## Seminar information archive

Seminar information archive ～05/25｜Today's seminar 05/26 | Future seminars 05/27～

#### Numerical Analysis Seminar

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

Finite element method with various types of penalty on domain/boundary (ENGLISH)

**Guanyu Zhou**(The University of Tokyo)Finite element method with various types of penalty on domain/boundary (ENGLISH)

[ Abstract ]

We are concerned with several penalty methods (on domain/boundary)

combining with finite element method to solve some partial differential equations. The penalty methods are very useful and widely applied to various problems. For example, to solve the Navier-Stokes equations in moving boundary domain, the finite element method requires to construct the boundary fitted mesh at every times step, which is very time-consuming. The fictitious domain method is proposed to tackle this problem. It is to reformulate the equation to a larger fixed domain, called the fictitious domain, to which we can take a uniform mesh independent on the original moving boundary. The reformulation is based on a penalty method on do- main. Some penalty methods are proposed to approximate the boundary conditions which are not easy to handle with general FEM, such as the slip boundary condition to Stokes/Navier-Stokes equations, the unilateral boundary condition of Signorini’s type to Stokes equations, and so on. It is known that the variational crimes occurs if the finite element spaces or the implementation methods are not chosen properly for slip boundary condition. By introducing a penalty term to the normal component of velocity on slip boundary, we can solve the equations in FEM easily. For the boundary of Signorini’s type, the variational form is an inequality, to which the FEM is not easy to applied. However, we can approximate the variational inequality by a variation equation with penalty term, which can be solve by FEM directly. In above, we introduced several penalty methods with finite element approximation. In this work, we investigate the well-posedness of those penalty method, and obtain the error estimates of penalty; moreover, we consider the penalty methods combining with finite element approximation and show the error estimates.

We are concerned with several penalty methods (on domain/boundary)

combining with finite element method to solve some partial differential equations. The penalty methods are very useful and widely applied to various problems. For example, to solve the Navier-Stokes equations in moving boundary domain, the finite element method requires to construct the boundary fitted mesh at every times step, which is very time-consuming. The fictitious domain method is proposed to tackle this problem. It is to reformulate the equation to a larger fixed domain, called the fictitious domain, to which we can take a uniform mesh independent on the original moving boundary. The reformulation is based on a penalty method on do- main. Some penalty methods are proposed to approximate the boundary conditions which are not easy to handle with general FEM, such as the slip boundary condition to Stokes/Navier-Stokes equations, the unilateral boundary condition of Signorini’s type to Stokes equations, and so on. It is known that the variational crimes occurs if the finite element spaces or the implementation methods are not chosen properly for slip boundary condition. By introducing a penalty term to the normal component of velocity on slip boundary, we can solve the equations in FEM easily. For the boundary of Signorini’s type, the variational form is an inequality, to which the FEM is not easy to applied. However, we can approximate the variational inequality by a variation equation with penalty term, which can be solve by FEM directly. In above, we introduced several penalty methods with finite element approximation. In this work, we investigate the well-posedness of those penalty method, and obtain the error estimates of penalty; moreover, we consider the penalty methods combining with finite element approximation and show the error estimates.

### 2014/10/17

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

A finite diameter theorem on RCD spaces (JAPANESE)

**Yu Kitabeppu**(Kyoto University)A finite diameter theorem on RCD spaces (JAPANESE)

[ Abstract ]

I will talk about a finite diameter theorem on RCD spaces of (possibly) infinite dimension. An RCD space is a generalization of a concept of a manifold with bounded Ricci curvature. Savar¥'e proves the "self-improving property" on RCD spaces via the Gamma calculus. Because of his work and Kuwada’s duality argument, we are able to get the L^{¥infty}-contraction of heat kernels. I will show the main result by combining the contraction property and a simple lemma.

I will talk about a finite diameter theorem on RCD spaces of (possibly) infinite dimension. An RCD space is a generalization of a concept of a manifold with bounded Ricci curvature. Savar¥'e proves the "self-improving property" on RCD spaces via the Gamma calculus. Because of his work and Kuwada’s duality argument, we are able to get the L^{¥infty}-contraction of heat kernels. I will show the main result by combining the contraction property and a simple lemma.

### 2014/10/15

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Classification of actions of compact abelian groups on subfactors with index less than 4 (ENGLISH)

**Koichi Shimada**(Univ. Tokyo)Classification of actions of compact abelian groups on subfactors with index less than 4 (ENGLISH)

### 2014/10/14

#### Seminar on Mathematics for various disciplines

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

Fluid flow and electromagnetic fields, from viewpoint of theoretical physics -- Is the Navier-Stokes Equation sufficient to describe turbulence at very high Reynolds numbers? -- (JAPANESE)

**Tsutomu Kambe**(University of Tokyo)Fluid flow and electromagnetic fields, from viewpoint of theoretical physics -- Is the Navier-Stokes Equation sufficient to describe turbulence at very high Reynolds numbers? -- (JAPANESE)

[ Abstract ]

There exists analogy between the fluid flow and electromagnetic fields with respect to their mathematical representations. This is reasonable because both are continuous physical fields having energy and momentum in space-time. In particular, fluid’s vorticity is analogous to magnetic field.

On the other hand, for simulation of atmospheric global motion on the giant computer Earth Simulator, many empirical physical parameters must be introduced in order to obtain realistic results for weather prediction, etc. This implies that the present system of equations of fluid flows may not be sufficient to describe fluid motions of large scales at very high Reynolds numbers. We consider whether the above-mentioned analogy is useful for improvement of the theory of turbulence at very high Reynolds numbers.

There exists analogy between the fluid flow and electromagnetic fields with respect to their mathematical representations. This is reasonable because both are continuous physical fields having energy and momentum in space-time. In particular, fluid’s vorticity is analogous to magnetic field.

On the other hand, for simulation of atmospheric global motion on the giant computer Earth Simulator, many empirical physical parameters must be introduced in order to obtain realistic results for weather prediction, etc. This implies that the present system of equations of fluid flows may not be sufficient to describe fluid motions of large scales at very high Reynolds numbers. We consider whether the above-mentioned analogy is useful for improvement of the theory of turbulence at very high Reynolds numbers.

#### Number Theory Seminar

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

A p-adic criterion for good reduction of curves (ENGLISH)

**Fabrizio Andreatta**(Università Statale di Milano)A p-adic criterion for good reduction of curves (ENGLISH)

[ Abstract ]

Given a curve over a dvr of mixed characteristic 0-p with smooth generic fiber and with semistable reduction, I will present a criterion for good reduction in terms of the (unipotent) p-adic étale fundamental group of its generic fiber.

Given a curve over a dvr of mixed characteristic 0-p with smooth generic fiber and with semistable reduction, I will present a criterion for good reduction in terms of the (unipotent) p-adic étale fundamental group of its generic fiber.

### 2014/10/11

#### Monthly Seminar on Arithmetic of Automorphic Forms

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

The "secondary spherical function" of the discrete series representations of $SU(3,1)$ (following the memo of Tadashi Miyazaki) (JAPANESE)

Toward Fourier expansion of automorphic forms on $Sp(2, R)$ (JAPANESE)

**Takayuki Oda**(Grad. School of Math.-Sci., Univ. of Tokyo) 13:30-14:30The "secondary spherical function" of the discrete series representations of $SU(3,1)$ (following the memo of Tadashi Miyazaki) (JAPANESE)

**Hideshi Takayanagi**(Sakushin-Gakuin University) 15:00-16:00Toward Fourier expansion of automorphic forms on $Sp(2, R)$ (JAPANESE)

### 2014/10/10

#### Colloquium

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

Etale cohomology of local Shimura varieties and the local Langlands correspondence (JAPANESE)

**Yoichi Mieda**(Graduate School of Mathematical Sciences, University of Tokyo)Etale cohomology of local Shimura varieties and the local Langlands correspondence (JAPANESE)

### 2014/10/08

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Realization of hyperbolic group $C^*$-algebras as decreasing intersection of Cuntz algebras $O_2$ (ENGLISH)

**Yuhei Suzuki**(Univ. Tokyo/RIMS, Kyoto Univ.)Realization of hyperbolic group $C^*$-algebras as decreasing intersection of Cuntz algebras $O_2$ (ENGLISH)

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Qualitative analysis of disease transmission dynamics for renewal equations (JAPANESE)

**Yoichi Enatsu**(Graduate School of Math. Sci. Bldg.)Qualitative analysis of disease transmission dynamics for renewal equations (JAPANESE)

### 2014/10/07

#### Tuesday Seminar on Topology

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

Transversality problems in string topology and de Rham chains (JAPANESE)

**Kei Irie**(RIMS, Kyoto University)Transversality problems in string topology and de Rham chains (JAPANESE)

[ Abstract ]

The starting point of string topology is the work of Chas-Sullivan, which uncovered the Batalin-Vilkovisky(BV) structure on homology of the free loop space of a manifold.

It is important to define chain level structures beneath the BV structure on homology, however this problem is yet to be settled.

One of difficulties is that, to define intersection products on chain level, we have to address the transversality issue.

In this talk, we introduce a notion of "de Rham chain" to bypass this trouble, and partially realize expected chain level structures.

The starting point of string topology is the work of Chas-Sullivan, which uncovered the Batalin-Vilkovisky(BV) structure on homology of the free loop space of a manifold.

It is important to define chain level structures beneath the BV structure on homology, however this problem is yet to be settled.

One of difficulties is that, to define intersection products on chain level, we have to address the transversality issue.

In this talk, we introduce a notion of "de Rham chain" to bypass this trouble, and partially realize expected chain level structures.

### 2014/10/03

#### Infinite Analysis Seminar Tokyo

13:30-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

KPZ equation and Macdonald process (JAPANESE)

Entanglement spectra in topological phases and coupled Tomonaga-Luttinger liquids (JAPANESE)

**Tomohiro SASAMOTO**(Department of Physics, Tokyo Institute of Technology) 13:30-15:00KPZ equation and Macdonald process (JAPANESE)

**Shunsuke FURUKAWA**(Department of Physics, the Tokyo University) 15:30-17:00Entanglement spectra in topological phases and coupled Tomonaga-Luttinger liquids (JAPANESE)

[ Abstract ]

The entanglement spectrum (ES) has been found to provide useful probes of topological phases of matter and other exotic strongly correlated states. For the system's ground state, the ES is defined as the full eigenvalue spectrum of the reduced density matrix obtained by tracing out the degrees of freedom in part of the system. A key result observed in various topological phases and other gapped systems has been the remarkable correspondence between the ES and the edge-state spectrum. While this correspondence has been analytically proven for some topological phases, it is interesting to ask what systems show this correspondence more generally and how the ES changes when the bulk energy gap closes.

We here study the ES in two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. In addition to having direct applications to ladder systems, this problem is closely related to the entanglement properties of two-dimensional topological phases. Based on the calculation for coupled chiral TLLs, we provide a simple physical proof for the correspondence between edge states and the ES in quantum Hall systems consistent with previous numerical and analytical studies. We also discuss violations of this correspondence in gapped and gapless phases of coupled non-chiral TLLs.

Reference: R. Lundgren, Y. Fuji, SF, and M. Oshikawa, Phys. Rev. B 88, 245137 (2013).

The entanglement spectrum (ES) has been found to provide useful probes of topological phases of matter and other exotic strongly correlated states. For the system's ground state, the ES is defined as the full eigenvalue spectrum of the reduced density matrix obtained by tracing out the degrees of freedom in part of the system. A key result observed in various topological phases and other gapped systems has been the remarkable correspondence between the ES and the edge-state spectrum. While this correspondence has been analytically proven for some topological phases, it is interesting to ask what systems show this correspondence more generally and how the ES changes when the bulk energy gap closes.

We here study the ES in two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. In addition to having direct applications to ladder systems, this problem is closely related to the entanglement properties of two-dimensional topological phases. Based on the calculation for coupled chiral TLLs, we provide a simple physical proof for the correspondence between edge states and the ES in quantum Hall systems consistent with previous numerical and analytical studies. We also discuss violations of this correspondence in gapped and gapless phases of coupled non-chiral TLLs.

Reference: R. Lundgren, Y. Fuji, SF, and M. Oshikawa, Phys. Rev. B 88, 245137 (2013).

### 2014/10/02

#### Operator Algebra Seminars

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Entropy and $L^2$-torsion (ENGLISH)

**Hanfeng Li**(SUNY Buffalo)Entropy and $L^2$-torsion (ENGLISH)

### 2014/10/01

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Back-and-forth between scattering theory and index theorems (ENGLISH)

**Serge Richard**(Nagoya Univ.)Back-and-forth between scattering theory and index theorems (ENGLISH)

### 2014/09/22

#### Infinite Analysis Seminar Tokyo

13:30-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Colored HOMFLY homology of knots and links (ENGLISH)

**Satoshi Nawata**(Theoretical Physics at NIKHEF)Colored HOMFLY homology of knots and links (ENGLISH)

[ Abstract ]

In this talk I will present structural properties of colored HOMFLY homology of knots and links. These rich properties of the categorification of the colored HOMFLY polynomial are obtained by using various methods: physics insights, representation theory of Lie super-algebras, double affine Hecke algebras, etc. This in turn enables computation of colored HOMFLY homology for various classes of knots and links and consequent computation of super-A-polynomial - the deformation of the classical A-polynomial. I will also explain recent results and special additional properties for colored Kauffman homology as well as the case of links. Although I will try to give a talk accessible to mathematicians, there is no proof and rigorousness in this talk.

In this talk I will present structural properties of colored HOMFLY homology of knots and links. These rich properties of the categorification of the colored HOMFLY polynomial are obtained by using various methods: physics insights, representation theory of Lie super-algebras, double affine Hecke algebras, etc. This in turn enables computation of colored HOMFLY homology for various classes of knots and links and consequent computation of super-A-polynomial - the deformation of the classical A-polynomial. I will also explain recent results and special additional properties for colored Kauffman homology as well as the case of links. Although I will try to give a talk accessible to mathematicians, there is no proof and rigorousness in this talk.

### 2014/09/19

#### Colloquium

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

William Thurston and foliation theory (ENGLISH)

**Etienne Ghys**(École normale supérieure de Lyon)William Thurston and foliation theory (ENGLISH)

[ Abstract ]

Between 1972 and 1976, William Thurston revolutionized foliation theory. Twenty years later, he described this period of his mathematical life in a remarkable paper « On proofs and progress in mathematics ». In this talk, I will begin by a general overview of some of Thurston's contribution to this theory. I will then describe some of the current development.

Between 1972 and 1976, William Thurston revolutionized foliation theory. Twenty years later, he described this period of his mathematical life in a remarkable paper « On proofs and progress in mathematics ». In this talk, I will begin by a general overview of some of Thurston's contribution to this theory. I will then describe some of the current development.

#### FMSP Lectures

14:30-16:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A bridge between knotted graphs and axiomatizations of groups (ENGLISH)

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

**Victoria Lebed**(Osaka City University, JSPS)A bridge between knotted graphs and axiomatizations of groups (ENGLISH)

[ Abstract ]

This talk will be devoted to a new algebraic structure called qualgebra. From the topological viewpoint, our construction is motivated by a study of knotted 3-valent graphs via combinatorially defined coloring invariants. From the algebraic viewpoint, it gives a part of an alternative axiomatization of groups, describing the properties of the conjugation operation and its interactions with the group multiplication. Explicit examples of qualgebras and associated graph invariants will be given.

[ Reference URL ]This talk will be devoted to a new algebraic structure called qualgebra. From the topological viewpoint, our construction is motivated by a study of knotted 3-valent graphs via combinatorially defined coloring invariants. From the algebraic viewpoint, it gives a part of an alternative axiomatization of groups, describing the properties of the conjugation operation and its interactions with the group multiplication. Explicit examples of qualgebras and associated graph invariants will be given.

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

### 2014/09/17

#### PDE Real Analysis Seminar

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

On the abstract evolution equations of hyperbolic type (JAPANESE)

**Kentarou Yoshii**(Faculty of Science Division I, Tokyo University of Science)On the abstract evolution equations of hyperbolic type (JAPANESE)

[ Abstract ]

This talk deals with the abstract Cauchy problem for linear evolution equations of hyperbolic type in a Hilbert space. We will discuss the existence and uniqueness of its classical solution and apply the results to linear Schrödinger equations with time dependent potentials.

This talk deals with the abstract Cauchy problem for linear evolution equations of hyperbolic type in a Hilbert space. We will discuss the existence and uniqueness of its classical solution and apply the results to linear Schrödinger equations with time dependent potentials.

### 2014/09/12

#### FMSP Lectures

10:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stochastic homogenization for first order Hamilton-Jacobi equations(III) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

**Hung V. Tran**(The University of Chicago)Stochastic homogenization for first order Hamilton-Jacobi equations(III) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

### 2014/09/10

#### FMSP Lectures

10:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stochastic homogenization for first order Hamilton-Jacobi equations(II) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

**Hung V. Tran**(The University of Chicago)Stochastic homogenization for first order Hamilton-Jacobi equations(II) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

### 2014/09/09

#### Tuesday Seminar of Analysis

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

Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

**Hatem Zaag**(CNRS / University of Paris Nord)Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

[ Abstract ]

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

#### FMSP Lectures

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

Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

**Hatem Zaag**(CNRS/University of Paris Nord)Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

[ Abstract ]

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

### 2014/09/08

#### FMSP Lectures

10:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stochastic homogenization for first order Hamilton-Jacobi equations(I) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

**Hung V. Tran**(The University of Chicago)Stochastic homogenization for first order Hamilton-Jacobi equations(I) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

### 2014/09/04

#### Lectures

12:10-13:00 Room #470 (Graduate School of Math. Sci. Bldg.)

"X-ray imaging of moving objects" (ENGLISH)

**Samuli Siltanen**(University of Helsinki, Finland)"X-ray imaging of moving objects" (ENGLISH)

### 2014/08/28

#### thesis presentations

10:00-11:15 Room #128 (Graduate School of Math. Sci. Bldg.)

On the C1 stabilization of homoclinic tangencies for diffeomorphisms in dimension three(3次元の微分同相写像に対するホモクリニック接触のC1安定化について) (JAPANESE)

**李 曉龍**(東京大学大学院数理科学研究科)On the C1 stabilization of homoclinic tangencies for diffeomorphisms in dimension three(3次元の微分同相写像に対するホモクリニック接触のC1安定化について) (JAPANESE)

### 2014/08/06

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

The stochastic SIS epidemic model in a periodic environment (ENGLISH)

**Nicolas Bacaer**(Insitut de Recherche pour le Developpement (IRD))The stochastic SIS epidemic model in a periodic environment (ENGLISH)

[ Abstract ]

In the stochastic SIS epidemic model with a contact rate a,

a recovery rate bT is such that (log T)/N converges to c=b/a-1-log(b/a) as N grows to

infinity. We consider the more realistic case where the contact rate

a(t) is a periodic function whose average is bigger than b. Then (log

T)/N converges to a new limit C, which is linked to a time-periodic

Hamilton-Jacobi equation. When a(t) is a cosine function with small

amplitude or high (resp. low) frequency, approximate formulas for C

can be obtained analytically following the method used in [Assaf et

al. (2008) Population extinction in a time-modulated environment. Phys

Rev E 78, 041123]. These results are illustrated by numerical

simulations.

In the stochastic SIS epidemic model with a contact rate a,

a recovery rate bT is such that (log T)/N converges to c=b/a-1-log(b/a) as N grows to

infinity. We consider the more realistic case where the contact rate

a(t) is a periodic function whose average is bigger than b. Then (log

T)/N converges to a new limit C, which is linked to a time-periodic

Hamilton-Jacobi equation. When a(t) is a cosine function with small

amplitude or high (resp. low) frequency, approximate formulas for C

can be obtained analytically following the method used in [Assaf et

al. (2008) Population extinction in a time-modulated environment. Phys

Rev E 78, 041123]. These results are illustrated by numerical

simulations.

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