## Seminar information archive

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

### 2013/03/18

#### Operator Algebra Seminars

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Ultraproducts of von Neumann algebras (JAPANESE)

**Hiroshi Ando**(IHES)Ultraproducts of von Neumann algebras (JAPANESE)

#### Colloquium

15:00-17:30 Room #050 (Graduate School of Math. Sci. Bldg.)

Value distribution theory and analytic function theory in several variables (JAPANESE)

My fifty years of differential equations (JAPANESE)

**NOGUCHI, Junjiro**(University of Tokyo) 15:00-16:00Value distribution theory and analytic function theory in several variables (JAPANESE)

**OSHIMA, Toshio**(University of Tokyo) 16:30-17:30My fifty years of differential equations (JAPANESE)

### 2013/03/15

#### Operator Algebra Seminars

15:45-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Infinitesimal 2-Yang-Baxter operators from a categorification

of the Knizhnik-Zamolodchikov connection (ENGLISH)

Twisted tensor product of $C^*$-algebras (ENGLISH)

**Lucio Cirio**(Univ. M\"unster) 15:45-16:45Infinitesimal 2-Yang-Baxter operators from a categorification

of the Knizhnik-Zamolodchikov connection (ENGLISH)

**Sutanu Roy**(Univ. G\"ottingen) 17:00-18:00Twisted tensor product of $C^*$-algebras (ENGLISH)

#### Numerical Analysis Seminar

10:00-12:15 Room #056 (Graduate School of Math. Sci. Bldg.)

Complex flow at the boundaries of branched models: numerical aspects (ENGLISH)

[ Reference URL ]

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

**Irene Vignon-Clementel**(INRIA Paris Rocquencourt )Complex flow at the boundaries of branched models: numerical aspects (ENGLISH)

[ Reference URL ]

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

#### Operator Algebra Seminars

14:30-15:30 Room #118 (Graduate School of Math. Sci. Bldg.)

The connected component of a compact quantum group (ENGLISH)

**Stefano Rossi**(Univ. Roma II)The connected component of a compact quantum group (ENGLISH)

#### Operator Algebra Seminars

13:15-14:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Two Amir-Cambern type theorems for $C^*$-algebras (ENGLISH)

**Jean Roydor**(Univ. Bordeaux)Two Amir-Cambern type theorems for $C^*$-algebras (ENGLISH)

#### PDE Real Analysis Seminar

10:30-11:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Geometric rigidity for incompatible fields and an application to strain-gradient plasticity (ENGLISH)

**Caterina Zeppieri**(Universität Münster)Geometric rigidity for incompatible fields and an application to strain-gradient plasticity (ENGLISH)

[ Abstract ]

Motivated by the study of nonlinear plane elasticity in presence of edge dislocations, in this talk we show that in dimension two the Friesecke, James, and Müller Rigidity Estimate holds true also for matrix-fields with nonzero curl, modulo an error depending on the total mass of the curl.

The above generalised rigidity is then used to derive a strain-gradient model for plasticity from semi-discrete nonlinear dislocation energies by Gamma-convergence.

The above results are obtained in collaboration with S. Müller (University of Bonn, Germany) and L. Scardia (University of Glasgow, UK).

Motivated by the study of nonlinear plane elasticity in presence of edge dislocations, in this talk we show that in dimension two the Friesecke, James, and Müller Rigidity Estimate holds true also for matrix-fields with nonzero curl, modulo an error depending on the total mass of the curl.

The above generalised rigidity is then used to derive a strain-gradient model for plasticity from semi-discrete nonlinear dislocation energies by Gamma-convergence.

The above results are obtained in collaboration with S. Müller (University of Bonn, Germany) and L. Scardia (University of Glasgow, UK).

### 2013/03/11

#### Operator Algebra Seminars

09:45-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Traces and Ultrapowers (ENGLISH)

Constructing subfactors with jellyfish (ENGLISH)

**Tristan Bice**(York Univ.) 09:45-10:45Traces and Ultrapowers (ENGLISH)

**David Penneys**(Univ. Toronto) 11:00-12:00Constructing subfactors with jellyfish (ENGLISH)

#### Operator Algebra Seminars

13:30-18:15 Room #123 (Graduate School of Math. Sci. Bldg.)

Classification and the Toms-Winter Conjecture (ENGLISH)

Free probability and planar algebras (ENGLISH)

Kirchberg $X$-algebras with real rank zero and intermediate cancellation (ENGLISH)

Classifying $C^*$-algebras up to W-stability (ENGLISH)

**Danny Hey**(Univ. Toronto) 13:30-14:30Classification and the Toms-Winter Conjecture (ENGLISH)

**Stephen Curran**(UCLA) 14:45-15:45Free probability and planar algebras (ENGLISH)

**Rasmus Bentmann**(Univ. Copenhagen) 16:00-17:00Kirchberg $X$-algebras with real rank zero and intermediate cancellation (ENGLISH)

**Luis Santiago-Moreno**(Univ. Oregon) 17:15-18:15Classifying $C^*$-algebras up to W-stability (ENGLISH)

#### GCOE Seminars

14:45-15:45 Room #123 (Graduate School of Math. Sci. Bldg.)

Free probability and planar algebras (ENGLISH)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~yasuyuki/mini2013-4.htm

**Stephen Curran**(UCLA)Free probability and planar algebras (ENGLISH)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~yasuyuki/mini2013-4.htm

### 2013/03/08

#### FMSP Lectures

10:30-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Knots, algorithms and linear programming: the quest to solve unknot recognition in polynomial time (ENGLISH)

**Benjamin Burton**(The University of Queensland, Australia)Knots, algorithms and linear programming: the quest to solve unknot recognition in polynomial time (ENGLISH)

[ Abstract ]

In this talk we explore new approaches to the old and difficult computational problem of unknot recognition. Although the best known algorithms for this problem run in exponential time, there is increasing evidence that a polynomial time solution might be possible. We outline several promising approaches, in which computational geometry, linear programming and greedy algorithms all play starring roles. We finish with a new algorithm that combines techniques from topology and combinatorial optimisation, which is the first to exhibit "real world" polynomial time behaviour: although it is still exponential time in theory, exhaustive experimentation shows that this algorithm can solve unknot recognition for "practical" inputs by running just a linear number of linear programs.

This is joint work with Melih Ozlen.

In this talk we explore new approaches to the old and difficult computational problem of unknot recognition. Although the best known algorithms for this problem run in exponential time, there is increasing evidence that a polynomial time solution might be possible. We outline several promising approaches, in which computational geometry, linear programming and greedy algorithms all play starring roles. We finish with a new algorithm that combines techniques from topology and combinatorial optimisation, which is the first to exhibit "real world" polynomial time behaviour: although it is still exponential time in theory, exhaustive experimentation shows that this algorithm can solve unknot recognition for "practical" inputs by running just a linear number of linear programs.

This is joint work with Melih Ozlen.

### 2013/03/07

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

Smoothing of sign test and approximation of its p-value (JAPANESE)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/16.html

**MAESONO, Yoshihiko**(Kyushu University)Smoothing of sign test and approximation of its p-value (JAPANESE)

[ Abstract ]

In this talk we discuss theoretical properties of smoothed sign test, which based on a kernel estimator of the underlying distribution function of data. We show the smoothed sign test is equivalent to the usual sign test in the sense of Pitman efficiency, and its main term of the variance does not depend on the distribution of the population, under the null hypothesis. Though smoothed sign test is not distribution-free, we can obtain Edgeworth expansion which does not depend on the distribution. This is a joint work with Ms. Mengxin Lu of Kyushu University.

[ Reference URL ]In this talk we discuss theoretical properties of smoothed sign test, which based on a kernel estimator of the underlying distribution function of data. We show the smoothed sign test is equivalent to the usual sign test in the sense of Pitman efficiency, and its main term of the variance does not depend on the distribution of the population, under the null hypothesis. Though smoothed sign test is not distribution-free, we can obtain Edgeworth expansion which does not depend on the distribution. This is a joint work with Ms. Mengxin Lu of Kyushu University.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/16.html

### 2013/03/06

#### Lectures

16:00-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

The rotation set around a fixed point for surface homeomorphisms. (ENGLISH)

**Frederic Le Roux**(Institut de Mathematiques de Jussieu, Universite Pierre et Marie Curie)The rotation set around a fixed point for surface homeomorphisms. (ENGLISH)

[ Abstract ]

We propose two definitions of a local rotation set. As applications, one

gets some criteria for the existence of periodic orbits, and a clear

explanation of Gambaudo-Le Calvez-Pecou's version of the Naishul theorem:

for surface diffeomorphisms, the rotation number of the derivative at a

fixed point which is not a sink nor a source is a topological invariant.

Tha local rotation set also provide an unexpected topological

characterization for the parabolic fixed points of holomorphic maps.

We propose two definitions of a local rotation set. As applications, one

gets some criteria for the existence of periodic orbits, and a clear

explanation of Gambaudo-Le Calvez-Pecou's version of the Naishul theorem:

for surface diffeomorphisms, the rotation number of the derivative at a

fixed point which is not a sink nor a source is a topological invariant.

Tha local rotation set also provide an unexpected topological

characterization for the parabolic fixed points of holomorphic maps.

### 2013/03/05

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

ベイズ予測に基いた波動関数の推定と純粋状態モデルの無情報事前分布 (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/15.html

**TANAKA, Fuyuhiko**(University of Tokyo)ベイズ予測に基いた波動関数の推定と純粋状態モデルの無情報事前分布 (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/15.html

#### GCOE Seminars

17:00-18:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Uniqueness for inverse boundary value problems by Dirichlet-to

-Neumann map on subboundaries (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Uniqueness for inverse boundary value problems by Dirichlet-to

-Neumann map on subboundaries (ENGLISH)

[ Abstract ]

We consider inverse boundary value problems for elliptic equations of second order and survey recent results on the uniqueness mainly by partial boundary data. In particular, in two dimensions, we show uniqueness results by means of Dirichlet data supported on an arbitrary subboundary $\\widetilde\\Gamma$ and Neumann data measured on $\\widetilde\\Gamma$. We describe the key idea for the proof: complex geometric optics solutions which are constructed by a Carleman estimate. Also we show the uniqueness by Dirichlet-to-Neumann map on subboundaries in three dimensions.

We consider inverse boundary value problems for elliptic equations of second order and survey recent results on the uniqueness mainly by partial boundary data. In particular, in two dimensions, we show uniqueness results by means of Dirichlet data supported on an arbitrary subboundary $\\widetilde\\Gamma$ and Neumann data measured on $\\widetilde\\Gamma$. We describe the key idea for the proof: complex geometric optics solutions which are constructed by a Carleman estimate. Also we show the uniqueness by Dirichlet-to-Neumann map on subboundaries in three dimensions.

### 2013/03/04

#### GCOE Seminars

17:00-18:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Resonant Light Scattering by Small Particles (ENGLISH)

**M.I. Tribelsky**(Landau Institute)Resonant Light Scattering by Small Particles (ENGLISH)

[ Abstract ]

The problem of light scattering by a small spherical particle is studied within the framework of the exact solution of the Maxwell equations. It is shown that if imaginary part of the dielectric permittivity of the particle is small enough, the problem exhibits sharp giant resonances with very unusual properties. Specifically, the characteristic values of the electric and magnetic fields inside the particle and in its immediate vicinity are singular in the particle size. In non-dissipative case these quantities do not have definite limits when the radius of the particle tends to zero. The field of the Poynting vector in the immediate vicinity of the particle includes singular points, whose number, types and positions are very sensitive to the changes in the incident light frequency. As an example a bifurcation diagram, describing the behavior of the singular points in the vicinity of the dipole resonance for a particle with a certain fixed size is discussed.

The problem of light scattering by a small spherical particle is studied within the framework of the exact solution of the Maxwell equations. It is shown that if imaginary part of the dielectric permittivity of the particle is small enough, the problem exhibits sharp giant resonances with very unusual properties. Specifically, the characteristic values of the electric and magnetic fields inside the particle and in its immediate vicinity are singular in the particle size. In non-dissipative case these quantities do not have definite limits when the radius of the particle tends to zero. The field of the Poynting vector in the immediate vicinity of the particle includes singular points, whose number, types and positions are very sensitive to the changes in the incident light frequency. As an example a bifurcation diagram, describing the behavior of the singular points in the vicinity of the dipole resonance for a particle with a certain fixed size is discussed.

### 2013/02/28

#### GCOE Seminars

16:00-17:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Stability of the determination of the surface impedance of an

obstacle from the scattering amplitude (ENGLISH)

**Mourad Choulli**(Univ. Lorraine)Stability of the determination of the surface impedance of an

obstacle from the scattering amplitude (ENGLISH)

[ Abstract ]

In this joint work with Mourad Bellassoued and Aymen Jbalia, we prove a stability estimate of logarithmic type for the inverse problem consisting in the determination of the surface impedance of an obstacle from the scattering amplitude. We present a simple and direct proof which is essentially based on an elliptic Carleman inequality.

In this joint work with Mourad Bellassoued and Aymen Jbalia, we prove a stability estimate of logarithmic type for the inverse problem consisting in the determination of the surface impedance of an obstacle from the scattering amplitude. We present a simple and direct proof which is essentially based on an elliptic Carleman inequality.

#### GCOE Seminars

17:00-18:00 Room #270 (Graduate School of Math. Sci. Bldg.)

$L_0$ optimization and Lagrange multiplier (ENGLISH)

**Kazufumi Ito**(North Carolina State Univ.)$L_0$ optimization and Lagrange multiplier (ENGLISH)

[ Abstract ]

$L_p$ optimization with $p ¥in[0,1)$ is investigated. The difficulty of natural lack of weak lower-semicontinuity is addressed and the Lagrange multiplier theory is developed. Existence results and necessary optimality conditions are obtained, and the semismooth Newton method using the primal-dual active set is developed. The theory and algorithm are demonstrated for the case of optimal control problems. A maximum principle is derived and existence of controls, in some cases relaxed controls, is proved.

$L_p$ optimization with $p ¥in[0,1)$ is investigated. The difficulty of natural lack of weak lower-semicontinuity is addressed and the Lagrange multiplier theory is developed. Existence results and necessary optimality conditions are obtained, and the semismooth Newton method using the primal-dual active set is developed. The theory and algorithm are demonstrated for the case of optimal control problems. A maximum principle is derived and existence of controls, in some cases relaxed controls, is proved.

### 2013/02/27

#### GCOE Seminars

10:00-11:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Sufficient optimality conditions for a semi-linear parabolic system related to multiphase steel production (ENGLISH)

**Dietmar Hoemberg**(Weierstrass Institute)Sufficient optimality conditions for a semi-linear parabolic system related to multiphase steel production (ENGLISH)

[ Abstract ]

Multiphase steels combine good formability properties with high strength and have therefore become important construction materials, especially in automotive industry. The standard process route is hot rolling with subsequent controlled cooling to adjust the desired phase mixture. In the first part of the talk a phenomenological model for the austenite ferrite phase transition is developed in terms of a nucleation and growth process, where the growth rate depends on the carbon concentration in austenite. The approach allows for further extensions, e.g., to account for a speed up of nucleation due to deformation of austenite grains. The model is coupled with an energy balance to describe the phase transitions on a run-out table after hot rolling. Here, the most important control parameters are the amount of water flowing per time and the feed velocity of the strip. The spatial flux profile of the water nozzles has been identified from experiments.

Since the process window for the adjustment of the phase composition is very tight the computation of optimal process parameters is an important task also in practice. This is discussed in the second part of the talk using a classical optimal control approach, where a coefficient in the Robin boundary condition acts as the control. I will discuss necessary and sufficient optimality conditions, describe a SQP-approach for its numerical solution and conclude with some numerical results.

(joint work with K. Krumbiegel and N. Togobytska, WIAS)

Multiphase steels combine good formability properties with high strength and have therefore become important construction materials, especially in automotive industry. The standard process route is hot rolling with subsequent controlled cooling to adjust the desired phase mixture. In the first part of the talk a phenomenological model for the austenite ferrite phase transition is developed in terms of a nucleation and growth process, where the growth rate depends on the carbon concentration in austenite. The approach allows for further extensions, e.g., to account for a speed up of nucleation due to deformation of austenite grains. The model is coupled with an energy balance to describe the phase transitions on a run-out table after hot rolling. Here, the most important control parameters are the amount of water flowing per time and the feed velocity of the strip. The spatial flux profile of the water nozzles has been identified from experiments.

Since the process window for the adjustment of the phase composition is very tight the computation of optimal process parameters is an important task also in practice. This is discussed in the second part of the talk using a classical optimal control approach, where a coefficient in the Robin boundary condition acts as the control. I will discuss necessary and sufficient optimality conditions, describe a SQP-approach for its numerical solution and conclude with some numerical results.

(joint work with K. Krumbiegel and N. Togobytska, WIAS)

### 2013/02/23

#### Monthly Seminar on Arithmetic of Automorphic Forms

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

Ramanujan circulant graphs and the Hardy-Littlewood conjecture (JAPANESE)

Generalized Whittaker functions on $G_{2}$ (JAPANESE)

**Miki Hirano**(Ehime University) 13:30-14:30Ramanujan circulant graphs and the Hardy-Littlewood conjecture (JAPANESE)

[ Abstract ]

TBA

TBA

**Hiroaki Narita**(Kumamoto University) 15:00-16:00Generalized Whittaker functions on $G_{2}$ (JAPANESE)

[ Abstract ]

TBA

TBA

### 2013/02/22

#### Operator Algebra Seminars

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

II$_1$ factors with a unique Cartan subalgebra (ENGLISH)

**Stefaan Vaes**(KU Leuven)II$_1$ factors with a unique Cartan subalgebra (ENGLISH)

#### FMSP Lectures

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

II_1 factors with a unique Cartan subalgebra (ENGLISH)

**Stefaan Vaes**(KU Leuven)II_1 factors with a unique Cartan subalgebra (ENGLISH)

#### GCOE Seminars

16:00-17:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Exact insensitizing controls for scalar wave equations and control of coupled systems (ENGLISH)

**Fatiha Alabau-Boussouira**(Université de Lorraine)Exact insensitizing controls for scalar wave equations and control of coupled systems (ENGLISH)

[ Abstract ]

The control of scalar PDE's such as the wave or heat equation is by now well-understood.

It consists in building a source term which can drive the solution from a given initial state to a final (reachable) desired state.

Exact insensitizing control are exact controls which should satisfy an additional requirement:

they should be robust to small unknown perturbations of the initial data. More precisely, they should, as exact controls, drive the solution to the desired state, but they also should insensitize a given measure of the solution to such perturbations. One can show that the existence of exact insensitizing controls for a scalar wave equation is equivalent to the exact controllability by a single control of a system of two wave equations coupled in cascade.

We shall present in this talk the challenging issues and give some recent results and perspectives for the exact insensitizing control of scalar wave equations. We shall also give some more general results on the controllability of coupled systems by a reduced number of controls.

The control of scalar PDE's such as the wave or heat equation is by now well-understood.

It consists in building a source term which can drive the solution from a given initial state to a final (reachable) desired state.

Exact insensitizing control are exact controls which should satisfy an additional requirement:

they should be robust to small unknown perturbations of the initial data. More precisely, they should, as exact controls, drive the solution to the desired state, but they also should insensitize a given measure of the solution to such perturbations. One can show that the existence of exact insensitizing controls for a scalar wave equation is equivalent to the exact controllability by a single control of a system of two wave equations coupled in cascade.

We shall present in this talk the challenging issues and give some recent results and perspectives for the exact insensitizing control of scalar wave equations. We shall also give some more general results on the controllability of coupled systems by a reduced number of controls.

#### GCOE Seminars

17:00-18:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Carleman estimates and Lipschitz stability for Grushin-type operators (ENGLISH)

**Piermarco Cannarsa**(Univ. Roma II)Carleman estimates and Lipschitz stability for Grushin-type operators (ENGLISH)

[ Abstract ]

The Baouendi-Grushin operator is an important example of a degenerate elliptic operator that has strong connections with almost-riemannian structures. It is also the infinitesimal generator of a strongly continuous semigroup on Lebesgue spaces with very interesting properties from the point of view of control theory. Such properties will be discussed in this lecture, starting with approximate and null controllability.

We will then address the inverse source problem for these operators deriving a Lipschitz stability result.

The Baouendi-Grushin operator is an important example of a degenerate elliptic operator that has strong connections with almost-riemannian structures. It is also the infinitesimal generator of a strongly continuous semigroup on Lebesgue spaces with very interesting properties from the point of view of control theory. Such properties will be discussed in this lecture, starting with approximate and null controllability.

We will then address the inverse source problem for these operators deriving a Lipschitz stability result.

### 2013/02/21

#### FMSP Lectures

15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

)

Monte Carlo Methods for Partial Differential Equations: Computing Permeability

(ENGLISH)

**Michael Mascagni**(Florida State University)

Monte Carlo Methods for Partial Differential Equations: Computing Permeability

(ENGLISH)

[ Abstract ]

We present a brief overview of Monte Carlo methods for the solution of elliptic and parabolic partial differential equations (PDEs). We begin with a review of the Feynman-Kac formula, and its use in the probabilistic representation of the solutions of elliptic and parabolic PDEs. We then consider some specific Monte Carlo methods used for obtaining the solution of simple elliptic partial differential equations (PDEs) as part of exterior boundary value problems that arise in electrostatics and flow through porous media. These Monte Carlo methods use Feynman-Kac to represent the solution of the elliptic PDE at a point as the expected value of functionals of Brownian motion trajectories started at the point of interest. We discuss the rapid solution of these equations, in complex exterior geometries, using both the "walk on spheres" and "Greens function first-passage" algorithms. We then concentrate on methods for quickly computing the isotropic permeability using the "unit

capacitance" and "penetration depth'' methods. The first of these methods, requires computing a linear functional of the solution to an exterior elliptic PDE. Both these methods for computing permeability are simple, and provide accurate solutions in a few seconds on laptop-scale computers. We then conclude with a brief look at other Monte Carlo methods and problems that arise on related application areas.

We present a brief overview of Monte Carlo methods for the solution of elliptic and parabolic partial differential equations (PDEs). We begin with a review of the Feynman-Kac formula, and its use in the probabilistic representation of the solutions of elliptic and parabolic PDEs. We then consider some specific Monte Carlo methods used for obtaining the solution of simple elliptic partial differential equations (PDEs) as part of exterior boundary value problems that arise in electrostatics and flow through porous media. These Monte Carlo methods use Feynman-Kac to represent the solution of the elliptic PDE at a point as the expected value of functionals of Brownian motion trajectories started at the point of interest. We discuss the rapid solution of these equations, in complex exterior geometries, using both the "walk on spheres" and "Greens function first-passage" algorithms. We then concentrate on methods for quickly computing the isotropic permeability using the "unit

capacitance" and "penetration depth'' methods. The first of these methods, requires computing a linear functional of the solution to an exterior elliptic PDE. Both these methods for computing permeability are simple, and provide accurate solutions in a few seconds on laptop-scale computers. We then conclude with a brief look at other Monte Carlo methods and problems that arise on related application areas.

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