2022 Mathematical Modeling

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Academic unit or major
Undergraduate major in Mathematical and Computing Science
Takayasu Misako  Takayasu Hideki 
Class Format
Lecture    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Mon7-8(W834)  Thr7-8(W834)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
Language used
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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 elementary 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 elementary mathematical models for probabilistic and/or nolinear dynamical phenomena.


Random variable, probability distribution, correlation, diffusion phenomena, Brownian motion, branching process, phase transition, transport phenomena, complex network

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills
Learn the basics of mathematical modeling of unknown phenomena

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 and basic models 1 Basic distributions such as exponential distribution and the normal distributions and corresponding mathematical models
Class 3 Observation of phenomena and basic models 2 Power law distributions and corresponding mathematical models
Class 4 Observation of phenomena and basic models 3 Nonlinear dynamics and corresponding mathematical models
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: Application 1 Models of financial time series are introduced as an application of diffusion
Class 8 Modeling of diffusion phenomena: Application 2 Microscopic agent-based models of financial markets are introduced as an application of diffusion
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 phase transition phenomena 1 Basic models of phase transition, basic properties and theoretical solutions
Class 12 Modeling of phase transition phenomena 2 Transport phenomena and congestion phase transition, and related models
Class 13 Modeling of phase transition phenomena 3 Self-organized criticality and related models
Class 14 Modeling of complex networks Complex networks and related models

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.



Reference books, course materials, etc.

To be distributed electronically when needed.

Assessment criteria and methods

Students' understanding will be assessed by reports.

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.

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