### 2016　Mathematical Modeling

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Undergraduate major in Mathematical and Computing Science
Instructor(s)
Takayasu Misako  Kabashima Yoshiyuki  Kanazawa Kiyoshi
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
Mon7-8(W834)  Thr7-8(W834)
Group
-
Course number
MCS.T315
Credits
2
2016
Offered quarter
2Q
Syllabus updated
2016/4/27
Lecture notes updated
-
Language used
Japanese
Access Index ### Course description and aims

For understanding uncertain and/or complex phenomena that are difficult to describe from the first principle, it is particularly important to appropriately formulate them as mathematical problems. This task is often termed "modeling". This course shows elemental mathematical techniques that are necessary for the modeling, illustrating representative probabilistic and/or nonlinear phenomena.

### Student learning outcomes

The goal is to acquire basic skills for engaging in advanced modeling of more complex phenomena by learning elemental mathematical models for probabilistic and/or nolinear phenomena.

### Keywords

Random variable, probability distribution, statistic, diffusion phenomena, Brownian motion, branching process, aggregation process, nonlinear phenomena, neural network model

### Competencies that will be developed

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

### Class flow

For each topic, we first show concrete target phenomena, and introduce how they are formulated as mathematical problems.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Introduction to modeling General introduction to observation, modeling, analysis and evaluation
Class 2 Observation of phenomena 1 Basic distributions and corresponding mathematical models
Class 3 Observation of phenomena 2 Basic statistical quantities and corresponding mathematical models
Class 4 Observation of phenomena 3 Stationarity, reversibility and irreversibility
Class 5 Modeling of diffusion phenomena 1 Macroscopic irreversibility of diffusion phenomenon, and derivation of diffusion equation
Class 6 Modeling of diffusion phenomena 2 Microscopic view of diffusion phenomenon and Brownian motion
Class 7 Modeling of diffusion phenomena 3 Markov process and mathematical description of Brownian motion
Class 8 Modeling of diffusion phenomena 4 Examples of diffusion phenomena
Class 9 Modeling of branching and aggregation phenomena 1 Branching process and its modeling
Class 10 Modeling of branching and aggregation phenomena2 Aggregation process and its modeling
Class 11 Modeling of nonlinear phenomena 1 Description by ordinary differential equation. Linear stability analysis around fixed points.
Class 12 Modeling of nonlinear phenomena 2 Nonlinear oscillators and Hopf bifurcation
Class 13 Modeling of nonlinear phenomena 3 Coupled oscillators. Understanding of mechanism of synchronization using coupled phase oscillators.
Class 14 Modeling of neural networks 1 Classification of neural network models. Mathematical characterization of abilities and limitations of perceptron.
Class 15 Modeling of neural networks 2 Auto associative memory model. Understanding of memory retrieval using the notion of Lyapunov function.

None.

### Reference books, course materials, etc.

To be distributed electronically when needed.

### Assessment criteria and methods

Students' understanding will be assessed by final exam.

### Related courses

• MCS.T211 ： Applied Calculus
• MCS.T203 ： Linear Algebra and Its Applications
• MCS.T223 ： Mathematical Statistics
• MCS.T212 ： Fundamentals of Probability

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

Basic knowledge and skills about linear algebra, calculus, probability theory, and statistics are required. 