## 過去の記録

過去の記録 ～03/31｜本日 04/01 | 今後の予定 04/02～

### 2014年04月09日(水)

#### 作用素環セミナー

16:30-18:00 数理科学研究科棟(駒場) 122号室

Index and determnant of n-tuples of commuting operators (ENGLISH)

**Ryszard Nest 氏**(Univ. Copenhagen)Index and determnant of n-tuples of commuting operators (ENGLISH)

### 2014年04月08日(火)

#### トポロジー火曜セミナー

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

On the number of commensurable fibrations on a hyperbolic 3-manifold. (JAPANESE)

Tea: 16:00 - 16:30 コモンルーム

**正井 秀俊 氏**(東京大学大学院数理科学研究科)On the number of commensurable fibrations on a hyperbolic 3-manifold. (JAPANESE)

[ 講演概要 ]

By work of Thurston, it is known that if a hyperbolic fibred

$3$-manifold $M$ has Betti number greater than 1, then

$M$ admits infinitely many distinct fibrations.

For any fibration $\\omega$ on a hyperbolic $3$-manifold $M$,

the number of fibrations on $M$ that are commensurable in the sense of

Calegari-Sun-Wang to $\\omega$ is known to be finite.

In this talk, we prove that the number can be arbitrarily large.

By work of Thurston, it is known that if a hyperbolic fibred

$3$-manifold $M$ has Betti number greater than 1, then

$M$ admits infinitely many distinct fibrations.

For any fibration $\\omega$ on a hyperbolic $3$-manifold $M$,

the number of fibrations on $M$ that are commensurable in the sense of

Calegari-Sun-Wang to $\\omega$ is known to be finite.

In this talk, we prove that the number can be arbitrarily large.

#### 統計数学セミナー

13:00-14:10 数理科学研究科棟(駒場) 052号室

Parametric estimation in fractional Ornstein-Uhlenbeck process (ENGLISH)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/00.html

**Alexandre Brouste 氏**(Universite du Maine, France)Parametric estimation in fractional Ornstein-Uhlenbeck process (ENGLISH)

[ 講演概要 ]

Several statistical models that imply the fractional Ornstein-Uhlenbeck (fOU) process will be presented: direct observations of the process or partial observations in an additive independent noise, continuous observations or discrete observations. In this different settings, we exhibit large sample (or high-frequency) asymptotic properties of the estimators (maximum likelihood estimator, quadratic variation based estimator, moment estimator, …) for all parameters of interest of the fOU. We also illustrate our results with the R package yuima.

[ 参考URL ]Several statistical models that imply the fractional Ornstein-Uhlenbeck (fOU) process will be presented: direct observations of the process or partial observations in an additive independent noise, continuous observations or discrete observations. In this different settings, we exhibit large sample (or high-frequency) asymptotic properties of the estimators (maximum likelihood estimator, quadratic variation based estimator, moment estimator, …) for all parameters of interest of the fOU. We also illustrate our results with the R package yuima.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/00.html

### 2014年03月19日(水)

#### 古典解析セミナー

16:00-17:00 数理科学研究科棟(駒場) 128号室

Discrete Schlesinger Equations and Difference Painlevé Equations (ENGLISH)

**Anton Dzhamay 氏**(University of Northern Colorado)Discrete Schlesinger Equations and Difference Painlevé Equations (ENGLISH)

[ 講演概要 ]

The theory of Schlesinger equations describing isomonodromic

dynamic on the space of matrix coefficients of a Fuchsian system

w.r.t.~continuous deformations is well-know. In this talk we consider

a discrete version of this theory. Discrete analogues of Schlesinger

deformations are Schlesinger transformations that shift the eigenvalues

of the coefficient matrices by integers. By discrete Schlesinger equations

we mean the evolution equations on the matrix coefficients describing

such transformations. We derive these equations, show how they can be

split into the evolution equations on the space of eigenvectors of the

coefficient matrices, and explain how to write the latter equations in

the discrete Hamiltonian form. We also consider some reductions of those

equations to the difference Painlevé equations, again in complete parallel

to the differential case.

This is a joint work with H. Sakai (the University of Tokyo) and

T.Takenawa (Tokyo Institute of Marine Science and Technology).

The theory of Schlesinger equations describing isomonodromic

dynamic on the space of matrix coefficients of a Fuchsian system

w.r.t.~continuous deformations is well-know. In this talk we consider

a discrete version of this theory. Discrete analogues of Schlesinger

deformations are Schlesinger transformations that shift the eigenvalues

of the coefficient matrices by integers. By discrete Schlesinger equations

we mean the evolution equations on the matrix coefficients describing

such transformations. We derive these equations, show how they can be

split into the evolution equations on the space of eigenvectors of the

coefficient matrices, and explain how to write the latter equations in

the discrete Hamiltonian form. We also consider some reductions of those

equations to the difference Painlevé equations, again in complete parallel

to the differential case.

This is a joint work with H. Sakai (the University of Tokyo) and

T.Takenawa (Tokyo Institute of Marine Science and Technology).

### 2014年03月14日(金)

#### GCOEセミナー

16:00-16:50 数理科学研究科棟(駒場) 118号室

A new finite difference scheme based on staggered grids for Navier Stokes equations (ENGLISH)

**Kazufumi Ito 氏**(North Carolina State Univ.)A new finite difference scheme based on staggered grids for Navier Stokes equations (ENGLISH)

[ 講演概要 ]

We develop a new method that uses the staggered grid only for the pressure node, i.e., the pressure gird is the center of the square cell and the velocities are at the node. The advantage of the proposed method compared to the standard staggered grid methods is that it is very straight forward to treat the boundary conditions for the velocity field, the fluid structure interaction, and to deal with the multiphase flow using the immersed interface methods. We present our analysis and numerical tests.

We develop a new method that uses the staggered grid only for the pressure node, i.e., the pressure gird is the center of the square cell and the velocities are at the node. The advantage of the proposed method compared to the standard staggered grid methods is that it is very straight forward to treat the boundary conditions for the velocity field, the fluid structure interaction, and to deal with the multiphase flow using the immersed interface methods. We present our analysis and numerical tests.

#### GCOEセミナー

17:00-17:50 数理科学研究科棟(駒場) 118号室

Efficient Domain Decomposition Methods for a Class of Linear and Nonlinear Inverse Problems (ENGLISH)

**Jun Zou 氏**(The Chinese University of Hong Kong)Efficient Domain Decomposition Methods for a Class of Linear and Nonlinear Inverse Problems (ENGLISH)

[ 講演概要 ]

In this talk we shall present several new domain decomposition methods for solving some linear and nonlinear inverse problems. The motivations and derivations of the methods will be discussed, and numerical experiments will be demonstrated.

In this talk we shall present several new domain decomposition methods for solving some linear and nonlinear inverse problems. The motivations and derivations of the methods will be discussed, and numerical experiments will be demonstrated.

### 2014年03月13日(木)

#### 講演会

10:15-11:45 数理科学研究科棟(駒場) 470号室

Almost sure triviality of the $C^1$-centralizer of random circle diffeomorphisms with periodic points (ENGLISH)

**Michele Triestino 氏**(Ecole Normale Superieure de Lyon)Almost sure triviality of the $C^1$-centralizer of random circle diffeomorphisms with periodic points (ENGLISH)

[ 講演概要 ]

By the end of the 80s, Malliavin and Shavgulidze introduced a measure on the space of C^1 circle diffeomorphisms which carries many interesting features. Perhaps the most interesting aspect is that it can be considered as an analog of the Haar measure for the group Diff^1_+(S^1).

The nature of this measure has been mostly investigated in connection to representation theory.

For people working in dynamical systems, the MS measure offers a way to quantify dynamical phenomena: for example, which is the probability that a random diffeomorphism is irrational? Even if this question have occupied my mind for a long time, it remains still unanswered, as many other interesting ones. However, it is possible to understand precisely what are the typical features of a diffeomorphism with periodic points.

By the end of the 80s, Malliavin and Shavgulidze introduced a measure on the space of C^1 circle diffeomorphisms which carries many interesting features. Perhaps the most interesting aspect is that it can be considered as an analog of the Haar measure for the group Diff^1_+(S^1).

The nature of this measure has been mostly investigated in connection to representation theory.

For people working in dynamical systems, the MS measure offers a way to quantify dynamical phenomena: for example, which is the probability that a random diffeomorphism is irrational? Even if this question have occupied my mind for a long time, it remains still unanswered, as many other interesting ones. However, it is possible to understand precisely what are the typical features of a diffeomorphism with periodic points.

#### GCOEセミナー

17:00-18:00 数理科学研究科棟(駒場) 122号室

STABILITY IN THE OBSTACLE PROBLEM FOR A SHALLOW SHELL (ENGLISH)

[ 参考URL ]

http://fmsp.ms.u-tokyo.ac.jp/bm02.pdf

**Bernadette Miara 氏**(Univ. Paris-Est)STABILITY IN THE OBSTACLE PROBLEM FOR A SHALLOW SHELL (ENGLISH)

[ 参考URL ]

http://fmsp.ms.u-tokyo.ac.jp/bm02.pdf

### 2014年03月12日(水)

#### 講演会

10:15-11:45 数理科学研究科棟(駒場) 470号室

Invariant distributions for circle diffeomorphisms of

irrational rotation number and low regularity (ENGLISH)

**Michele Triestino 氏**(Ecole Normale Superieure de Lyon)Invariant distributions for circle diffeomorphisms of

irrational rotation number and low regularity (ENGLISH)

[ 講演概要 ]

The main inspiration of this joint work with Andrés Navas is the beautiful result of Ávila and Kocsard: if f is a C^\\infty circle diffeomorphism of irrational rotation number, then the unique invariant probability measure is also the unique (up to rescaling) invariant distribution.

Using conceptual geometric arguments (Hahn-Banach...), we investigate the uniqueness of invariant distributions for C^1 circle diffeomorphisms of irrational rotation number, with particular attention to sharp regularity.

We prove that If the diffeomorphism is C^{1+bv}, then there is a unique invariant distribution of order 1. On the other side, examples by Douady and Yoccoz, and by Kodama and Matsumoto exhibit differentiable dynamical systems for which the uniqueness does not hold.

The main inspiration of this joint work with Andrés Navas is the beautiful result of Ávila and Kocsard: if f is a C^\\infty circle diffeomorphism of irrational rotation number, then the unique invariant probability measure is also the unique (up to rescaling) invariant distribution.

Using conceptual geometric arguments (Hahn-Banach...), we investigate the uniqueness of invariant distributions for C^1 circle diffeomorphisms of irrational rotation number, with particular attention to sharp regularity.

We prove that If the diffeomorphism is C^{1+bv}, then there is a unique invariant distribution of order 1. On the other side, examples by Douady and Yoccoz, and by Kodama and Matsumoto exhibit differentiable dynamical systems for which the uniqueness does not hold.

#### 数理人口学・数理生物学セミナー

15:00-17:00 数理科学研究科棟(駒場) 126号室

When size does matter: Ontogenetic symmetry and asymmetry in energetics

(ENGLISH)

http://staff.science.uva.nl/~aroos/

**Andre M. de Roos 氏**(University of Amsterdam)When size does matter: Ontogenetic symmetry and asymmetry in energetics

(ENGLISH)

[ 講演概要 ]

Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as “ontogenetic symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

[ 参考URL ]Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as “ontogenetic symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

http://staff.science.uva.nl/~aroos/

#### GCOEセミナー

15:00-17:00 数理科学研究科棟(駒場) 126号室

When size does matter: Ontogenetic symmetry and asymmetry in energetics (ENGLISH)

http://staff.science.uva.nl/~aroos/

**Andre M. de Roos 氏**(University of Amsterdam)When size does matter: Ontogenetic symmetry and asymmetry in energetics (ENGLISH)

[ 講演概要 ]

Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as “ontogenetic symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

[ 参考URL ]Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as “ontogenetic symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

http://staff.science.uva.nl/~aroos/

### 2014年03月11日(火)

#### GCOEセミナー

17:00-18:00 数理科学研究科棟(駒場) 118号室

Inverse problem for the waves : stability and convergence matters (ENGLISH)

**Lucie Baudouin 氏**(LAAS-CNRS, equipe MAC)Inverse problem for the waves : stability and convergence matters (ENGLISH)

[ 講演概要 ]

This talk aims to present some recent works in collaboration with Maya de Buhan, Sylvain Ervedoza and Axel Osses regarding an inverse problem for the wave equation. More specifically, we study the determination of the potential in a wave equation with given Dirichlet boundary data from a measurement of the flux of the solution on a part of the boundary. On the one hand, we will focus on the question of convergence of the space semi-discrete inverse problems toward their continuous counterpart. Several uniqueness and stability results are available in the literature about the continuous setting of the inverse problem of determination of a potential in the wave equation. In particular, we can mention a Lipschitz stability result under a classical geometric condition obtained by Imanuvilov and Yamamoto, and a logarithmic stability result obtained by Bellassoued when the observation measurement is made on an arbitrary part of the boundary. In both situations, we can design a numerical process for which convergence results are proved. The analysis we conduct is based on discrete Carleman estimates, either for the hyperbolic or for the elliptic operator, in which case we shall use a result of Boyer, Hubert and Le Rousseau. On the other hand, still considering the same inverse problem, we will present a new reconstruction algorithm of the potential. The design and convergence of the algorithm are based on the Carleman estimates for the waves previously used to prove the Lipschitz stability. We will finally give some simple illustrative numerical simulations for 1-d problems.

This talk aims to present some recent works in collaboration with Maya de Buhan, Sylvain Ervedoza and Axel Osses regarding an inverse problem for the wave equation. More specifically, we study the determination of the potential in a wave equation with given Dirichlet boundary data from a measurement of the flux of the solution on a part of the boundary. On the one hand, we will focus on the question of convergence of the space semi-discrete inverse problems toward their continuous counterpart. Several uniqueness and stability results are available in the literature about the continuous setting of the inverse problem of determination of a potential in the wave equation. In particular, we can mention a Lipschitz stability result under a classical geometric condition obtained by Imanuvilov and Yamamoto, and a logarithmic stability result obtained by Bellassoued when the observation measurement is made on an arbitrary part of the boundary. In both situations, we can design a numerical process for which convergence results are proved. The analysis we conduct is based on discrete Carleman estimates, either for the hyperbolic or for the elliptic operator, in which case we shall use a result of Boyer, Hubert and Le Rousseau. On the other hand, still considering the same inverse problem, we will present a new reconstruction algorithm of the potential. The design and convergence of the algorithm are based on the Carleman estimates for the waves previously used to prove the Lipschitz stability. We will finally give some simple illustrative numerical simulations for 1-d problems.

### 2014年03月10日(月)

#### GCOEセミナー

13:30-15:00 数理科学研究科棟(駒場) 123号室

The two-dimensional random walk in an isotropic random environment (ENGLISH)

**Erwin Bolthausen 氏**(University of Zurich)The two-dimensional random walk in an isotropic random environment (ENGLISH)

#### 講演会

15:15-16:45 数理科学研究科棟(駒場) 123号室

Energy fluctuations in the disordered harmonic chain (ENGLISH)

**Marielle Simon 氏**(ENS Lyon, UMPA)Energy fluctuations in the disordered harmonic chain (ENGLISH)

[ 講演概要 ]

We study the energy diffusion in the disordered harmonic chain of oscillators: the usual Hamiltonian dynamics is provided with random masses and perturbed by a degenerate energy conserving noise. After rescaling space and time diffusively, we prove that energy fluctuations evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and an equivalent definition through the Green-Kubo formula. Since the model is non gradient, and the perturbation is very degenerate, the standard Varadhan's approach is reviewed under new perspectives.

We study the energy diffusion in the disordered harmonic chain of oscillators: the usual Hamiltonian dynamics is provided with random masses and perturbed by a degenerate energy conserving noise. After rescaling space and time diffusively, we prove that energy fluctuations evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and an equivalent definition through the Green-Kubo formula. Since the model is non gradient, and the perturbation is very degenerate, the standard Varadhan's approach is reviewed under new perspectives.

#### GCOEセミナー

17:00-18:00 数理科学研究科棟(駒場) 118号室

ELASTIC WAVES IN STRONGLY HETEROGENEOUS PLATES (ENGLISH)

[ 参考URL ]

http://fmsp.ms.u-tokyo.ac.jp/bm01.pdf

**Bernadette Miara 氏**(Univ. Paris-Est)ELASTIC WAVES IN STRONGLY HETEROGENEOUS PLATES (ENGLISH)

[ 参考URL ]

http://fmsp.ms.u-tokyo.ac.jp/bm01.pdf

### 2014年03月06日(木)

#### GCOEセミナー

10:10-11:00 数理科学研究科棟(駒場) 002号室

このセミナーは 研究集会「異常拡散の数理とシミュレーション手法ならびに関連する課題」(主催・共催:数物フロンティア・リーディング大学院プログラム、卓越した大学院拠点形成支援補

The Mathematical Model for the Contamination Problems and Related Inverse Problems (ENGLISH)

このセミナーは 研究集会「異常拡散の数理とシミュレーション手法ならびに関連する課題」(主催・共催:数物フロンティア・リーディング大学院プログラム、卓越した大学院拠点形成支援補

**J. Cheng 氏**(Fudan Univ.)The Mathematical Model for the Contamination Problems and Related Inverse Problems (ENGLISH)

[ 講演概要 ]

In this talk, we discuss the motivation for the study of the abnormal diffusion models for the contamination problems. From the practical point of view, several inverse problems proposed and studied Theoretic results, for example, the uniqueness and stability are shown. The possibility of the application of these studies is mentioned.

In this talk, we discuss the motivation for the study of the abnormal diffusion models for the contamination problems. From the practical point of view, several inverse problems proposed and studied Theoretic results, for example, the uniqueness and stability are shown. The possibility of the application of these studies is mentioned.

#### GCOEセミナー

11:10-12:00 数理科学研究科棟(駒場) 002号室

このセミナーは 研究集会「異常拡散の数理とシミュレーション手法ならびに関連する課題」(主催・共催:数物フロンティア・リーディング大学院プログラム、卓越した大学院拠点形成支援補

Inverse problems with fractional derivatives in the space variable (ENGLISH)

このセミナーは 研究集会「異常拡散の数理とシミュレーション手法ならびに関連する課題」(主催・共催:数物フロンティア・リーディング大学院プログラム、卓越した大学院拠点形成支援補

**W. Rundell 氏**(Texas A&M Univ.)Inverse problems with fractional derivatives in the space variable (ENGLISH)

[ 講演概要 ]

This talk will look at some classical inverse problems where the highest order term in the spatial direction is replaced by a fractional derivative. There are again some surprising results over what is known in the usual case of integer order derivatives but also quite different from the case of fractional diffusion in the time variable. The talk will give some answers, but pose many more open problems.

This talk will look at some classical inverse problems where the highest order term in the spatial direction is replaced by a fractional derivative. There are again some surprising results over what is known in the usual case of integer order derivatives but also quite different from the case of fractional diffusion in the time variable. The talk will give some answers, but pose many more open problems.

#### GCOEセミナー

14:30-15:20 数理科学研究科棟(駒場) 002号室

このセミナーは 研究集会「異常拡散の数理とシミュレーション手法ならびに関連する課題」(主催・共催:数物フロンティア・リーディング大学院プログラム、卓越した大学院拠点形成支援補

Neutral-fractional diffusion-wave equation and some properties of its fundamental solution (ENGLISH)

このセミナーは 研究集会「異常拡散の数理とシミュレーション手法ならびに関連する課題」(主催・共催:数物フロンティア・リーディング大学院プログラム、卓越した大学院拠点形成支援補

**Y. Luchko 氏**(Beuth Technical University of Applied Sciences)Neutral-fractional diffusion-wave equation and some properties of its fundamental solution (ENGLISH)

[ 講演概要 ]

Recently, the so called neutral-fractional diffusion-wave equation was introduced and analysed in the case of one spatial variable. In contrast to the fractional diffusion of diffusion-wave equations, the neutral-fractional diffusion-wave equation contains fractional derivatives of the same order both in space and in time. The fundamental solution of the neutral-fractional diffusion-wave equation was shown to exhibit properties of both the solutions of the diffusion equation and those of the wave equation.

In the one-dimensional case, the fundamental solution of the neutral-fractional diffusion-wave equation can be interpreted as a spatial probability density function evolving in time. At the same time, it can be treated as a damped wave whose amplitude maximum and the gravity and mass centres propagate with the constant velocities that depend just on the equation order.

In this talk, the problems mentioned above are considered for the multi- dimensional neutral-fractional diffusion-wave equation. To illustrate analytical findings, some results of numerical calculations and plots are presented.

Recently, the so called neutral-fractional diffusion-wave equation was introduced and analysed in the case of one spatial variable. In contrast to the fractional diffusion of diffusion-wave equations, the neutral-fractional diffusion-wave equation contains fractional derivatives of the same order both in space and in time. The fundamental solution of the neutral-fractional diffusion-wave equation was shown to exhibit properties of both the solutions of the diffusion equation and those of the wave equation.

In the one-dimensional case, the fundamental solution of the neutral-fractional diffusion-wave equation can be interpreted as a spatial probability density function evolving in time. At the same time, it can be treated as a damped wave whose amplitude maximum and the gravity and mass centres propagate with the constant velocities that depend just on the equation order.

In this talk, the problems mentioned above are considered for the multi- dimensional neutral-fractional diffusion-wave equation. To illustrate analytical findings, some results of numerical calculations and plots are presented.

#### GCOEセミナー

16:20-17:10 数理科学研究科棟(駒場) 002号室

このセミナーは 研究集会「異常拡散の数理とシミュレーション手法ならびに関連する課題」(主催・共催:数物フロンティア・リーディング大学院プログラム、卓越した大学院拠点形成支援補

New kind of observations in an inverse parabolic problem (ENGLISH)

このセミナーは 研究集会「異常拡散の数理とシミュレーション手法ならびに関連する課題」(主催・共催:数物フロンティア・リーディング大学院プログラム、卓越した大学院拠点形成支援補

**M. Cristofol 氏**(Aix-Marseille Univ.)New kind of observations in an inverse parabolic problem (ENGLISH)

[ 講演概要 ]

In this talk, I analyze the inverse problem of determining the reaction term f(x,u) in reaction-diffusion equations of the form ¥partial_t u-D¥partial_{xx}u = f(x,u), where f is assumed to be periodic with respect to x in R. Starting from a family of exponentially decaying initial conditions u_{0,¥lambda}, I will show that the solutions u_¥lambda of this equation propagate with constant asymptotic spreading speeds w_¥lambda. The main result shows that the linearization of f around the steady state 0,¥partial_u f(x,0), is uniquely determined (up to a symmetry) among a subset of piecewise linear functions, by the observation of the asymptotic spreading speeds w_¥lambda.

In this talk, I analyze the inverse problem of determining the reaction term f(x,u) in reaction-diffusion equations of the form ¥partial_t u-D¥partial_{xx}u = f(x,u), where f is assumed to be periodic with respect to x in R. Starting from a family of exponentially decaying initial conditions u_{0,¥lambda}, I will show that the solutions u_¥lambda of this equation propagate with constant asymptotic spreading speeds w_¥lambda. The main result shows that the linearization of f around the steady state 0,¥partial_u f(x,0), is uniquely determined (up to a symmetry) among a subset of piecewise linear functions, by the observation of the asymptotic spreading speeds w_¥lambda.

### 2014年02月28日(金)

#### GCOEセミナー

13:00-14:00 数理科学研究科棟(駒場) 002号室

Probing for inclusions for heat conductive bodies. Time independent and time dependent cases (ENGLISH)

**Patricia Gaitan 氏**(Aix-Marseille University)Probing for inclusions for heat conductive bodies. Time independent and time dependent cases (ENGLISH)

[ 講演概要 ]

This work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1{3 for probing the discontinuous part of the conductivity from local temperature and heat ow measurements at the boundary. The ap- proach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is pos- sible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering loca- tions of discontinuities approximately from noisy data.

For moving inclusions, we consider an inverse boundary value problem for the heat equation on the interval (0; 1), where the heat conductivity (t; x) is piecewise constant and the point of discontinuity depends on time : (t; x) = k2 (0 < x < s(t)), (t; x) = 1 (s(t) < x < 1). Firstly we show that k and s(t) on the time interval [0; T] are determined from the partial Dirichlet- to-Neumann map : u(t; 1) ! @xu(t; 1); 0 < t < T, u(t; x) being the solu- tion to the heat equation such that u(t; 0) = 0, independently of the initial data u(0; x). Secondly we show that the partial Dirichlet-to-Neumann map u(t; 0) ! @xu(t; 1); 0 < t < T, u(t; x) being the solution to the heat equation such that u(t; 1) = 0, determines at most two couples (k; s(t)) on the time interval [0; T], independently of the initial data u(0; x).

This work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1{3 for probing the discontinuous part of the conductivity from local temperature and heat ow measurements at the boundary. The ap- proach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is pos- sible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering loca- tions of discontinuities approximately from noisy data.

For moving inclusions, we consider an inverse boundary value problem for the heat equation on the interval (0; 1), where the heat conductivity (t; x) is piecewise constant and the point of discontinuity depends on time : (t; x) = k2 (0 < x < s(t)), (t; x) = 1 (s(t) < x < 1). Firstly we show that k and s(t) on the time interval [0; T] are determined from the partial Dirichlet- to-Neumann map : u(t; 1) ! @xu(t; 1); 0 < t < T, u(t; x) being the solu- tion to the heat equation such that u(t; 0) = 0, independently of the initial data u(0; x). Secondly we show that the partial Dirichlet-to-Neumann map u(t; 0) ! @xu(t; 1); 0 < t < T, u(t; x) being the solution to the heat equation such that u(t; 1) = 0, determines at most two couples (k; s(t)) on the time interval [0; T], independently of the initial data u(0; x).

#### GCOEセミナー

11:00-12:00 数理科学研究科棟(駒場) 002号室

Minimal time for the null controllability of parabolic systems: the effect of the index of condensation of complex sequences (ENGLISH)

**Assia Benabdallah 氏**(Aix-Marseille University)Minimal time for the null controllability of parabolic systems: the effect of the index of condensation of complex sequences (ENGLISH)

### 2014年02月27日(木)

#### GCOEセミナー

17:00-18:00 数理科学研究科棟(駒場) 370号室

A Carleman estimate with discontinuous coefficients through an interface crossing the boundary, Part II: for a an anisotropic elliptic operator. (ENGLISH)

**Yves Dermenjian 氏**(Univ. of Marseille)A Carleman estimate with discontinuous coefficients through an interface crossing the boundary, Part II: for a an anisotropic elliptic operator. (ENGLISH)

### 2014年02月18日(火)

#### GCOEセミナー

17:00-18:00 数理科学研究科棟(駒場) 370号室

A Carleman estimate with discontinuous coefficients through an interface crossing the boundary, Part I: for a stratified parabolic operator. (ENGLISH)

**Yves Dermenjian 氏**(Univ. of Marseille)A Carleman estimate with discontinuous coefficients through an interface crossing the boundary, Part I: for a stratified parabolic operator. (ENGLISH)

### 2014年02月17日(月)

#### 講演会

16:30-17:30 数理科学研究科棟(駒場) 002号室

Canceled!! (本人の都合により,講演は取りやめとなりました.)

Canceled!! (ENGLISH)

Canceled!! (本人の都合により,講演は取りやめとなりました.)

**Ratnasingham Shivaji 氏**(The University of North Carolina at Greensboro)Canceled!! (ENGLISH)

[ 講演概要 ]

Canceled!!

Canceled!!

### 2014年02月15日(土)

#### 調和解析駒場セミナー

13:00-18:00 数理科学研究科棟(駒場) 128号室

このセミナーは,月に1度程度,不定期に開催されます.

Some Hilbert-type inequalities involving the Hardy operator (ENGLISH)

弱空間に於ける条件付き平均作用素の一様有界性とその応用 (JAPANESE)

このセミナーは,月に1度程度,不定期に開催されます.

**Batbold Tserendorj 氏**(National University of Mongolia) 13:30-15:00Some Hilbert-type inequalities involving the Hardy operator (ENGLISH)

[ 講演概要 ]

In this talk,

we give some new Hilbert-type inequalities with a homogeneous kernel,

involving the Hardy operator. As an application,

we define related Hilbert-type integral operator

and consider its norm.

The problem of determining the norm of such operator is equivalent to

the problem of the best possible constant factor

involved in the right-hand side of related inequality.

In such a way,

we obtain the norm and the best possible constant factor in some general

settings

with conjugate exponents. In particular,

we obtain generalizations of some recent results,

from the literature.

In this talk,

we give some new Hilbert-type inequalities with a homogeneous kernel,

involving the Hardy operator. As an application,

we define related Hilbert-type integral operator

and consider its norm.

The problem of determining the norm of such operator is equivalent to

the problem of the best possible constant factor

involved in the right-hand side of related inequality.

In such a way,

we obtain the norm and the best possible constant factor in some general

settings

with conjugate exponents. In particular,

we obtain generalizations of some recent results,

from the literature.

**菊池 万里 氏**(富山大学) 15:30-17:00弱空間に於ける条件付き平均作用素の一様有界性とその応用 (JAPANESE)

[ 講演概要 ]

Fatou property を持つ確率変数(可測関数)の Banach lattice を

Banach 関数空間と呼ぶ.Banach 関数空間 X から自然な方法で弱空間

w-X を定義するとき,あらゆる条件付き平均作用素が X から w-X への

作用素として一様有界になるための必要十分条件を紹介し,その応用

として,Doob の弱型不等式が成立する為の必要十分条件を導く.また,

時間が許せば,他の型のマルチンゲール不等式にも言及したい.

Fatou property を持つ確率変数(可測関数)の Banach lattice を

Banach 関数空間と呼ぶ.Banach 関数空間 X から自然な方法で弱空間

w-X を定義するとき,あらゆる条件付き平均作用素が X から w-X への

作用素として一様有界になるための必要十分条件を紹介し,その応用

として,Doob の弱型不等式が成立する為の必要十分条件を導く.また,

時間が許せば,他の型のマルチンゲール不等式にも言及したい.

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