Department of Mathematics
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Item Extension of Isotropic Solutions into Anisotropic Domain Via Gravitational Decoupling Approach(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) MOBEEN AMIN; CIIT/FA19-RMT-048/LHR; Dr. Muhammad Zubair; LHR TP 7437In this thesis, we study known charged and uncharged isotropic Heintzmann solution and Durgapal IV solution and extend it to its well anisotropic domain by manipulating gravitational decoupling through extended geometric defor- mation (EGD) approach. We employ linear transformation on both metric po- tentials and convert the single set of EFEs into two less-complicated subsys- tems. The first one is related to the standard Einstein-Maxwell field equations (EFEs), while the other one is related to additional source. The matching con- ditions at the stellar surface are discussed in detail, where outer geometry is represented by Reissner-Nordstrom solution in the presence and schwarzschild solution in the absence of charge and also evaluate the expressions for ther- modynamical quantities ρtot, ptot r and ptot t using mimic constraint and equation of state (EoS).We discuss the physical properties of the stellar model. Physical analysis is carried out using different physical indicators i.e, energy conditions, equilibrium equation, casualty condition, Herrera’s cracking concept, adiabatic index, compactness and surface redshift. In order to check the viability of the anisotropic version of the solution, we consider three different realistic stars, namely SAXJ1808.4 − 3652, SMCX − 4 and PSRJ1614 − 2230, and analyze the behavior of the model against different values of intensity parameter, i.e., γ = 0, 0.2 and 0.3 and also evaluate the graphical behavior of compact star PSRJ1614 − 2230 against different values of γ = 0.25, 0.55 and 0.75 for different model.Item Explicit Formulas and Bounds on General Randic Index for Complex Graph Models(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Sajjad Asghar; SP23-RMT-021; Dr. Maqsood Ahmad; LHR TP 9586In chemical graph theory, a topological index is a numeric value extracted from the graph ical model of the molecule that provides insight into the structural, physical, bio-active, and chemical properties of the corresponding chemical compound. Randi´c index is very influential, well investigated, and considered to be the prototype of degree-based indices. General Randi´c index Rγ(Γ) is defined as the sum over all edges of the molecular graph Γ with weights dΓ(x)×dΓ(y) γ when γ is a positive integer. Graph operations play a phenomenal role in obtaining new and complex graphs, either of general or of chemical in terest. It is pertinent to explore various graph operations to establish a relationship with the topological indices of the base graphs. In this article, we intend to derive some explicit ex pressions and bounds on the general Randi´c index when γ ∈ Z+ for various complex graph operations like the cartesian product, tensor product, lexicographic product, strong prod uct, co-normal product, corona product, splices, links, and Haj´ os construction of graphs. Besides, we compute the general Randi´c index by applying the formulas obtained for few small cases to check the validity and reliability of our results.Item 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 7914We 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.Item Generalized Hermite--Jensen--Mercer-Type Inequalities in Fractal Sense(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Saba Yousaf; CIIT/FA19-RMT-007/LHR; Dr. Saad Ihsan Butt; LHR TP 7445The most notable inequality pertaining convex functions is Jensen’s inequality which has tremendous applications in several fields. Mercer introduced an important variant of Jensen’s inequality called as Jensen-Mercer’s inequality. Fractal sets are useful tools for describing the accuracy of inequalities in convex functions. We establish a generalized Jensen–Mercer inequality and generalized Hermite-Hadamard–Mercer inequalities for a generalized con- vex function on a real linear fractal set Rς ( 0 < ς ≤ 1). Further, we also demonstrate some generalized Jensen–Mercer type inequalities by employing local fractional calculus. We establish two new lemmas involving local fractional integrals. By using these lemmas, we obtain several results related to generalized Hermite–Hadamard–Mercer type integral in- equalities for local differentiable generalized convex functions on real linear fractal space. Lastly, some applications related to Jensen–Mercer inequality, ς -type special means, and probability density functions are given. The present approach is efficient, reliable, and may motivate further research in this area.Item Non-Uniform Invariant Measure For Some Hyperbolic Maps.(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Ibrahim Abdulazeez; FA19-RMT-401; LHR TP 7377; Dr. Muhammad Aqeel AhmadThe study of the circumstances that suggest that a dynamical system has at least one physical measure is a natural approach to the wider study of the presence (or lack thereof) of physical measures. The topic of what kind of measures can constitute physical measures is related to this approach. When T-invariant, ergodic and ν << m, this is the simplest example of physical measure. Then according to the concept of absolutely continuity, ν(Bν ) = 1 implies m(Bν ) > 0 (where Bν is the the basin of attraction of ν) is physical follows immediately from Birkhoff’s ergodic theorem in this situation. In this instance, the existence of physical measures is thus reduced to the following. Does T admit absolutely continuous invariant ergodic measure. The major theme we address in this thesis is determining a.c.i.p for the LSV Map...Item Entropy Measure Of Degree Based Invariants …(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Maryam Saber; FA19-RMT-034; LHR TP 7381; Dr. Kashif AliChemical graph theory is a branch of mathematical chemistry that uses graph theory techniques mathematically create the chemical phenomena. In addition, chemical graph theory refers to the applications of graph theory to solve molecular chain. In the field of chemical research this theory has important applications. In this thesis, we find the some topological indices and the estimated values of these indices. We determine some entropy of degree based indices of different chemical structures especially the dendrimers and para-line graph and we research the entropies of the graph linked to a new information function accessible. We explore topological indices M1(G)andM2(G), the redefined form of Zagreb indices, the F-index F(G);R(G) index of the molecular chain especially the dendrimers. Furthermore, we compare the predicted values of these indices and give the graphical representation of these indices.Item 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 ButtIn 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.Item General Results for Fractional Integral Inequalities Via New Class of Convex Function.(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Shahid Shaokat; FA19-RMT-047; LHR TP 7378; Dr.Saad Ihsan ButtIn this work, we will present some new results allied with fractional integrals for twice-differentiable functions using new class of convex function and we will give some new applications along with some new definitions. For the development of these new integrals inequalities, we will use Holder integral inequality, Power Mean integral inequality, Holder- Iscan integral inequality and improved Power Mean integral inequality by using Riemann-Liouville fractional integral operator for new class of convex function. We will give some new inequalities for midpoint and trapezoid formula for new class of convex function. Hence, these new results will lead us to the generalization of prior results.Item Inflationary Parameters and Swampland Conjectures in Various Scenarios of Gravity(Library Information Services COMSATS University Islamabad Lahore Campus, 2022) Shama Sadiq; FA20-RMT-019; LHR TP 7923; Dr. Shamaila RaniThe purposeofthisthesisistoinvestigatethein°ationaryparameters and swamplandconjecturesinthepresenceofscalar¯eldandChaplygin gas models.Weinvestigatethein°ationaryparameterssuchasslowroll parameters, scalarpowerspectrum,tensorpowerspectrum,spectralin- dex, tensor-to-scalarratiointhepresenceofscalar¯eldandChaplygingas modelsandalsodiscusstherecentlyproposedswamplandconjectures.The in°ationary scenarioisassumedtobeworkingunderastandardscalar¯eld having¡asdecayratio.Thisdecayratiohasdependenceintheformof generic power-lawofscalar¯eld Á along withthermalbathtemperature T, that is,¡(Á; T) = CÁ Ta Áa¡1 , where CÁ is dimensionlessparameterand a is in°ation decayrate.Assumingthatourmodelbendsaccordingtostrong dissipativeregime(R À 1), westudythebackgroundandperturbativedy- namics aswellasin°ationaryobservableasthescalarpowerspectrum PR, scalar spectralindex ns, dissipativeratio R, thetensor-to-scalarratio r, run- ning ofthescalarspectralindex dns d ln k and generalizedratioofSwampland de-Sitter conjecture ¶ TV ¶ V T for threedi®erentpotentials V (Á) = 3H2inf 8¼ e 2Á p(Á+1) , V = M4µ Á Mp¶2µ1 + ¯ Á2 M2p ¶ and V = 3 4M2(1 ¡ ep 2 3Á)2 respectively.Item On Degree Based Banhatti Indices Of Some Graphs(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Faheem Yaseen; FA19-RMT-043; LHR TP 7444; Dr. Sarfraz AhmadIn the domains of electronic and electrical engineering, graph theory is crucial. It’s significant in signal processing, networking, communication theory, and a variety of other fields. A topological index (TI) is a numerical value that correlates chemical networks with physical and chemical qualities, as well as chemical reactivity. In this thesis, we compute first and second Banhatti Indices, first and second K-hyper Banhatti indices, first and second multiplicative K-Banhatti indices, first and second multiplicative K-hyper Banhatti indices, K-harmonic Banhatti indices, multiplicative K-harmonic Banhatti indices for certain graphs like as Hierachical hypercube network one and two, block shift network one and two, Octagonal network and line graph of wheel graph. My thesis comprising of four chapters. In the first chapter we define some fundamental definitions. In the 2nd chapter, we find Banhatti indices for Hierachical hypercube network one and two, block shift network one and two. In the 3rd chapter, we find Banhatti indices for Octagonal network and line graph of wheel graph. In the 4th chapter, we present the conclusion of our thesis. In the last, we present the references of our thesis.