## Seminar information archive

Seminar information archive ～02/23｜Today's seminar 02/24 | Future seminars 02/25～

#### Seminar on Probability and Statistics

16:50-17:40 Room #123 (Graduate School of Math. Sci. Bldg.)

New Functionals inequalities via Stein's discrepancies

**Giovanni Peccati**(Universite du Luxembourg)New Functionals inequalities via Stein's discrepancies

#### Seminar on Probability and Statistics

17:40-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Stochastic geometry and Malliavin calculus on configuration spaces

**Giovanni Peccati**(Université du Luxembourg)Stochastic geometry and Malliavin calculus on configuration spaces

[ Abstract ]

I will present some recent advances in the domain of quantitative limit theorems for geometric Poisson functionals, associated e.g. with random geometric graphs and random tessellations, obtained by means of Malliavin calculus techniques. One of our main results consists in a general (optimal) Berry-Esseen bound for stabilizing functionals, based on Stein’s method, iterated Poincaré inequalities and a variant of Mehler’s formula. Based on several joint works with S. Bourguin, R. Lachièze-Rey, G. Last and M. Schulte, as well as on the recent monograph that I co-edited with M. Reitzner.

I will present some recent advances in the domain of quantitative limit theorems for geometric Poisson functionals, associated e.g. with random geometric graphs and random tessellations, obtained by means of Malliavin calculus techniques. One of our main results consists in a general (optimal) Berry-Esseen bound for stabilizing functionals, based on Stein’s method, iterated Poincaré inequalities and a variant of Mehler’s formula. Based on several joint works with S. Bourguin, R. Lachièze-Rey, G. Last and M. Schulte, as well as on the recent monograph that I co-edited with M. Reitzner.

### 2016/10/28

#### Mathematical Biology Seminar

13:30-14:30 Room #126 (Graduate School of Math. Sci. Bldg.)

When is the allergen immunotherapy effective? (JAPANESE)

**Akane Hara**(Graduate School of Systems Life Sciences, Kyushu University)When is the allergen immunotherapy effective? (JAPANESE)

[ Abstract ]

Allergen immunotherapy is a method to treat allergic symptoms, for example rhinitis and sneezing in Japanese cedar pollen allergy (JCPA). In the immunotherapy of JCPA, patients take in a small amount of pollen over several years, which suppress severe allergic symptoms when exposed to a large amount of environmental pollen after the therapy. We develop a simple mathematical model to identify the condition for successful therapy. We consider the dynamics of type 2 T helper cells (Th2) and regulatory T cells (Treg) and both of them are differentiated from naive T cells. We assume that Treg cells have a much longer lifespan than Th2 cells, which makes Treg cells accumulate over many years during the therapy.

We regard that the therapy is successful if (1) without therapy the patient develops allergic symptoms upon exposure to the environmental pollen, (2) the patient does not develop allergic symptoms caused by the therapy itself, and (3) with therapy the patient does not develop symptoms upon exposure. We find the conditions of each parameter for successful therapy. We also find that the therapy of linearly increasing dose is able to reduce the risk of allergic symptoms caused by the therapy itself, rather than constant dose. We would like to consider application of this model to other kind of allergy, such as food allergy.

Allergen immunotherapy is a method to treat allergic symptoms, for example rhinitis and sneezing in Japanese cedar pollen allergy (JCPA). In the immunotherapy of JCPA, patients take in a small amount of pollen over several years, which suppress severe allergic symptoms when exposed to a large amount of environmental pollen after the therapy. We develop a simple mathematical model to identify the condition for successful therapy. We consider the dynamics of type 2 T helper cells (Th2) and regulatory T cells (Treg) and both of them are differentiated from naive T cells. We assume that Treg cells have a much longer lifespan than Th2 cells, which makes Treg cells accumulate over many years during the therapy.

We regard that the therapy is successful if (1) without therapy the patient develops allergic symptoms upon exposure to the environmental pollen, (2) the patient does not develop allergic symptoms caused by the therapy itself, and (3) with therapy the patient does not develop symptoms upon exposure. We find the conditions of each parameter for successful therapy. We also find that the therapy of linearly increasing dose is able to reduce the risk of allergic symptoms caused by the therapy itself, rather than constant dose. We would like to consider application of this model to other kind of allergy, such as food allergy.

### 2016/10/27

#### Applied Analysis

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Sign-changing solutions of the nonlinear heat equation with positive initial value

(ENGLISH)

**Fred Weissler**(Universite Paris 13)Sign-changing solutions of the nonlinear heat equation with positive initial value

(ENGLISH)

[ Abstract ]

#### Infinite Analysis Seminar Tokyo

15:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Non-unitary highest-weight modules over the $N=2$ superconformal algebra (JAPANESE)

**Ryou Sato**(Graduate School of Mathematical Scineces, The University of Tokyo)Non-unitary highest-weight modules over the $N=2$ superconformal algebra (JAPANESE)

[ Abstract ]

The $N=2$ superconformal algebra is a generalization of the Virasoro algebra having the super symmetry.

The character formulas associated with the unitary highest weight representations

are expressed in terms of the classical theta functions, and have the remarkable

modular invariance. Based on the method of the $W$-algebras,

Kac and Wakimoto, on the other hand, showed that the

characters for a certain class of non-unitary highest weight representations

can be written in terms of the mock theta functions associated with the affine ${sl}_{2|1}$.

Then they found a way to identify these formulas with

real analytic modular forms by using the correction terms given by Zwegers.

In this seminar, we explain a way to construct the above mentioned

non-unitary representations from the representations of the algebra affine ${sl}_{2}$,

based on the Kazama-Suzuki coset construction, namely not from the $W$-algebra method.

We also investigate the relations between the mock theta functions and the ordinary

theta functions, appearing in this method.

The $N=2$ superconformal algebra is a generalization of the Virasoro algebra having the super symmetry.

The character formulas associated with the unitary highest weight representations

are expressed in terms of the classical theta functions, and have the remarkable

modular invariance. Based on the method of the $W$-algebras,

Kac and Wakimoto, on the other hand, showed that the

characters for a certain class of non-unitary highest weight representations

can be written in terms of the mock theta functions associated with the affine ${sl}_{2|1}$.

Then they found a way to identify these formulas with

real analytic modular forms by using the correction terms given by Zwegers.

In this seminar, we explain a way to construct the above mentioned

non-unitary representations from the representations of the algebra affine ${sl}_{2}$,

based on the Kazama-Suzuki coset construction, namely not from the $W$-algebra method.

We also investigate the relations between the mock theta functions and the ordinary

theta functions, appearing in this method.

### 2016/10/25

#### Algebraic Geometry Seminar

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

Q-Gorenstein deformation theory and it applications to algebraic surfaces (English)

**Yongnam Lee**(KAIST/RIMS)Q-Gorenstein deformation theory and it applications to algebraic surfaces (English)

[ Abstract ]

The notion of Q-Gorenstein variety is important for the minimal model theory and the compact moduli theory of algebraic varieties in characteristic 0. Also Q-Gorenstein deformation theory can be applied to construct (simply connected) surfaces of general type with geometric genus 0 over the field of any characteristic. In this talk, some applications of Q-Gorenstein deformation theory to algebraic surfaces and some interesting examples related to Q-Gorenstein morphisms will be presented.

The notion of Q-Gorenstein variety is important for the minimal model theory and the compact moduli theory of algebraic varieties in characteristic 0. Also Q-Gorenstein deformation theory can be applied to construct (simply connected) surfaces of general type with geometric genus 0 over the field of any characteristic. In this talk, some applications of Q-Gorenstein deformation theory to algebraic surfaces and some interesting examples related to Q-Gorenstein morphisms will be presented.

#### Tuesday Seminar of Analysis

16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)

Asymptotics for the integrable discrete nonlinear Schr\"odinger equation (JAPANESE)

**Hideshi YAMANE**(Department of Mathematical Sciences, School of Science, Kwansei Gakuin University)Asymptotics for the integrable discrete nonlinear Schr\"odinger equation (JAPANESE)

### 2016/10/24

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Structure and equivalence of a class of tube domains with solvable groups of automorphisms (JAPANESE)

**Satoru Shimizu**(Tohoku University)Structure and equivalence of a class of tube domains with solvable groups of automorphisms (JAPANESE)

[ Abstract ]

In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of $\frak g(T_{\Omega})$ plays an important role, where $\frak g(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.

In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of $\frak g(T_{\Omega})$ plays an important role, where $\frak g(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.

#### Operator Algebra Seminars

16:45-18:15 Room #117 (Graduate School of Math. Sci. Bldg.)

Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

**Yusuke Isono**(RIMS, Kyoto Univ.)Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

### 2016/10/18

#### Tuesday Seminar on Topology

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Conformal field theory for extended W-algebras (JAPANESE)

**Yoshitake Hashimoto**(Tokyo City University)Conformal field theory for extended W-algebras (JAPANESE)

[ Abstract ]

This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of

"infinitesimal deformation of parameter" and Jack symmetric polynomials.

In this talk I will discuss the following subjects:

1. "factorization" in conformal field theory,

2. tensor structure of the category of M-modules and "module-bimodule correspondence".

This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of

"infinitesimal deformation of parameter" and Jack symmetric polynomials.

In this talk I will discuss the following subjects:

1. "factorization" in conformal field theory,

2. tensor structure of the category of M-modules and "module-bimodule correspondence".

### 2016/10/17

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Takaaki Nomura**(Kyushu University)(JAPANESE)

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Unitarizability, Maurey-Nikishin factorization and Polish groups of finite type (English)

**Hiroshi Ando**(Chiba Univ.)Unitarizability, Maurey-Nikishin factorization and Polish groups of finite type (English)

### 2016/10/12

#### Number Theory Seminar

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Filtered de Rham Witt complexes and wildly ramified higher class field theory over finite fields (joint work with Shuji Saito and Yigeng Zhao) (English)

**Uwe Jannsen**(Universität Regensburg, The University of Tokyo)Filtered de Rham Witt complexes and wildly ramified higher class field theory over finite fields (joint work with Shuji Saito and Yigeng Zhao) (English)

[ Abstract ]

We will consider abelian coverings of smooth projective varieties over finite fields which are wildly ramified along a divisor D with normal crossings, and will describe the corresponding abelianized fundamental group via modified logarithmic de Rham-Witt sheaves.

We will consider abelian coverings of smooth projective varieties over finite fields which are wildly ramified along a divisor D with normal crossings, and will describe the corresponding abelianized fundamental group via modified logarithmic de Rham-Witt sheaves.

### 2016/10/11

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Global solutions to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations (English)

**Nam Quang Le**(Indiana University)Global solutions to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations (English)

[ Abstract ]

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem under some restrictions on the sign or integrability of the affine mean curvature. In this talk, we explain how to remove these restrictions and obtain global solutions under optimal integrability conditions on the affine mean curvature. Our analysis also covers the case of Abreu's equation arising in complex geometry.

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem under some restrictions on the sign or integrability of the affine mean curvature. In this talk, we explain how to remove these restrictions and obtain global solutions under optimal integrability conditions on the affine mean curvature. Our analysis also covers the case of Abreu's equation arising in complex geometry.

#### Algebraic Geometry Seminar

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

On varieties with splittings of relative Frobenius morphisms of Albanese maps

**Sho Ejiri**(University of Tokyo)On varieties with splittings of relative Frobenius morphisms of Albanese maps

[ Abstract ]

Varieties with splittings of Frobenius morphisms are called F-split varieties, which satisfy strong properties such as Kodaira vanishing. However, some important varieties are not F-split. For example, an abelian variety is F-split if and only if its p-rank is maximum. In this talk, we discuss the class of varieties with splittings of relative Frobenius morphisms of Albanese maps, which includes abelian varieties. As a consequence of Sannai and Tanaka's characterization of ordinary abelian varieties, we see that this class also includes F-split varieties. Furthermore, for varieties in this class, we show that the Kodaira vanishing theorem holds, and that Albanese maps are algebraic fiber spaces.

Varieties with splittings of Frobenius morphisms are called F-split varieties, which satisfy strong properties such as Kodaira vanishing. However, some important varieties are not F-split. For example, an abelian variety is F-split if and only if its p-rank is maximum. In this talk, we discuss the class of varieties with splittings of relative Frobenius morphisms of Albanese maps, which includes abelian varieties. As a consequence of Sannai and Tanaka's characterization of ordinary abelian varieties, we see that this class also includes F-split varieties. Furthermore, for varieties in this class, we show that the Kodaira vanishing theorem holds, and that Albanese maps are algebraic fiber spaces.

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebra in genus zero (JAPANESE)

**Nariya Kawazumi**(The University of Tokyo)The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebra in genus zero (JAPANESE)

[ Abstract ]

In view of results of Goldman and Turaev, the free vector space over the free loops on an oriented surface has a natural Lie bialgebra structure. The Goldman bracket has a formal description by using a special (or symplectic) expansion of the fundamental group of the surface. It is natural to ask for a formal description of the Turaev cobracket. We will show how to obtain a formal description of the Goldman-Turaev Lie bialgebra for genus 0 using a solution of the Kashiwara-Vergne problem. A similar description was recently obtained by Massuyeau using the Kontsevich integral. Moreover we propose a generalization of the Kashiwara-Vergne problem in the context of the Goldman-Turaev Lie bialgebra. This talk is based on a joint work with A. Alekseev, Y. Kuno and F. Naef.

In view of results of Goldman and Turaev, the free vector space over the free loops on an oriented surface has a natural Lie bialgebra structure. The Goldman bracket has a formal description by using a special (or symplectic) expansion of the fundamental group of the surface. It is natural to ask for a formal description of the Turaev cobracket. We will show how to obtain a formal description of the Goldman-Turaev Lie bialgebra for genus 0 using a solution of the Kashiwara-Vergne problem. A similar description was recently obtained by Massuyeau using the Kontsevich integral. Moreover we propose a generalization of the Kashiwara-Vergne problem in the context of the Goldman-Turaev Lie bialgebra. This talk is based on a joint work with A. Alekseev, Y. Kuno and F. Naef.

#### Seminar on Probability and Statistics

16:50-18:00 Room #052 (Graduate School of Math. Sci. Bldg.)

### 2016/10/04

#### Algebraic Geometry Seminar

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

Higher order minimal families of rational curves and Fano manifolds with nef Chern characters (Japanese. Writing in English. )

**Taku Suzuki**(Waseda University)Higher order minimal families of rational curves and Fano manifolds with nef Chern characters (Japanese. Writing in English. )

[ Abstract ]

In this talk, we introduce higher order minimal families $H_i$ of rational curves

associated to Fano manifolds $X$. We prove that $H_i$ is also a Fano manifold

if the Chern characters of $X$ satisfy some positivity conditions. We also provide

a sufficient condition for Fano manifolds to be covered by higher rational manifolds.

In this talk, we introduce higher order minimal families $H_i$ of rational curves

associated to Fano manifolds $X$. We prove that $H_i$ is also a Fano manifold

if the Chern characters of $X$ satisfy some positivity conditions. We also provide

a sufficient condition for Fano manifolds to be covered by higher rational manifolds.

#### Colloquium

15:30-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Waning and boosting : on the dynamics of immune status (ENGLISH)

http://www.uu.nl/staff/ODiekmann

**Odo Diekmann**(Utrecht University)Waning and boosting : on the dynamics of immune status (ENGLISH)

[ Abstract ]

A first aim is to briefly review various mathematical models of infectious disease dynamics that incorporate waning and boosting of immunity. The focus will be on models that are described by delay equations, in particular renewal equations [1]. Concerning within-host dynamics, we limit ourselves to the rather caricatural models of Aron [2] and de Graaf e.a. [3].From a biomedical point of view the main conclusion is that a higher force of infection may lead to less disease,see [4] and the references given there.

[1] O.Diekmann, M.Gyllenberg, J.A.J.Metz, H.R.Thieme, On the formulation and analysis

of general deterministic structured population models. I. Linear theory, J. Math. Biol. (1998) 36 : 349 - 388

[2] J.L. Aron, Dynamics of acquired immunity boosted by exposure to infection, Math. Biosc. (1983) 64 : 249-259

[3] W.F. de Graaf, M.E.E. Kretzschmar, P.M.F. Teunis, O. Diekmann, A two-phase within host model for immune response and its application to seriological profiles of pertussis, Epidemics (2014) 9 : 1-7

[4] A.N. Swart, M. Tomasi, M. Kretzschmar, A.H. Havelaar, O. Diekmann, The protective effect of temporary immunity under imposed infection pressure, Epidemics (2012) 4 : 43-47

[ Reference URL ]A first aim is to briefly review various mathematical models of infectious disease dynamics that incorporate waning and boosting of immunity. The focus will be on models that are described by delay equations, in particular renewal equations [1]. Concerning within-host dynamics, we limit ourselves to the rather caricatural models of Aron [2] and de Graaf e.a. [3].From a biomedical point of view the main conclusion is that a higher force of infection may lead to less disease,see [4] and the references given there.

[1] O.Diekmann, M.Gyllenberg, J.A.J.Metz, H.R.Thieme, On the formulation and analysis

of general deterministic structured population models. I. Linear theory, J. Math. Biol. (1998) 36 : 349 - 388

[2] J.L. Aron, Dynamics of acquired immunity boosted by exposure to infection, Math. Biosc. (1983) 64 : 249-259

[3] W.F. de Graaf, M.E.E. Kretzschmar, P.M.F. Teunis, O. Diekmann, A two-phase within host model for immune response and its application to seriological profiles of pertussis, Epidemics (2014) 9 : 1-7

[4] A.N. Swart, M. Tomasi, M. Kretzschmar, A.H. Havelaar, O. Diekmann, The protective effect of temporary immunity under imposed infection pressure, Epidemics (2012) 4 : 43-47

http://www.uu.nl/staff/ODiekmann

### 2016/10/03

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Visualizing the radial Loewner flow and the evolution family (JAPANESE)

**Hirokazu Shimauchi**(Yamanashi Eiwa College)Visualizing the radial Loewner flow and the evolution family (JAPANESE)

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

$C^*$-tensor categories and subfactors for totally disconnected groups

(English)

**Yuki Arano**(Univ. Tokyo)$C^*$-tensor categories and subfactors for totally disconnected groups

(English)

#### Tokyo Probability Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Coupled KPZ equations and complex-valued stochastic Ginzburg-Landau equation (日本語)

**Masato Hoshino**(Graduate School of Mathematical Science, the University of Tokyo)Coupled KPZ equations and complex-valued stochastic Ginzburg-Landau equation (日本語)

### 2016/09/27

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

CAT(0) properties for orthoscheme complexes (JAPANESE)

**Shouta Tounai**(The University of Tokyo)CAT(0) properties for orthoscheme complexes (JAPANESE)

[ Abstract ]

Gromov showed that a cubical complex is locally CAT(0) if and only if the link of every vertex is a flag complex. Brady and MacCammond introduced an orthoscheme complex as a generalization of cubical complexes. It is, however, difficult to tell whether an orthoscheme complex is (locally) CAT(0) or not. In this talk, I will discuss a translation of Gromov's characterization for orthoscheme complexes. As a generalization of Gromov's characterization, I will show that the orthoscheme complex of locally distributive semilattice is CAT(0) if and only if it is a flag semilattice.

Gromov showed that a cubical complex is locally CAT(0) if and only if the link of every vertex is a flag complex. Brady and MacCammond introduced an orthoscheme complex as a generalization of cubical complexes. It is, however, difficult to tell whether an orthoscheme complex is (locally) CAT(0) or not. In this talk, I will discuss a translation of Gromov's characterization for orthoscheme complexes. As a generalization of Gromov's characterization, I will show that the orthoscheme complex of locally distributive semilattice is CAT(0) if and only if it is a flag semilattice.

### 2016/09/26

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

TBA

(English)

**Sorin Popa**(UCLA)TBA

(English)

#### FMSP Lectures

16:00-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Mathematical Aesthetic Principles and Nonintegrable Systems (ENGLISH)

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

**Murray Muraskin**(University of North Dakota, Grand Forks)Mathematical Aesthetic Principles and Nonintegrable Systems (ENGLISH)

[ Abstract ]

The discussion presents a study of a set of mathematical principles that can be classified as "aesthetic”and shows that these principles can be cast into a set of nonlinear equations. The system of equations is nonintegrable in general. New techniques to handle the nonintegrability feature are discussed. We then illustrate how this system of equations leads to sinusoidal solutions, sine within sine solutions, the phenomenon known as beats, random type oscillations, two and three dimensional lattices, as well as multi wave packet systems. The sinusoidal solutions occur when the arbitrary data associated with the equations causes the equations to be linearized. The sinusoidal behavior totally disappears once the integrability equations are satisfied, illustrating how important the nonintegrability concept is to the development.

[ Reference URL ]The discussion presents a study of a set of mathematical principles that can be classified as "aesthetic”and shows that these principles can be cast into a set of nonlinear equations. The system of equations is nonintegrable in general. New techniques to handle the nonintegrability feature are discussed. We then illustrate how this system of equations leads to sinusoidal solutions, sine within sine solutions, the phenomenon known as beats, random type oscillations, two and three dimensional lattices, as well as multi wave packet systems. The sinusoidal solutions occur when the arbitrary data associated with the equations causes the equations to be linearized. The sinusoidal behavior totally disappears once the integrability equations are satisfied, illustrating how important the nonintegrability concept is to the development.

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

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