## Seminar information archive

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

#### FMSP Lectures

16:30-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Representations of reductive groups and L-functions (I) (ENGLISH)

**Birgit Speh**(Cornell University)Representations of reductive groups and L-functions (I) (ENGLISH)

[ Abstract ]

This is an introduction to the theory of L-functions and in particular of the local L-factors of representations in real and complex groups. Some familiarity with infinite dimensional representations would be very helpful, but I will not assume any knowledge of number theory. We will start in the first lecture by considering L-functions for Groessen characters and classical automorphic forms, in other words for automorphic representations of G(1) and GL(2). This will motivate the definition of the local L-factors of representations of GL(1,R) and GL(2,R). Then we will discuss Rankin convolutions and define the L-factors for infinite dimensional tempered representations of GL(n,R).

This is an introduction to the theory of L-functions and in particular of the local L-factors of representations in real and complex groups. Some familiarity with infinite dimensional representations would be very helpful, but I will not assume any knowledge of number theory. We will start in the first lecture by considering L-functions for Groessen characters and classical automorphic forms, in other words for automorphic representations of G(1) and GL(2). This will motivate the definition of the local L-factors of representations of GL(1,R) and GL(2,R). Then we will discuss Rankin convolutions and define the L-factors for infinite dimensional tempered representations of GL(n,R).

### 2013/07/17

#### Kavli IPMU Komaba Seminar

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

Homological Mirror Symmetry for toric Calabi-Yau varieties (ENGLISH)

**Daniel Pomerleano**(Kavli IPMU)Homological Mirror Symmetry for toric Calabi-Yau varieties (ENGLISH)

[ Abstract ]

I will discuss some recent developments in Homological Mirror

Symmetry for toric Calabi-Yau varieties.

I will discuss some recent developments in Homological Mirror

Symmetry for toric Calabi-Yau varieties.

### 2013/07/16

#### Tuesday Seminar on Topology

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

On new models of real hyperbolic spaces (JAPANESE)

**Sumio Yamada**(Gakushuin University)On new models of real hyperbolic spaces (JAPANESE)

[ Abstract ]

In this talk, I will introduce several new realization of the real hyperbolic spaces, using classical tools. The constructions will involve aspects of convex geometry as well as projective geometry, and they are interesting from the view point of the history of mathematics. This work belongs to a joint project with Athanase Papadopoulos.

In this talk, I will introduce several new realization of the real hyperbolic spaces, using classical tools. The constructions will involve aspects of convex geometry as well as projective geometry, and they are interesting from the view point of the history of mathematics. This work belongs to a joint project with Athanase Papadopoulos.

#### Numerical Analysis Seminar

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

Numerical computation of motion of interface networks (JAPANESE)

[ Reference URL ]

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

**Karel Svadlenka**(Kanazawa University)Numerical computation of motion of interface networks (JAPANESE)

[ Reference URL ]

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

### 2013/07/11

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Primitive forms via polyvector fields (ENGLISH)

**Changzheng Li**(IPMU)Primitive forms via polyvector fields (ENGLISH)

[ Abstract ]

The theory of primitive forms was introduced by Kyoji Saito in early 1980s, which was first known in singularity theory and has attracted much attention in mirror symmetry recently. In this talk, we will introduce a differential geometric approach to primitive forms, using compactly supported polyvector fields. We will first introduce the notion of primitive forms, making it acceptable to general audience. We will use the example of the mirror Laudau-Ginzberg model of P^1 to illustrate such approach. This is my joint work with Si Li and Kyoji Saito.

The theory of primitive forms was introduced by Kyoji Saito in early 1980s, which was first known in singularity theory and has attracted much attention in mirror symmetry recently. In this talk, we will introduce a differential geometric approach to primitive forms, using compactly supported polyvector fields. We will first introduce the notion of primitive forms, making it acceptable to general audience. We will use the example of the mirror Laudau-Ginzberg model of P^1 to illustrate such approach. This is my joint work with Si Li and Kyoji Saito.

### 2013/07/10

#### Number Theory Seminar

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On ramification filtration of local fields of equal characteristic (JAPANESE)

**Yuri Yatagawa**(University of Tokyo)On ramification filtration of local fields of equal characteristic (JAPANESE)

### 2013/07/09

#### Tuesday Seminar on Topology

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

Smooth 3-manifolds in the 4-sphere (ENGLISH)

**Ryan Budney**(University of Victoria)Smooth 3-manifolds in the 4-sphere (ENGLISH)

[ Abstract ]

Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.

Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.

#### Tuesday Seminar of Analysis

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

A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian

(Joint work with Georgi Raikov) (ENGLISH)

**Tom\'as Lungenstrass**(Pontificia Universidad Catolica de Chile)A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian

(Joint work with Georgi Raikov) (ENGLISH)

[ Abstract ]

The Landau Hamiltonian describes the dynamics of a two-dimensional

charged particle subject to a constant magnetic field. Its spectrum

consists in eigenvalues of infinite multiplicity given by $B(2q+1)$, $q\\in Z_+$. We

consider perturbations of this operator by including a continuous

electric potential that decays slowly at infinity (as $|x|^{-\\rho}$, $0<\\rho<1$).

The spectrum of the perturbed operator consists of eigenvalue clusters

which accumulate to the Landau levels. We provide estimates for the

rate at which the clusters shrink as we move up the energy levels.

Further, we obtain an explicit description of the asymptotic density

of eigenvalues for asymptotically homogeneous long-range potentials in

terms of a mean-value transform of the associated homogeneous

function.

The Landau Hamiltonian describes the dynamics of a two-dimensional

charged particle subject to a constant magnetic field. Its spectrum

consists in eigenvalues of infinite multiplicity given by $B(2q+1)$, $q\\in Z_+$. We

consider perturbations of this operator by including a continuous

electric potential that decays slowly at infinity (as $|x|^{-\\rho}$, $0<\\rho<1$).

The spectrum of the perturbed operator consists of eigenvalue clusters

which accumulate to the Landau levels. We provide estimates for the

rate at which the clusters shrink as we move up the energy levels.

Further, we obtain an explicit description of the asymptotic density

of eigenvalues for asymptotically homogeneous long-range potentials in

terms of a mean-value transform of the associated homogeneous

function.

### 2013/07/08

#### Seminar on Geometric Complex Analysis

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

Cohomologies and deformations of solvmanifolds (JAPANESE)

**Hisashi Kasuya**(Tokyo Institute of Technology)Cohomologies and deformations of solvmanifolds (JAPANESE)

[ Abstract ]

$G$を単連結可解リー群とし, $G$はココンパクト離散部分群$\Gamma$を持つとする. この時, コンパクト等質空間$G/\Gamma$をsolvmanifoldと呼ぶ. 本講演では, solvmanifoldのde Rhamコホモロジー, Dolbeaultコホモロジー, Bott-Chernコホモロジーの計算法を紹介する. さらにその計算法を用いた, ホッジ理論と変形理論の研究を紹介する.

$G$を単連結可解リー群とし, $G$はココンパクト離散部分群$\Gamma$を持つとする. この時, コンパクト等質空間$G/\Gamma$をsolvmanifoldと呼ぶ. 本講演では, solvmanifoldのde Rhamコホモロジー, Dolbeaultコホモロジー, Bott-Chernコホモロジーの計算法を紹介する. さらにその計算法を用いた, ホッジ理論と変形理論の研究を紹介する.

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Elliptic genera and two dimensional gauge theories (ENGLISH)

**Richard Eager**(Kavli IPMU)Elliptic genera and two dimensional gauge theories (ENGLISH)

[ Abstract ]

The elliptic genus is an important invariant of two dimensional conformal field theories that generalizes the Witten index. In this talk, I will first review the geometric meaning of the elliptic genus and Witten's GLSM construction. Then I will explain how the elliptic genus can be computed directly from a two dimensional gauge theory using localization. The central example of this talk will be the quintic threefold. The GLSM description of the quintic threefold has both a large-volume sigma model description and a Landau-Ginzburg description. I will explain how the GLSM calculation of the index reproduces the old results in these two phases. Time permitting, further applications and generalizations will be discussed.

The elliptic genus is an important invariant of two dimensional conformal field theories that generalizes the Witten index. In this talk, I will first review the geometric meaning of the elliptic genus and Witten's GLSM construction. Then I will explain how the elliptic genus can be computed directly from a two dimensional gauge theory using localization. The central example of this talk will be the quintic threefold. The GLSM description of the quintic threefold has both a large-volume sigma model description and a Landau-Ginzburg description. I will explain how the GLSM calculation of the index reproduces the old results in these two phases. Time permitting, further applications and generalizations will be discussed.

#### Lectures

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Meiji University)

Aggregation mechanism of biological species : from microscopic

and macroscopic viewpoints (JAPANESE)

**Hirofumi Izuhara**(Meiji Institute for Advanced Study of Mathematical Sciences,Meiji University)

Aggregation mechanism of biological species : from microscopic

and macroscopic viewpoints (JAPANESE)

[ Abstract ]

There are a lot of organisms which form aggregation in nature. In order to describe the dynamics of such biological species, a particle model is often proposed, which is based on the random walk from the microscopic point of view. On the other hand, when we take population densities of biological species into account, the dynamics is expressed as partial differential equations. We see that different models are proposed according to the viewpoints which we are focusing on. In this talk, we take aggregation phenomena of biological species as an example, and introduce a relation between a microscopic particle model and a macroscopic partial differential equation model.

There are a lot of organisms which form aggregation in nature. In order to describe the dynamics of such biological species, a particle model is often proposed, which is based on the random walk from the microscopic point of view. On the other hand, when we take population densities of biological species into account, the dynamics is expressed as partial differential equations. We see that different models are proposed according to the viewpoints which we are focusing on. In this talk, we take aggregation phenomena of biological species as an example, and introduce a relation between a microscopic particle model and a macroscopic partial differential equation model.

#### FMSP Lectures

16:15-17:15 Room #270 (Graduate School of Math. Sci. Bldg.)

Two-dimensional Calderon problems for Navier-Stokes equations and Lame system (ENGLISH)

**Oleg Emanouilov**(Colorado State Univ.)Two-dimensional Calderon problems for Navier-Stokes equations and Lame system (ENGLISH)

[ Abstract ]

We will prove the uniqueness in determining viscosity in two-dimensional Navier-Stokes equations by Dirichlet-to-Neumann map.

Moreover, without any smallness assumption, we establish the uniqueness in determining two Lame coefficients in two-dimensional isotropic Lame system Dirichlet-to-Neumann map.

We will prove the uniqueness in determining viscosity in two-dimensional Navier-Stokes equations by Dirichlet-to-Neumann map.

Moreover, without any smallness assumption, we establish the uniqueness in determining two Lame coefficients in two-dimensional isotropic Lame system Dirichlet-to-Neumann map.

### 2013/07/05

#### FMSP Lectures

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

Geometric applications of Wasserstein distance,

Lecture (IV) Applications to differential geometry and foliations (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Walczak.pdf

**Szymon M. Walczak**(University of Lodz, Poland)Geometric applications of Wasserstein distance,

Lecture (IV) Applications to differential geometry and foliations (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Walczak.pdf

### 2013/07/04

#### Seminar on Probability and Statistics

14:50-16:00 Room #052 (Graduate School of Math. Sci. Bldg.)

低ランク行列推定におけるベイズ推定法の性質 (JAPANESE)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/02.html

**SUZUKI, Taiji**(Tokyo Institute of Technology)低ランク行列推定におけるベイズ推定法の性質 (JAPANESE)

[ Abstract ]

真のパラメータが低ランク行列の構造を持つような低ランク行列推定問題を考える. 低ランク行列推定問題の例としては,低ランク行列の一部が見えている時にその残りを 推定する行列補完の問題などがある.応用としてはユーザへの推薦システムなどがある. これまでの理論解析は主にスパース正則化を用いた経験誤差最小化を対象としてきたが, 本発表ではベイズ法を考え,その統計的性質を調べる.ベイズ法においては, 正則化付き経験誤差最小化による方法とは異なるやや緩い仮定のもと, ほぼ最適な収束レートが導けることを示す.また,テンソル型データ (多次元アレイデータ)へも同様の議論が拡張可能であることも述べる.

[ Reference URL ]真のパラメータが低ランク行列の構造を持つような低ランク行列推定問題を考える. 低ランク行列推定問題の例としては,低ランク行列の一部が見えている時にその残りを 推定する行列補完の問題などがある.応用としてはユーザへの推薦システムなどがある. これまでの理論解析は主にスパース正則化を用いた経験誤差最小化を対象としてきたが, 本発表ではベイズ法を考え,その統計的性質を調べる.ベイズ法においては, 正則化付き経験誤差最小化による方法とは異なるやや緩い仮定のもと, ほぼ最適な収束レートが導けることを示す.また,テンソル型データ (多次元アレイデータ)へも同様の議論が拡張可能であることも述べる.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2013/02.html

### 2013/07/03

#### Number Theory Seminar

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

A general formula for the discriminant of polynomials over $¥mathbb{F}_2$ determining the parity of the number of prime factors

(JAPANESE)

**Takehito Yoshiki**(University of Tokyo)A general formula for the discriminant of polynomials over $¥mathbb{F}_2$ determining the parity of the number of prime factors

(JAPANESE)

[ Abstract ]

In order to find irreducible polynomials over $\\mathbb{F}_2$ efficiently, the method using Swan's theorem is known. Swan's theorem determines the parity of the numberof irreducible factors of a polynomial $f$ over $\\mathbb{F}_2$ with no repeated root, by using the discriminant ${\\rm D}(\\tilde{f})\\pmod 8$, where $\\tilde{f}$ is a monic polynomial over $\\mathbb{Z}_2$ such that $\\tilde{f}=f\\pmod 2$. In the lecture, we will give the formula for the discriminant ${\\rm D}(\\tilde{f}) \\pmod 8$ for a polynomial $f$ over $\\mathbb{F}_2$ with no repeated root. By applying this formula to various types of polynomials, we shall get the parity of the number of irreducible factors of them.

In order to find irreducible polynomials over $\\mathbb{F}_2$ efficiently, the method using Swan's theorem is known. Swan's theorem determines the parity of the numberof irreducible factors of a polynomial $f$ over $\\mathbb{F}_2$ with no repeated root, by using the discriminant ${\\rm D}(\\tilde{f})\\pmod 8$, where $\\tilde{f}$ is a monic polynomial over $\\mathbb{Z}_2$ such that $\\tilde{f}=f\\pmod 2$. In the lecture, we will give the formula for the discriminant ${\\rm D}(\\tilde{f}) \\pmod 8$ for a polynomial $f$ over $\\mathbb{F}_2$ with no repeated root. By applying this formula to various types of polynomials, we shall get the parity of the number of irreducible factors of them.

### 2013/07/02

#### Numerical Analysis Seminar

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

Numerical plasma simulation for reactive plasma deposition (JAPANESE)

[ Reference URL ]

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

**Masaru Miyashita**(Sumitomo Heavy Industries, Ltd.)Numerical plasma simulation for reactive plasma deposition (JAPANESE)

[ Reference URL ]

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

### 2013/06/29

#### Harmonic Analysis Komaba Seminar

13:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Critical Sobolev embedding of function spaces and the real interpolation functor

(JAPANESE)

On weighted estimates for multilinear Fourier multipliers with Sobolev regularity

(JAPANESE)

**Yoshihiro Sawano**(Tokyo Metropolitan University) 13:30-15:00Critical Sobolev embedding of function spaces and the real interpolation functor

(JAPANESE)

[ Abstract ]

We consider the endpoint case of the Sobolev embedding.

It is well known that the function spaces such as Sobolev spaces are not embedded into L^¥infty in the critical case.

One of the remedies is the Brezis-Gallouet-Wainger type

estimate. However, such an estimate involve the log term

and it can not be regarded as the norm.

In this talk, by using the real interpolation functor, we propose another formulation. We compare

the existing result with our new results.

If time permits, we mention some related results.

We consider the endpoint case of the Sobolev embedding.

It is well known that the function spaces such as Sobolev spaces are not embedded into L^¥infty in the critical case.

One of the remedies is the Brezis-Gallouet-Wainger type

estimate. However, such an estimate involve the log term

and it can not be regarded as the norm.

In this talk, by using the real interpolation functor, we propose another formulation. We compare

the existing result with our new results.

If time permits, we mention some related results.

**Mai Fujita**(Osaka University) 15:30-17:00On weighted estimates for multilinear Fourier multipliers with Sobolev regularity

(JAPANESE)

### 2013/06/28

#### FMSP Lectures

16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Mathematical model for the electrodiffusion of ions, Lecture II (JAPANESE)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~fmsp/files/FMSPLectures_Mori.pdf

**Yoichiro Mori**(University of Minnesota)Mathematical model for the electrodiffusion of ions, Lecture II (JAPANESE)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~fmsp/files/FMSPLectures_Mori.pdf

#### Colloquium

16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

The Geometry of protain modelling (JAPANESE)

**Hiroki Kodama**(Graduate School of Mathematical Sciences, The University of Tokyo)The Geometry of protain modelling (JAPANESE)

### 2013/06/27

#### FMSP Lectures

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

Geometric applications of Wasserstein distance,

Lecture (III) Curvature of metric measure spaces II

(ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Walczak.pdf

**Szymon M. Walczak**(University of Lodz, Poland)Geometric applications of Wasserstein distance,

Lecture (III) Curvature of metric measure spaces II

(ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Walczak.pdf

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On volume formulae in terms of orthospectrum (JAPANESE)

**MASAI, Hidetoshi**(Tokyo Institute of Technology)On volume formulae in terms of orthospectrum (JAPANESE)

[ Abstract ]

Bridgeman-Kahn and Calegari derived formulae to compute the volumes of compact hyperbolic n-manifolds with totally geodesic boundary in terms of orthospectrum. Here the orthospectrum is the set of length of geodesics perpendicular to the boundary at both ends. The two formulae are obtained by apparently different methods. In this talk, we prove that the two volume formulae coincide. We also discuss some interesting relationship between two formulae. This work is a joint work with Greg McShane.

Bridgeman-Kahn and Calegari derived formulae to compute the volumes of compact hyperbolic n-manifolds with totally geodesic boundary in terms of orthospectrum. Here the orthospectrum is the set of length of geodesics perpendicular to the boundary at both ends. The two formulae are obtained by apparently different methods. In this talk, we prove that the two volume formulae coincide. We also discuss some interesting relationship between two formulae. This work is a joint work with Greg McShane.

#### FMSP Lectures

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Mathematical model for the electrodiffusion of ions, Lecture I (JAPANESE)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~fmsp/files/FMSPLectures_Mori.pdf

**Yoichiro Mori**(University of Minnesota)Mathematical model for the electrodiffusion of ions, Lecture I (JAPANESE)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~fmsp/files/FMSPLectures_Mori.pdf

### 2013/06/26

#### Number Theory Seminar

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

An explicit construction of point sets with large minimum Dick weight (JAPANESE)

**Kousuke Suzuki**(University of Tokyo)An explicit construction of point sets with large minimum Dick weight (JAPANESE)

[ Abstract ]

Walsh figure of merit WAFOM($P$) is a quality measure of point sets $P$ for quasi-Monte Carlo integration constructed by a digital net method. WAFOM($P$) is bounded by the minimum Dick weight of $P^¥perp$, where the Dick weight is a generalization of Hamming weight. In this talk, we give an explicit construction of point sets with large minimum Dick weight using Niederreiter-Xing sequences and Dick's interleaving construction. These point sets are also examples of low-WAFOM point sets.

Walsh figure of merit WAFOM($P$) is a quality measure of point sets $P$ for quasi-Monte Carlo integration constructed by a digital net method. WAFOM($P$) is bounded by the minimum Dick weight of $P^¥perp$, where the Dick weight is a generalization of Hamming weight. In this talk, we give an explicit construction of point sets with large minimum Dick weight using Niederreiter-Xing sequences and Dick's interleaving construction. These point sets are also examples of low-WAFOM point sets.

### 2013/06/25

#### Tuesday Seminar on Topology

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

Higher-order generalization of Fukaya's Morse homotopy

invariant of 3-manifolds (JAPANESE)

**Tadayuki Watanabe**(Shimane University)Higher-order generalization of Fukaya's Morse homotopy

invariant of 3-manifolds (JAPANESE)

[ Abstract ]

In his article published in 1996, K. Fukaya constructed

a 3-manifold invariant by using Morse homotopy theory. Roughly, his

invariant is defined by considering several Morse functions on a

3-manifold and counting with weights the ways that the theta-graph can

be immersed such that edges follow gradient lines. We generalize his

construction to 3-valent graphs with arbitrary number of loops for

integral homology 3-spheres. I will also discuss extension of our method

to 3-manifolds with positive first Betti numbers.

In his article published in 1996, K. Fukaya constructed

a 3-manifold invariant by using Morse homotopy theory. Roughly, his

invariant is defined by considering several Morse functions on a

3-manifold and counting with weights the ways that the theta-graph can

be immersed such that edges follow gradient lines. We generalize his

construction to 3-valent graphs with arbitrary number of loops for

integral homology 3-spheres. I will also discuss extension of our method

to 3-manifolds with positive first Betti numbers.

#### Numerical Analysis Seminar

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

Digital watermarking methods using the wavelet transforms and interval arithmetic (JAPANESE)

[ Reference URL ]

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

**Teruya Minamoto**(Saga University)Digital watermarking methods using the wavelet transforms and interval arithmetic (JAPANESE)

[ Reference URL ]

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

< Previous 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138 Next >