Elliptic Asymptotics of the first Painlevé equations

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A graph of the first
Painlevé transcendents
on the complex domain

Paul Painlevé (left)
was a former prime
minister of France. He
is the firstmathematician who
flew on the airplane.
He was a passenger
of Wilber Wright (right)
on 1908.

Content:
It is important to study connection problems of solutionsof differential equations between two points in many
fields of mathematical sciences. The global study on
ordinary linear differential equations is still developing.
Paul Painlevé studied nonlinear differential equations
with second order, which have no movable branch points(so called the Painlevé property). He and his pupil,
Gambier, classified all of such equations in six types
around 1900. Many nonlinear equations appeared inphysics has the Painlevé property, and we can solve
connection problems on such equations. We expect that
the Painlevé equations play the same important role innonlinear analysis as the Bessel functions or
hypergeometric functions play in linear equations. The
Painlevé equations are also obtained by monodromy
preserving deformations. We can show the correspondence between global data of linear equations andlocal data of the Painlevé functions and we can study
nonlinear connection problems or the nonlinear Stokes
phenomenon on the Painlevé differential or difference
equations.
Keywords：Classical Analysis, the Painlevé equations,monodromy problems
E-mail: ohyama@tokushima-u.ac.jp
Tel. +81-88-656-7541
Fax: +81-88-656-7541
HP: http://math0.pm.tokushimau.ac.jp/~ohyama/index_e.html

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By using the functional analysis approach, I study on the
structure of solutions and the decay estimate of energy for
nonlinear PDEs that describe the nonlinear phenomena. In a
study of nonlinear equations, detailed analysis of thecorresponding linear equation becomes essential. I examine
the relationship of the nonlinearity of the equations and thefunctional spaces to which the solutions belong, and I
investigate on global solvability in time of solutions or blowup problems for PDEs. In addition, I research on decay
estimates the energy function and the derivatives.
Nonlinear degenerate dissipative Kirchhoff equations are
nonlinear PDEs that describe the nonlinear wave phenomena.
Solutions of these equations have decay estimates of thesame polynomial order from above and blow.
Vlasov-Poisson systems are basic equations that describe
plasma phenomena, in particular, the solution of the VlasovPoisson-Fokker-Plank system is asymptotic to the solution of
the corresponding linear Fokker-Plank system.
Keywords ：Nonlinear Analysis, PDEs
Field ：Mathematical Sciences
E-mail ： k.ono@tokushima-u.ac.jp
Tel. +81-88-656-7218
Fax ： +81-88-656-7218
HP ： http://www-math.ias.tokushima-u.ac.jp/

Fig.1 log|ζ(s)| (ζ(s):Riemann zeta function)
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Fig.2 Irregular primes and indicesFig.3 The number of exceptional primes

Content:
The main subject of our research is the ideal class groups ofalgebraic number fields. We have particularly investigatedGreenberg‘s conjecture and Vandiver’s conjecture on theclass numbers of real cyclotomic fields by using computers.
Furthermore, we are also interested in new applications ofalgebraic systems such as algebraic number fields and elliptic
curves, which have strong connections with cryptography.
A lot of mathematicians have been interested in Riemannzeta function (cf. Fig.1). Its special values have deep relations
with the ideal class groups of cyclotomic fields (cf. Fig.2).
These relations are expressed as correspondences of theclass numbers of real cyclotomic fields and the indices of theircircular units in full ones.
Greenberg‘s conjecｔure states that their p-parts are boundedin the Zp-extension. Moreover, Vandiver's conjecture statesthat they are trivial for p-cyclotomic fields. We have beenstudied these conjectures by using arithmetic special
elements such as cyclotomic units, Gauss sums, p-adic Lfunctions and auxiliary prime numbers. As results, we could
find a lot of examples for which Greenberg's conjecture holds,
and a lot of exceptional prime numbers for the Iwasawa
invariants (cf. Fig.3).

Keywords：algebraic number field,
class number, elliptic curve, cryptography
E-mail: hirokit@tokushima-u.ac.jp
Tel. +81-88-656-7549
Fax: +81-88-656-7549
HP : https://math0.pm.tokushima-u.ac.jp//~hiroki/index.html

S = 2! 2c

Xmlogc(tj ; x) +j=1

Xn (xi ¡ ¹ i )2
Xm (cj ¡ c(t j ; x)) 2+ ! 2c2! 2i ¹ 2i
f c(t j ; x)g
i= 1
j=1

(a) Objective function in nonlinear optimization problem
C(t) = c(t; Vd ; Vmax ; K m ) : Concentration
8dX a (t)>>= ¡ ka X a (t)
>
<
dt
>
>
>
:

ti 5 t < ti+ 1dC(t)Vmax CF ka X a¡=dt
Vd
Vd (K m + C)

crack 88< D1
X a (t i ) =<: D i + lim X a (t)t! ti¡ 0

C(t i ) =:( i = 1; 2; ¢¢¢)

0

i= 1

lim C(t)i= 2

t! t i ¡ 0(b) Di®erential equations(phenytoin)
Fig. 1 Bayesian estimat ion for pharmacokinet ics

MT C(Minimum Toxic Concent rat ion)

T herapeutic Range

MEC(Minimum E®ect ive Concent rat ion)Fig. 2 Pro¯le of the concentration and the therapeutic range

Content:
Pharmacokinetics plays an important role in efficacy
and safety pharmacotherapy. The estimation of individual
pharmacokinetic parameters from a few concentration
data is desirable in quick therapy. Bayesian estimation
using the population pharmacokinetic parameters isuseful for the estimation of individual pharmacokinetic
parameters. Here, population pharmacokinetic parameters mean statistic including average, variance andcorrelation coefficient. The numerical calculation of
nonlinear optimization is essential to Bayesian estimationor computation for population pharmacokinetic parameters. In addition, the theoretical value of concentration
data may be given with a nonlinear differential equation.
The stable computation in nonlinear optimization forpharmacokinetics is difficult because of the strong
nonlinearity. I am developing a stable and high-precision
numerical method for nonlinear optimization problem in
population pharmacokinetics and Bayesian estimation.

Keywords： Numerical analysis, Nonlinear optimization
E-mail: takeuchi@tokushima-u.ac.jp
Tel. +81-88-656-7544
Fax: +81-88-656-7544

Content:

3,4

3,3

Fig.1： Sierpiński graphs.
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Υ

,4

Γ
Γ

Fig. 2： Applications of thesubdivided-line graphoperation to the completegraph .

Fig. 3： Universalized
Sierpiński graphs.

Sierpinski graphs, ,1,
2 are known to begraphs with self‐similar structures and their various properties
have been studied until now. It is also known that Sierpinski
graphs are isomorphic to WK‐recursive networks which havebeen proposed as interconnection networks for massively
parallel computers because of their remarkable extendability.
The purpose of this study is mainly to investigate theirstructural properties with applications to interconnection
networks.
In this study, we newly introduced the subdivided‐line graphoperation Γ and showed that, is obtained from1,
by applying Γ. Although,
can be obtained by
combining copies of1,
based on the definition,
the constructions by Γ help us to investigate structuralproperties of, directly from those of
1, . So far,
we have obtained results on structural properties ofsubdivided‐line graphs concerning interconnection networks
such as diameter, connectivity, edge‐disjoint Hamiltonian
cycles, several variants of dominating sets, completely
independent spanning trees, and book‐embeddings. Besides,
we newly defined the class of universalized Sierpinski graphs
apart from the class of subdivided‐line graphs, and have been
investigating their structural properties.
Keywords ：subdivided-line graphs,
Sierpinski graphs
E-mail: hasunuma@tokushima-u.ac.jp
Tel. +81-88-656-7216
Fax: +81-88-656-7216

universalizedcrack

Fig.1 Phase Portrait of Zeeman’s class 27

Content:
2D Lotka-Volterra equations cannot have limit cycles. Thatis, except for conservative systems, the limit set consists ofequilibria only. On the other hand, 3D Lotka-Volterra systems
allow various types of complicated dynamics.
As long as we restrict 3D competitive Lotka-Volterra systems,
the possibility of the dynamics is limited. Hirsch showed thatcompetitive systems have the order preserving property, andthere is an invariant manifold (called the carrying simplex)
which attracts all orbits except for the origin. Thus, in 3D
competitive systems, the Poincare-Bendixson theorem holds,
and therefore the limit set consists of equilibria, limit cycles
and heteroclinic orbit only. Zeeman has divided all possible
phase portraits of 3D competitive Lotka-Volterra systems into
33 classes and showed that six classes can have the Hopf
bifurcation. Hofbauer and So constructed an example with two
limit cycles in Zeeman's class 27.
In this study, we present a concrete example with multiple
limit cycles for 3D competitive Lotka-Volterra systems. For
instance, we obtain an example with three limit cycles in
Zeeman’s class 27 as shown in figures.

Keywords： differential equations, Hopf bifurcation, limit
cyclesE-mail: murakami@ias.tokushima-u.ac.jp>
Tel. +81-88-656-7221
Fax: +81-88-656-7221
Fig.2 Limit cycles of Zeeman’s class 27

<図表>

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Content:
The relation between differential structure and topologic
properties that the maps on the closed manifold has arestudied for a long time, and there is much result. For example,
the relationship between uniformly hyperbolic properties andexpansive properties and shadowing properties, non-uniformly
hypebolic properties is a hyperbolic of partially hyperbolic anddominated splitting and the phase shift of the relationship.
Recently, a new concept that the topological nature was recaptured from the measure theory point of view, such as
expansion of and follow-up property can be considered, it has
been investigated the relationship between the set with auniform hyperbolicity. Although research the current hasremained on the relationship between the uniformly hyperbolic
properties are expected to lead the relationship between nonuniformly hyperbolic properties by adding the viewpoint of
measure theory.
In this study, it is intended that I check the relations withthese properties for the meeting having non-uniformly
hyperbolic properties in addition to the uniformly hyperbolic
properties. Particularly, I think that I can find the connection ofa gauging theory-like property and a topologic property byclarifying the topologic property that the one Pesin set of themeeting having non-uniformly hyperbolic properties has.

Keywords：hyperbolic structure, stability, Pesin set
E-mail: moriyasu@tokushima-u.ac.jp
Tel. +81-88-656-7220
Fax: +81-88-656-7220

crackContent:
Mathematical optimization is defined as finding the bestsolution for mathematical problems formulating real world
problems, e.g. production planning, location, etc.
An important issue for applying mathematicaloptimization is “uncertainty”, which can divided into thefollowing two types: one is “randomness”, which isincluded in random factors, e.g. weather, economic
conditions, etc. The other is “fuzziness”, which isincluded in evaluation or judgment of human beings.
Because real world problems include both randomness
and fuzziness, I study modeling for mathematicaloptimization by applying “fuzzy random variables”,
representing them simultaneously.
Formulated mathematical problems often haveenormous decision variables and conditions withcomplex characteristics. Because of difficulty of solving
them strictly, we study evolutional computing, e.g. GA
and PSO, for finding their good solutions efficiently.

Keywords： Operations Research (OR),Soft Computing
E-mail: uno.takeshi@tokushima-u.ac.jp
Tel. +81-88-656-7294
Fax: +81-88-656-7294
HP : http://www-math.ias.tokushima-u.ac.jp/~uno/crackContent:
I am interested in the study of a comparison principle and astrong comparison principle for semicontinuous solutions of
nonlinear partial differential equations.
As partial differential equations I considered the minimal
surface equation, the prescribed mean curvature equation, thelevel set equation of the mean curvature flow equation, thelevel set equation of an anisotropic curvature equation and pLaplace diffusion equation. As well known the above
equations are degenerate and singular. Usually for such
equation, we cannot expect existence of classical solutions.
So I will consider such equations with viscosity solutions.
For elliptic equations:Comparison principle: Let u be a lower semicontinuous
supersolution, and let v be a upper semicontinuous
subsolution. On the boundary of the domain we considered if
u is greater than or equal to v, then it holds in the wholedomain.
Strong comparison principle: Assume in the whole domain u isgreater than or equal to v. If u touches v in a interior point of
the domain, then u is equivalent to v in the whole domain.

Keywords：partial differential equation, viscosity solution
E-mail: ohnuma@tokushima-u.ac.jp
Tel. +81-88-656-7225
Fax: +81-88-656-7225

<図表>A plane curve consisting ofコンタクト不良a smoothコンタクト不良
curve and three lines
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The pull-back of threelines under a Galois
cover branched along
the smooth curve

Case 1.
Several triangles

Case 2.
One polygon

This difference show difference of embedded
topology of plane curves.

Content:
It is known that two algebraic curves on the complex
projective plane (called plane curves) may have different
embedded topology if arrangement of their singularities
are different. Namely, one plane curve cannot be
deformed continuously to the other curve in the
projective plane. I study the criterion for distinguishing
the embedded topology of plane curves.
The complex dimension of complex projective plane is 2.
Hence the real dimension of the plane is 4. The difficultyof this study is that we do not know how to watch thewhole of the plane. Thus we need a language torepresent difference of embedded topology of plane
curves.
Recently, it is known that the “splitting” of plane curves
by pull-back under a Galois cover over the plane
represent difference of embedded topology of plane
curves. In this study, we define the invariant “splitting
graph” which is a language for representing the splittingof plane curves for Galois covers, and give a criterion for
distinguishing embedded topology of plane curves.
Keywords：Plane Curve,
Embedded Topology,
Galois Cover
E-mail: Shirane@tokushima-u.ac.jp
Tel. +81-88-656-7295
Fax: +81-88-656-7295

Time-Complexity
Polynomial Time

Exponential Time

𝑂𝑂(log 𝑛𝑛) 𝑂𝑂(𝑛𝑛)𝑂𝑂(2𝑛𝑛 ) 𝑂𝑂(22𝑛𝑛 )𝑂𝑂(𝑛𝑛2 )2

𝑂𝑂(𝑛𝑛𝑘𝑘 )𝑂𝑂(2𝑛𝑛 )（𝑛𝑛 is the size of input data and 𝑘𝑘 is the constant number.）

The most fundamental classification is the distinction
between problems whosecrack
growth rate in terms of time ispolynomial and problems whose growth rate isexponential.

P ≠ NP Problem
NP

The class NP is the set of decision
problems whose solutions can be
determined by a non-deterministic
Turing machine in polynomial time.

P

The class P consists of all
problems that can be efficiently
computed.

The P ≠ NP problem is whether P and NP are in fact the same.

Content:
Computational complexity theory is a branch of the theory ofcomputation in theoretical computer science that focuses onclassifying computational problems according to theirinherent difficulty, and relating those classes to each other.
One aspect of computational complexity is related to analgorithm for solving instances of a problem. The
computational complexity of an algorithm is a measure of
how many steps the algorithm will require in the worst casefor an instance or input of a given size. The number of steps ismeasured as a function of that size. Moreover, the theory ofcomputational complexity involves classifying problems
according to their inherent tractability or intractability, that is,whether they are “easy” or “hard” to solve. This classification
scheme includes the well-known classes P and NP; the terms“NP-complete” and “NP-hard” are related to the class NP.
In our research, given a problem, we clarify which class it isbelong to, and develop an efficient algorithm for solving it if
it is belong to the class P.
Keywords：computational complexity, algorithm
E-mail: shin@tokushima-u.ac.jp
Tel. +81-88-656-7223
Fax: +81-88-656-7223

<図表>
<図表>

Theory of complex differential
equationse.g. hypergeometric functions
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p-adic differential
equations

Arithmetic theory
e.g. hypergeometric functions overfinite fields

Content:
My research field is arithmetic geometry. This is a fieldthat investigates arithmetic properties based on ideas
from various areas of mathematics, primarily complex
geometry and complex analysis.
I am particularly interested in p-adic differential
equations. The key ingredient is the world of ``p-adic
numbers’’, which serves as a bridge between the worldof "characteristic 0" (i.e. adding 1 repeatedly neverresults in 0) and the world of "characteristic p" (i.e.
adding p copies of 1 results in 0, where p is a prime
number). Functions and differential equations in p-adic
numbers sometimes show an interesting feature: while
they look like objects in the complex number world, theyalso possess number-theoretic information.
I have studied general (single-variable, with an arbitrary
number of parameters) hypergeometric differential
equation in p-adic numbers. Then, I proved that thisequation has an information of an algebraic-numbertheoretic function which is called ``hypergeometricfunctions over finite fields’’.
Currently, I am studying various objects on p-adic
numbers, somehow related to hypergeometric functions,to investigate other interesting connections between
complex number world and p-adic number world.

Keywords：Arithmetic geometry
E-mail: miyatani@tokushima-u.ac.jp
Tel. +81-88-656-7546
Fax:
HP : https://math.miyatani.org/

Solvability of Hyperbolic Systems of Conservation Laws
lecturer Kuniya Okamoto

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コンタクト不良

lContent:
Hyperbolic systems of conservation laws described as
the first order quasilinear partial differential equations
have been extensively studied. The most remarkablefeature is that not only this type of equations do notpossess the smoothing effects but also the regularities ofclassical solutions will be lost in finite time even if theinitial data are smooth. We introduce the notion of weaksolutions which interprets the derivatives of solutions in
the generalized sense, then we need to allow thepresence of discontinuities in the solutions such as shockwaves and discuss the solvability in the wider class.
However, in contrast to the single conservation laws, thecase of systems has not yet been successfully solved
until recently, except for the case that the total variationof initial data is sufficiently close to the equilibrium. In
terms of the interaction potential estimates of Glimm
type, we study the approximate solvability of a system ofconservation laws and the stability of weak solutions
even if the total variations of initial data are not small forthe presence of large oscillations.
Keywords：Hyperbolic systems, Conservation laws
E-mail: okamoto-kuniya@tokushima-u.ac.jp
Tel. +81-88-656-9441
Fax: +81-88-656-9441
HP :Liouville theorem with growth condition

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Content:
Nonexistence of harmonic maps
Harmonic maps are critical points of the energyfunctional. A Liouville type theorem states non-existenceof harmonic maps. We showed that a Liouville type result
holds under some optimal growth condition.
Instability of minimal submanifolds
Minimal submanifolds are critical points of the volume
functional. The instability of minimal submanifolds aremeasured by Morse index. We are trying to derive anappropriate index estimate via the Betti numbers, which
are topological information of minimal submanifolds.
Heat equation along geometric flows
We are interested in time-dependent Riemannian
manifolds like Ricci flow. In particular, we study the
behavior of heat equations under such time-dependentsituations.

C
Non-trivial harmonic map does not exist

Keywords：Geometric Analysis
E-mail: kunikawa@tokushima-u.ac.jp
Tel. +81-88-656-7228
Fax:
HP: https://k-kunikawa.github.io/

顔写真
（省略可）

Stationary Navier-Stokes equations:−Δ𝒖𝒖 + (𝒖𝒖 ⋅ 𝛁𝛁)𝒖𝒖 + 𝛁𝛁𝑝𝑝 = 𝒇𝒇
�(E)
div 𝒖𝒖 = 0

𝒖𝒖: flow velocity, 𝑝𝑝: pressure ← unknown
𝒇𝒇: external force ← given
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Image: Function spaces of 𝒇𝒇

Content:

I am interested in the stationary Navier-Stokes
equations, which describes the behavior of a fluidwith no time variation of flow velocity. The purpose
is to find the borderline between the well-posedness(existence, uniqueness, and continuous dependenceof solutions for given external forces) and illposedness in terms of function spaces for solutions
and external forces.
In the case of the two-dimensional whole space, theanalysis of this equation is extremely difficult (due toa phenomenon known as Stokes' paradox). However,there is a few previous studies on the well-posedness
around special solutions (e.g., uniform, symmetric,
and rotational flows) and the ill-posedness around atrivial solution (zero). Based on these studies, I aim
to generalize the conditions for both well-posedness
and ill-posedness, and to construct a systematic
analysis method for the two-dimensional case.
Keywords ： fluid dynamics, partial differential
equations, functional analysis
E-mail: tsurumi.hiroyuki@tokushima-u.ac.jp
Tel. +81-88-656-7542
Fax: +81-88-656-7542

Content:<図表>

Tensor Triangular Geometry : Balmer (2005)

Forgetting the tensor structure

Triangular Geometry : Matsui (2021)

The left-hand sides contain geometric information on
original Noetherian scheme X and the right-hand sides
contain (tensor) triangulated category structure. Hence theabove results connect them.

One of the most classical and important problems incommutative algebra is to classify all modules up toisomorphisms. This problem is quite difficult and it is
known to be hopeless in general.
On the other hand, classifying subcategories (certainclasses of objects) of a given category is a moremanageable approach than classifying objects. So far,there are many results in this approach.
My research aims to connect geometric information on agiven commutative ring R or more generally a scheme X
and triangulated category structure of their derived
categories using classifications of subcategories. In 2005,
Balmer developed ``tensor triangular geometry” andproved that the tensor triangulated category structure of
the perfect derived category Dpf (X) determines theoriginal Noetherian scheme X. However, tensor
structures are too strong in general and there are various
important examples of triangulated categories without
natural tensor structures.
Recently, I have generalized his theory without using
tensor structure. As Balmer‘s theory is, this theory is
expected to apply in many areas in mathematics such asalgebraic geometry, representation theory of groups,
stable homotopy theory, and so on.

Keywords：Commutative ring/Scheme,
Derived Categories
E-mail: hmatsui@tokushima-u.ac.jp
Tel. +81-88-656-7296
Fax: +81-88-656-7296

<図表>Inverse image of stereo. proj.
ℂ ∋ 𝑔𝑔
2Re 𝑔𝑔 2Im 𝑔𝑔 −1 + 𝑔𝑔 2↦,,
1 + 𝑔𝑔 2 1 + 𝑔𝑔 2 1 + 𝑔𝑔 2
Lift it to 4-dimensional
Minkowski space

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𝕀𝕀3
ℝ2,1

𝕃𝕃3

ℝ3

We can describe spaces
in a unified way, including
the Euclidean space.

Applying a transformation…

We can construct various
smooth and discrete surfaces.

Content:
Differential geometry of surfaces is an importantresearch field with a long history that forms the basis ofmodern differential geometry. Recently, with thedevelopment of computer science and related fields, it
has been actively studied to reorganize and reconstructconventional differential geometry in a discrete setting.
I am working on the differential geometry of surfaces
and discrete surfaces. In the study of differential
geometry, there is a concept of curvatures that describe
how curved geometric objects such as curves andsurfaces are. The study of differential geometry withspecific curvature conditions is interesting because it
intersects with various mathematical studies.
In our recent work, we derived constructions of various
discrete surfaces by developing discrete surface theoryin 4-dimensional Minkowski space. This includes discrete
minimal surfaces in 3-dimensional Euclidean space This
result not only unifies the conventional constructions of
discrete surfaces, but also leads to the construction of
new discrete surfaces.
Keywords： discrete differential geometry,
integrable systems
E-mail: yasumoto.masashi@tokushima-u.ac.jp
Tel. +81-88-656-7297
Fax:
HP : https://sites.google.com/site/
homepageofmasashiyasumoto/home