Department of Mathematics
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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 Cesaro Means Method for Monotone Inclusion, Split Equilibrium and Fixed Point Problems in Hilber Spaces(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Hafiz Muhammad Waseem Farooq; FA19-RMT-023; LHR TP 7389; Dr. Muhammad Aqeel Ahmad KhanItem 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 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 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 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.Item Existence And Stability Of Bumps Of Bums As A …(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Anisa Zafar; FA19-RMT-012; LHR TP 7356; Dr. Muhammad YousafItem Degree Based Topological Indices of Some Chemical Graphs(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Zafar Iqbal; FA19-RMT-037; LHR TP 7434; Dr. Muhammad Kamran SiddiquiGraph 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 like as First and Second zagreb index, First and Second zagreb co-index, First and Second multiplicative zagreb index, Forgotten topological index, Hyper zagreb index, Atom bond connectivity index, Geometric arithmetic index, augmented zagreb index, Balaban index, general Randic index for ∝= 1, −1, 1 2 , − 1 2 and Redefined first, second and third zagreb index for some chemical graphs like as Hydroxy chloroquine, Chloroquine, H-Naphtalenic Nanosheet and Dexamethasone. My thesis comprising of four chapters. In the first chapter we will define some fundamental definitions and we will see the literature review. In the 2nd chapter, we will find indices for the chemical graph of Hydroxy chloroquine and Chloroquine and then we will draw the numerically and graphically comparison of different indices. In the 3rd chapter, we will find indices for the chemical graph of H-Naphtalenic nanosheet and Dexamethasone and then we will draw the numerically and graphically comparison of different indices. In the 4th chapter, we will present the conclusion of our thesis. At the last we will represent the reference of our thesis.Item Thermodynamics in Non-Zero Torsion Theory(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Nageen Pervaiz; FA19-RMT-001; LHR TP 7419; Dr. Shamaila RaniIn thisthesis,westudythecosmicevolutionoftheacceleratinguniverse in theEinstein-Cartan-Sciama-Kibbletheorybyusing°athomogeneous and isotropicmodel.Inthisregards,weconsidersomeparameterizations of theequationofstateparameter ! in termsofredshiftfunction.This parametrization maybeaTaylorseriesextensioninthescalefactor,a Taylorseriesextensionintheredshiftoranyothergeneralparametrization of !. Weusegraphicalbehaviortoexplorethecosmologicalparameters suchasdeceleration,Om-diagnostic,state¯nderandsquaredspeedofsound parameters. Also,weexaminethevalidityofthegeneralizedsecondlaw of thermodynamicsatapparenthorizontakingintoaccountBekenstein- Hawkingentropy.Item On the Construction of Higher Order Conservative Methods(Library Information Services COMSATS University Islamabad Lahore Campus, 2012) Zareen Akhtar; FA19-RMT-026; LHR TP 7417; Dr. Yousaf HabibFor the numerical solution of conservative differential equations such as Hamiltonian system or differential equations that arise in different physical phenomena with some structure, we use structure preserving numerical methods. In particular, symplectic Runge-Kutta methods, G-symplectic General Linear methods and Variational Integrators have been explored. In this thesis Variational Integrators for degenerate Lagrangian and non- degenerate Lagrangian have been discussed. For degenerate Lagrangian, we obtain multi-step numerical methods which suffer from parasitic corruption. In order to control the parasitism we write Variational Integrators as General Linear methods and then employ the projection technique to project the numerical solution onto the desired manifold