Department of Mathematics

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Subject: search.filters.subject.Dr. Adeel Farooq

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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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    Words for generators of some subgroups of the Baby Monster group
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Hafiz Sami Ullah; CIIT/FA19-RMT-110/LHR; Dr. Adeel Farooq; LHR TP 7620
    In this thesis we have computed the generators for some subgroups of the Baby monster group B[1], which is the second largest group in sporadic simple groups. Here we have S10 < S8(2) < Fi23 < B. To compute the generators for the subgroups of Baby Monster group we have used two softwares namely GAP and MeatAxe. Also we have computed the coxeter generators[2] satisfying coxeter relations for Y311, Y221, Y321, Y421, Y331, Y431, Y222, Y322, Y422, Y332 and Y432. Finally, we have expressed all the coxeter generators in terms of generators of Fi23 inside the Baby Monster monster group B.
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    The Fuzzy N-Leibniz Algebras
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Anjum Ali Sher; SP19-RMT-026; Dr. Adeel Farooq; LHR TP 7347
    In this thesis we have shown that, 1. The (m + 1)-Leibniz algebras are expressible in the form of m-magma. 2. The homomorphic pre-image of a fuzzy ideal of a m-Leibniz algebra is a fuzzy ideal. 3. If S a fuzzy subset of a m-Leibniz algebra A then, (i) S is either a fuzzy subalgebra or a fuzzy ideal of A, (ii) Every nonempty t-level subset U(s, t) is also a fuzzy subalgebra or ideal of A. 4. Homomorphic image of the fuzzy ideal of a m-Leibniz algebra having the supre- mum property is also a fuzzy Leibniz ideal. 5. If A is a m-Leibniz algebra then the maximal normal fuzzy subalgebra of A is a Boolean algebra.
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    Words for Maximal Subgroups in Certain Sporadic Simple Groups
    (Library Information Services, COMSATS University, Lahore Campus, 2018) Faisal Yasin,; FA13-PMATH-002; Dr. Adeel Farooq,; LHR TP 5946
    In this thesis we found generators for the Maximal Subgroups (MSs) of the Sporadic Sim ple Groups (SSGs) as words in the generators of the parent group. The idea of standard generators was presented by Wilson. He began a project known as online form of Atlas, ”http://brauer.maths.qmul.ac.uk/Atlas” in which he provides the representation and words for MSs of the SSGs. However there are some cases, which have to be dealed. We pursue the work initiated by R. A. Wilson of finding words for the MSs of SSG. Thus the only cases on the list of World-wide-web Atlas which need to be solved are Co1, HN, Fi23, Fi24, B and M. There are 14 MSs of Harada-Norton group HN. Generators of 11 MSs are already known. In this thesis we have provided generators of remaining 3 MSs. Similarly, we have provided words for four MSs of Fi23, words for sixteen MSs ofCo1, words for 11 MSsofFi24 and words for MSs of some linear groups. In World-Wide-Web Atlas of group representation there is only one copy of S5 of the MSs of L3(5), but we have found words for two non-conjugate copies of S5 in Table 15 of second chapter
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    Consensus Based Group Decision Making With Incomplete Reciprocal Fuzzy Preference Relations Using Product T-norm.
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Mustanser Hussain; FA17-RMT-035; Dr. Adeel Farooq; LHR TP 5661
    In this dissertation, a consensus-based method for group decision making (GDM) using product transitivity with incomplete fuzzy preference relations (IFPRs) is proposed. Additionally, an average aggregation operator has been used at the first level to estimate the missing preference values and construct the complete fuzzy preference relation (FPR). Then it is confirmed to be product consistent by using the transitive closure formula. Following this, weights of decision makers (DMs) are evaluated by merging consistency weights and predefined priority weights (if any). The consistency weights for the DMs are estimated through product consistency investigation of the information provided by each DM. The consensus process determines whether the selection procedure should be initiated or not. The hybrid comprises of a quitting process and feedback mechanism, and is used to enhance the consensus level amongst DMs in case of an inadequate state. The quitting process arises when some DMs decided to leave the course, and is common in GDM while dealing with a large number of alternatives. The feedback mechanism is the main novelty of the proposed technique which helps the DMs to improve their given preferences based on this consistency. At the end, a numerical example is deliberated to measure the efficiency and applicability of the proposed method after the comparison with some existing models under the same assumptions. The results show that proposed method can offer useful comprehension into the GDM process. ix
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    Geometry of Manifolds and the Gauss-Bonnet Theorem
    (Library Information Services, CUI Lahore, 2024) Ayodeji Simeon Farominiyi; CIIT/FA21-RMT-108/LHR; Dr. Adeel Farooq
    This thesis delves into the fascinating realm of smooth manifolds, investigating their intri- cate geometry through a comprehensive exploration of tangent spaces, connections, curva- ture and the Gauss-Bonnet theorem, a crucial result in the theory of surfaces. This theorem establishes a connection between a surface’s geometric and topological properties and acts as a model for similar statements that hold true in higher-dimensional contexts. The study is organized into three chapters, each addressing key aspects of this subject and culminating with an illustrative example featuring the sphere.Chapter 1 serves as an introduction to the theory of differential manifolds. It begins with a comprehensive overview of the underlying concepts, including topological spaces, charts, atlases, and smooth maps. The chapter provides a detailed examination of tangent spaces and tangent bundles, shedding light on the notion of derivatives and directional derivatives on manifolds. Vector fields and vector bundles are then introduced, enabling a deeper understanding of the interplay between smooth functions and smooth vector fields. The chapter, after the introduction of tensor products, also explores the concept of a Rie- mannian metric, highlighting its significance in defining lengths, angles, and inner products on manifolds. Additionally, differentiable forms are discussed, unveiling their role in cap- turing the geometric properties of manifolds.
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    Locating some Subgroups of the Baby Monster Group
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Abdul Sattar; FA19-RMT-029; LHR TP 7907; Dr. Adeel Farooq
    In this thesis, we have computed the generators for some subgroup of Y-Groups. These are the subgroups of the bimonster group whose generators satisfy the coxeter relations. It can be presented by Y-diagrams whose nodes represent the generators and satisfy the coxeter relations. We have computed the coxeter generators satisfying coxeter relations for Y111, Y211, Y411, Y421 and Y441. The generators of Y-group have been determined by computing successive centralizers. We have also surveyed some maximal subgroups of the Baby Monster Group.
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    Certain Results on m-polar Fuzzy Groups
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Musawwar Hussain; FA18-RMT-015; LHR TP 6455; Dr. Adeel Farooq
    In this thesis we have developed some fundamental results from group theory to fuzzy and m-polar fuzzy groups. We have defined some basic characteristics of fuzzy groups and have provided the m-polar fuzzy analogues of fuzzy subsets, groups, level sets, and cosets. We have investigated an equivalence relation to distinguish whether a m-polar fuzzy subgroup of a group G is equivalent to any other m-polar fuzzy subgroup of that same group G, or not. We determined the maximal chains and a general formula for quaternion and dihedral groups to give the cardinality of their fuzzy and m-polar fuzzy subgroups. We intro duced aninequality which gives quantitative relationship between maximal chains, level subgroups and fuzzy subgroups. Moreover, we mainly constructed the m-polar fuzzy analogues of isomorphism theorems and butterfly lemma.
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    On the Structure of Unitary Units of F3mC9 and F3m(C3 ×C3)
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) NAJAM UL ABBAS; CIIT/FA17-RMT-038/LHR; Dr. Adeel Farooq; LHR TP 5663
    In this thesis, we have constructed the group ring matrix of group algebras F3mC9 and F3m(C3 ×C3). We have found center of group algebras F3mC9 and F3m(C3 ×C3). On the basis of group ring matrix, the center of unitary unit group have been established and then order of unitary unit group has been computed. We have also studied the above group algebras for m = 1 computationally using the GAP package LAGUNA. The orders of normalized unit groups, unitary unitary group and unit group has been computed. Moreover, Augmented Ideals of the above group algebras has been studied and we have provided their basis.
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    Analysis of Certain Maximal Subgroups in the Monster
    (Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Samra Akram (FA23-RMT-036); Dr. Adeel Farooq; LHR TP 9782
    The Conjugacy Class Fusions of maximal subgroups 112 : (5×2·A5), (PSL2(11)×PSL2(11)) : 4,72 : SL2(7),PSL2(19) : 2, and PSL2(13) : 2 of the Monster group are thoroughly examined in this thesis, which builds upon the seminal work of Heiko Dietrich, Melissa Lee, and Tomasz Popiel. We’ve provided fresh perspectives on the subgroup structure and fusion patterns of M by examining and analyzing their findings. ix