Contents

General Information

References

Prerequisite

  • Real and complex analysis

Homeworks

  • Homework 1: Return by September 19, 2025 (Friday) 23:59
  • Homework 2: Return by September 26, 2025 (Friday) 23:59
  • Homework 3: Return by October 10, 2025 (Friday) 23:59
  • Homework 4: Return by October 17, 2025 (Friday) 23:59
  • Homework 5: Return by November 14, 2025 (Friday) 23:59

Schedule

  • The lectures are on Friday (09:10-12:00) at 志希070116.
Time Room Activities
5.9.2025 09:10-12:00 志希070116 Week 1: Fundamental theorem of ODE
12.9.2025 09:10-12:00 志希070116 Week 2: Some techniques for solving ODE, from ODE to PDE, Linear equations
19.9.2025 09:10-12:00 志希070116 Week 3: Homogeneous ODE with constant coefficients, fundamental matrix solution (Return Homework 1 by 23:59)
26.9.2025 09:10-12:00 志希070116 Week 4: Computations of the matrix exponential \(\exp : \mathbb{C}^{n\times n} \rightarrow {\rm GL}(n,\mathbb{C})\) (Return Homework 2 by 23:59)
3.10.2025 09:10-12:00 志希070116 Week 5: matrix logarithm, one parameter subgroup, 1-1 correspondence between matrix Lie group \(\mathscr{G}={\rm GL}(n,\mathbb{C}),{\rm U}(n,\mathbb{C}),{\rm SL}(n,\mathbb{C}),{\rm SU}(n,\mathbb{C})\) and Lie algebra \(\mathfrak{g}=\mathfrak{gl}(n,\mathbb{C}),\mathfrak{u}(n,\mathbb{C}),\mathfrak{sl}(n,\mathbb{C}),\mathfrak{su}(n,\mathbb{C})\)
10.10.2025 09:10-12:00 志希070116 Week 6 - No class: national day (Return Homework 3 by 23:59)
17.10.2025 09:10-12:00 志希070116 Week 7: ODE with variable coefficients (Return Homework 4 by 23:59)
click me to see the title and abstract of today's first talk (30 minutes)
Speaker. Wu, Pei-Wei
Title. Curvature 曲率
Abstract. 介紹何謂高斯曲率以及如何推導。
24.10.2025 09:10-12:00 志希070116 Week 8 - No class: Retrocession Day
31.10.2025 09:10-12:00 志希070116 Week 9: Higher order linear ODE, Strum-Liouville eigenvalue problem
click me to see the title and abstract of today's first talk (30 minutes)
Speaker. Kao, An-Hsien
Title. Krylov Subspace and Arnoldi
Abstract. This presentation introduces the Krylov subspace method, focusing on how the Arnoldi iteration is applied to solve eigenvalue problems for large, sparse matrices efficiently.
7.11.2025 09:10-12:00 志希070116 Week 10: Constant coefficient transport equation \(\partial_{t}u + c\partial_{x}u=0\) and its discretization, 1-dimensional wave equation \(\partial_{t}^{2}u - c^{2}\partial_{x}^{2}u=0\) on the whole line \(\mathbb{R}\)
click me to see the title and abstract of today's first talk (30 minutes)
Speaker. Song, Jia-Ying
Title. Introduction of Gradient Descent and Backpropagation
Abstract. The first part of this report discusses the concept and formulas of gradient descent, while the second part focuses on how backpropagation was derived and its connection to the use of gradient descent in the early development of machine learning.
14.11.2025 09:10-12:00 志希070116 Week 11 (Return Homework 5 by 23:59)
click me to see the title and abstract of today's first talk (30 minutes)
Speaker. Lin, Pei-Syuan
Title. Cantor set and its properties
Abstract. This presentation will cover the construction of the Cantor Set and proceed to introduce and prove its five properties: compact, nowhere dense, uncountability, measure zero, and perfect set.

click me to see the title and abstract of today's second talk (30 minutes)
Speaker. Lin, Pei-Syuan
Title. Introduction to the Cantor-Lebesgue Function
Abstract. This presentation constructs the Cantor-Lebesgue function and shows that it is continuous, monotone increasing, and has derivative zero almost everywhere. Furthermore, it is demonstrated that the function is singular and not absolutely continuous.
21.11.2025 09:10-12:00 志希070116 Week 12: reflection method, Duhamel's principle
28.11.2025 09:10-12:00 志希070116 Week 13: The auxiliary function \(M_h(x,r):=\frac{1}{|\partial B_{r}(x)|}\int_{\partial B_{r}(x)}h(y)\,\mathrm{d}y\) for all \(x\in\mathbb{R}^{n}\) and \(r>0\)
click me to see the title and abstract of today's first talk (30 minutes)
Speaker. Wang, Hsing-Yu
Title. Lagrangian mechanics
Abstract. A mechanical analysis different from Newtonian mechanics and the comparison of some systems analyzed by these two methods.
5.12.2025 09:10-12:00 志希070116 Week 14
click me to see the title and abstract of today's first talk (30 minutes)
Speaker. Hung, Chun Wei
Title. Hausdorff measure and Hausdorff dimension
Abstract. This presentation will construct the Hausdorff measure, introduce its properties, and present the concept of Hausdorff dimension.

click me to see the title and abstract of today's second talk (30 minutes)
Speaker. Hung, Chun Wei
Title. Calculate the dimension
Abstract. This presentation will introduce the box-counting dimension and self-similarity as tools for estimating and computing the Hausdorff dimension.

click me to see the title and abstract of today's third talk (30 minutes)
Speaker. Chan, Yung-Hsiang
Title. Laplace equation
Abstract. This presentation will Introduce Laplace equation and solve it using separation of variables.
12.12.2025 09:10-12:00 志希070116 Week 15
click me to see the title and abstract of today's first talk (30 minutes)
Speaker. Wu, Pei-Wei
Title. RSA Cryptography
Abstract. This presentation will introduce RSA Cryptography and its algorithm.

click me to see the title and abstract of today's second talk (30 minutes)
Speaker. Kao, An-Hsien
Title. Low Rank Approximation by Krylov Subspace Method
Abstract. This presentation reports our progress on extending the Golub–Kahan bidiagonalization from vector spaces to function spaces, which allows us to automatically build adaptive basis functions without using pre-selected bases like Fourier bases. We apply this method to the Herglotz wave function operator to solve ill-posed inverse problems.
19.12.2025 09:10-12:00 志希070116 Week 16
click me to see the title and abstract of today's first talk (30 minutes)
Speaker. Wang, Hsing-Yu
Title. \(n\)-dimensional balls and gamma function
Abstract. An introduction to volume and surface area of an \(n\)-dimensional ball, the definition and some properties of gamma function.

Completion

  • The course can be taken for credit by attending the lectures, returning written solutions (60%) in \(\LaTeX\) and giving (at least) 2 presentations (each 20%).