Department of Applied Mathematics and Physics (Dept. AMP)

Transcription

1 数理工学専攻 Department of Applied Mathematics and Physics (Dept. AMP) 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.)

2 8 groups: Current State of the Department 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 master course students each year 4-6 doctor course students each year

3 Research (keywords): Research and Education 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.

4 Group 1: Applied Mathematical Analysis Computer Science New Singular Value Decomposition Algorithms faster computing of singular value decomposition Mathematics Discrete Integrable systems Discrete and ultradiscrete system Orthogonal polynomials. Special functions Enumerative combinatorics. Graph 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.

5 Group 2: Discrete Mathematics 1. 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. Construction of Theoretical Foundation for New and Efficient Algorithms Discrete Mathematics, Graph Theory, Optimization Theory, Complexity Theory, Data Structure, Algorithm Design Network Design Scheduling Graph Drawing Inf. Visualization 2D, 3D Packing Polynomial Time Algorithms Approximation Algorithms Branch-and-Bound Method 3. Construction of Solver Systems: Formulate new mathematical models, and construct a solver system by integrating related algorithms effectively. Chemical Graphs H H C H C O Metaheuristics H

6 Packing and Related Problems 2D & 3D objects, container Sphere Approximation Nonlinear Optimization Metaheuristics Non-overlap layout 2D Irregular Strip Packing 3D Molecule Packing Road Label Layout

7 Group 3: System Optimization Research interests Application and Theory of Optimization There are a lot of objectives which we want to optimize. Finance Transportation Physical system People want to maximize (optimize) the profit. People try to minimize (optimize) the time or distance to the destination. The nature tends to minimize (optimize) the potential energy of materials. How can we find the optimal solution?

8 Group 4: Control Systems Theory New paradigm of Control theory constraint of channel capacity and data compression convex optimization and control theory behavior approaches Analysis and design of control system constrained control networked control hybrid system System Identification (Modeling) State-space representations differential equations transfer functions

9 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 Noise Output Data-processing Model Data-processing

10 Group 5: Physical Statistics 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.

11 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.

12 Group 6: Dynamical Systems Theory 1. Geometric Mechanics: classical and quantum 2. Hamiltonian Dynamics with computer simulation 3. Geometric model of quantum computation 4. Applications of differential geometry

13 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.

14 Group 7: Mathematical Finance Mathematical modeling/analysis of financial markets. Stochastic models are intensively used because of randomness in real financial market. Some Historical References: Nobel Prize in Economics: Markowitz, Miller, Sharpe (1990), (Black), Merton, Scholes (1997) Gauss Prize (by IMU, 2006) Ito

15 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)

16 Group 8: Applied Mathematical Modeling (Operated Jointly with Industry) Conceptual Model Visit Shop p Buy Agent i Agent j a y( t i) + b u( t i j y( t) = j) Numerical Model 3.5% 3.0% 2.5% Read Bulletin Board q Contributor 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. 2.0% 1.5% 1.0% 0.5% 0.0% -0.5% -1.0% -1.5% 02/11/5 02/11/12 02/11/19 02/11/26 02/12/3 02/12/10 02/12/17 02/12/24 02/12/31 03/1/7 03/1/14 03/1/21