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.
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
|Intercultural skills||Communication skills||✔ Specialist skills||✔ Critical thinking skills||✔ Practical and/or problem-solving skills|
For each topic, we first show concrete target phenomena, and introduce how they are formulated as mathematical problems.
|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 3||Models of price fluctuations in financial markets are introduced as an application of diffusion|
|Class 8||Modeling of diffusion phenomena 4||Microscopic agent-based models of price fluctuations in 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|
|Class 13||Modeling of phase transition phenomena 3||Self-organized criticality and related models|
|Class 14||Modeling of complex networks 1||Complex Networks, Adjacency matrix, characterization of networks|
|Class 15||Modeling of complex networks 2||Complex networks of business firms: modeling and simulation|
To be distributed electronically when needed.
Students' understanding will be assessed by final exam.
Basic knowledge and skills about linear algebra, calculus, probability theory, and statistics are required.