Thesis (Bachelor of Science in Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10361/28421

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Open Access
Modeling a differentiated goods bertrand duopoly under uncertain demand
(BRAC University, 2025-05) Rahman, Fardeen; Taha, Mashsharat Wasi; Rafayal Ahmed; Department of Mathematics and Natural Sciences
This 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.
Open Access
Elliptic curve isogenies and their embedding into homomorphisms of p-adic tate modules
(BRAC University, 2025-01) Chowdhury, Atonu Roy; Chowdhury, Syed Hasibul Hassan; Department of Mathematics and Natural Sciences
This 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.
Open Access
Simple quantum algorithms for k-mismatch problem
(BRAC University, 2024-04) Habib, Ruhan; Chowdhury, Syed Hasibul Hassan; Hassan, Syed Hasibul; Department of Mathematics and Natural Sciences
For 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 = m􀀀k 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).
Open Access
On the Seiberg-witten invariants of smooth 4-Manifolds
(BRAC University, 2024-10) Nazrul, Nian Ibne; Chowdhury, Syed Hasibul Hassan; Department of Mathematics and Natural Sciences
This 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.
Open Access
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 Rafiqul; Department of Mathematics and Natural Sciences
This 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.
Open Access
Predicting diabetes using machine learning: a comparative study of supervised classification models
(BRAC University, 2023) Pushpo, Mahzebin; Islam, Mohammad Rafiqul; Department of Mathematics and Natural Sciences
Diabetes 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.
Open Access
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 Islam; Department of Mathematics and Natural Sciences
This 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.
Open Access
Study on the numerical analysis of storm surges around the coastal zone of Bangladesh
(BRAC University, 2023-08) Joty, Musarrat Rashidi; Naher, Hasibun; Department of Mathematics and Natural Sciences
Storm 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.
Open Access
Numerical Modeling of storm surge for the coastal region of Bangladesh
(BRAC University, 2022) Chaudhury, Samara; Naher, Dr. Hasibun; Department of Mathematics and Natural Sciences
This 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.
Open Access
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. Mahboob; Department of Mathematics and Natural Sciences
In 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.
Open Access
Women in Science, Technology, Engineering and Mathematics (STEM) in Bangladesh
(BRAC University, 2018-10) Anwar, Faraque Muhammad; Naher, Dr. Hasibun; Department of Mathematics and Natural Sciences
This is a study to show the participation of female students in subjects related to science in the recent days compared to the previous days. To conduct the study, I have designed the study by showing the cultural barrier of the participation of women in education and how this is changing in time and how they are participating in the various science related fields in the current days by overcoming their cultural and social barriers. I have shown the Statistics of all education boards of Bangladesh in the two major Programs, S.S.C. and H.S.C. examinations of the years 2012 to 2017. For further study, I have shown a short overview of University of Dhaka which is one of the leading universities of Bangladesh. Afterwards, the study was followed by showing the participation of the Women in the different professions in the Women Association of Engineering, Planner and Architect (WAEPA) which was updated in the year 2015. To make the study stronger, I have shown the comparative study of the graduates of BRAC University which is in the top ranked among all private universities in Bangladesh. I have used the tables, bar and pie charts to show the comparisons between different years and male and female participants in different science related sectors.
Open Access
Exact solutions of fractional differential equations by using new generalized (G´=G)-expansion method
(BRAC University, 10/25/2018) Shafia, Humayra; Naher, Dr. Hasibun; Department of Mathematics and Natural Sciences
In this thesis, Exact Solutions of fractional differential equations by using (G'/G)- Expansion Method the nonlinear partial fractional differential equations are renewed to the nonlinear ordinary differential equations by using the fractional complex transformation. We apply the extended (G´=G)-expansion method to generate travelling wave solutions to the time and space fractional derivative nonlinear KdV equation. The obtained solutions reveal that the extended (G´=G)-expansion method is very effcient and competent mathematical tool for generating abundant solutions and can be used world class of nonlinear evolution fractional order equations.
Open Access
System of linear equations in Computed Tomography (CT)
(BRAC University, 2018-09) Rahman, A.F.M. Azmain Musfiq; Ahmed, Mohammad Maruf; Department of Mathematics and Natural Sciences
In X-rays the internal structures often overlap, thereby it reduces the chances of detecting any anomalies (if any) and hence the patient may be given an improper diagnosis or they will have to suffer due to participating in multiple scans, which is inevitable if their illness continues and the radiologists are unable to find a cause. In that aspect Computed Tomography (CT), uses multiple X-rays at variable angles directed at a section of the patient’s body, to acquire images (single X-ray shot gives image from a specific angle) of the entire cross section. These images are then taken by the scanner and the data (all the images) is then processed to create a detailed image of the cross section. This image is very detailed and so chances of detecting anomaly (if any) goes up. This thesis focuses on representing the scans of CT using a field of linear algebra, namely the System of Linear Equations to be more specific. That is to say, in this thesis we acquire a system of linear equations using the arbitrary data collected after sending and detecting X-ray beams at a patient’s cross section (slice), then solving it using Gaussian Elimination and the Matrix Inversion method to acquire roots or solution values which represent the substances present in the slice. These values are then cross referenced with a table that shows the range of values (linearly attenuated for this arbitrary scan) that represent various substances like bones, tissues (healthy or tumorous) and metallic objects that may or may not be present in the slice. Thus by using this table we can identify as to what the roots actually are within the slice.
Open Access
Review on cyclic group and affine cryptosystem and its application on cryptography
(BRAC University, 2018-09) Tasnim, Samiha; Ahmed, Mohammad Maruf; Department of Mathematics and Natural Sciences
In this thesis, I have tried to briefly describe the concept of cryptography and group theory and the relation between them. In short, Cryptography is regarded as a medium which enables communications to take place under secure parameters. The process of encryption and decryption which uses algorithm and key to convert plain texts into encrypted ones and vice versa makes such secure communication possible even with the presence of malicious third parties. A major portion of the study of cryptography deals with Group Theory. Group theory, perhaps the primary algebraic structure to be studied abstractly, is one of the most fundamental structures. A group is a finite or infinite set of elements together with a binary operation that satisfies the four basic properties of closure, associability, the identity property and the inverse property. In this thesis, I have put the Diffie Hellman’s protocol to demonstrate an application in which an outside client can privately communicate with members of a particular company without running the risk of important facts getting leaked elsewhere. Affine cryptosystem is used to encrypt and decrypt messages which uses the ℤ26 is also included in this thesis by which the client can send encrypted messages to any specific member he wants, and the receiver can also decrypt the message.
Open Access
Generalized inverse and it’s applications to the solutions of system of linear equations and semigroup
(BRAC University, 9/27/2018) Shiham, Muhtadi Nasrullah; Ahmed, Mohammad Maruf; Department of Mathematics and Natural Sciences
There are a number of versatile generalizations of the usual inverse matrix, referred to in this thesis as generalized inverse matrices. The definitions and properties of some of the common generalized inverse matrices are described, including methods for constructing them. A number of applications are discussed, including their use in solving consistent systems of linear equations which do not have the same number of equations as variables, or which have a singular coefficient determinant. A certain type of generalized inverse is shown to give the least‐squares solution of an inconsistent system of linear equations. Other applications are to systems of nonlinear equations, to integer solutions of systems of equations and to linear programming. The purpose of the thesis is to show that a singular or 𝑚×𝑛 matrix has a generalized inverse (g-inverse). A matrix 𝐴− is said to be a generalized inverse if it fulfils the condition 𝐴𝐴−𝐴=𝐴. The raw canonical system is used to find the generalized inverse. So we will be using theorems, examples and programming language to prove and solve G-inverse. This thesis will also consist of semigroups, contour integration and applications which is related generalized inverse.
Open Access
Mathematical models to explain the export of Bangladesh
(BRAC University, 5/24/2018) Haque, Mohaiminul; Islam, Dr. Mohammad Rafiqul; Department of Mathematics and Natural Sciences
Export is a very important medium for a Nation to earn foreign currencies and specially for a country like Bangladesh to maintain their economy and GDP (Gross Domestic Product). In Bangladesh economy, a very significant percentage depends on export specially the RMG export holds 62% of total export which is the most successful export item for them. They particularly do high export of RMG to European nation and USA, who have become successful export partners of Bangladesh. It is very important to monitor the Export of Bangladesh and for this, many mathematical models have been used earlier. In continuation, in this research three renowned mathematical models/methods have been applied on the very recent 27 years export data (secondary data) collected from library and website of Bangladesh Bureau of Statistics and Bangladesh respectively along with the error calculation using RMSE (root mean sum squared error) to choose the best method for explaining the Bangladesh export progress. The three renowned mathematical methods like Geometric method, Least square regression and Exponential method have been used to forecast after calculating the overall growth rate by each method respectively. Finally, Exponential method has been selected as best fitted model of Bangladesh Export among these three which helps to speed up the business decision as time is less consuming to understand the performance of Bangladesh.
Open Access
Analytical solution of non-linear partial differential equation by using the extended (𝐆′/𝐆) expansion method with non-linear auxiliary equation
(BRAC University, 3/1/2018) Karmaker, Bishwajit; Naher, Dr. Hasibun; Department of Mathematics and Natural Sciences
Among some new methods, these were introduced to find the exact solution of Non-Linear Partial Differential Equations (NLPDEs), (G'/G) expansion method proposed by Mingliang Wang, is straightforward and easy to handle as it gives rich new solutions. On the other hand, Solitons play a dynamic role in the field of engineering applications and, nonlinear science and it delivers more perception into the related nonlinear scientific occurrences by leading to forthcoming scientific research. So, to check the validity and effectiveness of our method we have implemented the extended (G'/G) expansion method to the (2+1) dimensional breaking soliton equation. The outcomes we have found here, are more common, successfully recovered the most of the earlier recognized results which have been established by other sophisticated methods. We have found some new results as well which will lead us to study some new phenomena in future. We have stated the travelling wave solutions here by three types of family. They are the hyperbolic family, the trigonometric family and, the rational family. The results along with the graphical illustration have revealed the high productivity of this algorithm with trustworthiness.
Open Access
Travelling wave and non-travelling wave solutions of nonlinear evolution equation via new generalized and Improved (G'/G) - expansion method
(BRAC University, 3/1/2018) Khan, Anika Tahsin; Naher, Dr. Hasibun; Department of Mathematics and Natural Sciences
In this thesis, we have applied the new extension of the generalized and improved ( G' / G) - expansion method in order to find the explicit solutions of non-travelling and travelling wave solutions of Fisher equation. Many new general and abundant solutions are obtained and they have been described in five different families in terms of hyperbolic functions, trigonometric functions and rational functions, involving many new and real parameters. It also shows that the method which has been applied is direct, concise and an effective tool for solving any kinds of nonlinear evolution equations, which arises frequently in applied mathematics, mathematical physics, engineering and in many scientific real time application. Furthermore we have presented the graphical solutions of the obtained solutions by using the computational software Maple.

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