M.Phil / MS
Permanent URI for this collectionhttps://repository.cuilahore.edu.pk/handle/123456789/52
This collection archives the complete set of theses produced by students of the COMSATS University Islamabad, Lahore Campus.
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Item On the LCM Lattice of Powers of Monomial Ideal in Three Variables(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) MUHAMMAD WAQAR BASHIR; CIIT/SP20-RMT-016/LHR; Dr. Sarfraz Ahmad; LHR TP 7630Commutative algebra, graph theory, algebraic geometry and combinatorics, semigroup rings and combinatorial optimization, integer programming, and other fields are all related with monomial algebra. Suppose K is a field, S = K[x1, ..., xn] is a polynomial ring with n number of variables, lcm(J), the LCM lattice of J, may be constructed for each mono- mial ideal J ⊂ S = K[x1, ..., xn]. We categorise the powers of monomial ideals in terms of their qualities in terms of LCM lattices. The lcm lattice of the ideal J created by a set of monomials is defined as a lattice with every member being a least common multiple of generating set of J. In particular, we discuss about powers of monomial ideal and staircase of monomial ideal in three variable. The powers and products of monomial ideals in poly- nomial rings are the subject of this thesis. On the polynomial ring K[x1, x2, x3], we derive conditions for the power of monomial ideals. The powers of monomial ideals are easier to calculate when the requirements are met.Item On Expected Values of Some Chemical Indices in the Random Phenylene Chains(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) MUNEEBA; CIIT/FA19-RMT-063/LHR; Dr. Sarfraz Ahmad; LHR TP 7426Chemical graph theory is an interdisciplinary field where the molecular structure of a chemical compound is analyzed as a graph. In recent years, numerous chemical indices have been proposed. In this thesis, we determine some topological indices and the ex- pected values of these indices. We explore some indices known as the Zagreb indices (including redefined forms of the Zagreb indices), the forgotten topological index, the reduced reciprocal Randic index and the reduced second Zagreb index of some random Phenylene and the random Cyclooctane chains. Later this, we look at the Zagreb Con- nection indices of some metal trihalides. Moreover, we make a comparison among the expected values of these indices and give a graphical representation respecting these ran- dom chains and metal trihalides.Item On Tchebyshev Subdivision of Simplicial Complexes(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Iqra Siddique; CIIT/FA16-RMT-035/LHR; Dr. Sarfraz Ahmad; LHR TP 5197In this work, we studied the Tchebyshev subdivision of a simplicial complex Δ as combinatorial and algebraic point of view. In particular, we have studied f-vector and h-vector transformation maps and properties of corresponding transformation matrices. We have also studied the impact of this subdivision on the algebraic properties of the corresponding Stanely Reisner Rings IΔ, including dimension, primary decomposition and Hilbert series. We extended these results to some other subdivisions of a simplicial complex Δ.Item Topological Descriptors of OTIS Interconnection, Block Shift and Hierarchical Hypercube Networks(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Iftikhar Ahmed; CIIT/FA14-PMATH-003/LHR; Dr. Sarfraz Ahmad; LHR TP 7453Topological Indices(TIs) are the numbers associated with the graphs of chemical compounds/networks that help us understand their properties. The aim of the present work is to compute TIs of OTIs swapped networks, Hierarical hypercube networks and block shift networks. Some of the TIs are computed directly and some are computed indirectly. For indirect computations the M-polynomial has been used. The M-polynomial contains key facts about degree dependent TIs, that is why it is so important. We compute Hosoya polynomials, Harary polynomials, Wiener index, modified Wiener index, Hyper Wiener index, Harary index, generalized Harary index, multiplicative Wiener index, M-polynomials, first Zagreb index, second Zagreb index, modified second Zagreb index, generalized Randi\'c index, inverse Randi\'c index, symmetric division index, Harmonic index, inverse sum index, augmented Zagreb index, Redefined first second and third Zagreb index and Zagreb type polynomials of networks mentioned above. We also computed irregularities, multiplicative version of different TIs . Our results can help the discovery of new medicines and in development of new software.Item To Optimize Selection of Charging Types for Electric Buses(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Ali Awais; SP23-RMT-003; Dr. Sarfraz Ahmad; LHR TP 9570Outranking techniques are a class of decision-making methods that excel in providing well suited solutions based on the pairwise comparison of alternatives across multiple criteria. These methods are particularly useful in complex decision-making scenarios where exper imental data is limited, information is imprecise, and human evaluations are subjective. In real-world applications, decisions often involve uncertainty and ambiguity, which tradi tional decision-making models struggle to handle. To address these challenges, spherical fuzzy sets have emerged as a powerful tool, offering a richer structure to capture the full spectrum of human opinions, including satisfaction, dissatisfaction, neutrality, and absti nence. This research presents a series of outranking methodologies within the spherical fuzzy model to address multi-criteria group decision-making (MCGDM) problems more effec tively. Specifically, two significant variants of the PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluation) family—Spherical Fuzzy PROMETHEE I and Spherical Fuzzy PROMETHEE II—are proposed and applied to the selection of an optimal charging type for electric buses. The PROMETHEE methods, enhanced by spher ical fuzzy sets, are particularly well-suited for dealing with uncertainty and imprecision in human judgments. The case study focuses on the charging infrastructure for electric buses, a critical aspect of the growing adoption of electric transportation. Different charging alternatives are eval uated based on multiple criteria, including cost-effectiveness, charging time, energy ef ficiency, and scalability of infrastructure. By using spherical fuzzy numbers, this study captures the imprecision and subjective nature of expert evaluations, providing a more ro bust and flexible decision-making framework. The results demonstrate how the Spherical Fuzzy PROMETHEE methods can effectively rank alternatives, helping decision-makers identify the most suitable charging type under uncertain and ambiguous conditions. This research contributes to the field of multi-criteria decision-making by introducing spher ical fuzzy PROMETHEE methods as an effective tool for addressing real-world decision problems where human judgment and imprecision play a significant role. The findings ix have important implications for both the electric transportation sector and broader appli cations in energy management, supply chain optimization, and other domains requiring robust decision support systemsItem Study of Solitary Wave Solutions of Some Nonlinear Evolution Equations(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Mehak Shahzad; FA22-RMT-034; Dr. Sarfraz Ahmad; LHR TP 9384We investigate soliton interactions within the framework of the Cubic-Quartic Fokas Lenells equation (CQFL) and Nonlinear Schrdinger Equation (NLSE) by employing the generalised Kudryashov approach (GK), Sine jordan equation approach and the Hirota bilinear method (HBM). Many integrable NLEEs possess the soliton solutions which re tains the speed as well as the shape, even after collision with each other. Several efficient techniques have been introduced in literature for obtaining soliton solutions of NLSEs and NLEEs for last two decades, e.g F-expansion approach and its modification, HBM, Sine-Gordon architectonic, and other approaches. The Hirota bilinear method (HBM) was introduced by Ryogo Hirota in 1970s, for solving integrable systems of PDEs and is still used an efficient mathematical method for studying propagation of wave packets. In this thesis, firstly we study soliton interactions by HBM and GKM of Cubic-Quartic Fokes lenell equation with linear gain or loss and time modulated dispersion. Due to its spatially varying coefficients property it has significance in the field of fluid dynamics, classical and quantum field theories, nonlinear optics and physics etc. The nonlinear Schrodinger equation with AC nonlinearity (NLSE-AC) will be studied with the assis- ¨ tance of GKM and SGE and sort out some new solitons and their interactions for our governing model. In this research , we investigate multiple soliton interactions and other solitary wave solutions (SWS) for a NLSE with AC nonlinearity (NLSE-AC). Due to its high order dispersion term. By controlling the parameters, we will obtain S, V, parabolic and anti parabolic, butterfly, bright and dark shaped solitons. Due to variance in grav itation these solitons play a significant role in lasers, magnetic devices and nonlinear phenomenaItem 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 Multiple Lump And Rogue Wave Solutions For …(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Muhammad Aamir Ashraf; FA19-RMT-092; LHR TP 7421; Dr. Sarfraz AhmadA solitary wave is a localised “wave of transcription” that appears from a balance between dispersive and nonlinear effects. A soliton is a specific type of solitary wave, since it maintains its original shape and speed when collides with other solitons. Nonlinear evolution equations (NLEEs) are the main classification of soliton solutions. In this work, our purpose is to attain lump, lump with one and two-kink soliton, periodic– waves, rogue–waves solution for various forms of NLEEs. We’ll also obtain M-shaped rational soliton, M-shaped interactions with kink and periodic waves, periodic-cross kink solutions, homoclinic breather and multi–wave solutions. In this thesis, we’ll study various forms of NLEEs to attained these types of solutions by the combinations of bilinear, hyperbolic, trigonometric, exponential, and hyperbolic functions. We also seen the degenerations of hybrid waves, through some fittingly bilinear forms and also discussed graphical behavior of our system.Item On Some Partitions Of The Multicomplexes(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Ali Shan; FA19-RMT-088; LHR TP 7418; Dr. Sarfraz AhmadThe study of monomial ideal of polynomial ring ’R’ at most with the pretty clean three variables is done. The polarisation I p is known as clean If and only if, the arbitrary monomial ideal ’I’ is also relatively clean. A proper revised characterisation is prettyclean along the monomialideals in theterms of arithmetic degree is presented. This means that a simplicial complex’s polarisation corresponds to shellable if and only if a multicomplex is shellable. In this thesis Stanley decomposition with respect to prime filtrations is discussed.Item On Vertex Mostar Index of the Boron and Polyomino Chains(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) NAJMA SHABIR; FA19-RMT-039; LHR TP 7420; Dr. Sarfraz AhmadA bond-additive topological invariant, known as the Mostar index, has been described as a measure of peripherality in networks and graphs. Later on, other index variations were created, including the edge Mostar index and the total Mostar index. Some experiments are done using vertex-Mostar index on boron nanotubes and polyomino chains but attained results are not good. Therefore, we show how a variation of the cut method described by Klavar, Gutman, and Mohar may be used to efficiently compute it for Honeycomb network H(m;n). We compute the vertex-Mostar index of Honeycomb network H(m;n). Moreover, We’ve also included results related the vertex-Mostar index of Honeycomb network H(m;n).