeper understanding of their geometric and algebraic nature. Through these explorations, we uncover the rich interplay between combinatorial geometry, alge- braic structures, and topological properties inherent in simplicial complexes, facilitating broader comprehension and application across mathematical domains.
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    Deviation of Rotating Black Hole in Loop Quantum Gravity from Kerr Black Hole
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) MADIHA KANWAL; CIIT/FA20-BSM-041/LHR; Dr. Muhammad Zubair; LHR TP 9916
    First and foremost, I would like to thank ALLAH Almighty (the most beneficent and most merciful) for giving me the strength, knowledge, ability and opportunity to undertake this research study and to preserve and complete it satisfactorily. My heartiest gratitude to Hazrat Muhammad (S.A.W) for His guidance to humanity and for material and spiritual uplift. I would like to express my deepest gratitude to my supervisor Dr. Muhammad Zubair, for his excellent guidance, interest, patience, and for providing me a decent atmo- sphere for the research. It was not an easy task but his extraordinary support made it so. I am also thankful to my senior research fellow Muhammad Ali Raza (PhD student) who always taught me with his humble attitude and with whom I collaborated and learned many things. Lastly, I would like to thank my parents and siblings, who always supported me and prayed for my success. Without their support, love and efforts, I would have not achieved all this. They gave me wings to fly.
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    Properties of Total Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) BISMA ATIQ; CIIT/FA20-BSM-007/LHR; Dr. Imran Ahmed; LHR TP 9913
    With every graph G (finite and undirected with no loops or multiple lines) there is associated a graph L(G), called the line-graph of G, whose points correspond in a one-to-one manner with the lines of G in such a way that two points of L(G) are adjacent if and only if the corresponding lines of G are adjacent. Let G = (V, E) be a graph. The Gallai total graph ΓT (G) of G is the graph, where V (ΓT (G)) = V ∪ E and uv ∈ E(ΓT (G)) if and only if (i) u and v are adjacent vertices in G, or (ii) u is incident to v or v is incident to u in G, or (iii) u and v are adjacent edges in G which do not span a triangle in G. The anti-Gallai total graph ∆T (G) of G is the graph, where V (∆T (G)) = V ∪ E and uv ∈ E(∆T (G)) if and only if (i) u and v are adjacent vertices in G, or (ii) u is incident to v or v is incident to u in G, or (iii) u and v are adjacent edges in G and lie on a same triangle in G. The Gallai middle graph ΓM(G) of a graph G = (V, E) is the graph whose vertex set is V ∪E and two edges ei, ej ∈ E are adjacent in ΓM(G), if they are adjacent edges of G and do not lie on a same triangle in G, or if e = uv ∈ E then e is adjacent to u and v in ΓM(G). The anti-Gallai middle graph ∆M(G) of G is the graph whose vertex set is V ∪E and two edges ei, ej ∈ E are adjacent in ∆M(G) if they are adjacent in G and lie on a same triangle in G, or if e = uv ∈ E then e is adjacent to u and v in ∆M(G).In this paper, we present Eulerian and Hamiltonian properties of Gallai and anti-Gallai middle graphs. In this paper, we investigate Total Graph, Gallai and anti-Gallai Total Graphs and Middle graph.