Applied Mathematics and Physics Course

Using Applied Mathematics and Physics to
Seek Solutions to Problems in Engineering/
Natural Systems
In the highly advanced information society of today, we encounter
various situations that involve modeling and analysis of complex andlarge-scale systems, their control, design, and operation.
In these situations, it is extremely important to uncover common
mathematical structures shared by those problems which are seemingly
unrelated, and to develop mathematical methods to solve them, in
addition to acquiring specialized knowledge of individual disciplines suchas information technology, electricity, mechanics, and chemistry.
From this viewpoint, the seven laboratories of the Course of Applied
Mathematics and Physics undertake leading-edge research on appliedmathematical analysis, discrete mathematics, system optimization,
control systems theory, applied mathematical modeling (adjunct unit),
physical statistics, and dynamical system theory.

35

Applied Mathematics and Physics Course

Universality of the Power law andEstablishing Risk Indicators
The discovery and application of a super generalized central limit theorem
Why are power laws so prevalent in nature and the universe? That is the mathematicalquestion that sparked this journey. Power laws, which are different to Gaussian
distributions, can be asymmetric and remarkably diverse. The mathematical basis for such
power laws can be traced back to a generalization of the central limit theorem, stating thatthe sum of random variables converges to a Gaussian distribution. While examining thismathematical basis, we constructed an algorithm to accurately estimate power laws from themassive amounts of data about the world that we now have access to. This has led us todevelop new data analysis methodology for identifying real-time risk indicators for financial
markets. As well as in financial markets, power laws also appear
in many aspects, including physical phenomena in the universe.
A power law is a common mathematical concept that sheds lightin fields as wide-ranging as statistical physics, probability theory,chaos theory, cosmology, financial markets, neuroscience, andnumber theory (Riemann zeta function). Join us on this voyage
of discovery! It has only just begun.

Ken Umeno

Professor, Applied Mathematics and Physics Course

Mathematics and Computers for Solving
Problems through Optimization

Applied Mathematics and Physics Course

Professor Umeno received his Doctoral Degree March 1995 from the University of Tokyo. In
April 1995, he was a special postdoctoral researcher at the Institute of Physical and Chemical
Research (RIKEN). From 1998 until he joined Kyoto University in 2012, he worked for Japan’ s
Ministry of Posts and Telecommunications (currently the National Institute of Information andCommunications Technology). From 2005 to 2010, he was the Laboratory Head of the Next
Generation Mobile Laboratory of RIKEN. He has been a Professor with Graduate School ofInformatics, Kyoto University, since 2012. His research specialty is chaos theory, complex
systems, and statistical mechanics. His recent invention is correlation analysis of GNSS datafor detecting ionospheric Precursors before large earthquakes and is currently investigating inuncovering the physical mechanism to generate electro-magnetic anomalies. Together
earthquake, he is also interested in physically inspired model of Financial Crisis as criticalphenomena of phase transition in statistical mechanics.

As the concept of “optimization” has become increasingly familiar to people, technicalterms such as “optimization” and “optimal solution” have made their way into everyday
language. In the field of applied mathematics, we first describe “everyday optimization” as an“mathematical optimization model” that enables computers to deal with. Then we develop
algorithms to find optimal solutions of such optimization models with large-scale andcomplex real-world applications.
Our laboratory deals mainly with continuous optimization, in which the candidate
optimal solutions are expressed as continuous variables.
Continuous optimization is an essential tool for deep learningand financial engineering. Why not come along and takeadvantage of the power of applied mathematics to help us shapea more optimal societyfor the benefit of many people?

Nobuo Yamashita

Professor, Applied Mathematics and Physics Course

Professor Yamashita received his Doctorate degree in March 1996 from Nara Institute of
Science and Technology. In April 1996 he was a research fellow at Japan Society for the
Promotion of Science. In August 1997 he was appointed an assistant professor at the Section ofApplied Mathematics and Physics, School of Engineering, Kyoto University. In April 2005 hebecame an associate professor at the department of Applied Mathematics and Physics,
Graduate School of Informatics, and In July 2014 he was promoted to professor. His research
specialty is continuous optimization, covering a broad range of fields which include, among
others, large-scale optimization, equilibrium problems, and nonlinear equations.

Graduate School of Informatics

36

Applied Mathematics and Physics Course

Outline■Group and Teaching Staff
Group

Teaching Staff

Applied Mathematical
Analysis

Satoshi Tsujimoto/Professor

Discrete Mathematics

Kazuya Haraguchi/Associate Professor

System Optimization

Nobuo Yamashita/Professor Ellen Hidemi Fukuda/Associate Professor
Hiroyuki Sato/Associate Professor Yuya Yamakawa/Assistant Professor

Control Systems Theory

Kenji Kashima/Associate Professor Kentaro Ohki/Assistant Professor

Physical Statistics

Ken Umeno/Professor Erica Uehara/Senior Lecturer Atsushi Iwasaki/Assistant Professor

Dynamical Systems

Kazuyuki Yagasaki/Professor Mitsuru Shibayama/Associate Professor
Yoshiyuki Yamaguchi/Assistant Professor

Applied Mathematical
Modeling (Adjunct Unit)

Yoichi Nonaka/Adjunct Professor
Yoshiyasu Takahashi/Adjunct Associate Professor (Hitachi Ltd.)■Curriculum of Applied Mathematics
and Physics Course
CurriculumofAppliedMathematicsandPhysicsCourseDoctoralProgram(Informatics)


rd

3 
nd

2 
st

1 


DoctoralThesis
Subjects provided by the Course (total 6 credits including 4 credits from seminars)

Seminar on Applied Mathematics and Physics, Advanced A, B, E (2 credits)
Seminar on Applied Mathematics, Advanced, E Seminar on Applied Mathematical Systems, Advanced, E
Seminar on Mathematical Physics, Advanced, E (2 credits each)ResearchGuidance

Master’sProgram(Informatics)
Master’sThesisndSubjectsprovidedbytheCourseAdvancedSubjects

SeminarSubjects

BasicSubjects(2creditseach)

RecommendedSubjectsprovidedbyother
Courses2 (Optional�12�credits�or�more�from�the�recommended�subjects�provided�by�the�other�Course.�8�or�more�credits�are�required�from�the�subjects�provided�
by�the�Course�and�“Computational�Science,�Introduction”�of�the�general�subject�provided�by�the�School.)�

- 7 -MathematicalAnalysis,Adv.,DiscreteMathematics,Adv.,ControlSystemsTheory,
Adv.,OptimizationTheory,Adv.,PhysicalStatistics,Adv.,DynamicalSystems,Adv.
IntroductiontoMathematicalFinance
(2creditseach)
FinancialEngineering,TopicsinAppliedMathematicsandPhysicsA
TopicsinAppliedMathematicsandPhysicsB (1crediteach)




st

1 

Priorto
admissionOperationsResearchAdv.,MathematicalPhysics,Adv.,SystemsAnalysis,Adv.

GeneralSubjectsprovidedbytheSchool(2creditseach) 
InterdisciplinarySubjectsofthePerspectivesinInformatics
(Mandatory2credits)
PerspectivesinInformatics1 PerspectivesinInformatics2�
PerspectivesinInformatics3E PerspectivesinInformatics4E�
PerspectivesinInformatics5E (2creditseach)



BasicMathematics

Calculus,Linearalgebra,etc.

Preferredto
understandthesubjectsontheright

SeminarinMathematicalAnalysis,Seminarin
DiscreteMathematics,SeminarinSystem
Optimization,SeminarinControlSystemsTheory
SeminarinPhysicalStatistics,SeminarinDynamical
Systems

PatternRecognitionAdv.E(IST),other11subjects

PerspectivesinPlatformStudies(2credits),ComputationalScience,Introduction(2credits),ComputationalScience,ExerciseA(1credit),
InformationandIntellectualProperty(2credits),Innovationand
Information(2credits),InformationAnalysisandManagement(2credits),InformationAnalysisandManagement,Exercise(1credit)
SocialContributionsthroughInformaticsE(1credit)Internshipinthe
FieldofInformaticsE(1credit)

AppliedMathematics

Complexfunctions,Fourieranalysis,
Numericalanalysis,Graphtheory,etc.

MathematicalSystems

Linearprogramming,Optimization,
Controltheory,etc.

Seminarsandexercises
forMaster’sThesis
(Mandatory10credits)

AdvancedStudyinSocial
Informatics2E(Assignedto
M2,5credits)
AdvancedStudyinSocial
Informatics1E�(Assignedto
M1,5credits)

Specific
subjects
providedbytheschool

MathematicalPhysics

Classicaldynamics,Differential
equation,Statisticalmechanics,etc.

Note:�Subjects�marked�with�the�letter�“E”�will�be�provided�in�English.

37AppliedMathematics and Physics Course:空箱，:玉の入った箱

図 3 “箱と玉の系” の時間発展

8

T. KATO, S. TSUJIMOTO, AND A. ZUK2 ソリトンの

この非常にシンプルな箱玉系は，離散可積分系を経由して「連続・離散・超離れる KdV 方程式と関係づく．次節以降で，Figure 1. BBS
この対応関係について解説していく．
Developing algorithms from integrable systems

Applied Mathematical Analysis

We carry out research in the areas of contemporary
soliton research and integrable system research, not only
regarding the applied analysis of orthogonal polynomials
and special functions that are closely associated withintegrable systems, but also regarding the application of
the mathematical methods developed by integrable
system studies to the solution of various problems
hitherto thought to be unrelated to integrable systems

5. BBS with carrier capacity( sLemma
u c h5.1.aThes boxnandum
r i c(BBS)a l withc acarrierl c ucapacityl a tcani obendepicted
a nbyd a l g o r i t h mball esystem
2. KdV 方程式の離散化
development). Our
Group
is
a
pioneer
ina this researchaaafield, and conducts studies into the applied analysis of
本節では，連続な時間・空間上の可積分系である
KdV 方程式
The (simple)
BBS is appeared as the limiting case of the above automaton with k → ∞.
integrable
systems in the development of algorithms and5.1. BBS with carrier capacity k  1.
の解析手法のごく簡単な紹介と，可積分性を保存する離散化おother new branches of mathematics from the perspectiveaa
いて考察する．
of computer science.
1

0

1

0

We have two operator recursions
(12)

a
a
2.1. KdV a方程式
a

0 0
0



0; k1

0

k

2



1

,

a1  a1;k1 [SatoshiTsujimoto]

0
a0

0
a1

.

We can describe theDiﬀerential
action of a 0 and a 1 on the binary sequences of length n by the matrices a 0( n ) and a 1( n ) .
From the definition of our automaton, they satisfy the following recurrence relations:

KdV(Korteweg-deVries)
方程式Waves
Shallow-water
Equation
a , + u
ut +a 6uu
0
x
xxx =


a0(0)  a0 ,

(0)
1

1


0
0
a ( n ) a1( n )  0, a1( n +1) .
a0
a0( n ) a1( n )0 Recurrence 0Computational Algorithm( n +1)は，無限遠方で十分速く u → 0 に収束するという境界条件のt, x に関する関数であり，u の添字は，各変つ．ここで，ut+1は，
11
uMn+1−1 uant =+ aδ 3+ at + −a 3 t+1u)，u4 xxx = ∂( uu/∂x．
例：ut = ∂u/∂tn
n+1

Relation
We can consider the following
transition operator:
(n )
k1

(n )
0

(n )∗
0

(n )
1

(n )∗
1

In section 7, we verify that these transition operators satisfy Markov properties.

Automaton
5.2. BBS with carrier
capacity k  2. In analogy to k  1 case, for k  2, we can consider the following
operators.

Box-Ball System

a0







a2





a0 a1
0 a2
0
a0;k2  linking, a1  a1;k2 and discrete, a2systems,
 a2;k2 
a0 
Theory0 0 continuous
a0 0
a1

Ultra-discrete soliton

and automata

0
a2



.

Discrete Mathematics
Exploring the complexity of discrete mathematics problems and developing algorithms
Topics in discrete mathematics, such as the graphs andnetworks used to represent systems, schedules toenhance the efficiency of production, and the logicalanalysis of large volumes of data, are closely related toapplications of research results. We explore thecomplexity of the computation used to solve these
problems; design exact and approximation algorithms;
develop tabu search algorithms, genetic algorithms andother metaheuristic algorithms; and apply them to solving
actual problems.
[Kazuya Haraguchi]

A puzzle in which you have to fit thepieces into a box of fixed width without
any of the rectangular pieces
overlapping and try to make the heightof the packed pie ces as low as possible.

Applied Mathematics and Physics Course

We have tree operator recursions

a1

The structural formula of a compound expected
to have a certain target value of heat of
combustion. It was constructed by formulating
and solving an inverse problem of a prediction
model learned from a database of compounds inthe form of a mixed-integer optimization problem.

Graduate School of Informatics

38

Applied Mathematics and Physics Course

OutlineSystem Optimization
Optimization is the keyword for solving problems
We conduct education and research regarding thetheory and methodology of system optimization,
which plays an important role as a mathematical
approach that is used to resolve many different kindsof practical problems. In particular, we develop
efficient mathematical optimization approaches toactual large-scale systems, complex nonlinear
systems, and systems with uncertainty, as well asbasic research regarding mathematical

Optimal solutions of an unconstrained problem

programming.
[Nobuo Yamashita , Ellen Hidemi Fukuda , Hiroyuki Sato , Yuya Yamakawa]Control Systems theory
Mathematical approaches to modeling and control
We carry out teaching and research regarding themathematical methodologies of modeling, analysis
and design of control systems, and their application
with the aim of developing practical and expansive
control theories. Our main research themes arerobust control, control systems with input/output
constraints, networked control systems, algebraic
system theory, mathematical optimization in control,
stochastic realization, system identification andquantum control theory.
[Kenji Kashima , Kentaro Ohki]
Conceptual diagram of a control system design

39

Applied Mathematics and Physics Course

Physical Statistics
The mathematical studies on dynamics of coupled multi-element network systems
and design theory of complex engineering systems
We aim to gain a mathematical and unified understandingof the complex and diverse phenomena that arise out of
the intense mutual interactions of multiple elements(units) in a system and apply this understanding to
information processing and design of complex engineeringsystems. For example, we will use stochastic process
theory, ergodic theory, statistical physics, dynamicalsystem theory, computer simulations, and large-scale

The distribution of vertices when a random network is
extended by applying an external field to both ends of the
network is computed from the graph Laplacian.

Polymer physics

Applied Mathematics and Physics Course

Chaos code for signal analysis
and multiuser
communications systemdata processing techniques to analyze information
processing and performance evaluation in neuralnetworks; the structure of the Internet and other complex
networks such as social media systems, and the
propagation of information within them; and the dynamical
properties of price change, stock markets and othereconomic phenomena.
[Ken Umeno , Erika Uehara , Atsushi Iwasaki]

Dynamical Systems
Looking into the world through dynamical systems theory
Our research purpose is to analyze complicated phenomena such as chaosand bifurcations in various systems appearing in natural science, engineeringand other disciplines using dynamical systems approaches, and apply them todevelop novel engineering technologies. For this purpose, we not only use
standard approaches but also establish groundbreaking theories in dynamicalsystems. Moreover, we utilize numerical approaches such as verifiable
computation rigorous numerics and large-scale numerical simulation, andstudy the nonintegrability of dynamical systems and differential equations,nonlinear waves in partial differential equations, periodic motions in the
n-body problem of classical mechanics, several problems in the kinetic theoryof many-body systems, design of spacecraft transfer trajectories, anddynamics and control of flying objects such as quadcopters.

4-body super-eight solution which is proven to exist
by using variational methods

Ω3
Ω4
ωz
ωy

ωx

Ω2

Ω1

[Kazuyuki Yagasaki , Mitsuru Shibayama , Yoshiyuki Yamaguchi]
Mathematical model of a quadcopter

Applied Mathematical Modeling Adjunct Unit (In collaboration with Hitachi, Ltd.)
Infusing information systems with intelligence
To make information systems useful to our day-to-day lives and industry atlarge, we need to be able to mathematically model both the behavior of people
and the movements of objects that these systems deal with. The form of thesemodels ranges from the conceptual to the numerically precise. We will
examine case studies from industry in our research of modeling technology,
including methods of using human knowledge (structural modeling) andmethods using actual data (multivariate analysis).
[Hitachi Ltd.: Yoichi Nonaka , Yoshiyasu Takahashi]

Modeling of a social infrastructure system

Graduate School of Informatics

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