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離散数理モデリングセミナー
過去の記録 ～04/26｜次回の予定｜今後の予定 04/27～
| 担当者 | 時弘哲治, ウィロックス ラルフ |
|---|
過去の記録
2018年11月20日(火)
15:00-16:30 数理科学研究科棟(駒場) 002号室
甘利俊一 氏 (理化学研究所)
情報幾何とその応用ー深層学習の解明に向けて
甘利俊一 氏 (理化学研究所)
情報幾何とその応用ー深層学習の解明に向けて
[ 講演概要 ]
情報幾何の基本的考えを説明した後で、深層学習にどう関係するのか、統計神経力学の立場から話をする。
情報幾何の基本的考えを説明した後で、深層学習にどう関係するのか、統計神経力学の立場から話をする。
2018年11月19日(月)
17:15-18:30 数理科学研究科棟(駒場) 056号室
Dinh T. Tran 氏 (School of Mathematics and Statistics, The University of Sydney)
Integrability for four-dimensional recurrence relations
Dinh T. Tran 氏 (School of Mathematics and Statistics, The University of Sydney)
Integrability for four-dimensional recurrence relations
[ 講演概要 ]
In this talk, we study some fourth-order recurrence relations (or mappings). These recurrence relations were obtained by assuming that they possess two polynomial integrals with certain degree patterns.
For mappings with cubic growth, we will reduce them to three-dimensional ones using a procedure called deflation. These three-dimensional maps have two integrals; therefore, they are integrable in the sense of Liouville-Arnold. Furthermore, we can reduce the obtained three-dimensional maps to two-dimensional maps of Quispel-Roberts-Thompsons (QRT) type. On the other hand, for recurrences with quadratic growth and two integrals, we will show that they are integrable in the sense of Liouville-Arnold by providing their Poisson brackets. Non-degenerate Poisson brackets can be found by using the existence of discrete Lagrangians and a discrete analogue of the Ostrogradsky transformation.
This is joint work with G. Gubbiotti, N. Joshi, and C-M. Viallet.
In this talk, we study some fourth-order recurrence relations (or mappings). These recurrence relations were obtained by assuming that they possess two polynomial integrals with certain degree patterns.
For mappings with cubic growth, we will reduce them to three-dimensional ones using a procedure called deflation. These three-dimensional maps have two integrals; therefore, they are integrable in the sense of Liouville-Arnold. Furthermore, we can reduce the obtained three-dimensional maps to two-dimensional maps of Quispel-Roberts-Thompsons (QRT) type. On the other hand, for recurrences with quadratic growth and two integrals, we will show that they are integrable in the sense of Liouville-Arnold by providing their Poisson brackets. Non-degenerate Poisson brackets can be found by using the existence of discrete Lagrangians and a discrete analogue of the Ostrogradsky transformation.
This is joint work with G. Gubbiotti, N. Joshi, and C-M. Viallet.
2018年06月25日(月)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Anton Dzhamay 氏 (University of Northern Colorado)
Gap Probabilities and discrete Painlevé equations
Anton Dzhamay 氏 (University of Northern Colorado)
Gap Probabilities and discrete Painlevé equations
[ 講演概要 ]
It is well-known that important statistical quantities, such as gap probabilities, in various discrete probabilistic models of random matrix type satisfy the so-called discrete Painlevé equations, which provides an effective way to computing them. In this talk we discuss this correspondence for a particular class of models, known as boxed plane partitions (equivalently, lozenge tilings of a hexagon). For uniform probability distribution, this is one of the most studied models of random surfaces. Borodin, Gorin, and Rains showed that it is possible to assign a very general elliptic weight to the distribution, with various degenerations of this weight corresponding to the degeneration cascade of discrete polynomial ensembles, such as Racah and Hahn ensembles and their q-analogues. This also correspond to the degeneration scheme of discrete Painlevé equations, due to Sakai. In this talk we consider the q-Hahn and q-Racah ensembles and corresponding discrete Painlevé equations of types q-P(A_{2}^{(1)}) and q-P(A_{1}^{(1)}).
This is joint work with Alisa Knizel (Columbia University)
It is well-known that important statistical quantities, such as gap probabilities, in various discrete probabilistic models of random matrix type satisfy the so-called discrete Painlevé equations, which provides an effective way to computing them. In this talk we discuss this correspondence for a particular class of models, known as boxed plane partitions (equivalently, lozenge tilings of a hexagon). For uniform probability distribution, this is one of the most studied models of random surfaces. Borodin, Gorin, and Rains showed that it is possible to assign a very general elliptic weight to the distribution, with various degenerations of this weight corresponding to the degeneration cascade of discrete polynomial ensembles, such as Racah and Hahn ensembles and their q-analogues. This also correspond to the degeneration scheme of discrete Painlevé equations, due to Sakai. In this talk we consider the q-Hahn and q-Racah ensembles and corresponding discrete Painlevé equations of types q-P(A_{2}^{(1)}) and q-P(A_{1}^{(1)}).
This is joint work with Alisa Knizel (Columbia University)
2018年01月17日(水)
17:00-18:45 数理科学研究科棟(駒場) 056号室
Samuel Colin 氏 (CBPF, Rio de Janeiro, Brasil) 17:00-17:50
Quantum matter bounce with a dark energy expanding phase (ENGLISH)
Mass of the vacuum: a Newtonian perspective (ENGLISH)
Samuel Colin 氏 (CBPF, Rio de Janeiro, Brasil) 17:00-17:50
Quantum matter bounce with a dark energy expanding phase (ENGLISH)
[ 講演概要 ]
The ``matter bounce'' is an alternative scenario to inflationary cosmology, according to which the universe undergoes a contraction, followed by an expansion, the bounce occurring when the quantum effects become important. In my talk, I will show that such a scenario can be unambiguously analyzed in the de Broglie-Bohm pilot-wave interpretation of quantum mechanics. More specifically, I will apply the pilot-wave theory to a Wheeler-DeWitt equation obtained from the quantization of a simple classical mini-superspace model, and show that there are numerical solutions describing bouncing universes with many desirable physical features. For example, one solution contains a dark energy phase during the expansion, without the need to postulate the existence of a cosmological constant in the classical action.
This work was done in collaboration with Nelson Pinto-Neto (CBPF, Rio de Janeiro, Brasil). Further details available at https://arxiv.org/abs/1706.03037.
Thomas Durt 氏 (Aix Marseille Université, Centrale Marseille, Institut Fresnel) 17:50-18:40The ``matter bounce'' is an alternative scenario to inflationary cosmology, according to which the universe undergoes a contraction, followed by an expansion, the bounce occurring when the quantum effects become important. In my talk, I will show that such a scenario can be unambiguously analyzed in the de Broglie-Bohm pilot-wave interpretation of quantum mechanics. More specifically, I will apply the pilot-wave theory to a Wheeler-DeWitt equation obtained from the quantization of a simple classical mini-superspace model, and show that there are numerical solutions describing bouncing universes with many desirable physical features. For example, one solution contains a dark energy phase during the expansion, without the need to postulate the existence of a cosmological constant in the classical action.
This work was done in collaboration with Nelson Pinto-Neto (CBPF, Rio de Janeiro, Brasil). Further details available at https://arxiv.org/abs/1706.03037.
Mass of the vacuum: a Newtonian perspective (ENGLISH)
[ 講演概要 ]
One could believe that special relativity forces us to totally renounce to the idea of an aether, but the aether reappears in general relativity which teaches us that space-time is structured by the local metrics. It also reappears in quantum field theory which teaches us that even at zero temperature space is filled by the quantum vacuum energy. Finally, aether reappears in modern astronomy where it was necessary to introduce ill-defined concepts such as dark matter and dark energy in order to explain apparent deviations from Newtonian dynamics (at the level of galactic rotation curves).
Newton dynamics being the unique limit of general relativistic dynamics in the classical regime, dark matter and dark energy can be seen as an ultimate, last chance strategy, aimed at reconciling the predictions of general relativity with astronomical data.
In our talk we shall describe a simple model, derived in the framework of Newtonian dynamics, aimed at explaining puzzling astronomical observations realized at the level of the solar system (Pioneer anomaly) and at the galactic scale (rotation curves), without adopting ad hoc hypotheses about the existence of dark matter and/or dark energy.
The basic idea is that Newtonian gravity is modified due to the presence of a (negative) density, everywhere in space, of mass-energy.
One could believe that special relativity forces us to totally renounce to the idea of an aether, but the aether reappears in general relativity which teaches us that space-time is structured by the local metrics. It also reappears in quantum field theory which teaches us that even at zero temperature space is filled by the quantum vacuum energy. Finally, aether reappears in modern astronomy where it was necessary to introduce ill-defined concepts such as dark matter and dark energy in order to explain apparent deviations from Newtonian dynamics (at the level of galactic rotation curves).
Newton dynamics being the unique limit of general relativistic dynamics in the classical regime, dark matter and dark energy can be seen as an ultimate, last chance strategy, aimed at reconciling the predictions of general relativity with astronomical data.
In our talk we shall describe a simple model, derived in the framework of Newtonian dynamics, aimed at explaining puzzling astronomical observations realized at the level of the solar system (Pioneer anomaly) and at the galactic scale (rotation curves), without adopting ad hoc hypotheses about the existence of dark matter and/or dark energy.
The basic idea is that Newtonian gravity is modified due to the presence of a (negative) density, everywhere in space, of mass-energy.
2017年11月01日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Basile Grammaticos 氏 (Université de Paris VII・XI)
Discrete Painlevé equations associated with the E8 group (ENGLISH)
Basile Grammaticos 氏 (Université de Paris VII・XI)
Discrete Painlevé equations associated with the E8 group (ENGLISH)
[ 講演概要 ]
I'll present a summary of the results of the Paris-Tokyo-Pondicherry group on equations associated with the affine Weyl group E8. I shall review the various parametrisations of the E8-related equations, introducing the trihomographic representation and the ancillary variable. Several examples of E8-associated equations will be given including what we believe is the simplest form for the generic elliptic discrete Painlevé equation.
I'll present a summary of the results of the Paris-Tokyo-Pondicherry group on equations associated with the affine Weyl group E8. I shall review the various parametrisations of the E8-related equations, introducing the trihomographic representation and the ancillary variable. Several examples of E8-associated equations will be given including what we believe is the simplest form for the generic elliptic discrete Painlevé equation.
2017年10月31日(火)
16:30-17:30 数理科学研究科棟(駒場) 126号室
Basile Grammaticos 氏 (Université de Paris VII・XI)
The end of the World (ENGLISH)
Basile Grammaticos 氏 (Université de Paris VII・XI)
The end of the World (ENGLISH)
[ 講演概要 ]
This is not a seminar on astrophysics or cosmology. I am not going to talk about something that will happen in billions of years. I will rather explain the menace to our civilisation and to the human species. Inspired from the works of several authors I will explain the existing risks. I will also present mathematical models which show that a general collapse is possible in the decades that follow.
This is not a seminar on astrophysics or cosmology. I am not going to talk about something that will happen in billions of years. I will rather explain the menace to our civilisation and to the human species. Inspired from the works of several authors I will explain the existing risks. I will also present mathematical models which show that a general collapse is possible in the decades that follow.
2017年10月31日(火)
15:30-16:30 数理科学研究科棟(駒場) 126号室
Fon-Che Liu 氏 (National Taiwan University)
A hierarchy of approximate regularity of functions (ENGLISH)
Fon-Che Liu 氏 (National Taiwan University)
A hierarchy of approximate regularity of functions (ENGLISH)
[ 講演概要 ]
A hierarchy of a certain weakly sensed regularity of functions defined on subsets of Euclidean n-space which originated from the well-known Lusin theorem that characterizes measurable functions in terms of approximate continuity will be introduced. Its intimate relations with the ordinary hierarchy of regularity in terms of order of continuous differentiability will be exposed and explained.
A hierarchy of a certain weakly sensed regularity of functions defined on subsets of Euclidean n-space which originated from the well-known Lusin theorem that characterizes measurable functions in terms of approximate continuity will be introduced. Its intimate relations with the ordinary hierarchy of regularity in terms of order of continuous differentiability will be exposed and explained.
2017年04月26日(水)
15:30-17:00 数理科学研究科棟(駒場) 056号室
土谷 洋平 氏 (神奈川工科大学)
nonlocalな古典可積分系に関する最近の話題 (JAPANESE)
土谷 洋平 氏 (神奈川工科大学)
nonlocalな古典可積分系に関する最近の話題 (JAPANESE)
[ 講演概要 ]
ここでいうnonlocalな可積分系とは、離散系のことではなく、特異積分変換項を持った微分方程式の可積分系をさします。そのような方程式に関しては、1.nonlocalな量子可積分系との関連について、3.テータ関数解について、3.佐藤理論から見た対称性についてといった話があります。最近の話題ということで、一応2.を中心にお話しようと思っています。技術的にはリーマン面上に特殊な第三種アーベル積分を構成する話で、地味に見えるかもしれませんが、きっと応用がある話だと思っています。
ここでいうnonlocalな可積分系とは、離散系のことではなく、特異積分変換項を持った微分方程式の可積分系をさします。そのような方程式に関しては、1.nonlocalな量子可積分系との関連について、3.テータ関数解について、3.佐藤理論から見た対称性についてといった話があります。最近の話題ということで、一応2.を中心にお話しようと思っています。技術的にはリーマン面上に特殊な第三種アーベル積分を構成する話で、地味に見えるかもしれませんが、きっと応用がある話だと思っています。
2017年02月09日(木)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Dinh Tran 氏 (University of New South Wales, Sydney, Australia)
Growth of degrees of lattice equations and its signatures over finite fields (ENGLISH)
Dinh Tran 氏 (University of New South Wales, Sydney, Australia)
Growth of degrees of lattice equations and its signatures over finite fields (ENGLISH)
[ 講演概要 ]
We study growth of degrees of autonomous and non-autonomous lattice equations, some of which are known to be integrable. We present a conjecture that helps us to prove polynomial growth of a certain class of equations including $Q_V$ and its non-autonomous generalization. In addition, we also study growth of degrees of several non-integrable equations. Exponential growth of degrees of these equations is also proved subject to a conjecture. Our technique is to determine the ambient degree growth of the equations and a conjectured growth of their common factors at each vertex, allowing the true degree growth to be found. Moreover, our results can also be used for mappings obtained as periodic reductions of integrable lattice equations. We also study signatures of growth of degrees of lattice equations over finite fields.
We study growth of degrees of autonomous and non-autonomous lattice equations, some of which are known to be integrable. We present a conjecture that helps us to prove polynomial growth of a certain class of equations including $Q_V$ and its non-autonomous generalization. In addition, we also study growth of degrees of several non-integrable equations. Exponential growth of degrees of these equations is also proved subject to a conjecture. Our technique is to determine the ambient degree growth of the equations and a conjectured growth of their common factors at each vertex, allowing the true degree growth to be found. Moreover, our results can also be used for mappings obtained as periodic reductions of integrable lattice equations. We also study signatures of growth of degrees of lattice equations over finite fields.
2016年12月19日(月)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Anton Dzhamay 氏 (University of Northern Colorado)
Discrete Painlevé equations on the affine A3 surface (ENGLISH)
Anton Dzhamay 氏 (University of Northern Colorado)
Discrete Painlevé equations on the affine A3 surface (ENGLISH)
[ 講演概要 ]
We explain how to construct the birational representation of the extended affine Weyl symmetry group D5 and consider examples of discrete Painlevé equations that correspond to certain translation elements in this group. One of the examples is the famous q-PV equation of Jimbo-Sakai. Some other examples are conjugated to it via explicit change of variables and we explain how representing translation elements as words in the group allows us to see the corresponding change of coordinates explicitly. We also show a new example of a discrete Painlevé equation that is elementary (short translation), but at the same time is different from the q-PVI equation.
We explain how to construct the birational representation of the extended affine Weyl symmetry group D5 and consider examples of discrete Painlevé equations that correspond to certain translation elements in this group. One of the examples is the famous q-PV equation of Jimbo-Sakai. Some other examples are conjugated to it via explicit change of variables and we explain how representing translation elements as words in the group allows us to see the corresponding change of coordinates explicitly. We also show a new example of a discrete Painlevé equation that is elementary (short translation), but at the same time is different from the q-PVI equation.
2016年12月17日(土)
10:00-18:00 数理科学研究科棟(駒場) 056号室
Anton Dzhamay 氏 (University of Northern Colorado) 10:00-10:50
Factorization of Rational Mappings and Geometric Deautonomization (ENGLISH)
From the QRT maps to elliptic difference Painlevé equations (ENGLISH)
The complete degeneration scheme of four-dimensional Painlevé-type equations (ENGLISH)
Degeneration of the Painlevé divisors (ENGLISH)
Rational approximation and Schlesinger transformation (ENGLISH)
Spaces of initial conditions for nonautonomous mappings of the plane (ENGLISH)
Anton Dzhamay 氏 (University of Northern Colorado) 10:00-10:50
Factorization of Rational Mappings and Geometric Deautonomization (ENGLISH)
[ 講演概要 ]
This talk is the first of two talks describing the joint project with Tomoyuki Takenawa and Stefan Carstea on geometric deautonomization.
The goal of this project is to develop a systematic approach for deautonomizing discrete integrable mappings, such as the QRT mappings, to non-automonous mappings in the discrete Painlevé family, based on the action of the mapping on the Picard lattice of the surface and a choice of an elliptic fiber. In this talk we will explain the main ideas behind this approach and describe the technique that allows us to recover explicit formulas defining the mapping from the known action on the divisor group (the factorization technique). We illustrate our approach by reconstructing the famous example of the q-PVI equation of Jimbo-Sakai from a simple QRT mapping.
Tomoyuki Takenawa 氏 (Tokyo University of Marine Science and Technology) 11:00-11:50This talk is the first of two talks describing the joint project with Tomoyuki Takenawa and Stefan Carstea on geometric deautonomization.
The goal of this project is to develop a systematic approach for deautonomizing discrete integrable mappings, such as the QRT mappings, to non-automonous mappings in the discrete Painlevé family, based on the action of the mapping on the Picard lattice of the surface and a choice of an elliptic fiber. In this talk we will explain the main ideas behind this approach and describe the technique that allows us to recover explicit formulas defining the mapping from the known action on the divisor group (the factorization technique). We illustrate our approach by reconstructing the famous example of the q-PVI equation of Jimbo-Sakai from a simple QRT mapping.
From the QRT maps to elliptic difference Painlevé equations (ENGLISH)
[ 講演概要 ]
This talk is the second part of the joint project with Anton Dzhamay and Stefan Carstea on geometric deautonomization and focuses on the elliptic case and the special symmetry groups. It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated.
In this talk we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Especially, in the case where the fiber is smooth elliptic, imposing certain restrictions on such non autonomous mappings, we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.
Hiroshi Kawakami 氏 (Aoyama Gakuin University) 13:30-14:20This talk is the second part of the joint project with Anton Dzhamay and Stefan Carstea on geometric deautonomization and focuses on the elliptic case and the special symmetry groups. It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated.
In this talk we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Especially, in the case where the fiber is smooth elliptic, imposing certain restrictions on such non autonomous mappings, we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.
The complete degeneration scheme of four-dimensional Painlevé-type equations (ENGLISH)
[ 講演概要 ]
In the joint work with H. Sakai and A. Nakamura, we constructed the degeneration scheme of four-dimensional Painlevé-type equations associated with unramified linear equations. In this talk I present the "complete" degeneration scheme of the four-dimensional Painlevé-type equations, which is constructed by means of the degeneration of HTL forms of associated linear equations.
Akane Nakamura 氏 (Josai University) 14:30-15:20In the joint work with H. Sakai and A. Nakamura, we constructed the degeneration scheme of four-dimensional Painlevé-type equations associated with unramified linear equations. In this talk I present the "complete" degeneration scheme of the four-dimensional Painlevé-type equations, which is constructed by means of the degeneration of HTL forms of associated linear equations.
Degeneration of the Painlevé divisors (ENGLISH)
[ 講演概要 ]
There are three types of curves associated with 4-dimensional algebraically completely integrable systems, namely the spectral curve, the Painlevé divisors, and the separation curve. I am going to explain these three curves of genus two taking examples derived from the isospectral limit of the 4-dimensional Painlevé-type equations and study the Namikawa-Ueno type degeneration.
Teruhisa Tsuda 氏 (Hitotsubashi University) 16:00-16:50There are three types of curves associated with 4-dimensional algebraically completely integrable systems, namely the spectral curve, the Painlevé divisors, and the separation curve. I am going to explain these three curves of genus two taking examples derived from the isospectral limit of the 4-dimensional Painlevé-type equations and study the Namikawa-Ueno type degeneration.
Rational approximation and Schlesinger transformation (ENGLISH)
[ 講演概要 ]
We show how rational approximation problems for functions are related to the construction of Schlesinger transformations. Also we discuss their applications to the theory of isomonodromic deformations or Painlevé equations. This talk is based on a joint work with Toshiyuki Mano.
Takafumi Mase 氏 (the University of Tokyo) 17:00-17:50We show how rational approximation problems for functions are related to the construction of Schlesinger transformations. Also we discuss their applications to the theory of isomonodromic deformations or Painlevé equations. This talk is based on a joint work with Toshiyuki Mano.
Spaces of initial conditions for nonautonomous mappings of the plane (ENGLISH)
[ 講演概要 ]
Spaces of initial conditions are one of the most important and powerful tools to analyze mappings of the plane. In this talk, we study the basic properties of general nonautonomous equations that have spaces of initial conditions. We will consider the minimization of spaces of initial conditions for nonautonomous systems and we shall discuss a classification of nonautonomous integrable mappings of the plane with a space of initial conditions.
Spaces of initial conditions are one of the most important and powerful tools to analyze mappings of the plane. In this talk, we study the basic properties of general nonautonomous equations that have spaces of initial conditions. We will consider the minimization of spaces of initial conditions for nonautonomous systems and we shall discuss a classification of nonautonomous integrable mappings of the plane with a space of initial conditions.
2016年11月28日(月)
17:15-18:30 数理科学研究科棟(駒場) 056号室
Alfred Ramani 氏 (IMNC, Universite de Paris 7 et 11)
Who cares about integrability ? (ENGLISH)
Alfred Ramani 氏 (IMNC, Universite de Paris 7 et 11)
Who cares about integrability ? (ENGLISH)
[ 講演概要 ]
I will start my talk with an introduction to integrability of continuous systems. Why is it important? Is it possible to give a definition of integrability which will satisfy everybody? (Short answer: No). I will then present the most salient discoveries of integrable systems, from Newton to Toda. Next I will address the question of discrete integrability. This will lead naturally to the question of discretisation (of continuous systems) and its importance in modelling. I will deal with the construction of integrable discretisations of continuous integrable systems and introduce the singularity confinement discrete integrability criterion. The final part of my talk will be devoted to discrete Painlevé equations. Due to obvious time constraints I will concentrate on one special class of these equations, namely those associated to the E8 affine Weyl group. I will present a succinct summary of our recent results as well as indications for future investigations.
I will start my talk with an introduction to integrability of continuous systems. Why is it important? Is it possible to give a definition of integrability which will satisfy everybody? (Short answer: No). I will then present the most salient discoveries of integrable systems, from Newton to Toda. Next I will address the question of discrete integrability. This will lead naturally to the question of discretisation (of continuous systems) and its importance in modelling. I will deal with the construction of integrable discretisations of continuous integrable systems and introduce the singularity confinement discrete integrability criterion. The final part of my talk will be devoted to discrete Painlevé equations. Due to obvious time constraints I will concentrate on one special class of these equations, namely those associated to the E8 affine Weyl group. I will present a succinct summary of our recent results as well as indications for future investigations.
2016年11月19日(土)
14:00-17:00 数理科学研究科棟(駒場) 056号室
長谷川貴之 氏 (富山高専) 14:00-15:15
数学で時間知覚を扱うことにより「時間とは何か？」という数千年来の未解決問題に攻めかかる (JAPANESE)
AKNS 形式における線形散乱問題の区分的近似解法とその応用 (JAPANESE)
長谷川貴之 氏 (富山高専) 14:00-15:15
数学で時間知覚を扱うことにより「時間とは何か？」という数千年来の未解決問題に攻めかかる (JAPANESE)
[ 講演概要 ]
時間の感覚器官を持っていないにもかかわらず，多くの生物は時間知覚を持っている．しかも原初的な知覚の一つとしてである．その時間知覚を，「ピーク法」という動物実験で得たデータを分析した．時間知覚研究は歴史的に浅い．John Gibbon (1934～2001) は正規分布を使って，確率論的説明を試みた．その後，Peter Richard Killeen とJ. Gregor Fetterman はガンマ密度関数を使って説明を試みた(1988)．また，Armand Machado はガンマ密度関数の正係数の線形和を使って説明を試みた(1997)．そこで講演者は，Machado の理論でコンピュータ・シミュレーションしているうち，負係数も許容してガンマ密度関数の線形和を使うことにより，ピアソンの積率相関係数・赤池情報量規準ともにGibbon の理論に遜色なく，しかも様々な心理要因を定量評価することができるものを探り当てた（[文献1]）．モデル中に神経の抑制機能を取り入れ，負係数の説明とした．具体的には，レバー押しの反応を表す時間t の関数R(t) を， と適切な実数係数を使って，とおき，ピーク法のデータに当てはめた．「スカラー特性(scalar property)」と呼ばれている動物の計時行動に見られる特性を，心理学でいう「移調」として説明できた．また，スカラー特性からの微妙なズレを定量的に見ることにより，実験での時間範囲では単一の時計が発動されていたということも示すことができた．
[文献1] Takayuki Hasegawa & Shogo Sakata, A model ofmultisecond timing behavior under peak-interval pro-cedures, Journal of Computational Neuroscience, 38-2,301-313 (2015)
藤嶋浩史 氏 (キャノン) 15:45-17:00時間の感覚器官を持っていないにもかかわらず，多くの生物は時間知覚を持っている．しかも原初的な知覚の一つとしてである．その時間知覚を，「ピーク法」という動物実験で得たデータを分析した．時間知覚研究は歴史的に浅い．John Gibbon (1934～2001) は正規分布を使って，確率論的説明を試みた．その後，Peter Richard Killeen とJ. Gregor Fetterman はガンマ密度関数を使って説明を試みた(1988)．また，Armand Machado はガンマ密度関数の正係数の線形和を使って説明を試みた(1997)．そこで講演者は，Machado の理論でコンピュータ・シミュレーションしているうち，負係数も許容してガンマ密度関数の線形和を使うことにより，ピアソンの積率相関係数・赤池情報量規準ともにGibbon の理論に遜色なく，しかも様々な心理要因を定量評価することができるものを探り当てた（[文献1]）．モデル中に神経の抑制機能を取り入れ，負係数の説明とした．具体的には，レバー押しの反応を表す時間t の関数R(t) を， と適切な実数係数を使って，とおき，ピーク法のデータに当てはめた．「スカラー特性(scalar property)」と呼ばれている動物の計時行動に見られる特性を，心理学でいう「移調」として説明できた．また，スカラー特性からの微妙なズレを定量的に見ることにより，実験での時間範囲では単一の時計が発動されていたということも示すことができた．
[文献1] Takayuki Hasegawa & Shogo Sakata, A model ofmultisecond timing behavior under peak-interval pro-cedures, Journal of Computational Neuroscience, 38-2,301-313 (2015)
AKNS 形式における線形散乱問題の区分的近似解法とその応用 (JAPANESE)
[ 講演概要 ]
ソリトン方程式に対しては一定の条件下で逆散乱法によって厳密解を構成できるが，代数的な方法によって具体的な解を連鎖的に生成できるのは，ピュアソリトンと呼ばれる理想的な初期波形の場合に限られる．一方，物理現象を記述する偏微分方程式の立場から実験状況を鑑みると，「任意の初期波形に対して十分時間がたった後の漸近的情報」を抽出することが重要である．だが，このような無限時間経過後の情報を直接発展方程式の数値積分によって得ることは困難なことが多い．本セミナーでは実験で観測される量として最も重要な「最終的に生成されるソリトン数」に着目し，シュワルツ級の任意一次元パルスに対して，近似的に所望の量を与える枠組みを二つ与える．
非ピュアソリトンは低振幅の輻射を放出しながら最終的にソリトンの形になっていくことが知られており，初期状態として複数の非ピュアソリトンを考えた場合はそれぞれのパルスから出る輻射成分が相互に干渉を起こす．このような相互干渉は比較的短時間のダイナミクスであるが，そのことが系の漸近的状態に決定的な影響を与えることがわかった．なお，本セミナーでは例示的にKdVやNLSEが属すクラスであるAKNS形式を用いるが，その他の形式にもこれらの方法は拡張可能である．
ソリトン方程式に対しては一定の条件下で逆散乱法によって厳密解を構成できるが，代数的な方法によって具体的な解を連鎖的に生成できるのは，ピュアソリトンと呼ばれる理想的な初期波形の場合に限られる．一方，物理現象を記述する偏微分方程式の立場から実験状況を鑑みると，「任意の初期波形に対して十分時間がたった後の漸近的情報」を抽出することが重要である．だが，このような無限時間経過後の情報を直接発展方程式の数値積分によって得ることは困難なことが多い．本セミナーでは実験で観測される量として最も重要な「最終的に生成されるソリトン数」に着目し，シュワルツ級の任意一次元パルスに対して，近似的に所望の量を与える枠組みを二つ与える．
非ピュアソリトンは低振幅の輻射を放出しながら最終的にソリトンの形になっていくことが知られており，初期状態として複数の非ピュアソリトンを考えた場合はそれぞれのパルスから出る輻射成分が相互に干渉を起こす．このような相互干渉は比較的短時間のダイナミクスであるが，そのことが系の漸近的状態に決定的な影響を与えることがわかった．なお，本セミナーでは例示的にKdVやNLSEが属すクラスであるAKNS形式を用いるが，その他の形式にもこれらの方法は拡張可能である．
