J.-P Chehab

P Garnier
email: pierre.garnier@u-picardie.fr2016

Y Mammeri

Numerical solution of the generalized Kadomtsev-Petviashvili equations with compact finite difference schemes

Keywords: Kadomtsev-Petviasvili equations, compact finite difference schemes, blow-up, transverse instabilities, Zaitsev solitons, line-soliton MS Codes. 35Q53, 65M06, 65M70, 35G25, 76B15

We propose compact finite difference schemes to solve the KP equations ut + uxxx + u p ux + λ∂ -1

x uyy = 0. When p = 1, this equation describes the propagation of small amplitude long waves in shallow water with weak transverse effects. We first present the numerical schemes which are compared to the Fourier spectral method. After establishing the numerical convergence, the scheme is validated. We then depict the behavior of solutions in the context of solitons instabilities and the blow-up.

Introduction

The Kadomtsev-Petviashvili equations u t + u xxx + uu x ± ∂ -1

x u yy = 0 were proposed to describe the propagation of long dispersive waves in shallow water taking into account weakly transverse effect in the y-direction [START_REF] Kadomtsev | On the stability of solitary waves in weakly dispersing media[END_REF][START_REF] Lannes | Consistency of the KP approximation[END_REF][START_REF] Lannes | Weakly transverse Boussinesq systems and the KP approximation[END_REF]. Here ∂ -1

x denotes the antiderivative and is defined by the Fourier transform

∂ -1 x u (ξ 1 , ξ 2 ) = û (ξ 1 , ξ 2 ) iξ 1 .
This antiderivative is introduced from the one-dimensional transport operator ∂ t +∂ x by considering weakly transverse perturbations, i.e. in the frequency region |ξ 2 /ξ 1 | 1. In this paper, we are concerned with the generalized Kadomtsev-Petviashvili equations u t + u xxx + u p u x + λ∂ -1 x u yy = 0,

with p ≥ 1. Here the equation is called KP-I when λ = -1 and KP-II when λ = 1.

It is well known that sufficiently smooth solutions vanishing at infinity preserve the following mass

+∞ -∞ +∞ -∞
u 2 (x, y, t)dxdy and energy

+∞ -∞ +∞ -∞ u p+2 (p + 1)(p + 2) - u 2 x 2 + λ(∂ -1
x u y ) 2 2 (x, y, t)dxdy.

Note also that one has a mass-zero constraint on x due to the equation itself, more precisely we have +∞ -∞ u(x, y, t)dx = 0, y ∈ R, t = 0, without any restraint on the initial data [START_REF] Molinet | Remarks on the mass constraint for KP type equations[END_REF]. This condition is of paramount importance in the construction of numerical schemes.

Numerical behavior of the KP equations has been intensely studied recently [START_REF] Hamidouche | Simulations numériques des équations de Kadomtsev-Petviashvili[END_REF][START_REF] Hamidouche | Numerical study of the solutions of the 3D generalized Kadomtsev-Petvisahvili equations for long times[END_REF][START_REF] Infeld | Numerical simulations of Kadomtsev-Petviashvili soliton interactions[END_REF][START_REF] Infeld | Instabilities and oscillations of one-and two-dimensional Kadomtsev-Petviashvili waves and solitons. II. Linear to nonlinear analysis[END_REF][START_REF] Klein | Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation[END_REF][START_REF] Klein | Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations[END_REF][START_REF] Klein | Fourth order time-stepping for small dispersion Kadomtsev-Petviashvili and Davey-Stewartson equations[END_REF][START_REF] Mammeri | Comparaison entre modèles d'ondes de surface en dimension 2[END_REF][START_REF] Wazwaz | A computational approach to soliton solutions of the Kadomtsev-Petviashvili equation[END_REF]. Due to the definition of the antiderivative, spectral methods has been privileged. This needs to impose periodic boundary conditions. In our work, compact finite difference schemes are proposed [START_REF] Lele | Compact finite difference schemes with spectral-like resolution[END_REF]. Their benefits are to easily adapt to all boundary conditions, and also to increase the order. Remark that, we can prove [START_REF] Mammeri | Unique continuation property for the KP-BBM equation[END_REF] that the antiderivative in a bounded domain is equivalently given by an integration in space

∂ -1 x u(x, y, t) = x -∞ u(x , y, t)dx .
The paper is organized as follows. Section 2 deals with the numerical schemes. It consists of building compact finite difference and spectral-compact schemes of arbitrary large order. These new schemes are validated in Section 3 with the Zaitsev soliton [START_REF] Zaitsev | Formation of stationary waves by superposition of solitons[END_REF]

ψ c (x, y, t) = 12α 2 1 -β cosh(αx -ωt) cos(δy) (cosh(αx -ωt) -β cos(δy)) 2 , β = δ 2 -3α 4 δ 2 , 3α 4 < δ 2 , ω = δ 2 + α 4 α , c = ω α .
as the exact solution of the KP-I equation. Finally, the robustness of the compact scheme is tested through three classical experiments and is compared to fully spectral method. First, the instability of the Zaitsev soliton is established using Schwartzian perturbations [START_REF] Klein | Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations[END_REF]. Then transverse instabilities of the Korteweg-de Vries line-soliton

Φ c (x, t) = (p + 1)(p + 2) 2 c 1/p sech 2/p p √ c 2 (x -ct) ,
under the KP-I flow is studied [START_REF] Alexander | On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation[END_REF][START_REF] De Bouard | Solitary waves of the generalized KP equations[END_REF][START_REF] Rousset | Transverse nonlinear instability of solitary waves for some hamiltonian PDEs[END_REF][START_REF] Rousset | Transverse nonlinear instability for two-dimensional dispersive models[END_REF][START_REF] Rousset | A simple criterion of transverse linear instability for solitary waves[END_REF][START_REF] Rousset | Transverse instability of the line solitary water waves[END_REF][START_REF] Rousset | Stability and instability of the KdV solitary wave under the KP-I flow[END_REF]. We conclude with the blow-up in finite time [START_REF] Liu | Blow-up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation[END_REF][START_REF] Liu | Strong instablity of solitary-wave solutions to a Kadomtsev-Petviashvili in three dimensions[END_REF][START_REF] Saut | Remarks on the generalized Kadomtsev-Petviashvili equations[END_REF][START_REF] Saut | Recent results on the generalized Kadomtsev-Petviashvili equations[END_REF] of the solution starting from the initial data

u 0 (x) = 3∂ xx e -α(x 2 +y 2 ) .

Time and space discretization of KP equations

The numerical schemes are introduced. We focus on the compact schemes. Then we remind the spectral scheme used in [START_REF] Hamidouche | Simulations numériques des équations de Kadomtsev-Petviashvili[END_REF][START_REF] Hamidouche | Numerical study of the solutions of the 3D generalized Kadomtsev-Petvisahvili equations for long times[END_REF][START_REF] Infeld | Numerical simulations of Kadomtsev-Petviashvili soliton interactions[END_REF][START_REF] Klein | Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation[END_REF][START_REF] Klein | Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations[END_REF]].

The time discretization

Let ∆t > 0 be the time step. In order to preserve the mean and the L 2 -norm, we use a Sanz-Serna scheme. It is written

u(t n+1 ) -u(t n ) ∆t + ∂ xxx u(t n ) + u(t n+1 ) 2 + 1 p + 1 ∂ x u(t n ) + u(t n+1 ) 2 p+1 + λ∂ -1 x ∂ yy u(t n ) + u(t n+1 ) 2 = 0. ( 2
)
Theorem 2.1. For solutions smooth enough, the scheme (2) conserves the mean and the L 2 -norm.

Proof. The mass conservation is showed by integrating (2) over space. The L 2 -norm conservation is obtained by

multiplying (2) with u(t n ) + u(t n+1 ) 2
, then integrating over space.

The space discretization

From (2), we write the discretization in space as

U n+1 -U n ∆t + A x,3 U n+1 + U n 2 + 1 p + 1 A x,1 U n+1 + U n 2 p+1 + λA x,-1 A y,2 U n+1 + U n 2 = 0, (3)
where A x,k , A y,k , is the matrix associated to the discretization of the k-th derivative in the x direction and the y direction, respectively. Here, U n denotes the approximation of u at time t n = n∆t, n ∈ N. We now mention different ways to compute these matrices, namely through compact schemes and Fourier transform.

Compact finite different schemes

We describe how to discretize the derivative and the antiderivative operators with high-order schemes. To simplify the readings, and, since two dimensional discretizations are easily obtained from the Kronecker product, we focus on the one dimensional case. We just remind that the Kronecker product of two matrices is the tensor product and has the same notation ⊗. Let f be a function defined in [-L, L], L > 0, and let h > 0 denote the space step. We set

x i = -L + h(i -1) and f i = f (x i ) for 0 ≤ i ≤ N .
Let us remind the principle of the compact finite difference method.

The classical order 2 central scheme that discretizes the first derivative, is given by

f i = f i+1 -f i-1 2h .
To improve the order, we consider more points [START_REF] Lele | Compact finite difference schemes with spectral-like resolution[END_REF]. For example, the scheme

α f i-1 + f i+1 + f i = b f i-2 -f i+2 4h + a f i-1 -f i+1 2h .
allows to reach order 6 by choosing suitable coefficients α, a and b from Taylor expansions. Indeed, the only possible values are a = 14 9 , α = 1 3 and b = 1 9 . To obtain order 2, the parameters has to verify the condition 1 + 2α = a + b e.g. a = 1, α = b = 0 gives the classical central scheme. A 4-th order can be provided by a = 3 2 , α = 1 4 and b = 0. We rewrite these schemes as a system P m F = Q m F where P m and Q m are the matrices defined by

P m =                 1 α α 1 α α α α (0) (0)                 , Q m = 1 2h                           0 a b 2 -a -b 2 0 a b 2 -a -b 2 -a -b 2 -b 2 a b 2 b 2 (0) (0)                          
, F and F are vectors associated to the function and its derivative. Then the matrix associated to the first derivative is P -1 m Q m . We emphasize that the coefficients in the upper right corner and the lower left corner correspond to the periodic bound conditions. Concerning other boundary conditions, which do not allow self centered scheme, we rather take an offcenter scheme. An admissible scheme for the first derivative is

f i + αf i+1 = 1 h (af i + bf i+1 + cf i+2 + df i+3 ) ,
allows to reach the fourth order.

In the same way, we write for the second derivative

α f i-1 + f i+1 + f i = b f i+2 -2f i + f i-2 4h 2 + a f i+1 -2f i + f i-1 h 2 ,
and the Taylor expansions provide

• order 2 1 + 2α = a + b. • order m ∈ {4, 6} 1 + 2α = a + b, 2α (2l)! = 2 4 l b + a (2l + 2)! , l = 1 . . . m 2 -1.
The following scheme to discretize the third derivative

α f (3) i-1 + f (3) i+1 + f (3) i = a f i+2 -f i-2 -2(f i+1 -f i-1 ) 2h 3 + b f i+3 -f i-3 -3(f i+1 -f i-1 ) 8h 3
gives

• an order 2 scheme if 1 + 2α + 2β = a + b + c.
• and an order m ∈ {4, 6} if

1 + 2α = a + b, 2α (2l)! = 2a 2 2(l+1) -1 + 3 4 b 3 2(l+1) -1 1 (2l + 3)! , l = 1 . . . m 2 -1.
It remains to compute the discretization of the antiderivative ∂ -1 x , which is not immediate. The matrix P -1 m Q m associated with the operator ∂ x as above beeing not invertible in general. More precisely, we have

P m F = Q m F
where F and Q m is not invertible. To close the system, we use the mass zero constraint of the KP equation [START_REF] Molinet | Remarks on the mass constraint for KP type equations[END_REF]. Hence, we impose F i = 0 and F i = 0. Then the last line of Q m and P m are replaced with 1 to obtain two new matrices Q m and P m

P m =                 1 α α 1 α α 1 1 (0) (0)                 , Q m = 1 2h                           0 a b 2 -a -b 2 0 a b 2 -a -b 2 -a -b 2 -b 2 b 2 1 1 (0) (0)                           .
This new matrix Q m becomes invertible (when N is odd) and

F = Q m -1 P m F , D x -1 = Q m -1 P m . Remark 2.2. If N is even, if C i denote the columns of Q m , the operation C 0 -C 1 + C 2 -C 3 • • • is null and Q m is not invertible. Gathering D x , D yy , D xxx and D x -1 , the discretizations of ∂ x , ∂ yy , ∂ xxx and ∂ -1
x respectively, we obtain the nonlinear system

U n+1 -U n ∆t + I y ⊗ D xxx U n+1 + U n 2 + 1 p + 1 I y ⊗ D x U n+1 + U n 2 p+1 + λ I y ⊗ D x -1 (D yy ⊗ I x ) U n+1 + U n 2 = 0. ( 4
)
It can be rewritten

I y ⊗ I x + ∆t 2 D xxx + λ ∆t 2 D yy ⊗ D x -1 U n+1 = I y ⊗ I x - ∆t 2 D xxx -λ ∆t 2 D yy ⊗ D x -1 U n - ∆t p + 1 I y ⊗ D x U n+1 + U n 2 p+1 .
A Picard fixed-point method is used at each time iteration to compute U n+1 .

Theorem 2.3. The scheme (4) preserves the mean and the mass zero constraint.

Proof. We verify that

i (U n+1 ) i = i (U n ) i .
We write the 1D discretization matrices as

D x = P -1 1 Q 1 , D yy = P -1 2 Q 2 , D xxx = P -1 3 Q 3 and D x -1 = Q 1 -1 P 1 . First i I y ⊗ D xxx U n+1 + U n 2 i = 0. Indeed, the matrix Q 3 is antisymmetric, P 3 is symmetric and ∀j i (Q 3 ) i,j = 0, i (P 3 ) i,j = C, C = 0.
Let us verify that i (D xxx X) i = 0 for any vector X. We set

Y = D xxx X = P -1 3 Q 3 X, thus P 3 Y = Q 3 X. We have i (Q 3 X) i = i k (Q 3 ) i,k X k = k i (Q 3 ) i,k X k = 0,
and

i (P 3 Y ) i = i k (P 3 ) i,k Y k = k i (P 3 ) i,k Y k = C k Y k .
We infer that

k Y k = i P -1 3 Q 3 X i = i (D xxx X) i = 0.
We therefore obtain i

I y ⊗ D xxx U n+1 + U n 2 i = 0.
In the same way, we have i

[(I y ⊗ D x ) X] i = 0 and i [(D yy ⊗ I x ) X] i = 0 for any vector X. Since I y ⊗ D x -1 (D yy ⊗ I x ) = D yy ⊗ D x -1 = (D yy ⊗ I x ) I y ⊗ D x -1 , ( 4
) implies i (U n+1 ) i = i (U n ) i .
The conservation of the L 2 -norm

I y ⊗ D x U n+1 + U n 2 p+1 , U n+1 + U n 2 = 0
is more difficult to reach with high order schemes. Nevertheless, the precision of compact schemes will ensure a good conservation of the L 2 -norm, as shown in the numerical simulations.

Remark 2.4. In order to solve the linear system AX = b, where

A = I y ⊗ I x + ∆t 2 D xxx + λ ∆t 2 D yy ⊗ D x -1 ,
we use a diagonal preconditioner and a GMRES method.

Fourier spectral method

Because of the definition of the antiderivative ∂ -1 x = F -1 1 iξ , the Fast Fourier Transform (FFT) is the most common way to solve the KP equation with periodic boundary conditions in space. We define the Fourier modes

k x = 2πk L x and k y = 2πl L y in the domain [-L x , L x ] × [-L y , L y ].
The Fourier transform applied to the KP equation,gives

∀k ∈ Z * , l ∈ Z dû ∆t (k, l, t) -i k 3 x -λ k 2 y k x û(k, l, t) + ik x p + 1 F u p+1 (k, l, t) = 0.
The mass zero constraint is translated as û(0, l, t) = 0. In order to conserve the L 2 -norm, a Sanz-Serna scheme is used again

ûn+1 -ûn ∆t -i k 3 x -λ k 2 y k x ûn+1 + ûn 2 + i k x p + 1 F u n + u n+1 2 p+1 = 0. ( 5
)
With the following matrices (where N x and N y are powers of 2)

K = 2π L diag 0, 1, • • • , N 2 -1, - N 2 , • • • , -1 , = x or y, K x -1 = 2π L x diag 0, 1, • • • , 1 Nx 2 -1 , - 2 N x , • • • , -1 ,
we solve thanks to a Picard fixed-point method

Ûn+1 -Ûn ∆t -i I y ⊗ K 3 x -λK 2 y ⊗ K x -1 Ûn+1 + Ûn 2 + i I y ⊗ K x -1 1 p + 1 F U n+1 + U n 2 p+1 = 0. ( 6
)
It is worth to notice that the matrix K x is not invertible. The average in x beeing zero, we can define the matrix K x -1 .

Mixed spectral-compact schemes

The wave, solution of the KP equation, propagates mainly in the x-direction. For realistic configurations or less consuming computational, it is natural to consider a large domain in the x-direction and to restrict the spatial domain in the y-direction. In this case, we combine FFT in the x-direction and compact schemes with Dirichlet and/or Neumann condition in y as follows

Ûn+1 -Ûn ∆t -i I y ⊗ K 3 Ûn+1 + Ûn 2 + i p + 1 (I y ⊗ K) F U n + U n+1 2 p+1 -iλ D yy ⊗ K -1 Ûn+1 + Ûn 2 = 0.
Gathering Û n+1 and Û n gives

I d -i ∆t 2 I y ⊗ K 3 + λ∆tD yy ⊗ K -1 Ûn+1 = I d + i ∆t 2 I y ⊗ K 3 + λ∆tD yy ⊗ K -1 Ûn + i ∆t p + 1 (I y ⊗ K) F U n + U n+1 2 p+1 .

Numerical results

After establishing the convergence, the schemes are tested and validated according to three well known behaviors of the KP equations, namely, the perturbations of Zaitsev soliton, transverse instabilities and blow-up in finite time [START_REF] Klein | Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations[END_REF]. The codes are written in python.

Order estimates and comparison

We consider the Zaitsev travelling waves as exact solution the KP-I equation with p = 1. It is defined by

Ψ(x, y, t) = 12α 2 1 -β cosh(αx -ωt) cos(δy) (cosh(αx -ωt) -β cos(δy)) 2 , (7)
where We compare the L 2 -norm error obtained by the schemes presented in section 2 with varying space discretization. The results are listed in Table 1. It is not surprising that the classical Fourier scheme gives the best result. Nevertheless, the error given by compact scheme is well decreasing and the results are quite good from the fourth order. To verify the mass zero constraint, we choose as initial datum (Figure 3)

β = δ 2 -3α 4 δ 2 , 3α 4 < δ 2 , ω = δ 2 + α 4 α , c = ω α .

Method

u 0 (x, y) = A(1 - 2sx
2 )e -sx 2 -ty 2 with s = 0.25, t = 7.5 and A = 5.

Here, the domain is [-25, 25] × [-5, 5] and dt = 10 -4 and other parameters are sum up in Table 2.

Method N x N y full 4 th order compact scheme 501 100 FFT in x and 4 th order compact scheme in y 2 9 100 full FFT 2 9 2 7

Table 2: Parameters for a Gaussian as initial datum. The first column denotes 4 th order compact scheme, the second column is 4 th order mixed scheme and third column represents the FFT.

Perturbations of the Zaitsev soliton

The Zaitsev soliton ( 7) is perturbed with a Gaussian function. More precisely, the simulation starts from the following initial datum

u 0 (x, y) = Ψ x + L x 2 , y, 0 + 6 x + L x 2 e -(x+ Lx 2 ) 2 -y 2 .
Here we take α = 1, β = 0.5 and δ = 3 in the domain [-25, 25] × [-5, 5]. We present in Figure 5 the simulation performed with 4 th order compact schemes with N x = 501, N y = 100 and dt = 10 -4 . The L 2 -norm, the mean are well conserved and the L ∞ -norm has a consistent behaviour (Figure 6). As expected [START_REF] Klein | Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations[END_REF], Figure 5 shows that peaks are appearing and the Zaitsev soliton develops into a lump.

Transverse instabilities of the line-soliton

We investigate the transverse instabilities [START_REF] Alexander | On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation[END_REF][START_REF] Rousset | Transverse nonlinear instability of solitary waves for some hamiltonian PDEs[END_REF][START_REF] Rousset | Transverse nonlinear instability for two-dimensional dispersive models[END_REF][START_REF] Rousset | A simple criterion of transverse linear instability for solitary waves[END_REF][START_REF] Rousset | Transverse instability of the line solitary water waves[END_REF][START_REF] Rousset | Stability and instability of the KdV solitary wave under the KP-I flow[END_REF] of the Korteweg-de Vries soliton. We consider the perturbed line-soliton u 0 (x, y) = 12 sech 2 x + 0.4 cos 2y Ly .

In Figure 7, we use 4 th order compact schemes with N x = 501, N y = 100 and dt = 10 -4 . In Figure 8, we can observe that the norms are well conserved and that the L ∞ -norm is increasing, translating that the line-soliton is unstable as shown in Figure 7 [START_REF] Hamidouche | Simulations numériques des équations de Kadomtsev-Petviashvili[END_REF][START_REF] Hamidouche | Numerical study of the solutions of the 3D generalized Kadomtsev-Petvisahvili equations for long times[END_REF][START_REF] Klein | Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations[END_REF].

Blow-up in finite time

To illustrate the blow-up in finite time of the KP-I equation when p = 2, we consider the initial datum u 0 (x, y) = 3∂ xx e -(x 2 +y 2 ) .

We use 6 th order compact schemes in the domain [-10, 10] × [-2.5, 2.5] with N x = 201, N y = 50 and dt = 10 -6 . Figure 9 shows that the minimum blows up. We see in Figure 10 that the L 2 -norm and the mean are conserved, whereas the L ∞ -norm becomes singular.

Conclusion

In this paper, we propose new compact schemes to solve the Kadomtsev-Petviashvili equations. We planned to do the simulations with periodic boundary conditions, these schemes can be easily adapted to other boundary conditions. This allows to find the expected results in terms of error, blow-up and transverse instabilities with high precision schemes.

  The computational domain is Ω = [-L x ; L x ] × [-L y ; L y ] = [-89.6; 89.6] × [-21; 21] and we take α = 0.0174, δ = π Ly = π 21 and c = 0.76 [6], and the time step dt = 10 -4 . The initial datum is represented in Figure 1.

Figure 1 :

 1 Figure 1: At left, 3-dimensional representation of the Zaitsev travelling wave. At right, 1-dimensional projection for y =0 (-), 5 (--), 10 (-•-).

Figure 2

 2 Figure 2 presents the numerical order obtained from mixed schemes with respect to N y .

Figure 2 :

 2 Figure 2: L 2 -norm error with respect to the discretization step computed from mixed schemes. Here we have mixed schemes.

Figure 3 :

 3 Figure 3: At left, the Gaussian initial datum with A = 5, s = 0.25 and t = 7.5. At right, the time evolution of the L 2 -norm and mean computed with the 4 th order compact scheme.

Figure 4 :

 4 Figure 4: Solutions of KP-I equations at time t = 0.5 (first row) and t = 1 (second row). The first column denotes 4 th order compact scheme, the second column is 4 th order mixed scheme and third column represents the FFT.

Figure 5 :

 5 Figure 5: At left, perturbed Zaitsev soliton as initial datum. At right, solution of the KP-I equation with p = 1 at time t = 5.

Figure 6 :

 6 Figure 6: L 2 -norm, mean and L ∞ -norm with respect to time of the solution of the KP-I equation starting from perturbed Zaitsev.

Figure 7 :

 7 Figure 7: At left, perturbed KdV soliton as initial datum. At right, solution of the KP-I equation with p = 1 at time t = 5.

Figure 8 :

 8 Figure 8: L 2 -norm, mean and L ∞ -norm with initial datum equal to a perturbed KdV soliton (cf Figure 7)

Figure 9 :

 9 Figure 9: At left, second derivative of a Gaussian as initial datum. At right, blow-up of the solution of KP-I with p = 2 at time t = 0.115.

Figure 10 :

 10 Figure 10: Evolution with time of the L 2 -norm, mean and L ∞ -norm with the second derivative of a Gaussian as initial datum.

Table 1 :

 1 L 2 -norm error for different settings

	N x N y L 2 -norm error


DGAC Jean Carlioz
email: jean.carlioz@aviation-civile.gouv.fr

Cybersecurity of embedded computers systems

Several articles have recently raised the issue of computer security of commercial flights by evoking the "connected aircraft, hackers target" or "Wi-Fi on planes, an open door for hackers ? " Or "Can you hack the computer of an Airbus or a Boeing ?". The feared scenario consists in a takeover of operational aircraft software that intentionally cause an accident.

Moreover, several computer security experts have lately announced they had detected flaws in embedded systems, that exploited could lead to the aircraft's takeover by the "hacker". This multiplication of alarming publications occurs in a context of proliferation of sophisticated computer attacks, targeting commercial enterprises to steal their sensitive information.

The world of aviation, if not offering the same exposure to the cyber threat as a public website, had however for some years its computer systems increasingly interconnected. Flight Management Computers "sealing" hitherto guaranteed by the autonomous nature of its trade (with the ground), now communicate with other systems. If this situation is not unique, it is natural that the growing number of interconnections leads observers to wonder about new vulnerabilities.

The attack points mentioned are schematically shown (red): (1) Internet connection (2) interconnects the two aircraft systems, (3) Flight system itself, (4) operatively connected with the ground

What is the threat?

A particularly competent "hacker" could target the "Flight Management System" (the aircraft computer system management) to disrupt the flight parameters. To succeed this attack, he should introduce a connection between the ground and the aircraft, for example by injecting a message containing false information deliberately erroneous and dangerous.

Press articles also cite a new danger: the companies now offer their passengers IT services and entertainment. These services are distributed on the plane by a network. The risk consists here on a malicious access to operational network if these two elements (entertainment and operational resources) are not separated. A passenger or a hacker from the Internet could potentially exploit this interconnection.

Likelihood of feared scenarios

If the terms of mentioned attacks are complex, some are possible. Thus, the interception and returning a false message by ground-air liaison ("4" in the diagram) is technically feasible. Furthermore, injection, by this same bias, a dangerous operational message is technically described, but could not be successful, even in test patterns.

Furthermore, the scenario of an attack by a passenger who crossed the interconnexion between secure computer entertainment system and the operating system of the aircraft ("3" in the diagram) is serious. This interconnexion is technically a software gateway, so fallible. However, manufacturers are working to strengthen the security in order to make such an impossible scenario. But today there is no "zero risk" as long as these security measures are not deployed. Some of the scenarios mentioned in the press are technically feasible, with significant resources, and growing interconnections of the systems make them potentially plausible.

Impact on flight safety

If the press interest in the topic of cyber threat is new, security experts have already repeatedly had to answer questions about the cyber vulnerabilities of avionics systems.

The drivers, such as controllers, are equipped and trained for their collaboration quickly detects deviation of the aircraft systems. In addition, these systems have the capacity to automatically detect such drift. If the attack occurred, and managed to "FMS" causing its drift, the pilot would immediately disengage it and follow the manual procedure. The conventional navigation means would remain available to the driver.

However, further analysis of the consequences of such an attack, that simultaneously affects multiple aircraft on approach, remains to be achieve.

Recent work by securing devices in response to the cyber threat

Manufacturers are actively working to maintain a very high level of security in their planes. For example, A380 & A350 are subject to a rigorous process that incorporates new scenario of "cyber-risk" in the update chain. In addition, a battery of auditssurrounded by precautions to ensure confidentialitywas conducted on the systems of these devices, concluding that a maximum level in the scale of protection. Those new process will retroactively be applied to older planes.

While recalling that "there is no zero risk", state agencies estimate that the recent Airbus aircrafts are now the most secure embedded complex systems in civil field.

Honeywell, producing the "FMS" (see annex), the operating system many aircraft models, integrated the attack scenarios to consolidate its product, since 2013. The product has not changed since the tests, Honeywell believes that FMS is not today more exposed.

Conclusion

Given the increasing interconnection of systems, some of the scenarios mentioned in the press would be technically possible, with significant resources.

However, even with a computer system failure, a crew is trained to keep control of the flight, according to procedures also known to air traffic controllers, navigation by conventional means.

In any case, the failure of one system alone would not impact flight safety. The air transport safety based on multiple redundancies, many backup devices and robust procedures with degraded modes which operators are well trained.

Nonetheless industrialists such as navigation services will integrate this new risk, working to limit impacts.

Annex

The components of the information "airplane" system

ADS-B

Modern aircraft, thanks to satellite positioning systems (such as GPS) know their position so much more accurate than ground control (radars having limited accuracy). The ADS-B system exploits this ability: the aircraft calculates its own position and regularly broadcasts by radio. Most modern aircraft are equipped with ADS-B (A 320-330-340-380, B-737-747-777, ...).

The ACARS

The ACARS is a digital data exchange system between an aircraft and the ground, supported by a ground network. It is used for the transmission of standard information from the floor (departure clearance, weather information, track changes, ...) or from the aircraft (technical condition, time of arrival).

The ACARS is based on radio transmissions, direct or via satellite. The "soil" network distributes the message between the radio station that received and the recipient. Thus, every company equipped with ACARS can communicate with its aircraft, anywhere in the world.

FMS

The "Flight Management System" is an embedded system that supports the strategic management of the flight (flight plan, loading ...) and its tactical management (position, embedded sensors, the crew orders). It consists of two computers (disengageable) that control the autopilot and provide information to the crew, flight parameters (Pilot Flight Director), navigation (Navigation Display), status of systems (Electronic Centralized Aircraft Monitoring), settings engines or alerts.

Only the ACARS system can address data from the ground to FMS.


Elemer Elad Rosinger
email: eerosinger@hotmail.com

Elemer Elad

Rosinger Bell

Elemér Elad Rosinger

Marie-Louise Nykamp

Bell Inequalities ?

Keywords:

est destinée au dépôt

the Bell Inequalities story. The timeliness of such an attempt here, needless to say not the first regarding the Bell Inequalities story, is again brought to the fore due to the no less than three most freshly claimed to be fundamental contributions to the Bell Inequalities story, [START_REF] Giustina | A signicant-loophole-free test of Bells theorem with entangled photons[END_REF][START_REF] Hensen | Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km[END_REF][START_REF] Shalm | A strong loophole-free test of local realism[END_REF], described and commented upon in some detail in [START_REF] Khrennikov | After Bell[END_REF].

1. The Bell Inequalities an unintended plagiarism of already longer known Pure Mathematics results ?

In [START_REF] Pitowsky | From George Boole to John Bell -the origins of Bells inequalities[END_REF], back in 1989, it was clearly and with abundant detail shown that the so called, and much celebrated Bell Inequalities happen to belong to a wider class of probabilistic type inequalities first established by George Boole and later considered and studied by a number of mathematicians. All that happened prior to and/or independently of quanta. In fact, the respective studies mentioned in [START_REF] Pitowsky | From George Boole to John Bell -the origins of Bells inequalities[END_REF] are of a purely mathematical nature, thus without absolutely any physical type considerations, be they classical or quantum. Indeed, Boole initiated all such inequalities in the Appendix of his much celebrated 1854 book "The Laws of Thought". Following it and considerably extending it, a purely mathematical study of a wide class of inequalities was presented in the work of a number of mathematicians and probabilists, see relevant details in [START_REF] Pitowsky | From George Boole to John Bell -the origins of Bells inequalities[END_REF]. Seemingly, John Bell, more than a whole century after Boole, and back in 1964, was not aware of any of the above when he worked out the inequalities associated nowadays with his name. But then, nowadays, such an oversight is simply, and not seldom, the rather innocent effect of the widely and wildly raging ultra-specialization in sciences ...

The 2011 Hans de Raedt, et. al., paper [3]

The paper [START_REF] De Raedt | Extended Boole-Bell Inequalities Applicable to Quantum Theory[END_REF] shows, among others, two facts which hardly anyone seems to know among quantum physicists, namely :

A) The Bell Inequalities are of a purely mathematical nature, involving nothing more than usual logic and a rather elementary algebra. In this regard, the relevant results of George Boole are also mentioned.

B) The conclusion of John Bell that certain quantum contexts violate the so called Bell Inequalities is an effect of a simple and trivial error in the statistical manipulation of data. Once that error is avoided, that claimed violation is no longer supported.

It follows therefore that, so far, one has not obtained any support for the Bell Inequalities being violated either by classical, or by quantum systems.

Therefore, in this regard, the Bell Inequalities are so far irrelevant in physics.

3. Fighting a conveniently set up "straw man" ... Amusingly, in recent foundational debates regarding quanta one of the conflicting views got labeled as "Bell was wrong", see for instance subsection 2.8 in [START_REF] Khrennikov | After Bell[END_REF].

However, as follows from the above, this labeling is but an erroneous setup in which a mere "straw man" is created in order to be quite easily attacked, and hopefully also demolished ... Indeed, as far as one can know in present mathematics and quanta, John Bell was not wrong either with the inequalities associated with him nowadays, or with the quantum systems he considered.

Instead, what was wrong was his particular statistical manipulation in which he made a simple and trivial error, an error which ever since is endlessly repeated by just about everybody involved in the issue ... In this way, as far as known nowadays, the error of John Bell simply cannot be found in the mathematics of the Bell Inequalities, or in the behaviour of entangled quanta !

No, not at all, since that error is in the simple and trivial manipulation of statistical data, as brought to light in [START_REF] De Raedt | Extended Boole-Bell Inequalities Applicable to Quantum Theory[END_REF].

Consequently, even those who claim -including John Bell himselfthat the Bell Inequalities were violated, are simply talking nonsense, since all that happened was that the respective claimed violation was obtained based alone on a simple and trivial error in the statistical handling of data. Thus the thing which is clearly violated and keeps being violated, are a few of the simple and trivial rules of statistics regarding finite sets of data.

That is, therefore, the only meaning so far of that label of "Bell was wrong" ! 4. Some further clarifications for the possible benefit of physicists ...

Let us stop for a short while, step back even if one single step only, and try and have a brief deeper look into logic, mathematics and physics, all of which are of course essentially involved in issues such as the alleged violation of the Bell Inequalities ... A more general such an approach regarding logic, mathematics and physics can be found in [START_REF] Rosinger | Five Departures in Logic, Mathematics, and thus -either we like it[END_REF][START_REF] Rosinger | Five Departures in Logic, Mathematics, and thus -Whether We Like It, or Not -in Physics as Well[END_REF] which, however, is not sine qua non in this paper.

a) As of today, it is not known whether usual mathematics, such as for instance that which alone is involved in establishing the mentioned Boole type inequalities, thus the Bell Inequalities as well, is contradiction free. On the other hand, the chances that it is so are not at all negligible in view of the fact that, after many millennia, no contradiction was ever found in mathematics.

b) If in physics, chemistry, or some other realm outside of mathematics, one finds a valid statement, say, S which contradicts a certain valid statement, say, M in mathematics, then the very same statement S will contradict all possible other valid statements M* in mathematics. Indeed, this fact is an immediate logical consequence of the assumptionaccepted so far -that mathematics is contradiction free, see Appendix 1. Consequently, if certain quantum considerations do violate the Bell Inequalities, then the very same quantum consideration must also vio-late all other valid mathematical statements, in particular, they must violate all such equations like 0 = 0, or 1 = 1, and so on ... Therefore, there is no point in endlessly insisting that the Bell Inequalities may have some most privileged and rather unique status, being so far the only purely mathematical statements which -allegedly -are violated by quanta.

c) In case in some realm, say, R outside of mathematics one may eventually find a valid statement which happens to contradict a valid mathematical statement, then one or more of the following situations could be possible : c1) usual logic may be inadequate for mathematics and/or for that realm R, c2) mathematics and/or that realm R may be contradictory, c3) the given mathematical modelling of that realm R may be unsatisfactory.

d) In case mathematics and/or quantum theory would happen to be contradictory in themselves, a way out is mentioned in Appendix 2. After all, it would make quite a few things involved sound so much more simple ...

And who knows, it may possibly make them also so much more clear ...

Appendix 1

We start by as simple a sketch of a proof as possible of the statement in b) in section 4. This proof is of a purely logical nature, and thus of a far more general scope than the issues related to the Bell Inequalities. The general background is the so called Predicate Calculus in Mathematical Logic. As is well known, usual axiomatic mathematical theories are formulated within that background.

We are interested in the following facts in Predicate Calculus.

Given two well formed formulas, or in short, wff-s, p and q, then p =⇒ q is also a wff. Furthermore, p =⇒ q is true, if and only if the wff (*1) ((non p) or q) is true.

Let us suppose that

(*2) s =⇒ (non b)
is true, for certain given wff-s s and b. Then in view of (*1), it follows that ((non s) or (non b)) is also true, since in general, non (non p) and p are simultaneously true, for every wff p.

Let now T be the set of all true wff-s. Then for every wff-s x and y, we have And now, a more detailed argument.

(*3) (x ∈ T , x =⇒ y) =⇒ y ∈ T is true.
Let us start by noticing the often missed fact that Axiomatic Mathematical Theories as mere Models ! And this should, among others, be made clear to physicists as well, since the axiomatizations of various branches of physics has hadand does have also today -a strong tendency, at least since Newton's "Principia Mathematica".

Indeed, ever since Euclid axiomatized Geometry more than two millennia ago, there has been a widespread and strong tacit tendency, and not only among mathematicians, to identify the respective specific, particular axiomatic theory with Geometry as such. No lesser a philosopher than Immanuel Kant (1724-1804) considered the resulting Euclidean Geometry to be the only possible one in the whole Creation. Therefore the shock in the early 1800s when non-Euclidean geometries have been discovered.

A similar phenomenon happened more than a century back with the Peano Axioms of the natural numbers which were supposed to express absolutely all the relevant properties of such numbers. And yet, we had to face the related shock of the Gödel Incompleteness Theorem which, in the early 1930s, came as a total surprise.

The moral, of course, is that, on one hand, we may have a concept like "geometry" or "number", for instance, while on the other hand, we can have one or another specific, particular axiomatic mathematical theory which aims to describe such a most abstract concept. Thus here we deal with a rather inevitable divide Abstract Concept ........................... Axiomatic T heory and the gap between these two sides may be hard to bridge, let alone eliminate. In conclusion, we have to note that

• Each Axiomatic Theory can only be a particular, specific MODEL of a given Abstract Concept

Let us now have a brief look at what is in fact an axiomatic mathematical theory, or more generally, an axiomatic theory.

One starts such an axiomatic theory with setting up a formal deductive system. Namely, let A be an alphabet which can be given by any nonvoid finite or infinite set. Then a procedure is given according to which one can in a nite number of steps effectively construct -by using the symbols in A -a set F of well formed formulas, or in short, wff-s.

Next, one chooses a nonvoid set R of logical deduction rules which operate as follows (A1.1)

F ⊇ P R -→ Q ⊆ F
that is, from any set P of wff-s which are the premises, it leads to a corresponding set Q of wff-s which are all the logical consequences of P .

Further, it will be convenient to assume that, for every set of well formed formulas P ⊆ F, we have (A1.2)

P ⊆ R(P ) = R(R(P ))
in other words, the premises P themselves are supposed to be among the logical consequences R(P ), and in addition, these logical consequences R(P ) of P contain all the possible logical consequences of P , thus the iteration of the logical deduction rules R does not produce further logical consequences of P . Clearly, condition (A1.2) does not lead to a loss of generality regarding R in (A1.1). Indeed, if the relation

∀ P ⊆ F : P ⊆ R(P )
is not satisfied, then this relation will obviously be satisfied by the modification of R given by R * (P ) = P R(P ). Also, if the relation

∀ P ⊆ F : R(R(P )) = R(P )
is not satisfied, then this relation will obviously be satisfied by the modification of R given by R * * (P ) = R(P ) R(R(P )) R(R(R(P ))) . . .

And now at last come the axioms which can be any nonvoid subset

A ⊆ F of wff-s.

With the above established, the respective axiomatic theory follows easily as being the smallest subset T ⊆ F with the properties

(A1.3) A ⊆ T (A1.4) T ⊇ P R -→ Q ⊆ T
in which case the wff-s in T are called the theorems of the given axiomatic theory defined by the axioms A.

In view of (A1.3), clearly, all axioms in A are also theorems. Now an essential fact is that the set T of theorems depends not only on the axioms in A, but also on the logical deduction rules R, and prior to that, on the set F of well formed formulas. Consequently, it is appropriate to write (A1.5) T F ,R (A) or more simply T R (A)

for the set T of theorems.

Here are some of the relevant questions which can arise regarding such axiomatic systems :

• are the axioms in A independent of one another ?

• are the axioms in A consistent, that is, do they not conflict with one another ?

• are the axioms in A complete ?

Independence means that for no axiom α ∈ A, do we have T R (A) = T R (B), where B = A \ {α}. In other words, the axioms in A are minimal in order to obtain the theorems in T R (A). This condition can be formulated equivalently, but more simply and sharply, by saying that for no axiom α ∈ A, do we have α ∈ T R (B), where B = A \ {α}.

As for consistency, it means that there is no theorem τ ∈ T R (A), such that for its negation non τ , we have non τ ∈ T R (A).

Completeness, in one of its possible formulations, means that, given any additional axiom β ∈ F \ A which is independent from A, that is, for which β / ∈ T R (A), then this extended set of axioms B = A {β} is inconsistent.

It is obvious, therefore, that in setting up axiomatic systems, there is a lot of freedom in choosing the alphabet A, the well formed formulas F, the logical deduction rules R and the axioms A, all of which influence the resulting theorems T . However, such a freedom is not necessarily a complete blessing when it comes to express all the possible relevant properties of such abstract concepts as "geometry", "numbers", and so on. Indeed, each particular such choice may not only miss on certain relevant properties, but may actually introduce some strange and unintended ones as well. In this regard, in modern times, it was the philosophy of neo-positivism, or the so called third positivism, which in the early 20th century brought to attention the fact that the very structure of language can significantly influence thinking and the results of thought, and in particular, can lead to pseudo-problems. Not much later, in linguistics, a similar idea arose with the Sapir-Whorf Hypothesis about the relativity of language.

A corresponding recognition in mathematics, as mentioned, started to emerge in the early 1800s, even if tentatively, with the non-Euclidean geometries, and was later confirmed by further modern developments of various axiomatic mathematical theories, a most important moment in this regard being Gödel's Incompleteness Theorem, in the early 1930s.

In conclusion, it is appropriate to realize even if in the day to day activity of the so called "working mathematicians" and "working physicists" it may still be disregarded that an axiomatic theory most likely fails to express all of the properties of the domain of mathematics or physics which it is supposed to model, and in fact, may even introduce inappropriate properties.

And now, based on the above, we can give a simple proof of the fact presented in b) in section 4 above.

Let us consider the usual mathematical theory -actually but not necessarily reducible to elementary algebra -which is involved in the considerations related to the Bell Inequalities. Then we have a corresponding alphabet A, set F of well formed formulas, logical deduction rules R, and lastly, set of axioms A. All of these will result in the corresponding set T R (A) of mathematical theorems. And obviously, one of these theorems is precisely the statement of the Bell Inequalities, a statement which, for convenience we shall denote by (A1.6) βι ∈ T R (A)

and we obtained a contradiction, since we have now that βι, α 0 , σ ∈ T R (A), and then, in view of (A1.7), T R (A) is contradictory, in view of the following property shared by all axiomatic sets of theorems

(A1.14) (ξ =⇒ χ) =⇒ χ ∈ T R (A)
for all ξ ∈ T R (A), χ ∈ F.

Appendix 2

It is remarkable that due to practical requirements in theoretical computer science, there have for more than two decades by now been developments in dealing with and making use of inconsistent axiomatic theories. Details in this regard can be found in [START_REF] Rosinger | Five Departures in Logic, Mathematics, and thus -either we like it[END_REF][START_REF] Rosinger | Five Departures in Logic, Mathematics, and thus -Whether We Like It, or Not -in Physics as Well[END_REF].

  5. A ... naive ... question to quantum physicists ... In view of b) in section 4 above, the following holds : The Bell Inequalities are violated by quanta ⇐⇒ ⇐⇒ The equation 0 = 0 is violated by quanta So then, may I kindly ask : Why do quantum physicists not keep making noise about the equation 0 = 0 being violated by the quanta ?

  Furthermore, let us suppose that in (*2) we have (*4) s, b ∈ T Then we show that the wff (*5) ∀ a ∈ T : s =⇒ (non a) is true. Indeed, assume that the wff in (*5) is not true, then ∃ a 0 ∈ T : non (s =⇒ (non a 0 )) hence (*1) implies the true wff ∃ a 0 ∈ T : s and a 0 It follows that b, a 0 , s ∈ T , and then (*2), (*3) give the contradiction that b, non b ∈ T .

Let us now consider a formulation of usual quantum theory which allows the mathematical modelling of the quantum contexts that -allegedly -lead to the violation of the Bell Inequalities. Then the mentioned mathematical modelling corresponds to a set S of well formed formulas which is a subset of F.

In the above framework, the alleged violation of the Bell Inequalities means the following :

There exists a well formed formula σ ∈ S, such that (A1.7)

σ =⇒ non βι

Then we show that we have

Indeed, we recall that for every p, q ∈ F, we have (A1.9) (p =⇒ q) ∈ T R (A) ⇐⇒ ((non p) or q) ∈ T R (A) hence (A1.10) non (p =⇒ q) ∈ T R (A) ⇐⇒ (p and (non q)) ∈ T R (A)

Let us now assume that (A1.8) does not hold, thus we have

for a certain α 0 ∈ T R (A). But then the relation (A1.10) gives (A1.12) σ, α 0 ∈ T R (A) which means that, in particular, we have (A1.13) σ ∈ T R (A)


Elemér Elad

Rosinger Chairman
email: eerosinger@hotmail.com

Marie-Louise Nykamp

Do Quanta Violate the Equation 0 = 0 ?

Ever since the celebrated 1964 paper of John Bell, the statement that "Quantum systems violate the Bell inequalities", [1,2], has a very large support among quantum physicists as well as others claiming some knowledge about quanta. Amusingly, it has so far escaped the general notice that, if indeed, quanta do violate that Bell inequalities, then -due to elementary facts of Logic -they must also violate all other valid mathematical relations, thus among them, the equation 0 = 0. Here the respective elementary facts of Logic are presented.

Preliminaries

Usual Quantum Theory, more precisely, the Quantum Mechanics of finite, non-relativistic quantum systems -as all other modern theories of Physics -leads to a system of mathematical equations and other mathematical statements which are supposed to be mathematically correct and logically non-contradictory with one another. Among them are those regarding the quantum systems which are supposed to violate the Bell inequalities. As is known, [START_REF] Bell | On the Einstein-Podolski-Rosen paradox[END_REF][START_REF] Bell | Speakable and Unspeakable in Quantum Mechanics[END_REF], the respective mathematical statements which are used to claim that quantum systems violate the Bell inequalities can be expressed in a finite number of such statements, thus together, they can be denoted, say, by the mathematical statement s.

On the other hand, and hardly, if at all, known among quantum physicists, the Bell inequalities are of a purely mathematical nature, that is, they can be proved based on rather elementary algebra and without absolutely any other considerations, be of a physical, philosophical, or other non-mathematical nature, [START_REF] Pitowsky | From George Boole to John Bell -the origins of Bells inequalities[END_REF][START_REF] Rosinger | George Boole and the Bell inequalities[END_REF][START_REF] Rosinger | The irrelevance of Bell inequalities in Physics : Comments on the DRHM paper[END_REF][START_REF] Rosinger | The Irrelevance of Bell Inequalities in Physics: Comments on the DRHM Paper[END_REF][START_REF] Rosinger | Bell Inequalities[END_REF]. Let us then denote, by b the mathematical statement of the Bell inequalities.

Last, let us denote by z the statement 0 = 0.

The Logical Setup

Now, the mentioned long ongoing and highly popular claim is that

(2.1)
s =⇒ non b

Based on that, we show that the seemingly far stronger implication also hold

(2.2) s =⇒ non a
where a can be any valid mathematical statement. Thus, we would have in particular that

(2.3) s =⇒ non z Now (2.2) amounts to (2.4) ( s =⇒ non b ) =⇒ ( ∀ a ∈ M : s =⇒ ( non a ) )
where M denotes a large enough set of valid mathematical statements in order to include the considerations regarding the present arguments.

Indeed, the proof of (2.4) goes as follows. We recall from Mathematical Logic that ( p =⇒ q ) ⇐⇒ ( ( non p) ∨ q ) Therefore, (2.4) is logically equivalent with

(2.5) ( s ∧ b ) ∨ ( ∀ a ∈ M : ( ( non s ) ∨ ( non a ) )
Let us now assume that (2.5) does not hold. Then we have the valid statement

(2.6) ( ( non s ) ∨ ( non b ) ) ∧ ( ∃ a 0 ∈ M : ( s ∧ a 0 ) )
However, since s is assumed to be valid, it follows that so is non b, since ( ( non s ) ∨ ( non b ) ) must be valid. On the other hand, the validity of ( ∃ a 0 ∈ M : ( s ∧ a 0 ) ) implies that a 0 is also valid.

In conclusion, both non b and a 0 are valid.

Now we note that we assumed b, a 0 ∈ M, hence we obtain b, non b ∈ M which is a contradiction, since one assumes that Mathematics is contradiction free.

As for the case in which Mathematics may turn out to be contradictory, possible ways to proceed can be found in [START_REF] Rosinger | Five Departures in Logic, Mathematics, and thus -either we like it[END_REF][START_REF] Rosinger | Five Departures in Logic, Mathematics, and thus -Whether We Like It, or Not -in Physics as Well[END_REF] and the literature mentioned there.

3. Strange ... Quantum related phenomena ...

The fact, again hardly, if at all, noted among quantum physicists, is that recently, it was shown in [START_REF] De Raedt | Extended Boole-Bell Inequalities Applicable to Quantum Theory[END_REF] that the Bell inequalities are actu-ally not violated by quantum systems, see also related details in [START_REF] Rosinger | The irrelevance of Bell inequalities in Physics : Comments on the DRHM paper[END_REF][START_REF] Rosinger | The Irrelevance of Bell Inequalities in Physics: Comments on the DRHM Paper[END_REF][START_REF] Rosinger | Bell Inequalities[END_REF]. Indeed, as it turns out, the long and widely held opinion that the Bell inequalities would be violated by quantum systems is due to nothing else but a rather simple and elementary mistake in the handling of finitely many statistical data.

But let us, for the moment, leave the previous remark aside ... Then one can still ask the following question :

Why do quantum physicists not keep making noise about the equation 0 = 0 being violated by quanta ? Such a rather ... blunt and eye catching ... question would make not a few things involved sound so much more simple ... And given such luck, it may also make them much more clear as well ... Meanwhile, a widespread celebration seems to emerge for what, allegedly, seem to be no less than three major results, [START_REF] Giustina | A signicant-loophole-free test of Bells theorem with entangled photons[END_REF][START_REF] Hensen | Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km[END_REF][START_REF] Shalm | A strong loophole-free test of local realism[END_REF], that are supposed to ... plug once and for all ... the apparent loopholes still existing in various proofs of how the quanta have for long been claimed to ... violate ... the Bell inequalities ...

Well, a somewhat more articulate outside observer may be inclined to quip as follows :

"Science is not done scientifically ..."

But then, why should we be surprised ? After all, art is seldom, if at all, done artfully ... Or love, lovingly ... Let alone, religion religiously ... However, as if to compensate for all the above less than perfect performances, we can note that ... violence is more often than not done violently ...


Pol Grasland-Mongrain
email: pol.grasland-mongrain@ens-cachan.org

Erika Miller-Jolicoeur

An Tang

Stefan Catheline

Guy Cloutier
email: guy.cloutier@umontreal.ca

Contactless Remote Induction of Shear Waves in Soft Tissues Using a Transcranial Magnetic Stimulation Device

This study presents the first observation of shear wave induced remotely within soft tissues. It was performed through the combination of a transcranial magnetic stimulation device and a permanent magnet. A physical model based on Maxwell and Navier equations was developed. Experiments were performed on a cryogel phantom and a chicken breast sample. Using an ultrafast ultrasound scanner, shear waves of respective amplitude of 5 and 0.5 micrometers were observed. Experimental and numerical results were in good agreement. This study constitutes the framework of an alternative shear wave elastography method.

Introduction

Propagation of elastic waves in solids has been described in various fields of physics, including geophysics, soft matter physics or acoustics. Elastic waves can be separated in two components in a bulk: compression waves, corresponding to a curl-free propagation; and shear waves, corresponding to a divergence-free propagation. Shear waves have drawn a strong interest in medical imaging with the development of shear wave elastography methods [START_REF] Muthupillai | Magnetic resonance elastography by direct visualization of propagating acoustic strain waves[END_REF], [START_REF] Armen P Sarvazyan | Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics[END_REF]. These methods use shear waves to measure or map the elastic properties of biological tissues. Shear wave speed measurement permits calculation of the tissue shear modulus. Shear wave elastography techniques have been successfully applied to several organs such as the liver [START_REF] Sandrin | Transient elastography: a new noninvasive method for assessment of hepatic fibrosis[END_REF], the breast [START_REF] Berg | Shear-wave elastography improves the specificity of breast US: the BE1 multinational study of 939 masses[END_REF], the arteries [START_REF] Schmitt | Ultrasound dynamic micro-elastography applied to the viscoelastic characterization of soft tissues and arterial walls[END_REF] and the prostate [START_REF] Ll Cochlin | Elastography in the detection of prostatic cancer[END_REF], to name a few examples. The brain has also been studied, and its elasticity is of strong interest for clinicians [START_REF] Yogesh K Mariappan | Magnetic resonance elastography: a review[END_REF], [START_REF] Scott A Kruse | Magnetic resonance elastography of the brain[END_REF]. For example, it has been shown that Alzeihmer's disease, hydrocephalus or multiple sclerosis are associated with changes in brain elastic properties [START_REF] Murphy | Decreased brain stiffness in Alzheimer's disease determined by magnetic resonance elastography[END_REF], [START_REF] Taylor | Reassessment of brain elasticity for analysis of biomechanisms of hydrocephalus[END_REF], [START_REF] Wuerfel | MR-elastography reveals degradation of tissue integrity in multiple sclerosis[END_REF].

Clinical shear wave elastography techniques currently rely on an external vibrator [START_REF] Muthupillai | Magnetic resonance elastography by direct visualization of propagating acoustic strain waves[END_REF], [START_REF] Sandrin | Transient elastography: a new noninvasive method for assessment of hepatic fibrosis[END_REF] or on a focused acoustic wave [START_REF] Nightingale | Acoustic radiation force impulse imaging: in vivo demonstration of clinical feasibility[END_REF], [START_REF] Armen P Sarvazyan | Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics[END_REF] as the shear wave source. However, these techniques are limited in situations where the organ of interest is located behind a strongly attenuating medium like the brain behind the skull and surrounded by the cerebrospinal fluid. While external shakers are able to transmit some shear waves, using acoustic, pneumatic, piezoelectric or electromagnetic actuators [START_REF] Scott A Kruse | Magnetic resonance elastography of the brain[END_REF], [START_REF] Latta | Convertible pneumatic actuator for magnetic resonance elastography of the brain[END_REF], [START_REF] Weaver | Magnetic resonance elastography using 3D gradient echo measurements of steady-state motion[END_REF], [START_REF] Braun | Electromagnetic actuator for generating variably oriented shear waves in MR elastography[END_REF], this approach can be uncomfortable for patients. Alternatively, acoustic waves may be transmitted through the skull to induce shear waves inside the brain, but the skull attenuates and deforms the acoustic beam, preventing efficient transmission of energy. Recently, it has also been shown that physiological body motion can be used, via blood pulsation [START_REF] Hirsch | In vivo measurement of volumetric strain in the human brain induced by arterial pulsation and harmonic waves[END_REF], [START_REF] Weaver | Brain mechanical property measurement using MRE with intrinsic activation[END_REF] or noise correlation [START_REF] Gallot | Passive elastography: shear-wave tomography from physiological-noise correlation in soft tissues[END_REF], [START_REF] Zorgani | Brain palpation from physiological vibrations using MRI[END_REF], but these methods still require further development before clinical application in the context of brain elastography.

Recently, it was demonstrated that the combination of an electrical current and a magnetic field could create displacements which propagate as shear waves in biological tissues [START_REF] Basford | Lorentz-forceinduced motion in conductive media[END_REF], [START_REF] Grasland-Mongrain | Imaging of shear waves induced by Lorentz force in soft tissues[END_REF]. If the electrical current is induced using a coil, this would allow the technique to remotely induce shear waves. In the case of brain elastography, this would allow inducing shear waves directly inside the brain.

To achieve this objective, we propose to use a transcranial magnetic stimulation (TMS) device [START_REF] Hallett | Transcranial magnetic stimulation and the human brain[END_REF]. This instrument is used to induce an electrical current directly inside the brain by using an external coil. TMS is currently employed by neurologists to study brain functionality [START_REF] Risto J Ilmoniemi | Transcranial magnetic stimulation-a new tool for functional imaging of the brain[END_REF] and by psychiatrists to treat depression [START_REF] Sakkas | Repetitive transcranial magnetic stimulation (rTMS) in a patient suffering from comorbid depression and panic disorder following a myocardial infarction[END_REF]. TMS is occasionally combined with magnetic resonance imaging (MRI) [START_REF] Devlin | Semantic processing in the left inferior prefrontal cortex: a combined functional magnetic resonance imaging and transcranial magnetic stimulation study[END_REF], [START_REF] Bohning | Mapping transcranial magnetic stimulation (TMS) fields in vivo with MRI[END_REF]; however, no study has yet reported the production of shear waves when combining TMS and magnetic fields.

This article first presents the physical model describing the generation of shear waves resulting from the combination of a remotely induced electrical current and a magnetic field. It describes experiments performed in poyvinyl alcohol cryogel and biological tissue samples.

A numerical study of the experiments is then presented. Results section shows a good consistency between experimental and numerical displacement maps. Some critical excitation parameters were investigated as well as dependence of the shear wave amplitude with the magnetic field and electrical current intensity. Practical implementation in a context of shear wave elastography of the brain is finally discussed.

Physical model

We set up the experiment illustrated in Figure 1-(A). The key components are as follows: a coil induces an electrical current j in the sample; a magnet creates a magnetic field B; an ultrasound probe tracks displacements u propagating as shear waves in the sample. X is defined as the main magnetic field axis, Z as the main ultrasound propagation axis, and Y an axis orthogonal to X and Z following the right-hand rule. The origin of coordinates (0,0,0) is located in the middle of the coil (i.e., between the two loops). For a circular coil centered in (0,0,0) of linear element dl crossed by an electrical current I(t), using Coulomb gauge (i.e., ∇.A = 0 where A is the magnetic potential vector), and negligible propagation time of electromagnetic waves, the electrical field E(r, t) along space r and time t is equal to [START_REF] David | Classical Electrodynamics[END_REF]:

E(r, t) = -∇Φ - dI dt N µ 0 4π dl r (1)
where Φ is the electrostatic scalar potential, N is the number of turns of the coil and µ 0 is the magnetic permeability of the coil material. In an unbounded medium, Φ is only due to free charges [START_REF] Grandori | Magnetic stimulation of the motor cortex-theoretical considerations[END_REF], that we supposed negligible in our case. Being additive, the total electrical field created by two or more coils is simply the sum of the contribution of each coil. The induced electrical current density j is retrieved using the local Ohm's law j = σE, where σ is the electrical conductivity of the medium.

The body Lorentz force f can then be calculated using the relationship f = j × B, where B is the magnetic field created by the permanent magnet. Considering the tissue as an elastic, linear and isotropic solid, Navier's equation governs the displacement u at each point of the tissue submitted to an external body force f [START_REF] Aki | Quantitative Seismology[END_REF]:

ρ d 2 u dt 2 = (K + 4 3 µ)∇(∇.u) + µ∇ × (∇ × u) + f ( 2
)
where ρ is the medium density, u the local displacement, K the bulk modulus and µ the shear modulus. Using Helmholtz decomposition u = ∇φ + ∇ × ψ, where φ and ψ are respectively a scalar and a vector field, two elastic waves can be retrieved: a compression wave, propagating at a celerity c k = (K + 4 3 µ)/ρ, and a shear wave, propagating at celerity c s = µ/ρ [START_REF] Armen P Sarvazyan | Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics[END_REF]. As ρ varies typically by a few percent between different soft tissues [START_REF] Richard | Foundations of Biomedical Ultrasound[END_REF], we can suppose an homogeneous density, and measuring c s allows to compute the shear modulus µ of the tissue.

Material and methods

In the experimental setup, pictured in Figure 1-(B), the electrical current was induced by a clinical TMS device using a 2x75 mm diameter coil (MagPro R100 device with C-B60 Butterfly coil, MagVenture, Farum, Danemark). The coil was placed 1 cm away from the medium, without any contact, and fixed to an independent support. The electrical current in the coil was in "monophasic" mode, i.e., a half cycle of 0.4 ms with a rising time of 70 µs, as illustrated in Figure 2-(A). Alternatively, "biphasic" mode, i.e., a full sinus cycle of 0.4 ms could be used, as illustrated in Figure 2-(B). According to the manufacturer's specifications, at 100% amplitude, current reached a magnitude of 149.10 6 A.s -1 in the coil (i.e., 30 kT.s -1 during rising time), leading to a peak transient magnetic field of 2 T.s -1 at the surface of the coil and of 0.74 T -1 (i.e., 12 kT.s -1 during rising time) at 20 mm in depth.

The magnetic field was induced by a 5x5x5 cm 3 N48 NdFeB magnet (model BY0Y0Y0, K&J Magnetics, Pipersville, PA, USA). The magnet was placed 1 cm away from the medium, without any contact, and fixed to a second independent support. In the medium location, the magnetic field intensity ranged from 100 to 200 mT, as measured by a gaussmeter (Model GM2, AlphaLab, Salt Lake City, UT, USA).

Main tested sample was a 4x8x8 cm 3 water-based tissue-mimicking phantom made with 5% polyvinyl alcohol (PVA), 0.1% graphite powder and 5% NaCl, giving a theoretical electrical conductivity of 7.5 S.m -1 . Three freezing/thawing cycles were applied to stiffen the material [START_REF] Fromageau | Estimation of polyvinyl alcohol cryogel mechanical properties with four ultrasound elastography methods and comparison with gold standard testings[END_REF]. The graphite powder (#282863 product, Sigma-Aldrich, Saint-Louis, MO, USA) was made of submillimeter particles, which presented a speckle pattern on ultrasound images. The sample was placed in a rigid plastic box of 2 mm thick layers with an opening on a side to introduce the ultrasound probe. The rigid box simulated a solid interface such as a skull and ensure also that any observed movement was not due to surrounding displacement of air. Alternatively, we used a similar phantom made of 5% PVA, 0.1% graphite powder and 2% NaCl, giving a theoretical electrical conductivity of 3.5 S.m -1 . A biological tissue sample was also tested. This tissue was a chicken breast sample bought in local grocery of approximately 3x5x5 cm 3 . It was degassed in a 20 o C saline water (0.9% NaCl) during two hours prior to the experiment.

Each sample was observed with a 5 MHz ultrasonic probe made of 128 elements (ATL L7-4, Philips, Amsterdam, Netherlands) coupled to a Verasonics scanner (Verasonics V-1, Redmond, WA, USA). The probe was in contact with the sample with an ultrasound coupling gel but was fixed on a third independent support. It was used in ultrafast mode [START_REF] Bercoff | Supersonic shear imaging: a new technique for soft tissue elasticity mapping[END_REF], to acquire 1000 frames per second using plane waves and Stolt's fk migration algorithm [START_REF] Garcia | Stolt's fk migration for plane wave ultrasound imaging[END_REF]. The Z component of the displacement in the sample was observed by performing cross-correlations between radiofrequency images with a speckle-tracking technique, using 128x5 pixels 2 cross-correlation windows [START_REF] Montagnon | Real-time processing in dynamic ultrasound elastography: A GPU-based implementation using CUDA[END_REF]. Noise was partly reduced using a low-pass frequency filter (cut-off frequency at 1 kHz). Time t = 0 ms was defined as the electrical burst emission.

Great care was taken to ensure that the three supports were not in contact and fixed separately. It could ensure that any vibration of one of the element could not be transmitted to the medium.

Numerical study

Additionally to the experiments, a three dimensional simulation of the experiments was performed using Matlab (Matlab 2010, The MathWorks, Natick, MA, USA). The numerical study was performed by (1) calculating the electrical current induced by the coil, (2) simulating the magnetic field created by the permanent magnet, (3) computing the resulting Lorentz force inside the medium, and finally (4) computing the propagation along space and time of the displacement due to the Lorentz force.

Using Equation 1 with two 75 mm diameter coils crossed by a 149.10 6 A.s -1 electrical current, representing the TMS coil used in the experiment, electrical field E was calculated in a 20x10x20 cm 3 volume (see [START_REF] Grandori | Magnetic stimulation of the motor cortex-theoretical considerations[END_REF] for details on mathematical solving). Using Ohm's law, the electrical current j was estimated assuming an electrical conductivity σ = 7.5 S.m -1 . No border effect has been taken into account. Induced electrical current in a XY plane at a depth of 2 cm with 2x2 mm 2 pixels is illustrated in Figure 3-(A), with colors indicating the absolute magnitude and arrows the direction. The electrical current reached a density of 4 kA.m -2 at the medium location.

A finite element software (Finite Element Magnetic Method [START_REF] Meeker | Finite Element Method Magnetics[END_REF]) was used to produce a two dimensional simulation of the magnetic field B. The magnetic field was supposed to be approximately constant in the sample along the Y axis. The magnetostatic problem was solved from equations ∇ × H = ∇ × M , ∇B = 0 and B = µ p H, with H magnetic field intensity, M magnetization of the medium, and µ p medium permeability. Medium was considered as linear, and space was meshed with approximately 0.5 cm 2 triangles. The software simulated a N48 NdFeB permanent magnet of 5x5 cm 2 placed in a 30x30 cm 2 surface of air. Resulting magnetic field in a XZ plane is illustrated in Figure 3-(B), with colors indicating the absolute magnitude and arrows the direction. The magnetic field ranged from 100 to 200 mT at the medium location.

The body Lorentz force f was computed from the cross-product of j and B. The resulting Lorentz force in a XZ plane with 2x2 mm 2 pixels is illustrated in Figure 3-C, with arrows indicating the Lorentz force vector and color its amplitude along Z -as the electrical current is induced in the XY plane and the magnetic field essentially along X direction, Lorentz force is mainly along Z direction. Lorentz force reached a magnitude of 600 N.m -3 in the medium location.

Finally, displacement u(r, t) was determined analytically along space (pixels of 2x2 mm 2 ) and time (steps of 1 ms) by solving Equation 2 with the Green operator [START_REF] Aki | Quantitative Seismology[END_REF]. It used a medium density ρ of 1000 kg.m -3 , a bulk modulus K of 2.3 GPa and a shear modulus µ of 16 kPa, corresponding to a shear wave speed of 4 m.s -1 .

Results

Z component maps of the displacements over time are illustrated in Figure 4, respectively 1, 2, 4, 8 and 12 ms after current emission, as given by the simulation (A), experiment in the PVA phantom (B) and experiment in the chicken breast sample (C). Initial displacements occurs where the Lorentz force has the highest magnitude, on the opposite side of the ultrasound probe, so displacement is not due to probe vibration. Displacements reached an amplitude of 5 µm in the phantom and 0.5 µm in the chicken sample. Displacement maps were harder to compute in the chicken breast sample, as electrical conductivity was lower and as speckle was of poorer quality. They propagated as shear waves, whose speed was 4 m.s -1 for the simulation, 4.0±1.0 m.s -1 for the PVA phantom and 3.5±1.0 m.s -1 for the chicken sample along Z axis. These values correspond respectively to a Young's modulus of 48±24 kPa for the PVA phantom and of 37±20 kPa for the chicken sample.

Figure 5-(A-B) illustrates Z-component map 6 ms after excitation, when excited in "monophasic" mode and in "biphasic" mode respectively. Average displacement in the region of interest is equal to 3.3 µm in the first case and 0.2 µm in the second case: only "monophasic" mode is able to induce observable shear waves.

Figure 5-(C-D) illustrates Z-component map 6 ms after excitation in a 5% salt medium and in a 2% medium respectively (note that (E) and (A) are identical). Average displacement in the region of interest is equal to 3.3 µm in the first case and 1.4 µm in the second case: when electrical conductivity of the medium decreases, shear wave amplitude decreases roughly by a same factor. Figure 5-(E-F) illustrates Z-component map 6 ms after excitation, when excited with a 100% and 50% amplitude in the coil respectively (note that (C) and (A) are identical). Average displacement in the region of interest is equal to 3.3 µm in the first case and 1.3 µm in the second case: shear wave amplitude is roughly divided by two when excitation amplitude is halved (according to the device panel).

Figure 5-(G-H) illustrates Z-component map 6 ms after excitation, when excited with a 100% and a -100% amplitude in the coil respectively, in the 2% salt medium (note that (G) and (D) are identical). Average displacement in the region of interest is equal to 1.3 µm in the first case and -1.4 µm in the second case: displacement amplitude is inverted when excitation is inverted.

The normalized amplitude of shear waves versus distance between the magnet and the PVA sample is illustrated in Figure 6-(A); versus distance between the coil and the PVA sample along the Z axis in Figure 6-(B); and versus distance between the center of the coil and the center of the ultrasound probe along X axis (0 being defined as the coil center aligned with the probe center) in Figure 6-(C). Amplitude of shear waves was measured as the mean squared displacement between 15 and 25 mm of the coil inside the medium, an arbitrary location where shear waves had high amplitudes. Amplitudes were normalized by the maximum measured, respectively at a distance of 4 mm between the magnet and the sample, 10 mm between the coil and the medium and 0 mm between the center of coil and the center of probe. We observed a decrease of the shear wave amplitude when the distance between the medium and the magnet increased, with an excellent agreement between experimental and numerical results. This was expected as the magnetic field decreased with distance. Similar observations can be made when the coil is drawn further from the sample. When we moved the coil along the X direction, we observed a strong maximum between two minima separated by 75 mm, corresponding to the length between the two centers of the TMS coil, which is also in agreement with the current density profile along this direction.

Discussions

Practical application

This study used an ultrasound device to image the sample and track shear waves, due to its high temporal resolution, availability and ease of use. However, for a clinical implementation such as brain elasticity imaging, MRI is more suited for tracking shear waves, as acoustic waves used in ultrasound imaging for shear wave tracking are attenuated by the skull. In a practical MRI implementation, no magnet would be necessary, and MRI-compatible coils should be used. As most clinical MRI scanners use 1.5 T or more magnetic fields, which is at least ten times higher than the one used in this study, displacement amplitude could thus be increased by a similar factor thus improving shear wave tracking. Magnetic resonance elastography is usually employing continuous shear wave excitations. However, induction of a continuous electrical current in the TMS coil may interfere with MRI measurements, so repetitive triggered transient excitations may be used.

Displacement amplitude

In our numerical study, Lorentz force magnitude reached about 600 N.m -3 for a 150 mT permanent magnetic field and a 7.5 S.m -1 medium. The literatures provide numerous measurements of grey and white matter electrical conductivity. This parameter is however difficult to measure and thus presents a high variability. It indeed varied from 0.02 to 2 S.m -1 depending on measurements [START_REF] Gabriel | Electrical conductivity of tissue at frequencies below 1 MHz[END_REF]. Using an average value of 0.2 S.m -1 , in a 1.5 T MRI system, the Lorentz force would reach a magnitude of about 160 N.m -3 . We can compare this value with the acoustic radiation force used for shear wave elastography. This force is calculated with the equation f ARF = 2αI/c, where α is the attenuation in the medium, I the ultrasound intensity and c the speed of sound. Using Nightingale et al. parameters [START_REF] Nightingale | On the feasibility of remote palpation using acoustic radiation force[END_REF] (α = 0.4 Np.cm -1 , I = 2.4 W.cm -2 , c = 1540 m.s -1 ), f ARF is about 1200 N.m -3 , which led in their experimental study to displacements from 2.9 µm. The Lorentz force reported in the current study is about one order of magnitude smaller but we could nevertheless observe displacements of 5 µm in the PVA phantom: it is indeed not only the amplitude, but also the shape and duration of excitation which contributes to the displacement magnitude.

Note that displacement reached an amplitude of 0.5 µm in the chicken sample. Electrical conductivity of muscle (longitudinal) is about 0.4 S.m -1 and is expected to decrease notably after animal death. So although placed in saline water, the effective conductivity of the sample can be expected to be quite lower than that of saline (1.8 S.m -1 ): this the probable explanation of the quite low amplitude displacement.

The excitation mode and duration also have an influence on the displacement amplitude. For example, the "biphasic" mode could not induce any observable displacement. This is probably due to the quick succession of positive and negative displacements with a mean at the noise level amplitude.

Finally, one could notice in the numerical study that displacements were slightly higher than the experimental values in the phantom. Various factors like viscosity and border effects, which were not included in our model, could explain this difference. Moreover, there were uncertainties about the electrical current amplitude and shape in the coil, as constructor values were used, and about the electrical conductivity of the medium, as this parameter is not entirely determined by the concentration of NaCl.

Source localization

In reported experiments, the shear wave source was 3 to 4 cm wide. With currently existing TMS coil geometries, it could hardly be lower than 1 cm. While this is higher than acoustic radiation force using a single point focus (1-2 mm), this last technique is hardly applicable in the brain because of the skull, as mentioned earlier. Compared to current magnetic resonance elastography methods using an external shaker, the shear wave source is far more localized. For whole brain elasticity measurements, having a source spreading on a few cm should not be a problem. For localized measurements with the proposed method, the shear wave source would be placed close to the region of interest, but not inside.

Safety of the method

Regarding the safety of the method, strong magnetic fields in MRI systems are considered biologically harmless [START_REF] Schenck | Safety of strong, static magnetic fields[END_REF]. About the potential harmful effects of the electrical current induced by the TMS coil, safety guidelines based on clinical reports have also been provided for the TMS technique [START_REF] Rossi | Safety, ethical considerations, and application guidelines for the use of transcranial magnetic stimulation in clinical practice and research[END_REF]. These guidelines have been respected in the current study: the excitation amplitude stayed within the manufacturer's limits with less than one activation per second (no harmful effect is expected with this configuration). Main concerns would be with repetitive bursts of short repetition periods, as it could increase local temperature in the brain and potentially be detrimental for the instrument. The combination of electrical current induced by the TMS coil with strong magnetic fields can produce displacements in the range of a few tenths of micrometers at most in biological tissues, but no harmful effects have been reported so far with shear waves of this amplitude [START_REF] Mj Skurczynski | Evaluation of experimental methods for assessing safety for ultrasound radiation force elastography[END_REF], [START_REF] Ec Ehman | Vibration safety limits for magnetic resonance elastography[END_REF]. Precautions linked to the proposed technique are consequently the same as those for TMS and MRI -mainly the absence of ferromagnetic materials in the body.

Figure 1 .

 1 Figure 1. (A) Scheme of the experiment. A coil is inducing remotely an electrical current (blue circles) in a sample. A magnet creates a magnetic field (pink arrows) in the sample. Combination of the electrical current and the magnetic field induces a Lorentz force (red arrows). This force creates displacements which propagate as shear waves (green waves) tracked through an ultrasound probe. (B) Experimental setup. The tested sample is a polyvinyl alcohol tissue-mimicking phantom (plastic box surrounding the phantom was removed for clarity). The electrical current was applied by the coil. The magnetic field was created by the magnet. Ultrasound images were acquired through the probe coupled to the sample.

Figure 2 .

 2 Figure 2. (A) Electrical current in the coil and induced electrical current when used in "monophasic" mode. (B) Electrical current in the coil and induced electrical current when used in "biphasic" mode. (Right) Scheme of the induction of electrical current in the medium by the TMS coil.

Figure 3 .

 3 Figure 3. (A) Electrical current induced by two 75 mm-diameter coils, in a XY plane in a 7.5 S.m -1 medium at 2 cm of the coil, as calculated analytically. Black lines are representing the electrical current lines and colors the magnitude. Electrical current reached a magnitude of 4 kA.m -2 in the medium location (dashed line). (B) Magnetic field as simulated by the Finite Element Magnetic Method software from a 5x5 cm 2 NdFeB magnet (dotted line). The magnetic field ranged from 100 to 200 mT at the medium location (dashed line). Black lines are representing the magnetic field lines and colors the magnitude. (C) Lorentz force in a XZ plane in the medium, as calculated from electrical current and magnetic field. Arrows are representing force vectors and colors the amplitude of the Z component. Lorentz force reached a magnitude of 600 N.m -3 in the medium location (dashed line).

Figure 4 .

 4 Figure 4. Z component maps of the displacement over time, respectively 1, 2, 4, 8 and 12 ms after current emission, as given by the simulation (A), the experiment on the PVA phantom (B) and on the chicken sample (C). A shear wave can be observed in the three cases, with hand drawn black arrows indicating propagation of the wave front.

Figure 5 .

 5 Figure 5.(A-B) Z component map with excitation with "monophasic" and "biphasic" mode respectively. Shear wave can be observed in "monophasic" mode, but no displacement occurs in "biphasic" mode. (C-D) Z component map with excitation in a 5% salt medium and a 2% salt medium respectively. Amplitude of displacement is approximately divided by the same factor as electrical conductivity. (E-F) Z component map with excitation with 100% amplitude and 50% amplitude respectively. Amplitude of displacements is roughly divided by two when amplitude of excitation is halved. (G-H) Z component map with excitation with 100% amplitude and -100% amplitude respectively. Amplitude of displacement is inverted when excitation amplitude is inverted.

Figure 6 .

 6 Figure 6. (A) Normalized magnitude of shear waves versus distance along X from the magnet, experimental (red markers) and numerical (red line) results. (B) Normalized magnitude of shear waves versus distance along Z from the coil, experimental (green markers) and numerical (green line) results. (C) Normalized magnitude of shear waves versus distance along X between the center of the coil and the center of the probe, experimental (blue markers) and numerical (blue line) results.

Acknowledgments

Part of the research has been founded by the Fondation pour la Recherche Médicale, the Natural Sciences and Engineering Research Council of Canada, the Fonds de Recherche du Québec en Santé and the Fondation de l'Association des Radiologistes du Québec. The authors would like to thank MagVenture and Dr Paul Lespérance for the loan of the TMS devices.