Serge Dumont

Imen Manoubi
email: imen.manoubi@yahoo.fr

Numerical solutions to a BBM-Burgers model with a nonlocal viscous term

Keywords: BBM-Burgers equation, decay rate, fractional derivatives, Gear scheme, nonlocal viscous model, quadrature methods, water waves

0 u(s) √ t-s ds is the Riemann-Liouville half derivative. In particular, we implement different numerical schemes to approximate the solution and its asymptotical behavior. Also, we compare our numerical results with those given in [1, 2] for similar models.

In this paper, we numerically investigate the BBM-Burgers equation with a nonlocal viscous term

u t + u x -βu txx + √ ν √ π ∂ ∂t t 0 u(s) √ t -s ds + γuu x = αu xx ,
where 1

INTRODUCTION

The mathematical modeling and analysis of water wave propagation are challenging topics. In their work, J. Bona et al. have derived a family of Boussinesq systems from the two-dimensional Euler equations for free-surface flow in [START_REF] Bona | Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory[END_REF]. Modeling the effects of viscosity on the propagation of long waves is an important challenge that has been investigated since the time of Stokes and has received a lot of interest in the last decade (see [START_REF] Kakutani | Effect of viscosity on long gravity waves[END_REF][START_REF] Ott | Damping of solitary waves[END_REF] and references therein). Besides, P. Liu and T. Orfila [START_REF] Liu | Viscous effects on transient long wave propagation[END_REF], D. Dutykh, and F. Dias [START_REF] Dutykh | Viscous potential free surface flows in a fluid layer of finite depth[END_REF] have independently derived viscous asymptotic models for transient long-wave propagation including viscous effects. These effects appear as nonlocal terms in the form of convolution integrals. The derivation of this model holds in 3 D and 2 D cases. Using a one-way wave reduction (see [START_REF] Bona | Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory[END_REF][START_REF] Diethelm | An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives[END_REF] for details), the authors in [START_REF] Chen | Decay of solutions to a water wave model with a nonlocal viscous dispersive term[END_REF] investigated a reduced nonlinear model that reads

1 u t + u x + βu xxx + √ ν √ π t 0 u t (s) √ t -s ds + γuu x = αu xx , (1)
where 1 √ π t 0 ut (s) √ t-s ds is the Caputo half-derivative. Here u is the horizontal velocity of the fluid, -αu xx is the usual diffusion, βu xxx is the geometric dispersion, √ ν √ π t 0 ut (s) √ t-s ds stands for the nonlocal diffusivedispersive term. The parameters β, ν, γ, and α are dedicated to balance the effects of viscosity and dispersion against nonlinear effects. Moreover, in the recent work [START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF], one of the authors has considered the following water wave model

u t + u x + βu xxx + √ ν √ π ∂ ∂t t 0 u(s) √ t -s ds + γuu x = αu xx , (2)
where 1 √ π

∂ ∂t t 0 u(s) √
t-s ds is the Riemann-Liouville half derivative. Particularly, it is proved the local and the global existence result and decay estimates for the integro-differential equation ( 2) when β = 0, ν = α = γ = 1 supplemented with the initial condition u 0 ∈ L 1 (R) ∩ L 2 (R). Precisely, the following theorem is stated Theorem 1.1 (I. Manoubi, [START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF]) Let u 0 ∈ L 2 (R), then there exists a unique local solution u ∈ C([0, T ); L 2

x (R)) of ( 2). Moreover for u 0 ∈ L 1 (R) ∩ L 2 (R), there exists a positive constant C 0 > 0 that depends on u 0 such that if �u 0 � L 1 (R) is small enough, there exists a unique global solution u ∈ C(R + ; L 2

x (R)) ∩ C 1/2 (R + ; H -2 x (R)) of ( 2) given by

u(t, x) =[K RL (t, •)�u 0 ](x) -N ⊛ u 2 (t, x), (3)
where K RL and N are given by

K RL (t, x) = 1 2 √ πt e -x 2 4t e -x -1 - 1 2 +∞ 0 e -µ 2 4t -µ|x| 2t -µ 2 dµ , and
N(t, x) = 1 4 √ πt ∂ x e -x 2 4t e -x -1 - 1 2 +∞ 0 e -µ 2 4t -µ|x| 2 -µ 2 dµ . with x -= |x|-x 2
= max(-x,0), � represents the usual convolution product and ⊛ is the time-space convolution product defined by

v ⊛ w(t, x) = t 0 R v(t -s, x -y)w(s, y)dsdy.
whenever the integrals make sense. In addition, we have the following estimate

max(t 1/4 , t 3/4 )�u(t, •)� L 2 x (R) + max(t 1/2 , t)�u(t, •)� L ∞ x (R) ≤ C 0 . ( 4
)
The proof of this theorem is presented in [START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF].

In addition, in the recent work [START_REF] Goubet | Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach[END_REF], the authors succeeded to remove the smallness condition on the initial data in Theorem 1.1. Moreover, they proved the weak convergence to zero of the solution.

Furthermore, in their recent work [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF], S. Dumont and J.-B Duval investigated numerically the decay rate for solutions to the following water wave model

u t + u x -βu txx + √ ν √ π t 0 u t (s) √ t -s ds + γuu x = αu xx , (5)
The approximation of time fractional operators has received a lot of interest during last decades for their wide application in fluid, in solid mechanics and in visco-elasticity. The formulation of a numerical stable scheme is crucial but also a difficult issue because of the nonlocal feature of such operators.

The classical methods used in the literature [START_REF] Menza | Approximations numériques d'équations de Schrödinger non linéaires et de modèles associés[END_REF][START_REF] Galucio | The G α -scheme for approximation of fractional derivatives: application to the dynamics of dissipative systems[END_REF][START_REF] Galucio | An adaptation of the Gear scheme for fractional derivatives[END_REF][START_REF] Koh | Application of fractional derivatives to seismic analysis of base-isolated models[END_REF][START_REF] Li | A fast time stepping method for evaluating fractional integrals[END_REF][START_REF] Shokooh | A comparison of numerical methods applied to a fractional model of damping materials[END_REF][START_REF] Zhang | Numerical algorithm for dynamic problems involving fractional operators[END_REF] consist in the approximation of these fractional operators using either convolution integrals or the so-called Gear scheme for fractional operators.

Recently, number of authors in the automatic community developed an alternative method, called the diffusive realization, which is devoted to causal pseudodifferential operators [START_REF] Montseny | Diffusion monodimensionnelle et intégration d'ordre 1/2[END_REF][START_REF] Montseny | Modèle simple d'amortisseur viscoélastique. Application à une corde vibrante[END_REF][START_REF] Montseny | Optimal models of fractional integrators and application to systems with fading memory[END_REF][START_REF] Staffans | Well-posedness and stabilizability of a viscoelastic equation in energy space[END_REF]. Different applications of this approach can be found in [START_REF] Audounet | A threshold phenomenon in the propagation of a point-source initiated flame[END_REF][START_REF] Helie | Diffusive reprentations for the analysis and simulation of flared acoustic pipes with viscothermal losse[END_REF][START_REF] Matignon | Asymptotic stability of linear conservative systems when coupled with diffusive systems[END_REF][START_REF] Montseny | Diffusive representation for pseudodifferentially damped nonlinear systems[END_REF]. The main idea of this method is to replace the nonlocal operator by a linear differential equation. The resulting diffusive model is infinite dimensional, but local in time. Hence, the new model is more easy to solve for both analytical and numerical points of view.

In this paper, we are interested in the following equivalent Benjamin-Bona-Mahony (BBM) model of ( 2)

u t + u x -βu txx + √ ν √ π ∂ ∂t t 0 u(s) √ t -s ds + γuu x = αu xx . (6)
We implement two numerical schemes to approximate the solution of ( 6). The first one is detailed in [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF][START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF][START_REF] Galucio | An adaptation of the Gear scheme for fractional derivatives[END_REF] and is based on the Gear scheme for the approximation of the Riemann-Liouville half-derivative. The second method is based on the diffusive realization of the nonlocal operator supplemented with a splitting scheme (see [START_REF] Diethelm | An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives[END_REF][START_REF] Lombard | Numerical modeling of nonlinear acoustic waves in a tube connected with Helmholtz[END_REF] and references therein). We perform numerical simulations on the solutions and on the decay rates for different values of the parameters β, ν, γ, and α. We compare between these schemes. Furthermore, we compare our numerical results with those given in [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF][START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF][START_REF] Goubet | Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach[END_REF].

Remark 1 We note that the well-posedness of the model ( 6) may be proved mathematically for initial data u 0 ∈ L 2 (R) using the diffusive realization of the half-order derivative and following the same steps as presented at [START_REF] Goubet | Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach[END_REF].

The outline of this article is as follows: in Section 2, we develop the dispersion relation of the model [START_REF] Liu | Viscous effects on transient long wave propagation[END_REF]. Then, in Section 3, we present a first numerical scheme of (6) and numerical results using the Gear scheme to approximate the nonlocal term. In Section 4, we perform a second numerical scheme based on the diffusive realization of the nonlocal term followed by several numerical simulations for model [START_REF] Liu | Viscous effects on transient long wave propagation[END_REF]. A comparison between the two schemes is also performed.

DISPERSION RELATION

We discuss, in this section, the dispersion relation for the linearized asymptotic model. Similarly to [START_REF] Chen | Decay of solutions to a water wave model with a nonlocal viscous dispersive term[END_REF], we take β = 1, γ = 0, α = ν and we consider a Laplace-Fourier analysis due to the presence of the nonlocal term.

Consider the linear BBM-Burgers equation

u t + u x -u txx = νu xx . ( 7
)
We seek a plane wave solution of the form u(t, x) = v(t)e ikx with v(0) = 0. Substituting this solution into [START_REF] Dutykh | Viscous potential free surface flows in a fluid layer of finite depth[END_REF], we get

(1 + k 2 )v t + (νk 2 + ik)v = 0. ( 8
)
We now apply the Laplace transform to [START_REF] Diethelm | An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives[END_REF]. We recall that the Laplace transform in time of a function v of exponential order α is defined by

L(v)(τ) =ṽ(s) = +∞ 0 v(t)e -tτ dt,
for all τ such that Re(τ)>α. Hence, we get

(1 + k 2 )τ + (νk 2 + ik) = 0. ( 9
)
The real part of τ namely, �(τ) =-νk 2 1+k 2 represents the dissipation relation. The imaginary part, denoted by ω =-Im (τ) = k 1+k 2 , represents the dispersion relation. In the sequel, we linearize (2). Thus, we get

u t + u x -u txx + √ ν √ π ∂ ∂t t 0 u(s) √ t -s ds = νu xx . (10)
Substituting the plane wave solution into (10), we get

(1 + k 2 )v t + √ ν √ π ∂ ∂t t 0 v(s) √ t -s ds + (νk 2 + ik)v = 0. ( 11
)
We now apply the Laplace transform to [START_REF] Menza | Approximations numériques d'équations de Schrödinger non linéaires et de modèles associés[END_REF]. Since v(0) = 0, we get

(1 + k 2 )τ + √ ντ + νk 2 + ik = 0. (12)
In order to solve equation [START_REF] Galucio | The G α -scheme for approximation of fractional derivatives: application to the dynamics of dissipative systems[END_REF], we consider the change of variables: τ = z 2 using the principal determination of the logarithm, such that �(z)>0. Hence

(1 + k 2 )z 2 + √ νz + νk 2 + ik = 0. ( 13
)
Equation 13 has two solutions. We consider the solution z such that �(z)>0, namely

z = - √ ν + ν -4(1 + k 2 )(νk 2 + ik) 2(1 + k 2 ) Then -z 2 =-τ = -ν 2(1 + k 2 ) 2 + √ ν 2(1 + k 2 ) 2 ν -4(1 + k 2 )(νk 2 + ik) + νk 2 + ik 1 + k 2
By restricting to the regime ν << k << 1, we obtain

ν -4(1 + k 2 )(νk 2 + ik) = √ -4ik 1 + o(1) = 2e -i sgn(k)π/4 |k|+o( |k|). (14)
Therefore,

-τ = -ν 2(1 + k 2 ) 2 + √ ν (1 + k 2 ) 2 1 √ 2 -i sgn (k) √ 2 |k| + νk 2 + ik 1 + k 2 + o √ ν|k| (1 + k 2 ) 2 . ( 15
)
This implies

-τ = -ν 2(1 + k 2 ) 2 + √ ν|k| √ 2(1 + k 2 ) 2 + νk 2 1 + k 2 + i k 1 + k 2 -sgn(k) √ ν|k| √ 2(1 + k 2 ) 2 + o √ ν|k| (1 + k 2 ) 2 . ( 16
)
After simplifications

Im (-τ) = k -sgn(k) √ ν|k| √ 2 + o √ ν|k| (1 + k 2 ) 2 . ( 17
)
We observe that -Im (τ) has only one nonlinear term: -sgn(k) √ ν|k| √ 2 which represents the nonlocal dispersion. This term is coming from the nonlocal viscous effect. We would like to point out, as in [START_REF] Chen | Decay of solutions to a water wave model with a nonlocal viscous dispersive term[END_REF] for the KdV equivalent model [START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF], that if ν << k << 1 the viscous dispersion is dominant with respect to the geometric dispersion coming from the term u txx . Actually, the viscosity provides dissipation that is of importance. Nevertheless, it depends on parameters. We will see in examples below (Table 2) that the term -u txx also plays a role.

A FIRST NUMERICAL SCHEME

This section deals with the numerical solution of the nonlinear equation

u t -βu txx + √ ν √ π ∂ ∂t t 0 u(s) √ t -s ds = αu xx -u x -γuu x , (18)
supplemented with an initial condition u 0 . Here, β, ν, α, and γ are non-negative parameters.

Presentation of the scheme

We develop here a first numerical scheme using the Gear scheme. For that purpose, we follow the approach proposed in [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF][START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF]. First, we present the outline of the Gear scheme developed by A.-C Galucio et al. in [START_REF] Galucio | An adaptation of the Gear scheme for fractional derivatives[END_REF]. Let u be a time dependent function known only in its discretized values u n at each time t n , where n is a positive integer. The function u n is approximated by u(t n ) with t n = n∆t, where ∆t, TABLE 1 The first five coefficients g n+1 of the formal power series ( 22) which is supposed to be fixed, is the time step. Furthermore, let us introduce a delay operator given by (δ -u) n = u n-1 . Let G be the Gear operator defined by

j α = 1/3 α = 1/2 α = 3/4 0 1 1 1 1 -4 9 -2 3 -1 2 -7 81 -1 18
G = 1 ∆t 3 2 I -2δ -+ 1 2 (δ -) 2 , ( 19
)
that approximates the first derivative of u with respect to time. Then the fractional differential operator G α is given by

G α = 1 ∆t α 3 2 α I - 4 3 δ -+ 1 3 (δ -) 2 α
, which is directly obtained by evaluating the α-power of [START_REF] Montseny | Modèle simple d'amortisseur viscoélastique. Application à une corde vibrante[END_REF]. Then, using Newton binomial formula to compute the term in brackets, we get

G α = 1 ∆t α 3 2 α ∞ j=0 j l=0 4 3 j 1 4 l (-1) j C j α (-1) l C l j (δ -) j+l , (20)
where (-1) j C j α is given in terms of the Gamma function

(-1) j C j α = Γ(j -α) Γ(-α)Γ(j + 1)
.

Then the α-derivative of u at each time t n can be approximated by

(G α u) n = 1 ∆t α 3 2 α ∞ j=0 g j+1 u n-j , (21)
where g j+1 are rational numbers. For illustrative purposes, we present the first five G α -coefficients in Table 1 for three values of α : 1 3 , 1 2 , and 3 4 . As it was mentioned in [START_REF] Galucio | The G α -scheme for approximation of fractional derivatives: application to the dynamics of dissipative systems[END_REF], the Gear operator [START_REF] Montseny | Modèle simple d'amortisseur viscoélastique. Application à une corde vibrante[END_REF] leads to a three-level step algorithm, backward in time and second order accurate, for the approximation of classical time derivatives. As a consequence, it is a first-order accurate for the approximation of the half derivative. It is worth to note that numerical tests on the convergence of the Gear scheme have been performed in [START_REF] Galucio | An adaptation of the Gear scheme for fractional derivatives[END_REF], that confirm this property.

Let us notice that since we consider functions u that are vanishing for t < 0, the infinite sum in [START_REF] Staffans | Well-posedness and stabilizability of a viscoelastic equation in energy space[END_REF] becomes finite

(G α u) n = 1 ∆t α 3 2 α n j=0 g j+1 u n-j . ( 22
)
To write the numerical scheme associated to [START_REF] Montseny | Diffusion monodimensionnelle et intégration d'ordre 1/2[END_REF] we follow [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF][START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF]. For the first iteration, namely for n = 0, we use a Crank-Nicolson discretization in time for the linear terms and a fixed-point method for the nonlinear term. To this end, we rewrite the nonlinear term as uu x = 1 4 (u 2 ) x + 1 4 (u 2 ) x . Then we approximate the first term of the right-hand side explicitly and the second one implicitly. Hence the approximate solution u 1 verifies the semidiscret scheme

(1 -β∂ xx ) u 1 -u 0 ∆t + 1 2 3ν 2∆t ((g 2 + g 1 )u 0 + g 1 u 1 ) = α u 1 xx + u 0 xx 2 - u 1 x + u 0 x 2 - γ 2
(u 0 ) 2 x + (u 1 ) 2 x 4 . (23)
Then, for n ≥ 1, we discretize the right-hand side of (18) as in [START_REF] Kalisch | Models for internal waves in deep water[END_REF]. We use a Crank-Nicolson discretization in time for linear terms and Adams-Bashforth discretization for the nonlinear term. Hence, the proposed discretized equation in time of [START_REF] Montseny | Diffusion monodimensionnelle et intégration d'ordre 1/2[END_REF] reads: for all n ≥ 1

(1 -β∂ xx ) u n+1 -u n ∆t + √ ν(G 1/2 u) n = α u n+1 xx + u n xx 2 - u n+1 x + u n x 2 - γ 2
(3u n ) 2 x -(u n-1 ) 2 x 4 , (24)
where

(G 1/2 u) n = 1 2 G 1/2 (u n+1 + u n ) = 1 2 3 2∆t n+1 j=0 g n+2-j u j + n j=0 g n+1-j u j .
In the case ν = 0, this scheme has local truncation error of order (∆t) 2 and a second-order convergence is observed (for more details, see [START_REF] Kalisch | Models for internal waves in deep water[END_REF]).

Applying the Fourier transform in space to ( 23)-( 24) provides the scheme:

(1 + βξ 2 )(û 1 -û 0 ) + 1 2 3ν∆t 2 ((g 2 + g 1 )û 0 + g 1 û1 ) =- ∆t(αξ 2 + iξ) 2 (û 1 +û 0 ) - iγ∆tξ 8 (u 0 ) 2 + (u 1 ) 2 . ( 25
)
for all n ≥ 1

(1 + βξ 2 )(û n+1 -û n ) + 1 2 3ν∆t 2 n+1 j=0 g n+2-j ûj + n j=0 g n+1-j ûj =- ∆t(αξ 2 + iξ) 2 (û n+1 +û n ) - iγ∆tξ 8 3 (u n ) 2 -(u n-1 ) 2 . ( 26
)
We note that this first numerical scheme is of order 1 in time. Moreover, this scheme has local truncation error of order ∆t (see [START_REF] Dutykh | Viscous potential free surface flows in a fluid layer of finite depth[END_REF][START_REF] Dutykh | Viscous-potential free-surface flows and long wave modelling[END_REF], so that first order convergence is expected.

Numerical results and discussion

Similarly to [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF], we choose an initial datum u 0 which provides an exact BBM soliton for α = ν = 0 and β = γ = 1.

u 0 (x) = 3(p -1)sech 2 1 2 p -1 p (x -x 0 ) . ( 27
)
We take x 0 = 100 and p = 2. For the numerics, we consider periodic boundary conditions in space over an interval [0, L] with L large enough. In all the simulations hereinafter, we take L = 800, the space step size h = 0.1 and the time step size ∆t = 0.1. We note that we expect the decay rate of the solution to be as

�u(t, •)� L 2 x ≈ Ct a or �u(t, •)� L ∞
x ≈ Ct a � for t large with a, a � < 0. Thus, we have the following estimates on the ratios

lim t→∞ R 2 = lim t→∞ log �u(t + ∆t,.)� L 2 x �u(t,.)� L 2 x log t + ∆t t -1 = a, lim t→∞ R ∞ = lim t→∞ log �u(t + ∆t,.)� L ∞ x �u(t,.)� L ∞ x log t + ∆t t -1 = a � .
In Figure 1, we justify the convergence in time of the numerical scheme ( 25)- [START_REF] Lombard | Numerical modeling of nonlinear acoustic waves in a tube connected with Helmholtz[END_REF]. To this end, we fix the parameters values to α = β = 1, γ = 0.5, and ν = 0.1. Since we do not know the analytical solution, we denote by u n Ref the reference solution when ∆t = 0.04. Let u n be the numerical solution when increasing the time step ∆t from 0.05 to 0.2. Then, we denote by E n (∆t) =� u n

Refu n � 2 the L 2 -norm of the error in time. We recall that the solutions are calculated up to time T = 100. We plot E n with respect to ∆t. We observe that the error E n is decreasing when the time step ∆t is decreasing. Besides, this figure deals with the order in time of the scheme (25)- [START_REF] Lombard | Numerical modeling of nonlinear acoustic waves in a tube connected with Helmholtz[END_REF] which is given by the slope of the curve. We see that the measured values are close to a straight line with slope 1.5. This means that the convergence with respect to the time step of discretization is faster for this example than the expected one, equal to O(∆t).

In Figure 2, we simulate the scheme ( 25)- [START_REF] Lombard | Numerical modeling of nonlinear acoustic waves in a tube connected with Helmholtz[END_REF] with parameters values ν = 1, β, α and γ = 0 or 1. We present the solution at time T = 500 and the ratios R 2 and R ∞ . We observe that the shapes of the numerical solutions are very close. We conclude that the influence of the parameters α, γ, and β is less significant than the nonlocal term on the values of the solution. However, these parameters influence the decay rates. Moreover, in the case β = 0, the decay rates of the numerical solution match the theoretical results, presented in [START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF], very well. In addition, when comparing with the equivalent Caputo model (5) investigated in [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF], we see that in our case the solution decreases significantly. In fact, in Figure 7 of [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF], the maximum values of the numerical solutions are between 0.35 and 0.45. However, in our case, the maximum values of the numerical solutions are between 0.01 and 0.015. A similar observation was done when considering the K.d.V-like equation with the Caputo and the Riemann-Liouville nonlocal terms. For more details, see [START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF][START_REF] Chen | Decay of solutions to a water wave model with a nonlocal viscous dispersive term[END_REF]. Also, we observe that the velocity of the wave (defined e.g., by the evolution of the maximum of the solution) when using Caputo term is greater than that with the Riemann-Liouville term. In fact, in Figure 7 of [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF], we observe that the solutions with Caputo term are centered at x = 255. However, in our case, the solutions are centered at x = 130.

In Figure 3, we simulate the solutions and the ratios R 2 and R ∞ when ν = 0 for different values of the parameters β, α, and γ. We observe that the wave moves faster than the case ν = 1 (see Figure 4). Moreover, the amplitude of the solutions in Figure 3 is greater than in the case ν = 1 in Figure 4. Also, we observe that in the absence of the nonlocal term, the parameter γ describing the nonlinear term plays an important role in this simulation.

However, the parameter β affects weakly the amplitude of the solution. In addition, numerical results of decay rates in Figure 3 match very well the theoretical ones for this case. For more details, we refer the reader to [START_REF] Chen | Decay of solutions to a water wave model with a nonlocal viscous dispersive term[END_REF][START_REF] Amick | Decay of solutions of some nonlinear wave equations[END_REF] and references therein.

In Figure 4, we study the influence of different parameters on the solution and on the decay rates in L 2 and in L ∞ norms. We observe that when the ν = 0.1, the wave moves faster than the case ν = 1. Similarly to the Figure 3, the parameter β does not play an important role in this simulation. In addition, comparing to the results given in [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF], we observe that the solution with the Caputo half-derivative moves faster than that with the Riemann-Liouville half-derivative.

In Table 2, we display the values of the decay rates in L 2 and in L ∞ norms for different values of the parameters when ∆t = h = 0.1. We observe that when β = 0, the numerical results match well the mathematical results established in [START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF] for Equation 2. In addition, when β � = 0 the decay rate in L 2 norm is about -0.75 and is around -1 for the L ∞ -norm. Hence, we deduce that the decay rates of ( 18) is close to that of (2) but it is different. This difference is due to the dispersion term.

Finally, we calculate the computational time elapsed to simulate the numerical scheme when varying the parameters values. Results are displayed in Table 2. We see that using the Gear scheme is relatively expensive in computation time. This is expected due to the nonlocal feature of the halfderivative term. We note that this point will be addressed more precisely in the next section.

A SECOND NUMERICAL SCHEME

In order to improve the precision and the efficiency of the numerical scheme used before, we construct in this section a second numerical scheme associated to [START_REF] Liu | Viscous effects on transient long wave propagation[END_REF] based on a splitting method as described in [START_REF] Lombard | Numerical modeling of nonlinear acoustic waves in a tube connected with Helmholtz[END_REF]. In order to construct this scheme, we use the so-called diffusive realization of the half-order derivative. We refer the interested readers to [START_REF] Montseny | Optimal models of fractional integrators and application to systems with fading memory[END_REF][START_REF] Manoubi | Modèle visqueux asymptotique pour la propagation d'une onde dans un canal[END_REF][START_REF] Montseny | Diffusive representation of pseudo-differential time operators[END_REF]. To this end, we denote by

I 1/2 u(t) = 1 √ π t 0 u(s) √
t-s ds, the Riemann-Liouville half-order integral and by

D 1/2 u(t) = 1 √ π d dt t 0 u(s) √
t-s ds, the Riemann-Liouville half-order derivative. We recall that a diffusive realization of I 1/2 u is described as follows

� ∂ t ψ(t, σ) =-σ 2 ψ(t, σ) + 2 π u(t), ψ(0, σ) = 0, ∀σ ≥ 0, I 1/2 u(t) = � +∞ 0 ψ(t, σ)dσ. ( 28
)
where σ is a new variable not physically relevant. Hence, a diffusive realization of the half-order derivative D 1/2 u(t) can be deduced by derivation as follows:

� ∂ t ψ(t, σ) =-σ 2 ψ(t, σ) + 2 π u(t), ψ(0, σ) = 0, ∀σ ≥ 0, D 1/2 u(t) = � +∞ 0 � 2 π u(t) -σ 2 ψ(t, σ) � dσ. ( 29
)
Then we extend the diffusive realization [START_REF] Amick | Decay of solutions of some nonlinear wave equations[END_REF] for the functions u depending on time and space as follows.

� ∂ t ψ(t, x, σ) =-σ 2 ψ(t, x, σ) + 2 π u(t, x), ψ(0, x, σ) = 0, ∀σ ≥ 0, D 1/2 u(t, x) = � +∞ 0 ( 2 π u(t, x) -σ 2 ψ(t, x, σ))dσ. (30)

Presentation of the model

The nonlocal model ( 6) can be written as a PDE-ODE coupled system, using [START_REF] Manoubi | Modèle visqueux asymptotique pour la propagation d'une onde dans un canal[END_REF], as follows

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂ t u(t, x) + ∂ x (u + γ 2 u 2 ) =- √ ν � ∞ 0 ( 2 π u(t, x) -σ 2 ψ(t, x, σ))dσ +αu xx (t, x) + βu txx (t, x), t > 0, x ∈ R ∂ t ψ(t, x, σ) =-σ 2 ψ(t, x, σ) + 2 π u(t, x), t > 0, x ∈ R, σ ≥ 0, ( 31
)
supplemented with the initial conditions

∀x ∈ R, ∀σ ≥ 0, ψ(0, x, σ) = 0, ∀x ∈ R, u(0, x) = u 0 (x).
In order to approximate the Riemann-Liouville half-order derivative in ( 6), we need to approximate the generalized integral in [START_REF] Montseny | Diffusive representation of pseudo-differential time operators[END_REF]. To this end, we use a quadrature formula with N m points. We note by w i the weights and by σ i the nodes (or abscissae) of the appropriate quadrature method used in the approximation. We get

D 1/2 u(t, x) � Nm � i=1 w i � 2 π u(t, x) -σ 2 i ψ(t, x, σ i ) � = Nm � i=1 w i � 2 π u(t, x) -σ 2 i ψ i (t, x)
� , Hence, the system ( 31) is written as a first order system as follows

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t u(t, x) + ∂ x (u + γ 2 u 2 ) =- √ ν � Nm i=1 w i ( 2 π u(t, x) -σ 2 i ψ i (t, x)) +αu xx (t, x) + βu txx (t, x), t > 0, x ∈ R, ∀i = 1, ••• , N m , ∂ t ψ i (t, x) =-σ 2 i ψ i (t, x) + 2 π u(t, x), t > 0, x ∈ R, (32)
endowed with the initial conditions

∀x ∈ R, ∀i = 1, ••• , N m , ψ i (0, x) = 0, ∀x ∈ R, u(0, x) = u 0 (x).
Now, we note by

U = (u, ψ 1 , ••• , ψ Nm ) T ,
the vector of (N m + 1) unknowns, by

F(U) = (u + γ 2 u 2 , 0, ••• ,0) T ,
and finally

S = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ - √ ν � Nm i=1 w i √ νw 1 σ 2 1 ••• ••• √ νw Nm σ 2 Nm 2 π -σ 2 1 0 ••• 0 2 π 0 -σ 2 2 . . . 0 . . . . . . . . . 2 π 0 ••• ••• -σ 2 Nm ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .
Thus, the problem (32) is written in the following form

∂ t U + ∂ x F(U) = S(U) + G 1 ∂ 2 x U -G 2 ∂ t ∂ 2 x U, (33)
where G 1 and G 2 are diagonal matrices of order N m + 1. In the sequel, we introduce the so-called splitting scheme.

The splitting method

Let ∆t > 0, for all n ≥ 0, we recall that t n = n∆t and

U n (x) ≈ U(n∆t, x).
From Equation 33, we consider the propagation equation

∂ t U + ∂ x F(U) = G 1 ∂ 2 x U -G 2 ∂ t ∂ 2 x U, (34)
and the diffusive equation

∂ t U = S(U). (35)
We note by H a (respectively, H b ) the discrete operator of the solution of (34) (respectively, the solution of ( 35)). Then a Strang Splitting method of order 2 ( [START_REF] Holden | Operator splitting for the KDV equation[END_REF][START_REF] Leveque | Numerical methods for conservation laws[END_REF]) between t n and t n+1 is used to solve respectively ( 34) and (35) as follows

U (1) = H b ∆t 2 U n , U (2)
= H a (∆t)U (1) ,

U n+1 = H b ∆ 2 U (2) . (36)
Here, the constructed operators H a and H b are stable and of order 2. Then, the scheme (36) provides an approximation of order 2 in time to the problem [START_REF] Leveque | Numerical methods for conservation laws[END_REF].

In the sequel, we present the discretization of ( 34) and (35).

The propagation equation [START_REF] Moler | Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later[END_REF]. Here u is a solution of the BBM equation. We perform a semidiscrete in time scheme: we use a Crank-Nicolson scheme for the linear part and Adams-Bashforth scheme (see [START_REF] Kalisch | Models for internal waves in deep water[END_REF]) for the nonlinear part. For the space discretization, we use standard Fourier methods.

First, we note that the first approximate solution û1 is performed using a fixed-point method that verifies the semidiscrete scheme (of order 2).

(1 + βξ 2 ) û1 -û 0 ∆t = û1 +û 0 2 (-αξ 2 -iξ) - iγξ 8 (u 0 ) 2 + (u 1 ) 2 . ( 37
)
Then for n ≥ 1, the discret scheme is given by

(1 + βξ 2 ) ûn+1 -û n ∆t = ûn+1 +û n 2 (-αξ 2 -iξ) - iγξ 8 (3 (u n ) 2 -(u n-1 ) 2 ). (38)
Diffusion equation (35). We can solve mathematically (35) as follows

H b ∆t 2 U = e S ∆t 2 U. (39)
In addition, for all N m > 0, the exponential of the matrix S ∆t 2 is calculated numerically using the "scaling and squaring" method with a (6, 6) Padé approximation [START_REF] Moler | Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later[END_REF] (corresponding with Matlab ® to the function "expm(S ∆t 2 )"). We recall that the (p, q) Padé approximation of e A is given by

R pq (A) = (D pq (A)) -1 N pq (A),
where

N pq (A) = p � j=0 (p + q -j)!p! (p + q)!j!(p -j)! A j , D pq (A) = p � j=0 (p + q -j)!q! (p + q)!j!(q -j)! (-A) j .
Hence, the numerical scheme (36) is a three-step scheme. We note that the scheme (38) as well as the Strang Splitting method are of order two. It follows that (36) is of order 2. This will be numerically verified in the sequel.

Quadrature method

In this subsection, we are interested in the choice of the 2N m coefficients w i and σ i of the diffusive representation given in [START_REF] Holden | Operator splitting for the KDV equation[END_REF]. These coefficients aim to approach the improper integrals in the form

+∞ 0 ψ(σ)dσ � Nm � i=1 w i ψ(σ i ). ( 40
)
For seek of convenience, we drop here the variables t and x. The choice of these coefficients is an important issue, because it affects directly the accuracy of the method and the efficiency of the approximation. We note that many methods developed in the literature are especially based on the Gauss quadrature. The choice of the quadrature method was thoroughly discussed in [START_REF] Manoubi | Modèle visqueux asymptotique pour la propagation d'une onde dans un canal[END_REF] when considering the KdV equation with the Riemann-Liouville half-derivative. Based on this work, we present the quadrature method that will be used in the remaining of this article. Gauss-Jacobi quadrature method. Here, we aim to approximate the generalized integral (40) with the Gauss-Jacobi quadrature method. It consists in transforming the domain of integration from [0, +∞[ to [-1, 1] then applying the Gauss-Jacobi quadrature to the resulting integral. Hence, we choose the following change of variables

σ = 1 -z 1 + z then dσ dz = -2 (1 + z) 2 .
It follows that

+∞ 0 ψ(σ)dσ = 1 -1 2 (1 + z) 2 ψ 1 -z 1 + z dz.
Moreover, letting

ψ(z) = 2 (1 + z) 2 ψ 1 -z 1 + z , TABLE 3
ψ(σ)dσ = 1 -1 ψ(z)dz � Nm i=0 µ i ψ(z i ),
where µ i (resp. z i ) are the weights (resp. the nodes) of the standard Gauss-Jacobi quadrature formula over [-1, 1]. For illustrative purposes, we present in Table 3 the weights and the nodes of this quadrature formula for N m = 8. By identification, we deduce that the quadrature coefficients in (40) are given by

w i = 2 (1 + z i ) 2 µ i , σ i = 1 -z i 1 + z i .
We note that this strategy was proposed by Diethelm in [START_REF] Diethelm | An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives[END_REF] for the approximation of Caputo fractional-order derivative.

Numerical results and discussion.

We begin with justifying the convergence in time of the splitting scheme (37)-(38). To this end, we choose h = 0.1, L = 800, N m = 20, T = 100, and α = β = γ = ν = 1. We denote by u n

Ref the reference solution when ∆t = 0.05 and u n the numerical solution for different time steps ∆t. Also, we denote by E n (∆t) =�u n Refu n � 2 the L 2 -error in terms of ∆t. The results are presented in Figure 5. We see that the error E n decreases when the time step ∆t decreases. Also, we may determine numerically the order of the scheme (37)-(38). In fact, the measured values are close to a straight line with slope 2 which means that E n (∆t) ≈ C∆t 2 where C is a constant. We conclude that the numerical scheme (37)-( 38) is of order 2 in time.

Then, we aim to study numerically the convergence of the splitting scheme (37)-(38) with respect to the number of Gauss-Jacobi quadrature points. To this end, we consider the parameters ∆t = h = 0.1, L = 800, T = 100 and ν = α = β = γ = 1. We denote by u n Ref the reference solution for N m = 20 and by u n the numerical solution for different values of N m . Some other numerical tests show that convergence is obtained with N m = 20 (see also [START_REF] Lombard | Numerical modeling of nonlinear acoustic waves in a tube connected with Helmholtz[END_REF]). Moreover, we denote by E n (N m ) =�u n Refu n � 2 the L 2error in terms of quadrature points. We plot, in Figure 6, the Error E n with respect to N m . We see that the measured values are close to a straight line with slope -11, which

means that E n (N m ) ≈ C 1 Nm 11
where C is a constant. Besides, since the error is less than 10 -5 from N m = 15, we choose this number of points to realize the simulations in the sequel. Now, we examine the convergence of the decay rates of the Gear scheme ( 25)-( 26) and the splitting scheme (37)-(38). To this end, we take the parameters values L = 800, ∆t = h = 0.1, α = β = 1, ν = 0.1, and γ = 0.5. We calculate the solutions up to time T = 100. We denote by R Gr p the ratio in norm L p of the Gear scheme ( 25)-( 26) and by R Sp p the ratio in norm L p of the splitting scheme (37)-(38) where p = 2 or ∞. Also, we denote by Err p = R Sp p -R Gr p the difference with respect to the time of the decay rate obtained by the two approximations. In Figure 7, we plot Err 2 and Err ∞ with respect to time t. We observe that both errors Err 2 and Err ∞ are decreasing to zero especially for large times for all the time step ∆t chosen, that is the main part of interest of our study. We conclude that the numerical schemes converges for large time. 2 and Err n ∞ when increasing the time step ∆t from 0.05 to 0.25. We observe that both errors decreases when increasing the time step ∆t. Moreover, we observe that the errors are less than 10 -3 for large times when ∆t = 0.1. Hence, we conclude that this time step is suffisant to get accurate numerical results.

Finally, we determine the computation time of solutions (in seconds) using the splitting scheme (37)-(38) for different parameters values when ∆t = h = 0.1. Results are displayed in Table 4. As it is expected, the time elapsed to calculate the numerical solution using the splitting scheme is reduced. It represents 1/5 of the time necessary to calculate solutions with the Gear scheme ( 25)- [START_REF] Lombard | Numerical modeling of nonlinear acoustic waves in a tube connected with Helmholtz[END_REF]. We deduce that the use of the splitting scheme performs a numerical solution more accurate and in a relatively shorter time of computation.

CONCLUSION

In this article, we have constructed two numerical schemes to approximate the solutions and the decay rates to an asymptotical water wave model where the nonlocal viscous term is described by the Riemann-Liouville half derivative. We compare our numerical results to those given in [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with nonlocal viscosity[END_REF][START_REF] Manoubi | Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative[END_REF][START_REF] Goubet | Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach[END_REF]. We show that using the diffusive realization of the nonlocal operator supplemented with a splitting scheme leads to a very interesting gain of time computing of the numerical solution. This numerical scheme enables us to approximate nonlocal models in a shorter time when comparing with classical methods as the Gear scheme. A challenging issue is to address analytically the asymptotical behavior of the initial value problem. This question will be the subject of a future work.

FIGURE 1

 1 FIGURE 1 Error of the time discretization using the Gear scheme [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 3 FIGURE 4 1 [

 341 FIGURE 3 Numerical solutions and the ratios R 2 and R ∞ using the Gear scheme when ν = 0 and ∆t = 0.1 [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 5

 5 FIGURE 5 Error of the time discretization using the splitting scheme [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 6

 6 FIGURE 6 Error in terms of N m using the Splitting scheme [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 7

 7 FIGURE 7 Differences of the decay rates between the splitting and the Gear schemes when N m = 15, α = β = 1, ν = 0.1, γ = 0.5, and ∆t = 0.1 [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 8

 8 FIGURE 8 Differences of the decay rates between a solution of reference with ∆t = 0.05s and solutions obtained with larger time steps, using splitting scheme where N m = 15, α = 2, ν = 0.5, and γ = β = 1 [Color figure can be viewed at wileyonlinelibrary.com]

TABLE 2

 2 Decay rates of the solutions when varying the parameters of the Gear scheme with ∆t = h = 0.1

	Viscosity	Dispersive	Nonlinear	Diffusive	L 2	L ∞	Computational
	ν	term β	term γ	term α	decay rate	decay rate	time (sec)
	1	1	0	1	-0.73	-0.98	2214.46
	1	1	1	1	-0.72	-0.96	2215.47
	1	0	0	0	-0.76	-1.03	2258.77
	0	1	1	1	-0.25	-0.52	190.12
	0	1	0	1	-0.25	-0.5	193.49
	0	0	1	1	-0.25	-0.5	191.24
	1	0.1	1	1	-0.72	-0.95	2142.32
	0.1	1	0	1	-0.79	-1.12	2181.37
	1	0	0.1	0.1	-0.75	-1.02	2158.70

  Weights and abscissae of the Gauss-Laguerre and Gauss-Jacobi quadrature formulas for N m = 8

	Gauss-Jacobi
	w i	z i	σ i
	128.3897	-0.9603	49.3650
	10.7575	-0.7967	8.8361
	2.7870	-0.5255	3.2153
	1.0879	-0.1834	1.4493
	0.5179	0.1834	0.6899
	0.2696	0.5255	0.3110
	0.1378	0.7967	0.1132
	0.0527	0.9603	0.0203
	we get
	+∞
	0

TABLE 4

 4 Decay rates of the solutions when varying parameters of the Splitting scheme with ∆t = h = 0.1

	Viscosity	Dispersive	Nonlinear	Diffusive	L 2	L ∞	Computational
	ν	term β	term γ	term α	decay rate	decay rate	time (sec)
	1	1	0	1	-0.74	-0.98	400.95
	1	1	1	1	-0.73	-0.97	401.8
	1	0	0	0	-0.76	-1.03	408.67
	0	1	1	1	-0.25	-0.52	190.12
	0	1	0	1	-0.25	-0.5	193.49
	0	0	1	1	-0.25	-0.5	191.24
	1	0.1	1	1	-0.73	-0.96	399.78
	0.1	1	0	1	-0.84	-1.16	394.59
	1	0	0.1	0.1	-0.76	-1.02	393.23

ACKNOWLEDGMENTS

The authors would like to express their appreciation to Referees for their constructive remarks and suggestions. Also, the authors are grateful to Professor Olivier Goubet for his useful comments on section 2. This work was partially supported by a grant from the Simons Foundation.

ORCID

Imen Manoubi http://orcid.org/0000-0003-2243-1330

Funding information

This work was partially supported by a grant from the Simons Foundation. √ π ∂ ∂t t


Francesco Beccuti
email: francesco.beccuti@unito.it

A tale of four cities: reflections of master's students in mathematics on a visual word problem

Keywords: Teaching and learning of logic, reasoning and proof, Curricular and institutional issues concerning the teaching of mathematics at university level, Visualization, Imagination, Intuition

A group of master's students in mathematics was asked to reflect on a visual word problem. No one of the students identified the correct solution and defended instead an incorrect one. The results are explained by the difficulty of the problem itself (reflecting into the students' overall difficulty in imagining its solution) as well as by the students' tendency to overgeneralize. Interestingly, some of the students reach a contradictory statement (which they do not dismiss or acknowledge as such) as a consequence of the effort to accommodate their own mathematical reasoning with what they perceive to be a normative characterization of the problem coming from the lecturer. I conclude by discussing psycho-pedagogical considerations on imagination and intuition with related issues of university curriculum reform.

INTRODUCTION

While in general research in tertiary education is extending beyond the level of undergraduate studies (Artigue, 2021, p. 14;cf. also [START_REF] Winsløw | Research on university mathematics education[END_REF], there seems to be very little or no research at all on students of graduate programs or courses in pure and applied mathematics (Winsløw & Rasmussen, 2020, p. 883-884) and specifically on master's students in mathematics. This is possibly a consequence of the two-years master's programs in mathematics being a relatively new phenomenon in Europe. Thus, most research on postsecondary mathematics education appears to concentrate either on undergraduate programs/courses in mathematics and related disciplines or else on graduate programs specifically designed for teachers. The present study contributes to this under-researched field by investigating how master's students in mathematics reflect on an unusual visual problem.

THEORETICAL FRAMEWORK AND RESEARCH QUESTION

Research in visualization within mathematics education originated in the work of Alan Bishop and was later carried out by various authors: see [START_REF] Presmeg | Visualization and Learning in Mathematics Education[END_REF] for a compendium. In this paper, I will follow the mainstream lineage of research developed by Abraham [START_REF] Arcavi | The role of visual representations in the learning of mathematics[END_REF] and Norma [START_REF] Presmeg | Research on visualization in learning and teaching mathematics[END_REF] albeit explicitly stressing on some hopefully clarifying preliminary definitions inspired from the work of psychologist Efraim [START_REF] Fischbein | Intuition in science and mathematics: An educational approach[END_REF] as well as from the writings of mathematicians such as Felix Klein, David Hilbert and Henry Poincaré. These definitions will constitute the framework for carrying out the analysis of the case-study presented below. This framework can be understood as a systematization of the traditional understanding of cognitive steps happening during a working mathematician's process of proving.

Vision may be defined unambiguously as the faculty by which we directly see things which are there for us to see. On the other hand, imagination is the faculty by which we see what is not there to see (in mathematics this usually happens in connection with some properties one wants to prove or show). It may be divided into passive imagination (the act of representing to oneself something prompted to us from an outside source) and active imagination (the act of representing to oneself something not prompted from the outside). Furthermore, intuition is the faculty by which we generalize the properties that we see or imagine. [1] Notice that the definition of intuition given above (essentially derived from [START_REF] Fischbein | Intuition in science and mathematics: An educational approach[END_REF] is somewhat more specific than the usual meaning given to the term "intuition" (mostly found within philosophy of mathematics or mathematicians' introspective accounts) which is generally an umbrella term used by authors to characterize any informal way of grasping mathematical truths outside of formal reasoning. Indeed, for the great majority of authors "intuitive reasoning" is nothing but a synonym of "informal reasoning". Notice also that for simplicity and adherence to tradition, I take in this paper a clear a priori distinction between informal and formal reasoning, albeit agreeing with the philosophical stance taken by [START_REF] Giardino | Intuition and Visualization in Mathematical Problem Solving[END_REF] that the two forms of reasoning are really inextricably intertwined. Notice also that the literature has traditionally distinguished between internal and external acts of visualization. [START_REF] Presmeg | Research on visualization in learning and teaching mathematics[END_REF] assumes this distinction as unproblematic by adopting the Piagetian view that any act of external visualization depends on internal mental images. I do not want to delve into this issue here, but I would like to remark that the distinction must be made at the level of imagination, i.e., the distinction does not concern vision (always external) and intuition (always internal).

To get a concrete grasp of these definitions and to simultaneously give an example of how these can be applied to analyze mathematical processes, let us look at the usual proof of the following proposition: the opposite sides of a parallelogram are congruent to each other. Provided that we indeed know what a parallelogram is, we can draw it (as an act of passive imagination) by tracing two pairs of parallel lines as in Figure 1.a. At this point we can see the parallelogram 𝐴𝐵𝐶𝐷 as a direct act of vision. Furthermore, in order to prove the proposition, we may (actively) imagine the segment 𝐴𝐶 (Figure 1.b) and consider the angles that this new segment forms with the lines. We are then able to conclude that angles 𝐷𝐴𝐶 and 𝐴𝐶𝐵 are congruent to each other (Figure 1.c) as well as angles 𝐶𝐴𝐵 and 𝐴𝐶𝐷 (Figure 1.d). Thus, triangles 𝑫𝑨𝑪 and 𝑨𝑪𝑩 are congruent (by known properties of congruence). Therefore, the opposite sides of the initial parallelogram are congruent to each other. Finally, it is by intuition that we realize that the property thus proved is not linked to the particular parallelogram considered, but holds in general for all parallelograms. Now, in the passage above the crucial imaginative step (the proverbial "idea" one must have) is to consider the segment 𝐴𝐶 and reduce the proof of the proposition to the proof of the congruence of triangles 𝐷𝐴𝐶 and 𝐴𝐶𝐵. In this example the segment or shape that one needs to imagine in order to complete the proof is almost evident to the trained eyes of a mathematician. However, this may not be the case for inexperienced pupils and, similarly, even experienced mathematicians might have trouble when solving a problem involving a difficult imaginative step which is not cued by the figure or diagram naturally representing the problem.

In this paper I will concentrate on a problem of this kind: the four cities problem, which I will describe below. My research question will thus be the following.

How do master's students in mathematics reason about the four cities problem?

Other than complementing the literature on university mathematics education, answering to this question will also contribute in general to the literature on students engaged in problem solving, which seems to have concentrated primarily on students of compulsory schools (cf. [START_REF] Verschaffel | Word problems in mathematics education: a survey[END_REF])

THE FOUR CITIES PROBLEM AND RESEARCH CONTEXT

The problem below was given to 28 students enrolled in a master's program in mathematics at the University of Turin, Italy. This is a competitive program focusing on pure and applied mathematics. The main requirement for entering the program is to have completed a three-year bachelor's in mathematics with good marks. Such bachelors, in the Italian university system (which does not offer a major-minor arrangement of credits but focuses instead almost entirely on mathematics) usually revolve around learning mathematical content knowledge in the form of theorems and proofs which were customarily tested by means of problem sets usually revolving on the application of these. The problem was given as part of a voluntary assignment within the students' first course in mathematics education. This is an elective course that students usually take in the first year of the master's program. During the course, other visual mathematical problems were presented, but this problem was the one which caused the most difficulties to the students.

Problem: Four cities are placed at the four corners of a square and an engineer wants to design a road which connects them. What path she has to choose in order to use the least amount of materials?

The problem is equivalent to the problem of finding the minimal path which connects the vertices of a square. The optimal solution to the problem is presented in red in Figure 2.d (modulo a 90-degree rotation), while Figure 2.a, 2.b and 2.c show paths which indeed connect the four cities but are not minimal.

[2]

The following hint was given by the lecturer right after the statement of the problem: "the solution is not the path consisting of the square itself" in order to help the students exclude right away the path presented in Figure 2.a. However, this hint may have instead prompted some confusion since a portion of the students interpreted it "normatively", so to speak, as we will see below. After this, no other communication took place between the students and the lecturer. The participants were asked to form groups of three or four. In each group one student (the observer) had to write a report observing and elaborating on the way she and her colleagues reasoned about the problem. In the spirit of [START_REF] Arzarello | A cognitive analysis of dragging practises in Cabri environments[END_REF], the students were invited to visualize the problem by means of a dynamic-geometry software (Geogebra) and given about one hour to solve the problem collaboratively.

METHOD

The data for this study consist of the written reports and the Geogebra files produced by the groups. A preliminary categorization of the groups was performed in terms of the final answer they gave to the problem.

Then the data were analysed by means of the framework described above. As seen in the case of the parallelogram theorem, a procedure of proof relying on visual representation can be segmented into steps linked to the given definitions of vision imagination and intuition. In that case, the analysis referred to an unproblematic and correct proof. However, nothing really prevents to apply it to any other correct or incorrect mathematical proof or proof attempt.

Thus, for each group of students, I analysed the respective texts in connection with the Geogebra files they produced in search for instances text and figures signalling acts of vision, imagination and intuition as connected to the logical structure of the argumentation they provided. In particular, the Geogebra files contained traces of both the students' acts of passive imagination (e.g., draw the initial square) as well as to their attempts to concretize products of their active imagination. These in turn were signalled by corresponding textual expressions describing attempts to add to their drawing new lines or figures. Finally instances of acts of intuition were similarly mostly signalled by textual data describing attempts at deduction and generalization.

In presenting the textual data, I translated the relevant passages from Italian as literally as possible.

RESULTS

Only one of the eight groups hinted at the correct solution. However, the students in this group admitted that one of them had already seen the problem before and hence they were excluded from the study. Among the remaining groups, four suggested that the solution was the one depicted in Figure 2.c: "the diagonals", while three groups suggested that the solution of the problem was the one depicted in Figure 2.a, "the square", and thus had to conceptually "accommodate" the aforementioned hint, as I will discuss below.

Analysis of two reports concluding that the solution is "the diagonals"

Let us now examine the reports of two representative groups (here called A and B) of the former portion of students. The remaining groups (C and D) had similar reports to Group B. Indeed, Group A drew a square together with its diagonals, and just wrote the following laconic sentence.

The minimal path to unite the 4 cities is through the bisectors of the quadrilateral, given the fact that a straight segment is always the shortest way to unite 2 points.

Here the students enact a false deduction, or an over-generalization as described in [START_REF] Fischbein | Intuition in science and mathematics: An educational approach[END_REF]: since the shortest way to unite two points is a straight line, then the shortest way to unite four points is simply two straight lines. This report does not furnish us with any clue as to how they arrived to consider the path consisting of the diagonals, or "the bisectors", as they say here.

On the other hand, a quotation from Group B's report may let us understand how these other students arrived at the same conclusion: the observer writes that his colleagues

[…] decided to represent the diagonals of the quadrilateral, since they thought that the best idea was that of starting from the properties offered by the quadrilateral […] they [then] asked themselves if there did not exist a path better than the one just deduced […] they then decided to construct a second quadrilateral and conjoin the vertices, not by the diagonals, but by segments located in a different way […] In conclusion both the girls agreed, in light of their reasoning and the tests performed, that the minimal path was the one represented by the diagonals of the quadrilateral.

Thus, the students in Group B chose the diagonals because they were "offered by the quadrilateral" itself. In other words, the imaginative step connected with the decision to conjoin opposite points in the parallelogram example above was, as they seem to mean, suggested or cued by the figure itself. Notice that these students also in the end overgeneralized as they considered two different configurations (their own drawings in the software are displayed in Figure 3.a and 3.b) and noticed that the path consisting of the diagonals was shorter than those, and then concluded that the former is shorter than any possible path. Interestingly, notice how the path displayed in Figure 3.a is not too distant from a correct solution.

Analysis of two reports concluding that the solution is (approximately) the square

What about the remaining three groups? As said before, the students in these groups stated that the solution the problem was the square itself but were puzzled by the hint which straightforwardly told them that this was not the case. Of course, the hint was specifically given in order to prompt students to think about other non-obvious solutions and help them in identifying the correct one. However, some of the students instead gave an argument for affirming that the solution was the square itself and then, since this had been excluded by the lecturer, proceeded to propose an "approximate solution" to the problem. Let us see how by analysing the reports of two representative groups (here called E and F) of this latter portion of students. The remaining Group G had a similar report as to Group F. Indeed, the observer of Group E wrote that her colleagues considered the diagonals first but then

This passage suggests that students in Group E were possibly also imposing to the problem the limitation that in order to visit all the cities a hypothetical traveller must not touch the same city two times (an interpretation which is at odds with the realistic setting within which the problem was presented). In any case, they were convinced that the solution to their interpretation of the problem had to be the square. Since this solution was ruled out by the hint, they then reasoned as follows.

[…] they thought of creating polylines [delle spezzate], not necessarily coinciding with the diagonals, which best approximated the perimeter so that their point of intersection lied on the square's axis. Initially they considered these just on two sides of the square while on the others they considered the diagonals. In order to understand if the minimal path was that formed by the polylines on two sides and the diagonals on the other two or rather was the path consisting of polylines on all four sides they decided […] to calculate which one was shorter […] Therefore […] their final conjecture was that of choosing, as minimal path, the one consisting of polylines which best approximate the square's perimeter.

What happened here? It appears that first the students did not imagine that other paths are possible, and as a consequence this led to a comparison whose result they thus generalized. Indeed, they constructed using Geogebra the two configurations displayed in Figure 4.a and 4.b below, then they calculated their respective perimeters (notice that for Figure 4.a this includes the dotted diagonals) and finally conjectured that the solution should be the latter. Furthermore, what is perhaps most striking here is the struggle the students experienced in formulating the consequences of their conjecture. Since the latter points to the fact that the solution should be the square itself, but since also the lecturer had ruled out this possibility, these students are then forced to conclude an impossibility: the minimal path is the one which best approximates the square itself, despite the fact that no such unique path exists! Similarly, Group F started by considering the diagonals of the square but then

[…] came the idea of creating a square in the centre [of the original square] whose side can vary between 0 and l [the side of the square] linking any of its vertex to one and only one city […] At this point, they used Geogebra to represent this situation (as displayed in Figure 5 where point 𝑅 can vary over side 𝑄𝑁) and concluded that

[…] point 𝑅 must be as close as possible to point 𝑁 for having the minimal path: this means that the two squares must have roughly the same side […] The minimal path is given by the approximation of the square having as vertices the four cities.

Here again for these students "the" solution is "the minimal path" consisting of the best approximation to the square itself, despite this being a mathematical impossibility. In conclusion, these data may be explained by the difficulty of the problem itself (which in turns reflect into the students' difficulty to imagine the solution) as well as by the students' tendency to overgeneralize connected to false deduction. Furthermore, many of the students' imagination seems to have been crucially impeded by the fact that the square itself cued a false solution: in the words of (some of) the students, the square "offered" the diagonals as the solution for presumably all Groups A, B, C and D. A similar phenomenon is probably safe to assume having played a role also for Groups E, F and G in convincing them that the square itself (or its "approximation") was the solution. This fact is not surprising perhaps and possibly just tells us something about the difficulty of the problem chosen.

On the other hand, the proneness of master's students in mathematics to overgeneralize is more interesting given the fact that this phenomenon is present in a form or another in the reports of both the first and the second portions of students. This behaviour could be taken to be desirable and proper when conjecturing. However, the substantial easiness with which these students passed from the particular to the general could be regarded as problematic as pertaining to master's students of mathematics (i.e., students who supposedly are at the pinnacle of mathematical instruction). As a linguistic observation, notice the awkward usage of the term "deduction" in the quoted passage of Group B, where "deduced" is used as a synonym of "conjectured" or simply "thought of".

Moreover, with respect to Groups E, F and G, we have seen how the students prefer to conclude an impossibility rather than dismissing their own mathematical reasoning or rather than dismissing what they perceive to be a normative characterization of the problem coming from the lecturer. This is in itself an interesting phenomenon which could be interpreted as signalling a formal and "ritualistic" way of dealing with mathematical problems. Furthermore, if further replicated with different (but analogous) problems and by means of a larger group of students, similar results may point to the fact that for some university students in mathematics what they perceive to be a correct mathematical argument together with what they perceive to be a normative statement coming from the lecturer has stronger epistemic (or perhaps only cognitive) primacy over acknowledging the contradictory nature of their conclusions.

Finally, I do not mean to understand these preliminary conclusions in purely psychological terms as dependent solely on the students' internal faculties, them being completely separated from the context in which these students were immersed.

[3] On the contrary, the students' difficulties I have outlined in this paper render possibly evident that the kind of mathematical training to which these students were exposed fails (at least in this case) as a training for problem solving involving a strong conjecturing component. More empirical data agreeing with these results may point to the fact that the type of mathematical rationality into which these students have been steeped in is very different from the kind of rationality which would ideally be that of a working mathematician. Indeed, a deeper and larger study into these phenomena would be required to reach more than tentative conclusions on this matter. Such study could perhaps suggest the need to develop the university mathematics curriculum in favour of a greater exposure of students to problems involving a stronger conjecturing component. It remains however an open question whether such exposure may in general succeed in the training of imagination and intuition and, in general, whether these faculties are susceptible to be trained at all. The latter in turn is a psychopedagogical matter over which further investigation would be needed.

NOTES

1. In accordance with the scholarship on visualization within mathematics education (cf. [START_REF] Arcavi | The role of visual representations in the learning of mathematics[END_REF], one can thus generally understand visualization in mathematics as all that concerns the faculties/properties/abilities above, i.e., the mode by which we bring mathematical objects at the attention of our senses, we manipulate them and we reflect on them, internally (i.e., in the mind) or externally via some material support, traditionally by hand-drawing on paper or, nowadays, by means of software-generated images.

2. An a priori analysis of the problem would be too long to give here. It is plausible to think that a path very similar to the one which is the solution of the problem must be reached by a unique act of imagination, which arguably seems to not be decomposable into simpler mental actions.

3. For instance, the behavior of Groups E, F and G just summarized may be perhaps explained by a difficulty in reasoning outside of the didactic contract the students assume to be in place (on this concept see [START_REF] Brousseau | Didactic Contract in Mathematics Education[END_REF], connected to a difficulty in questioning the authority of the teacher.

Figure 1 :

 1 Figure 1: visual steps involved in the proof of the proposition

Figure 2 :

 2 Figure 2: possible paths linking four vertices of a square

Figure 3 :

 3 Figure 3: Geogebra protocols from Group B

Figure 4 :

 4 Figure 4: Geogebra protocols from Group E

Figure 5 :

 5 Figure 5: Geogebra protocol from Group F CONCLUSION

After some reflection, they discussed on the fact that by considering the diagonals of the square, in order to visit all the cities, they necessarily needed to use one of the sides. Given the impossibility of doing this, they abandoned this idea.

AKNOWLEDGEMENTS

I would like to thank Ornella Robutti for the wise guidance and discussions which led to this study. She had the idea to test these students with the four cities problem. I would also like to thank Cristina Sabena, Ferdinando Arzarello and again Ornella Robutti for valuable comments and corrections on previous versions of this paper.


Viviane Durand-Guerrier
email: viviane.durand-guerrier@umontpellier.fr

Analysing proofs of the Bolzano-Weierstrass theorem A mean to improve proof skills and understanding of completeness

Keywords: Teaching and learning of logic, reasoning and proof, Epistemological studies of mathematical topics, Bolzano-Weierstrass Theorem, Proofs 'analysis, Teaching and learning of analysis and calculus

In this communication, we will analyse several proofs of the Bolzano-Weiersrass theorem. We will first present the method by Bolzano in the memoire of the Intermediate Value Theorem that is known to have inspired the proof by Weierstrass, then we will analyse proofs available for both the set version proved by Weierstrass, and the sequence version. We will emphasize the diversity of modes of reasoning on the one hand, on objects and completeness characterisation on the second hand. We will finally suggest scenarios for undergraduates or for prospective teachers.

INTRODUCTION

As accounted by [START_REF] Bergé | Students' perceptions of the completeness property of the set of real numbers[END_REF], undergraduate students having followed four courses on the set of real numbers might face still difficulties with task requiring a sound understanding of completeness. This motivates the search for activities able to contribute to this understanding. This communication falls within a wider research project aiming at identifying didactic means to improve the teaching and learning of the set of real numbers as a complete ordered set, considering the crucial role of epistemology in didactics of mathematics, and focusing on proof and proving. Our main didactic hypothesis is that logical analysis of proof fulfils three main functions: to control validity; to understand the strategy of the author of the proof; to contribute both to the development of proof and proving skills and to the appropriation of the mathematical content at stake. We have already discussed in other papers 1/ the potentiality of fixed-point theorems of increasing functions of a real subset in itself [START_REF] Durand-Guerrier | Conceptualization of the continuum an educational challenge for undergraduate students[END_REF]; 2/ the relevance of working on proofs of completeness in various settings of real numbers sets [START_REF] Durand-Guerrier | Working on proof as contribution to conceptualisation -The case of R-completeness[END_REF]; 3/ The fecundity of approaching the real exponential function in the more general frame of real function satisfying the algebraic relation 𝑓(𝑥 + 𝑦) = 𝑓(𝑥)𝑓(𝑦) [START_REF] Durand-Guerrier | Real exponential in discreteness-density-completeness contexts[END_REF]; 4/ the dialectical relationship between truth and proof in the case of the emergence of the Intermediate Value Theorem (IVT), while comparing Bolzano and Cauchy approaches (Durand-Guerrier, 2022). In this communication, we are extending our analysis to the Bolzano-Weierstrass Theorem (BWT) which plays an important role in Analysis. In his memoire on the IVT published in 1817, Bolzano used a method by dissection to prove (in modern term) the existence of a least upper bound for a bounded above set. Later, Weierstrass relied on the method described by Bolzano to prove the theorem that is named after the two mathematicians [START_REF] Oudot | Le théorème de Bolzano-Weierstrass[END_REF].

Nowadays, we found in the international literature various proofs of this theorem. We will argue that analysing these proofs might contribute to deepen both proofs skills and conceptual understanding of the concept of ℝ-completeness, considering both the version for sets and the version for sequences:

1. BWT for sets: Every infinite and bounded set of real numbers has at least an accumulation point. 2. BWT for sequences: Every bounded sequence has a convergent subsequence.

We first present and analyse the Bolzano dissection method and four BWT proofs. Then, we propose some guidelines for designing a didactical engineering.

PROOFS OF THE BOLZANO-WEIERSTRASS THEOREM

Although the name of Bernard Bolzano is associated to the name of Karl Weierstrass, some authors claim that Bolzano neither prove, nor even enounce the theorem (e.g., [START_REF] Oudot | Le théorème de Bolzano-Weierstrass[END_REF]. However, it is agreed that Bolzano used a dissection method in a lemma for the proof of the Intermediate Value Theorem (IVT), which inspired Weierstrass to prove that any infinite bounded subset of the set of real numbers has an accumulation point. (e.g., [START_REF] Oudot | Le théorème de Bolzano-Weierstrass[END_REF]. We present and briefly analyse 1/ the dissection method by Bolzano; 2/ the main ideas of the proof by Weierstrass of the BWT for sets; 3/an alternative proof of the BWT for sets in a paper by Mamona-Downs; 3/ two classical proofs of the BWT for sequences in a textbook.

In Durand-Guerrier and Arsac (2009, p. 152), we provided evidence that analysing logically a mathematical proof requires both logical and mathematical competencies. We have identified relevant questions that we will use as a lens for analysing the proofs presented in this section: what are the data and hypothesis? which objects are introduced along the proof, and with which aim? Which are the modes of reasoning, the explicit and implicit assumptions? In the case of BWT, we will also focus on the axioms of completeness, and on the recourse to potential infinity versus actual infinity.1

The dissection method by [START_REF] Bolzano | Rein Analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegen gesetztes Resultat gewâhren, wenigstens eine reelle Wurzel der Gleichung liege[END_REF] In the paragraph 12 of the proof of the IVT, Bolzano enounce and prove the theorem2 :

Theorem. If a property M does not belong to all values of a variable 𝑥, but does belong to all values which are less than a certain 𝑢, then there is always a quantity 𝑈 which is the greatest of those of which it can be asserted that all smaller 𝑥 has property M. (Russ, 1980, p. 174) He initiated the proof by introducing a positive quantity 𝑫 such that there is at least an element among those less than 𝒖 + 𝑫 that does not hold the property 𝑴 (such a quantity exists by hypothesis). Doing this, he implicitly introduces a real interval [𝒖, 𝒖 + 𝑫[ in which he will search the quantity 𝑼. Then he considers sequentially the quantities 𝒖 + 𝑫 𝟐 𝒎 , with 𝒎 a null or positive integer. In other words, he introduces a geometrical sequence with ratio 𝟏 𝟐 . It is sure, by the choice of 𝑫, that for 𝒎 = 𝟎 there is at least an element not satisfying the property 𝑴 among the element less than 𝒖 + 𝑫 𝟐 𝟎 . In case where whatever the value of 𝒎, there exists always an element not satisfying the property M along the element less than u+ 𝑫 𝟐 𝒎 , then he concludes that 𝒖 is the greatest of those quantities of which it can be asserted that all smaller 𝒙 has property M. The proof of this assertion (not provided in this part of the memoire) relies on the fact that given a real number 𝒗 > 𝒖, there exists an element between 𝒖 and 𝒗 that does not has the property; indeed there exists a positive integer 𝒌 such that 𝒖 < 𝒖 +

𝑫

$𝟐 𝒎&𝒓$𝟏 [, and the process is iterated. There are then two possible issues: either there is a step where the value of 𝑼 is reached, and the iterations stop; or the iterations do not stop and hence, it provides an increasing sequence in geometric progression of ratio 𝟏 𝟐 , with first term 𝒖 + 𝑫 𝟐 𝒎 , each term being less than 𝒖 + 𝑫 𝟐 𝒎$𝟏 , with the difference between two consecutive terms decreasing to 0. This way corresponds to the modern method by dichotomy. Bolzano used then a previous lemma ( §9) to conclude of the existence of a quantity 𝑼 which is the greatest of those of which it can be asserted that all smaller 𝒙 has property 𝑴.

Lemma. If, therefore, some given series has the property that each term is finite, but the change which it undergoes on every further continuation is smaller than any given quantity, provided only that the number of terms taken in the first place is large enough, then there is always one and only one constant quantity which comes as close to the value of this series as desired, if it is continued far enough. (Russ, 1980, p. 173).

In this proof, the data are a property M, a variable 𝒙 and a constant 𝒖; and the hypothesis: M does not belong to all values of 𝒙, but does belong to all values which are less than u. Then there are several objects thar are introduced: a constant 𝑫, a geometric sequence, and along the proof, some specific terms of the sequences identified by indexes of the power of 2; finally, a geometrical progression is introduced. To conclude to the existence, Bolzano refers to what we name nowadays the Cauchy criteria of convergence for sequence, one of the possible axioms of completeness. It is noticeable that as Bolzano did not create the set of real numbers (Freudenthal, 1971, p. 387), the proof in the lemma 9 relies on an assumption that he could not prove (an axiom). In this proof, Bolzano recourse to potential infinity through iteration process.

The proof by Weierstrass of the BWT for sets

According to [START_REF] Oudot | Le théorème de Bolzano-Weierstrass[END_REF], while in contemporaneous textbooks the BWT is most often given for sequences, Weierstrass proved the version for infinite sets. He first defined the notion of accumulation point for an infinite set.

Definition: an accumulation point of a given subset of the set of real numbers is a point such that each pointed neighbourhood has a non-empty intersection with the set.

He then proved by dichotomy, a method inspired by Bolzano, the theorem for sets: Every infinite and bounded set of real numbers has at least an accumulation point.

Here is a summary of the proof (translated from [START_REF] Oudot | Le théorème de Bolzano-Weierstrass[END_REF].

Proof -If a set 𝐴 is bounded, then it is included in a closed interval [𝑚, 𝑀].
As 𝐴 is infinite, cutting the interval through its midpoint, we get two intervals so that at least one contains an infinite number of points of 𝐴. The process is then iterated on an interval with infinite numbers of points of 𝐴 (there exists at least one), leading to the construction of a sequence of nested intervals with length less than % & ) at the step 𝑚. The conclusion follows because the set of real numbers being complete, it satisfies the property of nested intervals with length tending to 0: there exists a unique element belonging to each interval. This element is an accumulation point for 𝐴.

In this proof, the data is a set, with two hypothesises: the set is bounded; the set is infinite. Then an interval [𝑚, 𝑀] is introduced, and a process of dichotomy is performed. It is noticeable that the method is simpler than the method by Bolzano, thanks to the explicit introduction of an interval, that allows the recourse to actual infinite.

A proof of the BWT for sets in an educational paper

In a paper published in 2010, Mamona-Downs suggested that providing students opportunities to contrast the convergence behaviour of a sequence and the accumulation points of the underlying set of the sequence is worthwhile for undergraduates. In the paper, she provides a proof of BWT with recourse to the supremum property (p. 283).

"Theorem (Bolzano-Weierstrass): Let 𝑆 be a real set that is both infinite and bounded. Then 𝑆 has an accumulation point 4 .

As a preliminary step to the proof, the author explains that assuming the problem is solved, a candidate appears (the supremum of a well-chosen set, as shown in the proof below).

Proof: Consider the set 𝐿: {𝑥 ∈ [𝑎, 𝑏] ∶ [𝑎, 𝑥] contains none or a finite number of elements of 𝑆 less than 𝑥}. 𝐿 is non-empty and is bounded by 𝑏. Thus sup (𝐿) exists. Denote sup (L) by 𝑟.

Take any 𝜖 > 0, and consider the interval (𝑟 -𝜀, 𝑟 + 𝜖). Now [𝑎, 𝑟 -𝜖] contains none or a finite number of elements of 𝑆, and [𝑎, 𝑟 + 𝜀 ] contains an infinite number of elements of S. Then the interval ( 𝑟 -𝜀, 𝑟 + 𝜖) also contains an infinite number. This implies ( 𝑟 -𝜀, 𝑟 + 𝜖) contains an element of S 5 , so 𝑟 indeed is an accumulation point for 𝑆."

In this proof, the data is a set, with two hypothesises: the set is bounded; the set is infinite. There is a first introduction of an interval, relying on the hypothesis that the set is bounded, followed by the introduction of the set 𝐿, which is the left part of a cut, in Dedekind sense, of the interval [𝑎, 𝑏]. This choice is motivated by the preliminary step because the supremum linked to this cut has been identified as a candidate to be an accumulation point. The axiom for completeness is the existence of a supremum for any bounded subset of the set of real numbers. Once done, the author names 𝑟 the supremum and introduces a family of intervals centred in 𝑟. She uses implicitly that the complement of a finite subset in an infinite set is infinite, to assert that at each step, there is an interval with infinitely many elements of S; hence each interval of the family contains infinitely many elements of 𝑆; this proves that 𝑟 is an accumulation point of 𝑆.

Two classical proofs of the BWT for sequences

In this section, we present two classical proofs of the BWT out of a textbook for undergraduates [START_REF] Bartle | Introduction to real analysis[END_REF]. The two proofs of BWT are in the section 3.4 entitled "Subsequences and the Bolzano-Weierstrass theorem". The authors first enounce and prove a theorem guaranteeing the existence of monotone subsequence for any real sequence (p.78). We present the proof of this theorem because it is used in their first proof of BWT. If 𝑋 = (𝑥 ! ) is a sequence of real numbers, then there is a subsequence of 𝑋 that is monotone.

Proof. For the purpose of this proof, we will say that the mth term 𝑥 𝑚 is a "peak" if 𝑥 𝑚 ≥ 𝑥 𝑛 for all 𝑛 such that 𝑛 ≥ 𝑚 (That is, 𝑥 𝑚 is never exceeded by any term that follows it in the sequence.) Note that, in a decreasing sequence, every term is a peak, while in an increasing sequence, no term is a peak.

We will consider two cases, depending on whether 𝑋 has infinitely or finitely many peaks.

Case 1: X has infinitely many peaks. In this case, we list the peaks by increasing subscripts: 𝑥 𝑚 1 , 𝑥 𝑚 2 , … , 𝑥 𝑚 𝑘 , ….. Since each term is a peak, we have

𝑥 𝑚 1 ≥ 𝑥 𝑚 2 ≥ ⋯ ≥ 𝑥 𝑚 𝑘 ≥ ⋯ .
Therefore, the subsequence (𝑥 𝑚 ) of peaks is a decreasing subsequence of 𝑋.

5 Implicitly there is at least an element different from 𝑟.

Case 2: 𝑋 has a finite number (possibly zero) of peaks. Let these peaks be listed by increasing subscripts 𝑥 𝑚 1 , 𝑥 𝑚 2 , … , 𝑥 𝑚 𝑟 . Let 𝑠 1 = 𝑚 𝑟 + 1 the first index beyond the last peak.

Since 𝑥 𝑠 1 is not a peak, there exists 𝑠 2 > 𝑠 1 such that 𝑥 𝑠 1 < 𝑥 𝑠 2 . Since 𝑥 𝑠 2 is not a peak, there exists 𝑠 3 > 𝑠 2 such that 𝑥 𝑠 2 < 𝑥 𝑠 3 . Continuing in this way, we obtain an increasing subsequence $𝑥 𝑠 𝑘 % of 𝑋.

It is not difficult to see that a given sequence may have one subsequence that is increasing, and another subsequence that is decreasing." (op. cit. page 78).

In this proof, the data is a real sequence; there is no additional hypothesis. The notion of "peak" is introduced and depending on whether or not there are infinitely many peaks (actual infinity), one can construct either a decreasing subsequence or a finite sequence of peaks. In the former case, we have a proof of the theorem; in the latter case, an iterative process (potential infinity) provides an increasing sequence, and the theorem is proved. It is noticeable that this proof use only the order properties; the ℝcompleteness is not involved, so this theorem holds for example for rational sequences, while BWT does not. Both actual infinity and potential infinity are involved.

The authors move then to the BWT theorem for sequences for which they provide two proofs, the first one relying on the previous theorem.

" 3.4.8 The Bolzano-Weierstrass Theorem. A bounded sequence of real numbers as a convergent subsequent.

First proof -It follows from the Monotone Sequence Theorem that if 𝑋 = (𝑥 ! ) is a bounded sequence, then it has a subsequence 𝑋′ = D𝑥 ) ' E that is monotone. Since this subsequence is also bounded, it follows from the monotone convergence theorem 3.3.2 that the subsequence is convergent." (op. cit. p. 78-79).

Here the data is a sequence of real numbers; the hypothesis is that it is bounded. Thanks to theorem 3.4.7, the authors introduce a monotone subsequence of the given sequence. They use, without enouncing it, the assertion that a subsequence of a bounded sequence is bounded. The conclusion relies on theorem 3.2.2. (A monotone sequence of real numbers is convergent if and only if it is bounded). The completeness of the set of real numbers is required for the proof of the reverse implication. This theorem is among those that can be chosen as axiom of completeness. It was at the origin of the creation of irrational numbers by Dedekind, who proved it in his ordered complete system of real numbers [START_REF] Dedekind | Essays on the theory of numbers[END_REF]. In this textbook, the axiom of completeness is the supremum one: Every nonempty set of real numbers that has an upper bound has also a supremum (i.e. a least upper bound) (2.3.6, p.37).

The second proof relies on arguments close to those of the proof by Weierstrass for the version for sets by introducing the set of values of the sequence.

"Second proof -Since the set of values {𝑥 𝑛 : 𝑛 ∈ 𝑁} is bounded, this set is contained in an interval 𝐼 1 = [𝑎, 𝑏]. We take 𝑛 1 : = 1.

We now bisect 𝐼 1 into two equal subintervals 𝐼′ 1 and 𝐼′′ 1 , and divide the set of indices {𝑛 ∈ ℕ ∶ 𝑛 > 1} into two parts:

𝐴 1 = {𝑛 ∈ ℕ ∶ 𝑛 > 𝑛 1 , 𝑥 𝑛 ∈ 𝐼′ 1 } 𝐵 1 = {𝑛 ∈ ℕ ∶ 𝑛 > 𝑛 1 , 𝑥 𝑛 ∈ 𝐼′′ 1 }
If 𝐴 1 is infinite, we take 𝐼 2 = 𝐼′ 1 and let 𝑛 2 be the smallest natural number in 𝐴 1 . If 𝐴 1 is a finite set, then 𝐵 1 must be infinite, and we take 𝐼 2 = 𝐼′′ 1 and let 𝑛 2 be the smallest natural number in 𝐵 1 .

We now bisect 𝐼 2 into two equal subintervals 𝐼′ 2 and 𝐼′′ 2 , and divide the set of indices {𝑛 ∈ 𝑁 ∶ 𝑛 > 𝑛 + } into two parts:

𝐴 2 = {𝑛 ∈ ℕ ∶ 𝑛 > 𝑛 2 , 𝑥 𝑛 ∈ 𝐼′ 2 } 𝐵 2 = {𝑛 ∈ ℕ ∶ 𝑛 > 𝑛 2 , 𝑥 𝑛 ∈ 𝐼′′ 2 }
If 𝐴 2 is infinite, we take 𝐼 3 = 𝐼′ 2 and let 𝑛 3 be the smallest natural number in 𝐴 2 . If 𝐴 2 is a finite set, then 𝐵 2 must be infinite, and we take 𝐼 3 = 𝐼′′ 2 and let 𝑛 3 be the smallest natural number in 𝐵 2.

We continue in this way to obtain a sequence of nested intervals 𝐼 1 ⊇ 𝐼 In this proof, the data and the hypothesis are the same than for the first proof. The authors introduce the set of values of the sequence; as the sequence is bounded, the set is also; then, an interval is introduced to initiate the dichotomy6 , that provides both a sequence of nested intervals with length converging to 0, and a subsequence of the initial sequence. By the corresponding axiom of completeness, there exists a unique point in the intersection, that is proved to be the limit of the sequence. Comparing this proof with Weierstrass's one enlightens the links between the set version and the sequence version of BWT, and the links with axioms of ℝ-completeness.

The analysis of these five proofs confirms the potential of the BWT to work with undergraduates or prospective teachers on various characterisations of completeness, and on their links. In the second section, we provide some paths for designing a didactical engineering aiming at developing proof and proving skills together with contribution to a better understanding of the completeness of the set of real numbers.

GUIDELINES FOR DESIGNING A DIDACTICAL ENGINEERING

As Mamona-Downs suggested, we consider that introducing undergraduates or secondary mathematics prospective teachers to both versions of BWT and designing a didactical engineering 7 around the analysis of the various proofs would offer the opportunity to contrast the convergence behaviour of a sequence and the accumulation points of the underlying set of the sequence, and to contribute to the cognitive development of proof's skills [START_REF] Dreyfus | Cognitive development of proof[END_REF].

A possible scenario for undergraduates or prospective teachers

The scenario we now present has not yet been implemented but is based on our experience as a university teacher of undergraduates and as a teacher's trainer for prospective secondary mathematics teachers. The analysis of the five proofs in the first section is part of the a priori analysis; by lack of place, we will not go deeper in it in this section, but only indicates the guidelines of the proposal.

Asynchronous work -Reading the paper by [START_REF] Oudot | Le théorème de Bolzano-Weierstrass[END_REF]. This paper is online on the French website "CultureMaths" and provides historical and contextual elements.

First activity -In small groups, reading and analysing the excerpt of the proof by Bolzano presented above, including the lemma of §9. Possible questions are:

1. What are the data, the hypothesis, the objects introduced along the proof? 2. In which respect the method developed in the proof is related to the nowadays proof by dichotomy? 3. Considering your own knowledge, indicate in which conditions does the lemma of §9 used in the proof apply. 4. According to you, under which theoretical assumptions the proof is valid? Second activity8 -In small subgroups -Half of the subgroups works on the two proofs of the sequence version of the theorem provided by Bartle and Sherter (2000); the others work on two proofs of the set version -the one presented in [START_REF] Oudot | Le théorème de Bolzano-Weierstrass[END_REF], and the one by Mammona-Downs (2010). The students are requested to write down an account of their works. Possible questions are:

1. What are the data, the hypothesis, the objects introduced along the proof? 2. Which are the theorems / axioms that are used explicitly or implicitly? 3. Which mode or reasonings do you identify in the two proofs. 4. According to you, under which theoretical assumptions the proof is valid? Third activity -Collective work 1. Sharing the analysis by oral presentation with slides.

2. Discussing the following questions: where does the assumptions on completeness of the set of real numbers is called for in the different proofs? Are there some intermediate results that would hold in uncompleted ordered fields?

Possible further activities 1. Design and implementation of an algorithm for methods by dichotomy of the proof by Bolzano of the existence of the monotone subsequence and comparison with algorithmic methods in computer science [START_REF] Meyer | Recherche binaire et méthode de dichotomie, comparaison et enjeux didactiques à l'interface mathématiques -informatique[END_REF].

  𝑫𝟐 𝒌 < 𝒗. Opposite, if for a certain rank m, for the first time all elements less than 𝒖 + 𝑫 𝟐 𝒎 holds the property 𝑴, then he reinitiated the process with 𝒖 + 𝑫 𝟐 𝒎 playing the role of 𝒖, and 𝒖 + 𝑫 𝟐 𝒎$𝟏 playing the role of 𝒖 + 𝑫. This is motivated by the fact that at this step, the quantity 𝑼 should be searched in the interval [𝒖 + Bolzano notes that the difference between the two quantities is 𝑫 𝟐 𝒎 .The process is then reiterated, considering the sequence 3 8𝒖 +

	𝑫 𝟐 𝒎 +

𝑫 𝟐 𝒎 , 𝒖 + 𝑫 𝟐 𝒎$𝟏 [. 𝑫 𝟐 𝒎&𝒏 9. Then either 𝑼 = 𝒖 + 𝑫 𝟐 𝒎 and the theorem is proved; or there is a value 𝒓 such that 𝑼 belongs to [𝒖 + 𝑫 𝟐 𝒎 + 𝑫 𝟐 𝒎&𝒓 , 𝒖 + 𝑫 𝟐 𝒎

  1 ⊇ ⋯ ⊇ 𝐼 𝑘 ⊇ ⋯, and a subsequence (𝑥 𝑛 𝑘 ) of 𝑋 such that 𝑥 𝑛 𝑘 ∈ 𝐼 𝑘 for 𝑘 ∈ ℕ. Since the length of 𝐼 𝑘 is equal to 𝑘-1 , it follows from Theorem 2.5.3 that there is a (unique) common point 𝜉 ∈ 𝐼 1 for all 𝑘 ∈ ℕ. Moreover since 𝑥 𝑛 𝑘 and 𝜉 both belong to 𝐼 𝑘 , we have*𝑥 𝑛 𝑘 -𝜉* ≤ 𝑏 -𝑎2 𝑘-1 whence it follows that the subsequence (𝑥 𝑛 𝑘 ) of 𝑋 converge to 𝜉." (op. cit. p.79)

	𝑏-𝑎
	2

This paper will not consider the logical issues involved in the use of quantifiers and connectives.

For the theorem, its proof and the comments by Bolzano on this theorem, we use the English Translation by[START_REF] Russ | A translation of Bolzano's paper on the intermediate value theorem[END_REF].

In this proof, Bolzano does not use the term sequence, and the notion of interval; it remains implicit.

The author indicates that there is no loss of generality f we suppose that S is a subset of a closed interval,[a, b] say.

The proof is done in case this set of values is infinite

For didactical engineering in university mathematics education, seeGonzales-Martins et al. (2014)

An alternative for prospective secondary mathematics teachers, is to ask them to provide proofs of the BWT.

CONCLUSION

In this paper, we have tried to highlight the opportunities offered by the Bolzano-Weierstrass Theorem, an important theorem in Analysis with a lot of applications, to promote both proof skills and understanding of ℝ-completeness. We have presented guidelines for a didactical engineering to be further implemented and refined.