Department of Mathematics

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    Analysing Security of Pseudorandom Sequences under Computational Constraints
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Amna Nazir; CIIT/SP20-RMT-037/LHR; Dr. Adeel Farooq; LHR TP 7657
    In this thesis, we evaluated Pseudo Random Number Generators (PRNGs) on the basis of a test called Next Bit Test (NBT). To see if a PRNG is suitable for specific purpose, we tested it to see the results. On testing various PRNGs, we got different results that are being discussed in detail. On the basis of our analysis on different PRNGs, we recommend the Logged Fibonacci generator extended to the subtract and borrow generator that holds the properties of NBT and is acceptable to be used in cryptography.
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    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 7630
    Commutative 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.
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    Multi-person Decision Making Based on Triangular Fuzzy Preference Relations
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Ghulam Mohyudin; CIIT/SP20-RMT-015/LHR; Dr. Atiq ur Rehman; LHR TP 7629
    A multi-person decision making (MPDM) model is proposed when the experts evaluate their opinions through triangular fuzzy numbers. First, it is pointed out that the preference relations with triangular fuzzy numbers are inconsistent in nature. In order to distinguish the typical consistency, the concept of Lukasiewicz Transitivity consistency is proposed for triangular fuzzy matrices. The properties of triangular fuzzy numbers are studied in detail. Second, using preference values, a triangular fuzzy preference relation with Lukasiewicz Transitivity consistency is constructed. Third, a novel compatibility degree among triangular fuzzy preference relations is defined. There is a selection phase to rank all the alternatives for choosing the best of them. Finally, a new algorithm for the multi-person decision making problem with triangular fuzzy preference relations using Lukasiewicz Transitivity is presented. A numerical example is carried out to illustrate the proposed definitions and algorithm.
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    Degree Based Topological Co-Indices of Chemical Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Nazir Hussain; CIIT/SP20-RMT-021/LHR; Dr. Muhammad Kamran Siddiqui; LHR TP 7631
    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 have evaluated some degree based topological co-indices, namely, the general Randic co-index for ∝= 1, −1, 1 2 , − 1 2 , the atom bond connectivity co- index, the geometric arithmetic co-index, the first and second Zagreb co-indices, the first and second Zagreb co-indices, the forgotten co-index, the hyper Zagreb co-index, the Balaban co-index, the first and second multiple Zagreb co-indices, the redefined first, second and third Zagreb co-indices for some chemical and lattices graphs like as line graph of Metal Organic Networks, Triangular Benzenoids Gn ⱯϵN, Zigzag edge cornoid fused with starphene zcs(k,l,m), Hony comb network (Hcn), Hydro chloroquine HCQ Conjugated Hydroxyethyl. My thesis comprising of eight chapters. In the first chapter we will define some fundamental definitions. In the 2nd chapter, we will find degree based topological Co- indices for line graph of Metal Organic Networks. In the 3rd chapter, we will find degree based topological Co- indices for Triangular Benzenoids Gn ⱯϵN. In the 4th chapter, we will find degree based topological Co- indices for Zigzag edge cornoid fused with starphene zcs(k,l,m). In the 5th chapter, we will find degree based topological Co- indices for Hony comb network (Hcn). In the 6th chapter, we will find degree based topological Co- indices for Hydro chloroquine HCQ Conjugated Hydroxyethyl. In the 7th chapter, we will present the conclusion of our thesis. In the last, we will present the references of our thesis
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    Construction of new numerical scheme based on implicit method for two-population neural field model.
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Khalid Bilal; CIIT/SP20-RMT-023/LHR; Dr . Muhammad Yousaf Bhatti; LHR TP 7632
    The brain is the most complex organ of the human body. Neuron is the basic unit of brain. Signals are transmitted from one part of the brain to the other parts in the form of impulses. This mechanism is translated by different mathematical models. In this work we have explored the two-population neural field (TPNF) model which describes the activity of excitatory and inhibitory populations of neurons. This model is explored without external inputs, with spatial and with spatio-temporal external inputs. For numerical study of the model explicit Runge-Kutta method of order four is used. We developed a new numerical scheme based on implicit RK4 method and have studied the evolution of the solutions of the model for no external inputs and with external inputs.
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    Consensus-Based Group Decision Making Using Hesitant Fuzzy Preference Relations
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Syeda Mifzalah Bukhari; CIIT/SP20-RMT-019/LHR; Dr. Atiq ur Rehman; LHR TP 7650
    In this thesis we presented a consensus-based technique for dealing with multi-person decision making (MPDM) employing hesitant fuzzy preference relations (HFPRs) that aren't in the standard format. At the most basic level, we suggested a Lukasiewicz transitivity (TL-transitivity)-based technique for obtaining normalized hesitant fuzzy preference relations (NHFPRs), after which a consensus-based model is built. Then, in order to build TL-consistent HFPRs and symmetrical matrices, a transitive closure formula is established.
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    Effects of Modified Entropies on Geometric Thermodynamics of Black Holes
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Aqsa Tanveer; CIIT/SP20-RMT-011/LHR; Dr. Shamaila Rani; LHR TP 7647
    In this thesis, we analyze the thermodynamics of five-dimensional Schwarz- schild AdS black hole in AdS5 × S5 spacetime in the presence of Tsallis entropy. Since the cosmological constant Λ is considered as thermodynam- ical pressure with volume as its conjugate, but this explanation cannot be employed in AdS/CFT correspondence. In this study, we associate cosmo- logical constant Λ in boundary Gauge theory with the number of colors N and chemical potential is taken to be its thermodynamic conjugate. The two geometric parameters in the AdS black hole, r and L are substituted for two thermodynamic parameters in the micro-canonical ensemble, which are considered to be entropy S and N2. Moreover, we evaluate several thermodynamical geometry formulations, including Weinhold, Ruppeiner, and Quevedo and derive associated scalar curvatures for five-dimensional Schwarzschild AdS black hole
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    On Group Labeling of Antiprism Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Mazhar Hussain; CIIT/SP20-RMT-038/LHR; Dr. Hani Shaker; LHR TP 7640
    Assume a graph G with n vertices and Λ be any abelian group having n elements. Group distance magic labeling of G is a bijective map f: V → Λsuch that ∃μ∈Λ for that ∑ ∈ ( ) ( ) = ∀ v ∈ V, where are the vertices adjacent to v. In this thesis , we find the × × × , × × × , × × × , × × × , × × × , × × × , group distance magic labeling for antiprism graph !
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    Numerical And Exact Solutions For Various Nonlinear
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Syed Oan Abbas Bukhari; CIIT/SP20-RMT-039/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 7641
    A soliton is a particular type of solitary wave, that is retrieved as a result of integrable NLPDE. There are different type of solutions like analytic and approximate solutions exist. An exact solution is a symbolic representation of a value which solves a given equation exactly, while a numerical solution is numerical representation of a value which solves a given equation exactly or approximately. Numerical solutions are very usefull when we are unable to retrieve analytic solution. It is also used for comparison with analytic solutions. A lot of analytic and numer- ical methods exists in literature to solve the NLPDEs such as Bernouli’s equation approach, extended tanh method, modified simple equation method and RK methods etc. In this work, our purpose is to attain bell-type, kink and anti kink type, singular solitons and power series solutions for various forms of NLEEs. We’ll also obtain Weierstrass elliptic func- tions, Jacobi elliptic functions in terms of hyperbolic, trigonometric and rational functions. In this thesis, we’ll study various forms of NLEEs to attained these types of solutions with the help of various method like sine-cosine method (SCM), csch method, tan-cot method, sub-ode technique and residual power series method (RPSM). We will also compare our exact solutions with numerical solutions and then represent these solutions graphically.
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    Generalized Convexity and Related Integral Inequalities on Fractal Sets
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Arslan Razzaq; CIIT/SP20-RMT-034/LHR; Dr.Saad Ihsan Butt; LHR TP 7637
    In this proposal we will present the generalized convexity and related integral inequali- ties on fractal sets Rϖ ( 0 < ϖ ≤ 1). Thus, these developments allow us to develop new bounds for integral inequalities. We will give generalized Hermite-Hadmard (H-H) type inequalities in fractal sense. In this research work we will present some new results by em- ploying local fractional calculus for twice differentiable functions and we will give some new applications along with some new definitions. For the development of these new in- tegral inequalities, we will use generalized H¨older-integral inequalities and Power mean integral inequalities by using local fractional calculus. We will give some new inequalities for midpoint and trapezoid formula for new class of local fractal calculus. Hence, these new results will lead us to generalization of prior results.