### 2017　Mathematics of Discrete Systems

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Instructor(s)
Nakamura Kiyohiko  Watanabe Sumio  Kabashima Yoshiyuki  Murofushi Toshiaki  Takayasu Misako
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
Tue5-6(G221)  Fri5-6(G221)
Group
-
Course number
ART.T451
Credits
2
2017
Offered quarter
2Q
Syllabus updated
2017/3/17
Lecture notes updated
-
Language used
English
Access Index ### Course description and aims

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.

### Student learning outcomes

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.

### Keywords

Logic, set, probability, estimation, Monte Carlo methods, Markov chain, timesSeries analysis, random walk, graph, network

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills

### Class flow

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

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

none

### Reference books, course materials, etc.

S. Lipschutz & M. Lipson, “Schaum's Outline of Theory and Problems of Discrete Mathematics”, Revised 3rd ed., McGraw-Hill: ISBN 9780071615860

### Assessment criteria and methods

Students are assessed on exercises (84%) in class and examination (16%).

### Related courses

• ART.T455 ： Modeling of Discrete Systems

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

none 