This course gives an overview of 1) sets and logics and 2) probability and statistics, as an introduction to discrete systems followed by 3) Monte Carlo method and related issues and 4) basic concepts and applications of time series analysis. Finally, it addresses 5) graphs and networks. This course aims to teach students mathematical basis and typical methods to construct and analyze models of discrete systems.
By the end of this course, students will be able to: 1) Describe discrete systems mathematically and 2) Analyze and design discrete systems by using computers.
Logic, set, probability, estimation, Monte Carlo methods, Markov chain, timesSeries analysis, random walk, graph, network
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
In every class, the instructor will explain the topics specified in the course schedule using practical examples. Students are given exercise problems related to the lecture given that day to solve. To prepare for the class, students should read the course schedule section and check what topics will be covered. Required learnings should be completed outside of the classroom for preparation and review purposes.
|Course schedule||Required learning|
|Class 1||Propositional logic: proposition, logical formula, truth table, logical laws||Understand the contents covered by the lecture.|
|Class 2||Predicate logic: predicate, quantified proposition, multiply quantified proposition, restricted quantification, unique existential quantification||Understand the contents covered by the lecture.|
|Class 3||Sets and mappings: set, set operations, Cartesian product, power set, family of sets, mapping, surjection, injection, bijection, inverse mapping, composite mapping||Understand the contents covered by the lecture.|
|Class 4||Probability space and random variable: Definitions of probability space and random variable. convergence of sequence of random variables||probability space, random variable, and convergence.|
|Class 5||Limit Theorems: The law of large numbers, central limit theorem, and semi-circle law||Applications of several laws|
|Class 6||Statistical Estimation: Bayes, maximum likelihood, and evaluation||Bayesian and Maximum likelihood estimation and their evaluation|
|Class 7||Monte Carlo Methods||The problems to be solved, uniform sampling, importance sampling, rejection sampling|
|Class 8||Markov Chain Monte Carlo (MCMC) methods||Markov chain and stationary distribution, detailed balance, Metropolis method, Gibbs sampling|
|Class 9||Extensions of MCMC methods||Simulated annealing, replica exchange method, thermodynamic integration|
|Class 10||Time Series Analysis||Stationary and Non-Stationary Processes, Wiener–Khinchin Theorem, Autoregressive Models|
|Class 11||Introduction to Random Walk||To be annouced in class|
|Class 12||Application of Random Walk||To be annouced in class|
|Class 13||Graph theory: directed graphs, order relations, adjacency matrix||To be annouced in class|
|Class 14||Weighted graphs and the shortest paths||To be annouced in class|
|Class 15||Flow networks and maximum flow||To be annouced in class|
S. Lipschutz & M. Lipson, “Schaum's Outline of Theory and Problems of Discrete Mathematics”, Revised 3rd ed., McGraw-Hill: ISBN 9780071615860
Students are assessed on exercises (84%) in class and examination (16%).