Discrete Mathematics APM_1S003_EP (Year 1) explores the world of discrete mathematics which is a fundamental concept in many different areas of science and advanced mathematics. The first part of the course introduces basic notions in number and groups theory, and goes on with the study of the group of permutations. The second part of the course is devoted to probability theory on finite sets, the basics of graph theory and an introduction to Markov chains on finite spaces.
The aim of this course is to provide students with a working knowledge of basic mathematical algorithms and associated computer programming. We will cover several notions such as representation of numbers, rootfinding, polynomial approximation, numerical integration, and error analysis. A significant portion of the course will be devoted to implementation and experiments using Jupyter Notebooks with Python.
Grading
The grading will be based on the following elements:
- Very short tests (approx. 5min) at the beginning of the second lecture of each chapter, to ensure that you remember the main notions introduced during the first lecture of the chapter.
- A final exam (2h), during which you will have to complete a notebook with both theoretical answers (Markdown and Latex), code (Python), and illustrations. Authorized material during the exam: the notebooks used in the course that are provided on Moodle, as well as personnal class/lab notes.
- A final exam (2h), during which you will have to complete a notebook with both theoretical answers (Markdown and Latex), code (Python), and illustrations. Authorized material during the exam: the notebooks used in the course that are provided on Moodle, as well as personnal class/lab notes.
The final grade will be computed as follows:
max(final exam, 2/3*final exam + 1/3*short tests).
Prerequisites: MAA101, MAA102
The course.
The course aims both at making you discover mathematical modeling and how mathematics can bring some interesting answers. At this occasion, you will discover some mathematics concepts/techniques/tools motivated by questions arising from the problems coming from the «real world »
The lecture.
Every two weeks, a new lecture is done (on Friday). Each lecture will deal with a new situation of modeling and introduce some mathematical notion or techniques to study the model. Some of the topics will see are:
- The prediction problem and optimization of regular functions
- Connection sequences, differential equations and optimizations
- The problem of controlling a dynamical system and differential equations
- Best time to stop a game
The exercises sessions (TD)/challenges.
Every two weeks (the one when there is no lecture), you will have a TD. You should think about the challenge/exercise before TD. You can try to solve it partially but do not need to write something or search all questions. The TDs will be devoted to discussion about the challenge and correction of some questions.
Methodology for the grade
Grade= Max (F, 50% F + 50% Project) where
F= 50% Exam + 50% Participation
Project
It is a project which consists at picking a subject that leads to a mathematical modeling. In no more then 7 pages, you will describe and explain how to solve the problem by using some mathematical tools. You can also draw some graphics for illustration. A short defense could be organized if necessary.
Participation
The teacher in charge of the TDs will give you a grade regarding your involvement during the exercise sessions.
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