Foundation In 1950s, as a subdivision of engineering with

much emphasis on applications. Dept. AMP was founded in 1959 to study interdisciplinary & traversal areas, basics & fundamentals. (rich background in math. & phys.)

Department of Applied Mathematicsand Physics (Dept. AMP)

数理工学専攻

8 groups: Applied Mathematical Analysis Discrete Mathematics System Optimization Control Systems Theory Physical Statistics Dynamical Systems Theory Mathematical Finance (Instutute of Economic Research) Applied Mathematical Modeling (Operated jointly with Industry)

22 faculty members 22-26 master course students each year 4-6 doctor course students each year

Current State of the Department

Research and Education

Research (keywords):Control, Optimization, Discrete mathematics, Dynamical system, Computational science, Simulations, Finance, Econo-physics, ...

Educational aims:Flexible conception and high competence for searching solutions with profound attainments in mathematics and physics and computer sciences.

Group 1: Applied Mathematical Analysis

New Singular Value Decomposition Algorithms

faster computing of singular value decompositionComputer Science

Discrete and ultradiscrete systemOrthogonal polynomials. Special functionsEnumerative combinatorics. Graph

Mathematics

Discrete Integrable systems

Research subjects are integrable systems and their applications to engineering science. Based on the theory of the discrete integrable systems, we are capable of developing new numerical algorithms.

Construction of Theoretical Foundation for New and Efficient Algorithms

Discrete Mathematics, Graph Theory, Optimization Theory, Complexity Theory, Data Structure, Algorithm Design

Chemical Graphs

2D, 3D Packing

Scheduling

Graph DrawingInf. Visualization

Network Design

CC

H H

H

OH

Polynomial TimeAlgorithms

Branch-and-Bound Method

ApproximationAlgorithms

Metaheuristics

Group 2: Discrete Mathematics1. Establishment of Theoretical Foundation: Develop the theoretical foundation on optimization theory and complexity theory by applying the results in graph theory and discrete mathematics.2. Design and Analysis of Algorithms: Design new efficient algorithms under the algorithm frameworks suitably selected according to the complexity hardness and required solution quality.

3. Construction of Solver Systems: Formulate new mathematical models,and construct a solver system by integrating related algorithms effectively .

2D Irregular Strip Packing

3D Molecule Packing Road Label Layout

Packing and Related Problems

2D & 3D objects, container

Non-overlap layout

Sphere Approximation Nonlinear Optimization Metaheuristics

There are a lot of objectives which we want to optimize.

Finance People want to maximize (optimize) the profit.

Transportation People try to minimize (optimize)the time or distance to the destination.

Physical system The nature tends to minimize (optimize)the potential energy of materials.

How can we find the optimal solution?

“Application” and “Theory “of Optimization

Research interests

Group 3: System Optimization

Group 4: Control Systems Theory

New paradigm of Control theoryconstraint of channel capacity and data compression

convex optimization and control theorybehavior approaches

Analysis and design of control systemconstrained controlnetworked control

hybrid system

System Identification(Modeling)

State-space representationsdifferential equations

transfer functions

Modeling and System Identification

State-space representations, differential equations, and transfer functions are models of dynamical systems. In this study, we are interested in deriving dynamical models from input-output (or output) data. Techniques such as prediction error methods, subspace methods, and stochastic realization are primal tools. Recent study includes modeling of continuous-time systems, time varying systems, and nonlinear systems.

Input

System

Model

NoiseOutput

Data-processingData-processing

A conceptual illustration of a multi-element coupled system. Each unit influences one another with different coupling strength, which is described as a weighted undirected arrow. The causality of effect is represented as a blue weighted arrow.

Group 5: Physical Statistics

A conceptual illustration of a scale free network, a kind of complex network. A black filled circle represents a node and a curve between two nodes a link. Colored nodes (blue, red, and green one), which are called hubs, have a lot of neighbors. Such a network, where a few nodes have a lot of neighbors and most nodes have a few neighbors emerges frequently in socio-economic and biological systems, and ecosystems.

Group 6: Dynamical Systems Theory

1. Geometric Mechanics: classical and quantum2. Hamiltonian Dynamics with computer simulation3. Geometric model of quantum computation4. Applications of differential geometry

Model of falling cat

The falling cat can be modeled by jointed two cylinders with two torques as control inputs under the condition of the vanishing total angular momentum.

Mathematical modeling/analysis of financial markets. Stochastic models are intensively used because of “randomness” in real financial market.

Group 7: Mathematical Finance

Some Historical References:Nobel Prize in Economics:

Markowitz, Miller, Sharpe (1990), (Black), Merton, Scholes (1997)

Gauss Prize (by IMU, 2006)Ito

Examples:Black-Scholes Stock Price model:

dS(t)= S(t) { μdt + σdW(t) }stochastic differential equation(W: Brownian motion, source of

randomness)

Derivatives pricing/hedging:A stochastic control problem* PDE approach* Probabilistic approach

(+ stochastic numerics)

Group 8: Applied Mathematical Modeling(Operated Jointly with Industry)

Information systems valued for our better life and superior productivity shall be equipped with mathematical models describing dynamical behavior of human and objects in the systems. Ranging from conceptual to precise numerical forms, modeling technology is studied including utilization of expert knowledge (structural modeling) as well as observed data (multivariate analysis) with practical industrial case studies.

■Conceptual Model

■Numerical Model∑ ∑ −+−= )()()( jtubityaty ji

Agent i

Agent j

Shop p BulletinBoard q

Visit

Buy Contributor

Read

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

02/11/5

02/11/1

2

02/11/1

9

02/11/2

6

02/12/3

02/12/1

0

02/12/1

7

02/12/2

4

02/12/3

1

03/1/7

03/1/14

03/1/21