Thesis (Bachelor of Science in Mathematics)
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Item Modeling a differentiated goods bertrand duopoly under uncertain demand(BRAC University, 2025-05) Rahman, Fardeen; Taha, Mashsharat Wasi; Rafayal AhmedThis thesis explores the design of information structures in a Bertrand duopoly with differentiated products under demand uncertainty. Specifically, we consider a setting in which two firms compete in prices while facing a common uncertain demand parameter, which we modeled as a random variable over the unit interval. Drawing inspiration from the framework of Bayesian persuasion, we examine how posterior beliefs, induced through noisy signals, affect equilibrium payoffs. By comparing the expected profits under prior and posterior beliefs, we show—using Jensen’s inequality — that firms achieve strictly higher expected payoffs under the prior belief than under the posterior when the governing interaction terms lie in a certain interval. This result illustrates that more precise information can be strategically disadvantageous in a Bayesian game with continuous action and type spaces. Furthermore, we extend this finding to an n-player symmetric setting, where we discover that the aforementioned interval shrinks as the number of competing firms increases. These insights contribute to a deeper understanding regarding the role and value of private information in oligopoly pricing games.Item Elliptic curve isogenies and their embedding into homomorphisms of p-adic tate modules(BRAC University, 2025-01) Chowdhury, Atonu Roy; Chowdhury, Syed Hasibul HassanThis thesis investigates the interplay between the morphism spaces of elliptic curves and their associated Tate modules. Specifically, we focus on proving the injectivity of the map Hom (E1,E2) ⊗ Zp → Hom (Tp (E1) , Tp (E2)) , where E1 and E2 are elliptic curves, and Tp (E) denotes the p-adic Tate module associated with the elliptic curve E. The result connects the algebraic structure of morphisms over elliptic curves with the module-theoretic properties of their Tate modules. We employ tools from algebraic geometry and p-adic number theory, focusing on the role of endomorphism rings and Tate modules. This work contributes to understanding how information about elliptic curve morphisms is preserved and reflected in the realm of p-adic arithmetic.Item Simple quantum algorithms for k-mismatch problem(BRAC University, 2024-04) Habib, Ruhan; Chowdhury, Syed Hasibul Hassan; Hassan, Syed HasibulFor many problems, quantum computation has significant advantage compared to classical computers. One of the most surprising example of this is the Grover Search Algorithm, which can search through any unstructured data structure of size n with O ( p n) queries. This algorithm gives quadratic speed-up to many searchbased problems. Another important example is the Shor’s factorization algorithm, which solves the factorizatio problem with ~O (n3) operations. No polynomial-time classical algorithm for factorization is known to this date. In particular, quantum computing has given significant speed-ups to many string-based problems. String processing is an active and interesting field, with many applications in fields such as bioinformatics. It is also of significant theoretical interest: a subquadratic classical solution (or lack thereof) to string problems such as can shed light to the existence (or lack) of solutions to quite a few problems. If SETH (Strong Exponential Time Hypothesis) is true, string problems such as LCS do not have strongly subquadratic time (O (n2 ) for some > 0) classical solutions. However, we can use quantum computing to get faster solutions to these problems: LCS has a quantum algorithm with ~O n2=3 [13] query complexity in contrast to the classical known best of ~O (n2). In this thesis, we work on the k-mismatch problem. Here, given a pattern and a text, we have to find if the text has a substring with a Hamming distance of at most k from the pattern. This is a “fault-tolerant” version of the well-known string search problem. This important problem has been extensively studied in classical setting but had not been studied through quantum computation until 2022 by Jin and Nogler [16]. In that paper, they provided an ~O (k p n)-time quantum algorithm and showed that the problem has a quantum query lower-bound of p kn . They posed the question of whether there is a quantum algorithm with better query complexity than ~O k3=4n1=2+o(1) . In 2024, Kociumaka, Nogler, and Wellnitz [18] found an algorithm with optimal query complexity ~O ( p kn) and time complexity ~O p n=m( p km + k2) . This thesis provides simple quantum algorithms for two variants of the k-mismatch problems. The first is when r = mk is small. For this case, we have a ~O ( p rmn)-time quantum algorithm. The second algorithm solves the problem in an approximate manner, given a parameter > 0. It returns an occurrence as a match only if it is a (1 + )k-mismatch. If it does not return any occurrence, then there is no k-mismatch. This algorithm has a time complexity of ~O ( 1 p mn=k).Item On the Seiberg-witten invariants of smooth 4-Manifolds(BRAC University, 2024-10) Nazrul, Nian Ibne; Chowdhury, Syed Hasibul HassanThis thesis reviews Seiberg-Witten Gauge theory and the Seiberg-Witten invariants of smooth 4D manifolds. After reviewing some preliminaries on Clifford Algebras, Spinbundles, Dirac Operators, We go into discussing a system of mildly nonlinear partial differential equations on a U(1) bundle which are commonly known as Seiberg-Witten equations. We discuss its properties, consider their solution space and then quotient it by the equivalence due to gauge transformations. The moduli space that we get after moding on the space of solutions has some nicer properties as compared to Donaldson’s. In the last chapter, we briefly talk about the Witten conjecture which makes a connection between the Seiberg-Witten Invariants and the Donaldson invariants. Many physicists argue that using S-duality, SW theory and Donaldson theory can be viewed as the two extreme cases (one N → ∞, and the other N → 0) of a common theory, but S-duality is not yet mathematically understood fully rigorously. Even with seminal progresses regarding proving this conjecture which is widely believed to be true by many professional physicists- it still remains to be proven true in the general sense. This thesis acts as a review of these ideas as an introduction to Seiberg-Witten theory.Item Statistical analysis of network data flows and predictions using statistical and machine learning regression models(BRAC University, 2024-05) Boateng, Albert; Rahim, Maheen Mehjabeen; Islam, Mohammad RafiqulThis paper presents a statistical analysis of measurements relating to network’s data flows and predictions using statistical and machine learning regression models. The study’s objective is to use statistical methods and machine learning regression models to analyze and make predictions on a spatio-temporal traffic volume dataset obtained by Dr. Liang Zhao (Emory University), from sensors along two major highways in Northern Virginia and Washington, D.C. This work aims to answer some fundamental questions related to the network such as: 1. What statistical inferences and descriptive analysis can be made on the network’s data flow? 2. How can one obtain the Routine Matrix of the Network from the Adjacency Matrix? 3. How can one employ various techniques, such as Regularization and Singular Value Decomposition (SVD), to solve the singularity or ill posed nature of the network in the Traffic Matrix Estimation?, and 4. How can one apply Machine Learning regression models, such as Support Vector Regressor (SVR) and XGBoost Regressor, to make predictions on the Network’s flow volume? Concepts in this work or paper can be practically applied on other real world networks to analyze and make predictions on the network’s data flow.Item Predicting diabetes using machine learning: a comparative study of supervised classification models(BRAC University, 2023) Pushpo, Mahzebin; Islam, Mohammad RafiqulDiabetes is a primary worldwide health concern that can develop at any age and has serious consequences. It results from imbalanced glucose levels in the body. As well as being a long-term disease, it has other associated risks, from life-threatening problems to financial loss. So, it is essential to correctly detect this condition as soon as possible to mitigate further complications. Due to developments in medical technology, many tools are available today for diagnosing diseases. To ensure faster predictions and diagnosis of patients, one such tool known as machine learning (ML) algorithms is used. It is a section of Artificial Intelligence (AI) that replicates a human's learning process to train a system. In this study, the algorithms used to predict diabetes patients are supervised classification ML algorithms like Logistic Regression, K-Nearest Neighbor, Naïve Bayes, Decision Tree, and Random Forest. The data used is primary data, which is collected from Bangladeshi adults from different age groups. It consists of all the demographic data, medical history, and family information necessary for the study. The dataset is collected and cleaned for repetition and errors. From these data, diabetes status is taken as the dependent variable, and the associated risk factors are the independent variable. Then, the model is deployed using the RapidMiner tool. The confusion matrices for each model are also produced, and a comparative analysis is carried out. After evaluating their performances, the highest accuracy achieved was 94.62% and 94.23%. From these findings, the best model can be determined. This selection of the ideal model is useful because it will help in the proper and timely identification of patients in the future in the healthcare sector so that treatment can be done to curb the disease.Item Statistical analysis of network data flows and predictions using statistical and machine learning regression models(BRAC University, 2024-05) Boateng, Albert; Rahim, Maheen Mehjabeen; Mohammad Rafiqul IslamThis paper presents a statistical analysis of measurements relating to network’s data flows and predictions using statistical and machine learning regression models. The study’s objective is to use statistical methods and machine learning regression models to analyze and make predictions on a spatio-temporal traffic volume dataset obtained by Dr. Liang Zhao (Emory University), from sensors along two major highways in Northern Virginia and Washington, D.C. This work aims to answer some fundamental questions related to the network such as: 1. What statistical inferences and descriptive analysis can be made on the network’s data flow? 2. How can one obtain the Routine Matrix of the Network from the Adjacency Matrix? 3. How can one employ various techniques, such as Regularization and Singular Value Decomposition (SVD), to solve the singularity or ill posed nature of the network in the Traffic Matrix Estimation?, and 4. How can one apply Machine Learning regression models, such as Support Vector Regressor (SVR) and XGBoost Regressor, to make predictions on the Network’s flow volume? Concepts in this work or paper can be practically applied on other real world networks to analyze and make predictions on the network’s data flow.Item Study on the numerical analysis of storm surges around the coastal zone of Bangladesh(BRAC University, 2023-08) Joty, Musarrat Rashidi; Naher, HasibunStorm surges are a natural phenomenon preceding tropical cyclones, taking place in the coastal zones. Often times they are the most dangerous part of the cyclones, causing the most amount of damage to the population and property of the affected area. While it is quite difficult to predict the heights of surges caused by tropical cyclones accurately in real time, having access to more information regarding surge levels can help save lives and resources. This study is a review paper focused on the Bay of Bengal region, specifically the coastal zone of Bangladesh. It contains derivations of equations and methods as well as definitions of some important aspects of storm surge prediction models. There is a discussion on numerical modeling methods of predicting cyclone track, intensity and the associated storm surges along with the analysis of the data obtained and the results of some relevant papers. The findings here will hopefully summarize and help interested researchers to gain an idea on this field and what to focus on for future work. A deeper understanding of such models can assist governments and relevant authorities in helping communities by improving their disaster-management strategies in high-risk areas to protect coastal populations by having accurate, real-time information on upcoming storm surge forecasts.Item Numerical Modeling of storm surge for the coastal region of Bangladesh(BRAC University, 2022) Chaudhury, Samara; Naher, Dr. HasibunThis study investigates the estimated water levels during a cyclonic storm due to the interaction of tide and surge which also includes air bubble entrainment along the coastal areas of Bangladesh. Thus, a two-dimensional, vertically integrated hydrodynamic model in Cartesian coordinates has been developed. The model equations are solved using finite difference schemes implementing a staggered C-grid. Nested scheme methods are used to incorporate the complexity of the coast in order to not waste central processing unit time. Along the northeast corner of the innermost scheme or the very fine mesh scheme (VFMS), the Meghna river discharge is considered and the coastal and island boundaries are approximated via proper stair steps. A stable tidal condition over the area of interest is produced by making the sea level oscillate with the major tidal constituent M2 through the southern open boundary of the parent scheme or the coarse mesh scheme (CMS). The model is used to compute the water levels due to tide-surge interaction including the air bubble effect for the April 1991 cyclone. The model results are found to compare well with the observations from Bangladesh Inland Water Transport Authority. Therefore, the model is found to adequately simulate water levels which is in the presence of air bubbles. Additionally, it can also be observed that water levels are affected by factors such as river discharge and inverse barometer among others.Item PCR based analysis of single nucleotide polymorphsim in beta-casein A1 and A2 gene of bovine in Bangladesh(BRAC University, 2019-03) Rahman, Olema Taj; Hossain, M. MahboobIn the present study "PCR-based analysis of single nucleotide polymorphism in beta-casein A1 and A2 gene of bovine in Bangladesh" was conducted to differentiate between beta-casein-containing types A1 and A2. Casein contributes80% of the bovine milk protein and has four fractions (alpha S1-CN, alpha S2-CN, beta-CN, and k-CN). Beta casein contributes 25-35% of milk protein and many variants reported in various cattle breeds (A1, A2, A3, B, C, D, E, F, G, H1, H2 and I). The beta-casein variants A1 and A2 differ in the position of 67thamino acid, the substitution of proline in type A2 with Histidine(in A1) is primarily due to the replacement of nucleotide "C" with nucleotide "A" in the corresponding position of nucleotide. For the detection of polymorphsim of A1, A2 beta-casein gene from genomic DNA, thirty cattle (including both local and cross-bred) were selected. Allele-Specific PCR and Amplification Created Restriction Site PCR amplified the beta-casein gene. AS-PCR, ACRS-PCR and subsequent agarose gel electrophoresis could differentiate between A1, A2 types of beta-casein genes in these animals. The results of the screening showed three animal genotypes in these 30 animals. The number of animals with genotypes A1A1, A2A2 and A1A2 are 5, 13 and 12 respectively. The A2 and A1 allele frequencies are 0.63 and 0.37 respectively.