## Seminar information archive

Seminar information archive ～08/17｜Today's seminar 08/18 | Future seminars 08/19～

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Simply connected elliptic surfaces (JAPANESE)

**Kentaro Mitsui**(University of Tokyo)Simply connected elliptic surfaces (JAPANESE)

[ Abstract ]

We characterize simply connected elliptic surfaces by their singular fibers in any characteristic case. To this end, we study orbifolds of curves, local canonical bundle formula, and resolutions of multiple fibers. The result was known for the complex analytic case. Our method can be applied to the rigid analytic case.

We characterize simply connected elliptic surfaces by their singular fibers in any characteristic case. To this end, we study orbifolds of curves, local canonical bundle formula, and resolutions of multiple fibers. The result was known for the complex analytic case. Our method can be applied to the rigid analytic case.

### 2012/05/22

#### Tuesday Seminar of Analysis

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Viscosity solutions for nonlinear elliptic-parabolic problems (ENGLISH)

**Norbert Pozar**(Graduate School of Mathematical Sciences, The University of Tokyo)Viscosity solutions for nonlinear elliptic-parabolic problems (ENGLISH)

[ Abstract ]

We introduce a notion of viscosity solutions for a general class of

elliptic-parabolic phase transition problems. These include the

Richards equation, which is a classical model in filtration theory.

Existence and uniqueness results are proved via the comparison

principle. In particular, we show existence and stability properties

of maximal and minimal viscosity solutions for a general class of

initial data. These results are new even in the linear case, where we

also show that viscosity solutions coincide with the regular weak

solutions introduced in [Alt&Luckhaus 1983]. This talk is based on a

recent work with Inwon Kim.

We introduce a notion of viscosity solutions for a general class of

elliptic-parabolic phase transition problems. These include the

Richards equation, which is a classical model in filtration theory.

Existence and uniqueness results are proved via the comparison

principle. In particular, we show existence and stability properties

of maximal and minimal viscosity solutions for a general class of

initial data. These results are new even in the linear case, where we

also show that viscosity solutions coincide with the regular weak

solutions introduced in [Alt&Luckhaus 1983]. This talk is based on a

recent work with Inwon Kim.

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

The DtN finite element method and the Schwarz method for multiple scattering problems (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Daisuke Koyama**(The University of Electro-Communications)The DtN finite element method and the Schwarz method for multiple scattering problems (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

#### Tuesday Seminar on Topology

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Gamma Integral Structure in Gromov-Witten theory (JAPANESE)

**Hiroshi Iritani**(Kyoto University)Gamma Integral Structure in Gromov-Witten theory (JAPANESE)

[ Abstract ]

The quantum cohomology of a symplectic

manifold undelies a certain integral local system

defined by the Gamma characteristic class.

This local system originates from the natural integral

local sysmem on the B-side under mirror symmetry.

In this talk, I will explain its relationships to the problem

of analytic continuation of Gromov-Witten theoy (potentials),

including crepant resolution conjecture, LG/CY correspondence,

modularity in higher genus theory.

The quantum cohomology of a symplectic

manifold undelies a certain integral local system

defined by the Gamma characteristic class.

This local system originates from the natural integral

local sysmem on the B-side under mirror symmetry.

In this talk, I will explain its relationships to the problem

of analytic continuation of Gromov-Witten theoy (potentials),

including crepant resolution conjecture, LG/CY correspondence,

modularity in higher genus theory.

### 2012/05/21

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Characterizations of projective spaces and hyperquadrics

(JAPANESE)

**Taku Suzuki**(Waseda University)Characterizations of projective spaces and hyperquadrics

(JAPANESE)

[ Abstract ]

After Mori's works on Hartshorne's conjecture, many results to

characterize projective spaces and hyperquadrics in terms of

positivity properties of the tangent bundle have been provided.

Kov\\'acs' conjecture states that smooth complex projective

varieties are projective spaces or hyperquadrics if the $p$-th

exterior product of their tangent bundle contains the $p$-th

exterior product of an ample vector bundle. This conjecture is

the generalization of many preceding results. In this talk, I will

explain the idea of the proof of Kov\\'acs' conjecture for varieties

with Picard number one by using a method of slope-stabilities

of sheaves.

After Mori's works on Hartshorne's conjecture, many results to

characterize projective spaces and hyperquadrics in terms of

positivity properties of the tangent bundle have been provided.

Kov\\'acs' conjecture states that smooth complex projective

varieties are projective spaces or hyperquadrics if the $p$-th

exterior product of their tangent bundle contains the $p$-th

exterior product of an ample vector bundle. This conjecture is

the generalization of many preceding results. In this talk, I will

explain the idea of the proof of Kov\\'acs' conjecture for varieties

with Picard number one by using a method of slope-stabilities

of sheaves.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Local cohomology and hypersurface isolated singularities I (JAPANESE)

**Shinichi TAJIMA**(University of Tsukuba)Local cohomology and hypersurface isolated singularities I (JAPANESE)

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Topological Strings on Elliptic Fibrations (ENGLISH)

**Emanuel Scheidegger**(The University of Freiburg)Topological Strings on Elliptic Fibrations (ENGLISH)

[ Abstract ]

We will explain a conjecture that expresses the BPS invariants

(Gopakumar-Vafa invariants) for elliptically fibered Calabi-Yau

threefolds in terms of modular forms. In particular, there is a

recursion relation which governs these modular forms. Evidence comes

from the polynomial formulation of the higher genus topological string

amplitudes with insertions.

We will explain a conjecture that expresses the BPS invariants

(Gopakumar-Vafa invariants) for elliptically fibered Calabi-Yau

threefolds in terms of modular forms. In particular, there is a

recursion relation which governs these modular forms. Evidence comes

from the polynomial formulation of the higher genus topological string

amplitudes with insertions.

### 2012/05/19

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

TBA (JAPANESE)

TBA (JAPANESE)

**Takashi Taniguchi**(Kobe Univeristy) 13:30-14:30TBA (JAPANESE)

**Masao Tsuzuki**(Sophia University) 15:00-16:00TBA (JAPANESE)

### 2012/05/18

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

PAC-Bayesian Bound for Gaussian Process Regression and Multiple Kernel Additive Model (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/04.html

**SUZUKI, Taiji**(University of Tokyo)PAC-Bayesian Bound for Gaussian Process Regression and Multiple Kernel Additive Model (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/04.html

### 2012/05/16

#### Number Theory Seminar

16:40-17:40 Room #002 (Graduate School of Math. Sci. Bldg.)

On uniform bound of the maximal subgroup of the inertia group acting unipotently on $¥ell$-adic cohomology (JAPANESE)

**Naoya Umezaki**(University of Tokyo)On uniform bound of the maximal subgroup of the inertia group acting unipotently on $¥ell$-adic cohomology (JAPANESE)

[ Abstract ]

For a smooth projective variety over a local field,

the action of the inertia group on the $¥ell$-adic cohomology group is

unipotent if it is restricted to some open subgroup.

In this talk, we give a uniform bound of the index of the maximal open

subgroup satisfying this property.

This bound depends only on the Betti numbers of $X$ and certain Chern

numbers depending on a projective embedding of $X$.

For a smooth projective variety over a local field,

the action of the inertia group on the $¥ell$-adic cohomology group is

unipotent if it is restricted to some open subgroup.

In this talk, we give a uniform bound of the index of the maximal open

subgroup satisfying this property.

This bound depends only on the Betti numbers of $X$ and certain Chern

numbers depending on a projective embedding of $X$.

### 2012/05/15

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Strichartz estimates for Schr\\"odinger equations with variable coefficients and unbounded electromagnetic potentials (JAPANESE)

**MIZUTANI, Haruya**(Research Institute for Mathematical Sciences, Kyoto University)Strichartz estimates for Schr\\"odinger equations with variable coefficients and unbounded electromagnetic potentials (JAPANESE)

[ Abstract ]

In this talk we consider the Cauchy problem for Schr\\"odinger equations with variable coefficients and unbounded potentials. Under the assumption that the Hamiltonian is a long-range perturbation of the free Schr\\"odinger operator, we construct an outgoing parametrix for the propagator near infinity, and give applications to sharp Strichartz estimates. The basic idea is to combine the standard approximation by using a time dependent modifier, which is not in the semiclassical regime, with the semiclassical approximation of Isozaki-Kitada type. We also show near sharp Strichartz estimates without asymptotic conditions by using local smoothing effects.

In this talk we consider the Cauchy problem for Schr\\"odinger equations with variable coefficients and unbounded potentials. Under the assumption that the Hamiltonian is a long-range perturbation of the free Schr\\"odinger operator, we construct an outgoing parametrix for the propagator near infinity, and give applications to sharp Strichartz estimates. The basic idea is to combine the standard approximation by using a time dependent modifier, which is not in the semiclassical regime, with the semiclassical approximation of Isozaki-Kitada type. We also show near sharp Strichartz estimates without asymptotic conditions by using local smoothing effects.

### 2012/05/14

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Duality in the unit circle and the ring of p-adic intergers and van der Corput series (JAPANESE)

**Hiroshi KANEKO**(Tokyo University of Science)Duality in the unit circle and the ring of p-adic intergers and van der Corput series (JAPANESE)

### 2012/05/11

#### Colloquium

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Moduli spaces and symplectic derivation Lie algebras (JAPANESE)

**SAKASAI Takuya**(University of Tokyo)Moduli spaces and symplectic derivation Lie algebras (JAPANESE)

[ Abstract ]

First we overview Kontsevich's theorem describing a deep connection between homology of certain infinite dimensional Lie algebras (symplectic derivation Lie algebras) and cohomology of various moduli spaces. Then we discuss some computational results on the Lie algebras together with their applications (joint work with Shigeyuki Morita and Masaaki Suzuki).

First we overview Kontsevich's theorem describing a deep connection between homology of certain infinite dimensional Lie algebras (symplectic derivation Lie algebras) and cohomology of various moduli spaces. Then we discuss some computational results on the Lie algebras together with their applications (joint work with Shigeyuki Morita and Masaaki Suzuki).

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

Efficient Discretization of Stochastic Integrals (JAPANESE)

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/03.html

**FUKASAWA, Masaaki**(Department of Mathematics, Osaka University)Efficient Discretization of Stochastic Integrals (JAPANESE)

[ Abstract ]

Sharp asymptotic lower bounds of the expected quadratic variation of discretization error in stochastic integration are given. The theory relies on inequalities for the kurtosis and skewness of a general random variable which are themselves seemingly new. Asymptotically efficient schemes which attain the lower bounds are constructed explicitly. The result is directly applicable to practical hedging problem in mathematical finance; it gives an asymptotically optimal way to choose rebalancing dates and portofolios with respect to transaction costs. The asymptotically efficient strategies in fact reflect the structure of transaction costs. In particular a specific biased rebalancing scheme is shown to be superior to unbiased schemes if transaction costs follow a convex model. The problem is discussed also in terms of the exponential utility maximization.

[ Reference URL ]Sharp asymptotic lower bounds of the expected quadratic variation of discretization error in stochastic integration are given. The theory relies on inequalities for the kurtosis and skewness of a general random variable which are themselves seemingly new. Asymptotically efficient schemes which attain the lower bounds are constructed explicitly. The result is directly applicable to practical hedging problem in mathematical finance; it gives an asymptotically optimal way to choose rebalancing dates and portofolios with respect to transaction costs. The asymptotically efficient strategies in fact reflect the structure of transaction costs. In particular a specific biased rebalancing scheme is shown to be superior to unbiased schemes if transaction costs follow a convex model. The problem is discussed also in terms of the exponential utility maximization.

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/03.html

### 2012/05/08

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Infinite examples of non-Garside monoids having fundamental elements (JAPANESE)

**Tadashi Ishibe**(The University of Tokyo, JSPS)Infinite examples of non-Garside monoids having fundamental elements (JAPANESE)

[ Abstract ]

The Garside group, as a generalization of Artin groups,

is defined as the group of fractions of a Garside monoid.

To understand the elliptic Artin groups, which are the fundamental

groups of the complement of discriminant divisors of the semi-versal

deformation of the simply elliptic singularities E_6~, E_7~ and E_8~,

we need to consider another generalization of Artin groups.

In this talk, we will study the presentations of fundamental groups

of the complement of complexified real affine line arrangements

and consider the associated monoids.

It turns out that, in some cases, they are not Garside monoids.

Nevertheless, we will show that they satisfy the cancellation condition

and carry certain particular elements similar to the fundamental elements

in Artin monoids.

As a result, we will show that the word problem can be solved

and the center of them are determined.

The Garside group, as a generalization of Artin groups,

is defined as the group of fractions of a Garside monoid.

To understand the elliptic Artin groups, which are the fundamental

groups of the complement of discriminant divisors of the semi-versal

deformation of the simply elliptic singularities E_6~, E_7~ and E_8~,

we need to consider another generalization of Artin groups.

In this talk, we will study the presentations of fundamental groups

of the complement of complexified real affine line arrangements

and consider the associated monoids.

It turns out that, in some cases, they are not Garside monoids.

Nevertheless, we will show that they satisfy the cancellation condition

and carry certain particular elements similar to the fundamental elements

in Artin monoids.

As a result, we will show that the word problem can be solved

and the center of them are determined.

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Pressure Oscillation Problem of MPS time evolution scheme for incompressible Navier-Stokes equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Motofumi Hattori**(Kanagawa Institute of Technology )Pressure Oscillation Problem of MPS time evolution scheme for incompressible Navier-Stokes equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

#### Lectures

14:40-16:10 Room #470 (Graduate School of Math. Sci. Bldg.)

Embedding spaces and string topology (JAPANESE)

**Keiichi Sakai**(Shishu University)Embedding spaces and string topology (JAPANESE)

[ Abstract ]

There are several similarities between the topology of embedding spaces and that of (free) loop space.

In this talk I will review the similarities, with a focus on "string topology" for embedding spaces.

There are several similarities between the topology of embedding spaces and that of (free) loop space.

In this talk I will review the similarities, with a focus on "string topology" for embedding spaces.

### 2012/05/07

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

The second metric variation of the total $Q$-curvature in conformal geometry (JAPANESE)

**Yoshihiko Matsumoto**(University of Tokyo)The second metric variation of the total $Q$-curvature in conformal geometry (JAPANESE)

[ Abstract ]

Branson's $Q$-curvature of even-dimensional compact conformal manifolds integrates to a global conformal invariant called the total $Q$-curvature. While it is topological in two dimensions and is essentially the Weyl action in four dimensions, in the higher dimensional cases its geometric meaning remains mysterious. Graham and Hirachi have shown that the first metric variation of the total $Q$-curvature coincides with the Fefferman-Graham obstruction tensor. In this talk, the second variational formula will be presented, and it will be made explicit especially for conformally Einstein manifolds. The positivity of the second variation will be discussed in connection with the smallest eigenvalue of the Lichnerowicz Laplacian.

Branson's $Q$-curvature of even-dimensional compact conformal manifolds integrates to a global conformal invariant called the total $Q$-curvature. While it is topological in two dimensions and is essentially the Weyl action in four dimensions, in the higher dimensional cases its geometric meaning remains mysterious. Graham and Hirachi have shown that the first metric variation of the total $Q$-curvature coincides with the Fefferman-Graham obstruction tensor. In this talk, the second variational formula will be presented, and it will be made explicit especially for conformally Einstein manifolds. The positivity of the second variation will be discussed in connection with the smallest eigenvalue of the Lichnerowicz Laplacian.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Algebro-geometric characterization of Cayley polytopes (JAPANESE)

**Atsushi Ito**(University of Tokyo)Algebro-geometric characterization of Cayley polytopes (JAPANESE)

[ Abstract ]

A lattice polytope is called a Cayley polytope if it is "small" in some

sense.

In this talk, I will explain an algebro-geometric characterization of

Cayley polytopes

by considering whether or not the corresponding polarized toric

varieties are covered by lines, planes, etc.

We can apply this characterization to the study of Seshadri constants,

which are invariants measuring the positivity of ample line bundles.

That is, we can obtain an explicit description of a polarized toric

variety whose Seshadri constant is one.

A lattice polytope is called a Cayley polytope if it is "small" in some

sense.

In this talk, I will explain an algebro-geometric characterization of

Cayley polytopes

by considering whether or not the corresponding polarized toric

varieties are covered by lines, planes, etc.

We can apply this characterization to the study of Seshadri constants,

which are invariants measuring the positivity of ample line bundles.

That is, we can obtain an explicit description of a polarized toric

variety whose Seshadri constant is one.

#### GCOE Seminars

14:30-16:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Distributional behaviors of time-averaged observables in anomalous diffusions (subdiffusion and superdiffusion) (ENGLISH)

**Takuma Akimoto**(Keio university, Global environmental leaders program)Distributional behaviors of time-averaged observables in anomalous diffusions (subdiffusion and superdiffusion) (ENGLISH)

[ Abstract ]

In anomalous diffusions attributed to a power-law distribution,

time-averaged observables such as diffusion coefficient and velocity of drift are intrinsically random. Anomalous diffusion is ubiquitous phenomenon not only in material science but also in biological transports, which is characterized by a non-linear growth of the mean square displacement (MSD).

(subdiffusion: sublinear growth, super diffusion: superlinear growth).

It has been known that there are three different mechanisms generating subdiffusion. One of them is a power-law distribution in the trapping-time distribution. Such anomalous diffusion is modeled by the continuous time random walk (CTRW). In CTRW, the time-averaged MSD grows linearly with time whereas the ensemble-averaged MSD does not. Using renewal theory, I show that diffusion coefficients obtained by single trajectories converge in distribution. The distribution is the Mittag-Leffler (or inverse Levy) distribution [1,2].

In superdiffusion, there are three different mechanisms. One stems from positive correlations in random walks; the second from persistent motions in random walks, called Levy walk; the third from very long jumps in random walks, called Levy flight.

If the persistent time distribution obeys a power law with divergent mean in Levy walks, the MSD grows as t^2 whereas the mean of positions is zero. When an external bias is added in Levy walks, the response to bias (velocity of drift) appears in the distribution, which is what we term a distributional response [3]. The distribution is the generalized arcsine distribution.

These distributional behaviors open a new window to dealing with the average (ensemble or time average) in single particle tracking experiments.

[1] Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett. 101, 058101 (2008).

[2] T. Miyaguchi and T. Akimoto, Phys. Rev. E 83, 031926 (2011).

[3] T. Akimoto, Phys. Rev. Lett. 108, 164101 (2012)

In anomalous diffusions attributed to a power-law distribution,

time-averaged observables such as diffusion coefficient and velocity of drift are intrinsically random. Anomalous diffusion is ubiquitous phenomenon not only in material science but also in biological transports, which is characterized by a non-linear growth of the mean square displacement (MSD).

(subdiffusion: sublinear growth, super diffusion: superlinear growth).

It has been known that there are three different mechanisms generating subdiffusion. One of them is a power-law distribution in the trapping-time distribution. Such anomalous diffusion is modeled by the continuous time random walk (CTRW). In CTRW, the time-averaged MSD grows linearly with time whereas the ensemble-averaged MSD does not. Using renewal theory, I show that diffusion coefficients obtained by single trajectories converge in distribution. The distribution is the Mittag-Leffler (or inverse Levy) distribution [1,2].

In superdiffusion, there are three different mechanisms. One stems from positive correlations in random walks; the second from persistent motions in random walks, called Levy walk; the third from very long jumps in random walks, called Levy flight.

If the persistent time distribution obeys a power law with divergent mean in Levy walks, the MSD grows as t^2 whereas the mean of positions is zero. When an external bias is added in Levy walks, the response to bias (velocity of drift) appears in the distribution, which is what we term a distributional response [3]. The distribution is the generalized arcsine distribution.

These distributional behaviors open a new window to dealing with the average (ensemble or time average) in single particle tracking experiments.

[1] Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett. 101, 058101 (2008).

[2] T. Miyaguchi and T. Akimoto, Phys. Rev. E 83, 031926 (2011).

[3] T. Akimoto, Phys. Rev. Lett. 108, 164101 (2012)

### 2012/05/02

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A measurable group theoretic solution to von Neumann's Problem (after Gaboriau and Lyons) (JAPANESE)

**Yuhei Suzuki**(Univ. Tokyo)A measurable group theoretic solution to von Neumann's Problem (after Gaboriau and Lyons) (JAPANESE)

### 2012/05/01

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Minimal models, formality and hard Lefschetz property of

solvmanifolds with local systems (JAPANESE)

**Hisashi Kasuya**(The University of Tokyo)Minimal models, formality and hard Lefschetz property of

solvmanifolds with local systems (JAPANESE)

### 2012/04/27

#### Seminar on Probability and Statistics

15:00-16:10 Room #006 (Graduate School of Math. Sci. Bldg.)

Convergence conditions on step sizes in temporal difference learning (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/02.html

**NOMURA, Ryosuke**(Graduate school of Mathematical Sciences, Univ. of Tokyo)Convergence conditions on step sizes in temporal difference learning (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/02.html

### 2012/04/25

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Bost-Connes system and class field theory (JAPANESE)

**Takuya Takeishi**(Univ. Tokyo)Bost-Connes system and class field theory (JAPANESE)

### 2012/04/24

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Quantum mechanics and numerical analysis (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Hideaki Ishikawa**(Semiconductor Leading Edge Technologies, Inc.)Quantum mechanics and numerical analysis (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

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