https://scholarworks.unist.ac.kr/handle/201301/59 Wed, 08 Apr 2026 00:26:44 GMT 2026-04-08T00:26:44Z https://scholarworks.unist.ac.kr/handle/201301/88190 Title: Singular Layer Physics-Informed Neural Networks for Stiff Reaction-Diffusion Equations on Smooth Convex Domains Author(s): Ngon, Eaint Phoo Abstract: In this thesis, we develop and apply a novel machine learning approach, the Singular Layer Physics- Informed Neural Networks (sl-PINN), for singularly perturbed elliptic boundary value problems. These problems, which arise in the analysis of reaction-reaction systems, are challenging to approximate nu- merically due to the boundary layers and sharp transitions. We address these challenges by adding boundary layer correctors into the physics-informed neural networks framework, improving its ability to approximate solutions in smooth domains with curved boundaries. Our method is shown to effec- tively capture the solutions, offering a significant improvement over traditional numerical methods. The effectiveness of sl-PINN is validated through numerical experiments, in which it accurately solves stiff reaction-diffusion equations with high computational efficiency. Major: Department of Mathematical Sciences Thu, 31 Jul 2025 15:00:00 GMT https://scholarworks.unist.ac.kr/handle/201301/88190 2025-07-31T15:00:00Z https://scholarworks.unist.ac.kr/handle/201301/88189 Title: The Behavior of Euler–Kronecker Constants of Quadratic Fields Author(s): Kim, Somin Abstract: This thesis is a review of the work by Professor Lamzouri on the distribution of Euler–Kronecker con- stants associated with quadratic fields [5]. To study the behavior of the Euler–Kronecker constant gamma_Q(sqrtD), we construct a probabilistic random model gamma_rand(X) that reflects its key characteristics. We analyze the large moments of gamma_Q(sqrt D) and compute the Laplace transforms of both gamma_Q( sqrtD) and gamma_rand(X). Through these tools, we examine the distribution of the random model and, in turn, gain insights into the distribution of the actual constants over quadratic fields. Major: Department of Mathematical Sciences Thu, 31 Jul 2025 15:00:00 GMT https://scholarworks.unist.ac.kr/handle/201301/88189 2025-07-31T15:00:00Z https://scholarworks.unist.ac.kr/handle/201301/86519 Title: A Central Limit Theorem Related to Continued Fraction Expansions of Two Close Real Numbers Author(s): Lee, Jong-Dae Abstract: This thesis generalizes the central limit theorem initially proposed by C. Faivre, which pertains to continued fraction expansions. For a given real number x in the interval [0,1), the quantity kn(x) represents the maximum number of consecutive matching digits in the continued fraction expansions of two numbers y and z, where y and z are the lower and upper decimal approximations of x at the n-th decimal place, respectively. Faivre demonstrated that the appropriately normalized distribution of kn(x) converges to the standard normal distribution as n goes to infinity. Building on this, the study introduces kε(x), defined as the maximum number of matching digits in the continued fraction expansions of x and x + ε , where ε > 0 and ε → 0. The thesis establishes that the normalized distribution of kε(x) adheres to the central limit theorem, thereby generalizing Faivre’s results. Crucially, this work emphasizes that the analysis does not depend on comparing different ex- pansion systems; instead, the focus lies on the asymptotic behavior of two points x and x + ε as they converge arbitrarily close to each other. Major: Department of Mathematical Sciences Fri, 31 Jan 2025 15:00:00 GMT https://scholarworks.unist.ac.kr/handle/201301/86519 2025-01-31T15:00:00Z https://scholarworks.unist.ac.kr/handle/201301/86366 Title: Wave Breaking and Blow-up Profile in the Camassa-Holm equation Author(s): Yoon, Jeongsik Major: Department of Mathematical Sciences Fri, 31 Jan 2025 15:00:00 GMT https://scholarworks.unist.ac.kr/handle/201301/86366 2025-01-31T15:00:00Z