Honors Mathematics A-E2
| Numbering Code | U-LAS10 20017 LE55 | Year/Term | 2022 ・ Second semester |
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| Number of Credits | 2 | Course Type | Lecture |
| Target Year | Mainly 1st year students | Target Student | For science students |
| Language | English | Day/Period | Tue.3 |
| Instructor name | Karel SVADLENKA (Graduate School of Science Associate Professor) | ||
| Outline and Purpose of the Course |
This course provides opportunities to learn mathematics in more depth for highly motivated students. It supplements Calculus A and Linear Algebra A, and takes these basic courses as starting point to treat more advanced related topics. |
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| Course Goals | In addition to learning modern mathematics and proofs, students can learn how to discuss and present mathematical topics in English through this course. | ||
| Schedule and Contents |
Below is a list of themes that may be covered. The actual topics of the lecture will be determined upon investigating the interests and level of the participating students. The selected topics will be covered during 15 lectures, including one feedback session. 1. Topics in set theory (tentatively 5~9 weeks) 1.1 Equivalence relations, quotients and order relations 1.2 Axioms of Zermelo-Fraenkel set theory 1.3 Cantor-Schroeder-Bernstein theorem 1.4 Cantor's diagonal argument 2.Fundamental theory of real numbers (tentatively 3~6 weeks) 2.1 Peano’s axioms 2.2 Construction of Z, Q and R 3. Convex analysis (tentatively 2~5 weeks) 3.1 Convex sets and their properties 3.2 Convex functions and their properties 4. Topics in Hilbert spaces (tentatively 2~5 weeks) 4.1 Inner product spaces 4.2 Riesz representation theorem 4.3 Self-adjoint and normal operators 4.4 The spectral theorem 5. Numerical linear algebra (tentatively 2~5 weeks) 5.1 Singular decomposition of a matrix, least square approximation, QR decomposition 5.2 Diagonally dominant matrices 5.3 Basic iterative methods for linear systems 5.4 Conjugate gradient method |
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| Evaluation Methods and Policy |
The evaluation of the course will take into account the following criteria: (1) homework and presentation of students during the lectures (40%) (2) midterm and final examination (60%) However, according to the situation, the evaluation may be based only on (1) homework and presentation of students during the lectures (100%) . The method of evaluation will be made precise at the first lecture. |
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| Course Requirements | Calculus A and Linear Algebra A. Students are strongly encouraged to take Calculus B and Linear Algebra B in parallel (or prior) to this course. | ||
| Study outside of Class (preparation and review) |
As in every math course, students should read notes carefully and repeatedly after the class, solve exercise problems and try to find alternative proofs, counterexamples, etc. After many hours of such practice you may get an intuitive understanding of the materials covered. |
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| References, etc. |
Naive set theory, Paul R. Halmos, (Springer, 1974), ISBN:978-0-387-90092-6, e-bookあり
Other references will be announced during the class according to the selected topics. |
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