Manuel Fernando

Fernando Cortez

BLOW-UP FOR THE b-FAMILY OF EQUATIONS

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Blow-up for the b-family of equations

Introduction

The bi-Hamiltonian structure of certain evolution equations leads to various remarkable features such as infinitely many symmetries and conserved quantities, and in some cases to the exact solvability of these equations [START_REF] Magri | A Simple Model of the integrable Hamiltonian Equation[END_REF][START_REF] Olver | On the Hamiltonian structure of evolution equations[END_REF]. Examples include the KdV equation [START_REF] Caudrey | A new hierarchy of Korteweg-de Vries equations, Proceedings of the Royal Society of London[END_REF] and the Benjamin-Ono equation [START_REF] Benjamin | Internal waves of permanent form in fluids of great depth[END_REF]. Years later, R. Camassa and D. Holm [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF] in their studies of completely integrable dispersive shallow water equation tackled the following equation, (C-H)

u t + ku x -u xxt + 3uu x = uu xxx + 2u x u xx , x ∈ R, t > 0.
where u can be interpreted as a horizontal velocity of the water at a certain depth and k as the dispersion parameter. The equation (C-H) also has been derived independently by B. Fuchssteiner and A. Fokas in [START_REF] Fuchssteiner | Symplectic structures, their Bäcklund transformations and hereditary symmetries[END_REF]. When k = 0 (dispersionless case), the equation (C-H) possess soliton solutions peaked at their crest (often named peakons) [START_REF] Brandolese | Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF][START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF][START_REF] Camassa | A new integrable shallow water equation[END_REF]. Equation (C-H) is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler's equation in the shallow water regime. Like the KdV equation, the Camassa-Holm equation (C-H) describes the unidirectional propagation of waves at the surface of shallow water under the influence of gravity [START_REF] Camassa | An integrable shallow water equation with peaked solitons[END_REF][START_REF] Caudrey | A new hierarchy of Korteweg-de Vries equations, Proceedings of the Royal Society of London[END_REF]. The equation (C-H) is physically relevant as it also describes the nonlinear dispersive waves in compressible hyperelastic rods [START_REF] Brandolese | Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF][START_REF] Brandolese | Blowup issues for a class of nonlinear dispersive wave equations[END_REF][START_REF] Dai | Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod[END_REF]. It is convenient to rewrite the Cauchy problem associated with the dispersionless case of (C-H) in the following weak form:

(1.1)

   u t + uu x + ∂ x p * u 2 + u 2 x 2 = 0, x ∈ A, t > 0, u(0, x) = u 0 (x) x ∈ A,
where p(x) is the fundamental solution of the operator 1 -∂ 2 x in A. If A = R, we refer (1.1) as the non-periodic Camassa-Holm equation and p = 1 2 e -|x| , x ∈ R in this case. If otherwise that A = T = R/Z is unit circle, we refer (1.1) as the periodic Camassa-Holm equation, and p = cosh(x-[x]- 1 2 ) 2 sinh( 12 ) in this case. It is know that both the non-periodic and periodic Camassa-Holm equations are locally well-posed (in the sense of Hadamard) in the Sobolev space H s , with s > 3 2 . See [START_REF] Danchin | A few remarks on Camassa-Holm equation[END_REF][START_REF] Lellis | Low-regularity solutions of the periodic Camassa-Holm equation[END_REF][START_REF] Rodríguez-Blanco | On the Cauchy problem for the Camassa-Holm equation[END_REF]. There is an abundance of the literature about the issue of the finite time blowup (see [2-4, 11, 12, 25, 29, 33]) and the related issue of the global existence of strong solutions ( [START_REF] Constantin | Wave breaking for nonlinear nonlocal shallow water equations[END_REF][START_REF] Constantin | Global weak solutions for a shallow water equation[END_REF][START_REF] Gui | On the global existence and wave-breaking criteria for the two-component Camassa-Holm system[END_REF]).

On the other hand, Degasperis and Procesi [START_REF] Degasperis | Asymptotic integrability, in Symmetry and Perturbation Theory[END_REF], in their search of new integrability properties inside a wide class of equations, were led to consider the following integrable equation:

(D-P) u t -u txx + 4uu x = 3u x u xx + uu xxx .
As before, it is convenient to rewrite the Cauchy problem, using the same notations

(1.2) u t + uu x + ∂ x p * 3 2 u 2 = 0, x ∈ A, t > 0, u(0, x) = u 0 (x), x ∈ A.
A few years later, equation (1.2) as been proved to be relevant in shallow water dynamics, see [START_REF] Constantin | The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations[END_REF][START_REF] Constantin | Nonlinear Water Waves with Applications to WaveCurrent Interactions and Tsunamis[END_REF][START_REF] Dullin | Korteweg-de Vries and other asymptotically equivalent equations for shallow water waves[END_REF]. Both the Camassa-Holm equation and the the Degasperis-Procesi equation (D-P) possess a bi-Hamiltonian structure (see [START_REF] Degasperis | Asymptotic integrability, in Symmetry and Perturbation Theory[END_REF]). The local well-posedness in H s , with s > 3 2 for the Cauchy non periodic problem was elaborated in [START_REF] Zhaoyang | On the Cauchy problem for an integrable equation with peakon solutions[END_REF], and [START_REF]Global weak solutions for a new periodic integrable equation with peakon solutions[END_REF] for the Cauchy periodic problem. With respect to blow-up criteria on the line we refer to [START_REF] Degasperis | Asymptotic integrability, in Symmetry and Perturbation Theory[END_REF][START_REF] Liu | Global existence and blow-up phenomena for the Degasperis-Procesi equation[END_REF][START_REF] Guo | Blow-up and global solutions to a new integrable model with two components[END_REF][START_REF] Zhou | Blow-up phenomenon for the integrable Degasperis-Procesi equation[END_REF] and, for the unit torus, to [START_REF]Global weak solutions for a new periodic integrable equation with peakon solutions[END_REF][START_REF] Guo | Blow-up and global solutions to a new integrable model with two components[END_REF]. For the existence globally of the solution, see [START_REF] Liu | Global existence and blow-up phenomena for the Degasperis-Procesi equation[END_REF][START_REF] Zhaoyang | On the Cauchy problem for an integrable equation with peakon solutions[END_REF][START_REF] Guo | Blow-up and global solutions to a new integrable model with two components[END_REF]. Despite sharing some properties with the Camassa-Holm equation, the Degasperis-Procesi has its own peculiarities. A specific feature is that (D-P) admits, beside peakons (i.e., soliton solutions of the form u(t, x) = ce -|x-ct| , c > 0) also shock peakon solitons (i.e., solutions at the form u = 1 t+k sign(x)e -|x-ct| , k > 0). For more details see [START_REF] Escher | Global weak solutions and blow-up structure for the Degasperis-Procesi equation[END_REF][START_REF] Henry | Infinite propagation speed for the Degasperis-Procesi equation[END_REF][START_REF] Lundmark | Formation and dynamics of shock waves in the Degasperis-Procesi equation[END_REF]. After these premises, we will now focus on the Cauchy problem for the spatially periodic b-family of equations:

(1.3) u t -u txx + (b + 1)uu x = bu x u xx + uu xxx , x ∈ T, t > 0, u(0, x) = u 0 (x), x ∈ T,
where T is the unit torus. Here b is a real parameter, and u(x, t) stands for a horizontal velocity. The b-family of equations can be derived as the family of asymptotically equivalent shallow water wave equations that emerges at quadratic-order accuracy for any b = 1 by an appropriate Kodama transformation [START_REF] Degasperis | Asymptotic integrability, in Symmetry and Perturbation Theory[END_REF][START_REF] Dullin | Korteweg-de Vries and other asymptotically equivalent equations for shallow water waves[END_REF]. Again, when b = 2 and b = 3 (1.3) became (C-H) and (D-P) respectively. These values are the only values for which (1.3) is completely integrable. The Cauchy problem for the b-family of equations is locally well posed in the Sobolev space H s for any s > 3 2 , [START_REF] Christov | On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis-Procesi equation[END_REF][START_REF] Escher | Well-posedness, blow-up phenomena, and global solutions for the b-equation[END_REF][START_REF] Liu | Global existence and blow-up phenomena for the Degasperis-Procesi equation[END_REF][START_REF] Saha | Blow-Up results for the periodic peakon b-family of equations[END_REF]. In [START_REF] Chen | The Höder continuity of the solution map to the b-family of equations in weak topology[END_REF] it is proved that the solution map of the b-family of equations is Holder continuous as a map from bounded sets of H s (R), s > 3 2 with the H r (R) (0 ≤ r < s) topology, to C([0, T ], H r (R)). While that in [START_REF] Christov | Non-uniform continuity of periodic Holm-Staley b-family of equations[END_REF], the authors proved that the solution map is not uniformly continuous. Their proof relies on a construction of smooth periodic traveling waves with small amplitude. J. Escher and J. Seiler [START_REF] Escher | Well-posedness, blow-up phenomena, and global solutions for the b-equation[END_REF] showed that the periodic b-family of equations can be realized as Euler equation on the Lie group Diff ∞ (T) of all smooth and orientation-preserving diffeomorphisms on the unit torus, if b = 2 (C-H equation). The global existence theory of the solution of (1.3) is discussed in [START_REF] Escher | Well-posedness, blow-up phenomena, and global solutions for the b-equation[END_REF][START_REF] Liu | Global existence and blow-up phenomena for the Degasperis-Procesi equation[END_REF][START_REF] Saha | Blow-Up results for the periodic peakon b-family of equations[END_REF][START_REF]Global weak solutions for a new periodic integrable equation with peakon solutions[END_REF]. In this paper we rather focus on blow-up criteria as well in estimates about the lifespan of the solutions. The blowup problem for the b-family of equations has been already treated, e.g. in [START_REF] Christov | On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis-Procesi equation[END_REF][START_REF] Escher | Well-posedness, blow-up phenomena, and global solutions for the b-equation[END_REF][START_REF] Li | Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation[END_REF][START_REF] Saha | Blow-Up results for the periodic peakon b-family of equations[END_REF][START_REF] Guo | Blow-up and global solutions to a new integrable model with two components[END_REF][START_REF]On solutions to the Holm-Staley b-family of equations[END_REF]: in these references the condition on the initial datum u 0 leading to the blowup typically involves the computation of some global quantities (the Sobolev norm u 0 H 1 , or some other integral expressions of u 0 ). Motivated by the recent paper [START_REF] Brandolese | Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF] (where earlier blowup results for the Camassa-Holm equations were unified in a single theorem) we address the more subtle problem of finding a local-in-space blowup criterion for the b-family of equations, i.e., a blowup condition involving only the properties of u 0 in a neighborhood of a single point x 0 ∈ T.

Loosely, the contribution of this paper can be stated as follows: if the parameter b belongs to a suitable range (including the physically relevant cases b = 2 and b = 3), then then there exists a constant β b > 0 such that if

u 0 (x 0 ) < -β b |u 0 (x 0 )| ,
in at least one point x 0 ∈ T, then the solution arising from u 0 ∈ H s (T) (s > 3 2 ) must blow-up in finite time.

This paper is organized as follows. In the next section we start by introducing the relevant notations and function spaces, recalling a few basic results. Then we precisely state and prove our main theorem. An important part of our work will be devoted to the computations of sharp bounds for the constant β b and the lifespan of the solution. The smallest b > 1 to which our main theorem applies is computed numerically in the last part of the paper.

Blow-up for the periodic b-family of equations

It is convenient to rewrite the periodic Cauchy problem (1.3) in the following weak form (see [START_REF] Saha | Blow-Up results for the periodic peakon b-family of equations[END_REF]):

(2.1)

       u t + uu x + ∂ x p * b 2 u 2 + 3-b 2 u 2 x = 0, x ∈ T, t > 0, u(0, x) = u 0 (x), x ∈ T u(t, x) = u(t, x + 1) t ≥ 0, where (2.2) p(x) = cosh(x -[x] -1 2 ) 2 sinh 1 2 ,
is the fundamental solution of the operator 1 -∂ 2

x and [•] stands for the integer part of 3 2 satisfies (2.1) then we call u a strong solution to (2.1). If u is a strong solution on [0, T ) for every T > 0, then is called global strong solution of (2.1).

x ∈ R. If u ∈ C([0, T ), H s (T)) ∩ C 1 ((0, T * ), H s-1 (T)), with s >
If u 0 ∈ H s (T), s > 3 2 , an application of Kato's method [START_REF] Kato | Quasi-linear equations of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations[END_REF] leads to the following local well-posedness result: Theorem 2.1 (See [START_REF] Saha | Blow-Up results for the periodic peakon b-family of equations[END_REF]). For any constant b, given u 0 ∈ H s (T), s > 3 2 , then there exists a maximal time T * = T * ( u 0 H s ) > 0 and a unique strong solution u to (2.1), such that

(2.3) u = u(•, u 0 ) ∈ C([0, T * ), H s (T)) ∩ C 1 ([0, T * ), H s-1 (T)).
Moreover, the solution depends continuously on the initial data, i.e. the mapping

u 0 → u(•, u 0 ) : H s (T ) → C([0, T * ); H s (T )) ∩ C 1 ([0, T * ); H s-1 (T )) is continuous.
Remark 2.2. The maximal lifespan of the solution in Theorem 2.1 may be chosen independently of s in the following sense: [START_REF] Christov | On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis-Procesi equation[END_REF][START_REF] Saha | Blow-Up results for the periodic peakon b-family of equations[END_REF]. Moreover, by using the Theorem 2.1 and energy estimates, the following precise blow-up scenario of the solution to (2.1) can be obtained.

If u = u(•, u 0 ) ∈ C([0, T * ), H s (T))∩C 1 ([0, T * ), H s-1 (T)) to (2.1) and u 0 ∈ H s (T) for some s = s, s > 3 2 , then u = u(•, u 0 ) ∈ C([0, T * ), H s (T)) ∩ C 1 ([0, T * ), H s -1 (T)) and with same T * . In particular, if u 0 ∈ ∩ s≥0 H s , then u ∈ C([0, T * ), H ∞ (T)). See
Theorem 2.3 (See [9, 36]). Assume b ∈ R and u 0 ∈ H s (T), s > 3 2 . Then blow up of the strong solution u = u(•, u 0 ) in finite time occurs if only if (2.4) lim t→T * inf{(2b -1) inf x∈R [u x (t, x)]} = -∞
Before presenting our contribution, we will review a few known blow-up theorems with respect to (2.1).

Theorem 2.4 (See [START_REF] Saha | Blow-Up results for the periodic peakon b-family of equations[END_REF]). Let 5 3 < b ≤ 3 and

T u 3 x (0) dx < 0. Assume that u 0 ∈ H s (T), s > 3
2 , u 0 ≡ 0, and the corresponding solution u(t) of (2.1) has a zero for any time t ≥ 0. Then, the solution u(t) of the equation (2.1) blows-up finite time.

The next blow-up theorem uses the fact that if u(t, x) is a solution to (2.1) with initial datum u 0 , then -u(t, -x) is also a solution to (2.1) with initial datum -u 0 (-x). Hence due to the uniqueness of the solutions, the solution to (2.1) is odd as soon as the initial datum u 0 (x) is odd.

Theorem 2.5 (See [START_REF] Christov | On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis-Procesi equation[END_REF]). Let 1 < b ≤ 3. Assume that u 0 ∈ H s (T), s > 3 is odd and u 0 (0) < 0. Then the corresponding solution to Eq (2.1) blows-up in finite time.

Notations. For any real β, let us consider the 1-periodic function

(2.5) w(x) = p(x) + β∂ x p(x)
where p is the kernel introduced in (2.1) and ∂ x p denotes the distributional derivative on R, that agrees in this case with the classical i.e pointwise derivative on R \ Z. Notice that the non-negativity condition w ≥ 0 is equivalent to the inequality cosh(1/2) ≥ ±β sinh(1/2), i.e., to the condition

- e + 1 e -1 ≤ β ≤ e + 1 e -1 .
Throughout this section, we will work under the above condition on β. Let us now introduce the following weighted Sobolev space:

E β = {u ∈ L 1 loc (0, 1) : u 2 E β = 1 0 w(x)(u 2 + u 2 x )(x) dx < ∞}, (2.6)
where the derivative is understood in the distributional sense. Notice that E β agrees with the classical Sobolev space H 1 (0, 1) when |β| < e+1 e-1 , as in this case w is bounded and bounded away from 0, and the two norms • E β and • H 1 are equivalent. The situation is different for β = ± e+1 e-1 as E β is strictly larger that H 1 (0, 1) in this case. Indeed, we have

w(x) = 2e (e -1) 2 sinh(x), x ∈ (0, 1), if β = e+1 e-1 ; (2.7)
The elements of E (e+1)/(e-1) , after modification on a set of measure zero, are continuous on (0, 1], but may be unbounded for x → 0 + (for instance, |log(x/2)| 1/3 ∈ E (e+1)/(e-1) ). In the same way,

w(x) = 2e (e -1) 2 sinh(1 -x), x ∈ (0, 1), (if β = -e+1 e-1 ); (2.8)
after modification on a set of measure zero, the elements of E -(e+1)/(e-1) are continuous on [0, 1), but may be unbounded for x → 1 -. Let us now introduce the closed subspace E β,0 of E β defined as the closure of C ∞ c (0, 1) in E β . The elements of E β,0 satisfy the weighted Poincaré inequality below: Lemma 2.6. For all |β| ≤ e+1 e-1 , there exists a constant C > 0 such that

(2.9) ∀v ∈ E β,0 , 1 0 w(x) v 2 (x) dx ≤ C 1 0 w(x) v 2 x (x) dx.
Proof. This demonstration is found in [START_REF] Brandolese | On permanent and breaking waves in hyperelastic rods and rings[END_REF].

We need some notations.

Definition 2.7. For any real constant b = 1 and β, let J(b, β) ≥ -∞, be defined by (2.10)

J(b, β) = inf 1 0 (p + β∂ x p) b 2 u 2 + 3 -b 2 u 2 x dx; u ∈ H 1 (0, 1), u(0) = u(1) = 1 and
(2.11)

β b = inf β > 0 : β 2 + 2 |b -1| J(b, β) - b 2 ≥ 0 .
Notice that a priori 0 ≤ β b ≤ +∞, as the set on the right-hand side could be empty.

Main results.

Let us now formalize the goal of this paper.

Theorem 2.8. Let b ∈]1, 3] be such that β b is finite. Let u 0 ∈ H s (T) be with s > 3 2 and assume that there exists x 0 ∈ T, such that

(2.12) u 0 (x 0 ) < -β b |u 0 (x 0 )| .
then the corresponding solution u of (2.1) in C([0, T * ), H s (T))∩C 1 ([0, T * ), H s-1 (T)) arising from u 0 blows up in finite time. Moreover, the maximal time T * verifies

T * ≤ 2 (b -1) (u 0 (x 0 )) 2 -β 2 b u 2 0 (x 0 ) . (2.13)
Remark 2.9. Notice that the Theorem 2.8 relies on the condition that β b is finite. In section 2, we will prove that one indeed has β b < +∞, as soon as b is outside a very small neighborhood of 1. On the other hand, as we will see later on, for 1 < b < 1.0012 . . ., β b = +∞ and Theorem 2.8 does not apply in such range.

For the proof of Theorem 2.8, we need the following propositions.

Proposition 2.1. We have

(2.14) J(b, β) > -∞ ⇔              |β| ≤ e+1 e-1 , b ≤ 3, b 3-b > -1 C β , where C β > 0 is the best Poincaré constant in inequality (2.9).
Proof. Putting u = v + 1 and observing that 1 0 w(x) dx = 1, we see that

(2.15) J(b, β) = b 2 + inf{T (v) : v ∈ H 1 0 (0, 1)},
where

(2.16) T (v) = 1 0 w(x) b 2 (v 2 + 2v) + 3 -b 2 v 2 x (x) dx.
Assume that J(b, β) > -∞. In order to show |β| ≤ e+1 e-1 , we refer to the proof of proposition 3.3. in [START_REF] Brandolese | On permanent and breaking waves in hyperelastic rods and rings[END_REF]. In order to prove b ≤ 3, we consider |β| ≤ e+1 e-1 and

(2.17)

u n (x) = 1 + 1 2 sin(n2πx) ⇒ u n (x) = nπ cos(n2πx).
For each n ∈ N u n ∈ H 1 (0, 1), u n (1) = u n (0) = 1. Thus there is a constant c 1 > 0 independent of n, such that

∀n ∈ N 0 ≤ b 2 1 0 w(x)u 2 n (x) dx ≤ c 1 ,
and 3 -b 2 1 0 w(x)(u n ) 2 (x) dx → -∞,
because b > 3 and then J(b, β) = -∞. In order to prove the third inequality, we only have to treat the case b < 0. Applying the inequality (2.18)

1 0 w(x) b 2 (n 2 v 2 + 2nv) + 3 -b 2 n 2 v 2 x (x) dx ≥ J(b, β) - b 2 ,
valid for all v ∈ H 1 0 (0, 1) and all n ∈ N and letting n → ∞, we get

1 0 w(x) b 2 v 2 + 3 -b 2 v 2 x (x) dx ≥ 0.
We deduce:

1 0 w(x)v 2 (x) dx ≤ - 3 -b b 1 0 w(x)v 2 x (x) dx.
Then we get b 3-b ≥ -1 C β . But the equality case b 3-b = -1 C β can be excluded, as otherwise we could find a sequence v n such that ((b/2)

1 0 ωv 2 n )/((3 -b) 1 0 ω(v n ) 2
x ) converges to 1 and such that bωv n → -∞: for such a sequence we have

T (v n ) ∼ 1 0 bωv n → -∞, contradicting the assumption J(b, β) > -∞.
Conversely, assume that |β| ≤ e+1 e-1 . By the weighted Poincairé inequality (2.9), we can consider an equivalent norm in E β,0 :

(2.19) v E β,0 = 1 0 w(x)v x (x) dx. Since b 3-b > -1 C β , the symmetric bilinear form (2.20) B(u, v) = 1 0 w(x) b 2 uv + 3 -b 2 u x v x (x) dx,
is coercive on the Hilbert space E β,0 . Applying the Lax-Milgram theorem yields the existence and uniqueness of a minimizer v ∈ E β,0 for the functional T . But H 1 0 (0, 1) ⊂ E β,0 , so in particular, we get J(b, β) > -∞. Moreover, if |β| < e+1 e-1 , then recalling E β,0 = H 1 0 (0, 1) we see that J(b, β) is in fact a minimun, achieved at û = 1 + v ∈ H 1 (0, 1).

The next lemma provides some useful information about J(b, β).

Lemma 2.10. The function (b, β) → J(b, β) ∈ R ∪ {-∞} defined for all (b, β) ∈ R 2 is concave with respect to each one of its variables, and is even with respect to the variable

β. Also for all b ∈ R and |β| ≤ e+1 e-1 , -∞ ≤ J(b, e+1 e-1 ) ≤ J(b, β) ≤ J(b, 0) ≤ b 2 .
Proof. The proof is similar to that of the proposition 3.4. in [START_REF] Brandolese | On permanent and breaking waves in hyperelastic rods and rings[END_REF] The next lemma motivates the introduction of quantity the J(b, β) in relation with the b-family of equations. Proposition 2.2. Let (α, β) ∈ R 2 and u ∈ H 1 (T), we get

∀x ∈ T, (p + β∂ x p) * b 2 u 2 + 3 -b 2 u 2 x ) (x) ≥ J(b, β) u 2 (x).
Proof. Let α = α(b, β) be some constant. Because of the invariance under translation, we get that the inequality

(2.21) (p + β∂ x p) * b 2 u 2 + 3 -b 2 u 2 x ) (x) ≥ α u 2 (x),
holds true for all u ∈ H 1 (T) and all x ∈ T if and only if the inequality

(2.22) (p + β∂ x p) * b 2 u 2 + 3 -b 2 u 2 x ) (1) ≥ α u 2 (1),
holds true for all u ∈ H 1 (T). But on the interval ]0, 1[, (p + β∂ x p)(1 -x) = (p -β∂ x p)(x).

Then we get (2.23)

(p + β∂ x p) * b 2 u 2 + 3 -b 2 u 2 x ) (1) = 1 0 (p -β∂ x p) b 2 u 2 + 3 -b 2 u 2 x (x) dx.
Normalizing to obtain u(1) = 1, we get that the best constant α in inequality (2.21) satisfies α = J(b, -β) = J(b, β).

The next proposition provides a first lower bound estimate of J(b, β),

when b ∈ [-1, 3]. Proposition 2.3. Let -1 ≤ b ≤ 3 and |β| ≤ e+1 e-1 .
Then, if u ∈ H 1 (0, 1) such that u(1) = u(0), we get

(p ± β∂ x p) * b 2 u 2 + 3 -b 2 u 2 x ≥        δ b u 2 , if |β| ≤ 1 δ b 2 [(e + 1) -|β| (e -1)]u 2 , if 1 ≤ |β| ≤ e+1 e-1
, where

(2.24) δ b = √ 3 -b 4 3(1 + b) - √ 3 -b .
Remark 2.11. Notice that δ b ≥ 0 if and only if for 0 ≤ b ≤ 3.

Proof. It is sufficient to consider the case 0 ≤ β ≤ e+1 e-1 . We make the convolution estimates for (p + β∂ x p), the convolution estimates for (p -β∂ x p) being similar. First observe that:

(2.25) ∀x ∈ R p(x) = e x-1 2 -[x] 4 sinh 1 2 + e -x+ 1 2 +[x] 4 sinh 1 2 = p 1 (x) + p 2 (x).
We start with the estimate of p 1 * (a 2 u 2 + u 2 x )(1), with a ∈ R to be determined later. We get

p 1 * (a 2 u 2 + u 2 x )(1) = 1 4 sinh( 1 2 ) 1 0 e 1 2 -ξ (a 2 u 2 + u 2 x )(ξ) dξ ≥ -a 4 sinh( 1 2 ) 1 0 e 1 2 -ξ (2uu x )(ξ) dξ = -a 4 sinh( 1 2 ) (e -1 2 -e 1
2 )u 2 (1) -1 4 sinh( 12 )

1 0 e 1 2 -ξ au 2 dξ = a 2 u 2 (1) -p 1 * (au 2 )(1).
Hence

p 1 * ((a 2 + a)u 2 + u 2 x )(1) ≥ a 2 u 2 (1),
and because of the invariance under translations, we get (2.26)

p 1 * ((a 2 + a)u 2 + u 2 x ) ≥ a 2 u 2 .
Similarily:

p 2 * (a 2 u 2 + u 2 x )(1) = 1 4 sinh( 1 2 ) 1 0 e ξ-1 2 (a 2 u 2 + u 2 x )(ξ) dξ ≥ a 4 sinh( 1 2 ) 1 0 e ξ-1 2 (2uu x )(ξ) dξ = a 4 sinh( 1 2 ) (e 1 2 -e - 1
2 )u 2 (1) -1 4 sinh( 12 )

1 0 e ξ-1 2 au 2 dξ = a 2 u 2 (1) -p 2 * (au 2 )(1).
Hence, again using the invariance under translations, we get (2.27)

p 2 * ((a 2 + a)u 2 + u 2 x ) ≥ a 2 u 2 .
Choose a such that a 2 + a = b 3-b . This is indeed possible if -1 ≤ b < 3 (if b = 3, the proposition is trivial and there is nothing to prove). We get:

p 1 * b 2 u 2 + 3 -b 2 u 2 x ≥ δ b 2 u 2 , (2.28) p 2 * b 2 u 2 + 3 -b 2 u 2 x ≥ δ b 2 u 2 . (2.29)
Now, from the identity p = p 1 + p 2 and ∂ x p = p 1 -p 2 , that holds both in the distributional and in the a.e. pointwise sense, we get (2.30)

p + β∂ x p = (1 + β)p 1 + (1 -β)p 2 .
If 0 ≤ β ≤ 1, then from (2.28) and (2.30), we deduce

(2.31) (p + β∂ x p) * b 2 u 2 + 3 -b 2 u 2 x ≥ [(1 + β) + (1 -β)] δ b 2 u 2 = δ b u 2 .
We proved as follows. From We deduce, using (2.28): Proof of Theorem 2.8. By the well-posedness result in H s (T), with s > 3/2, the density of H 3 (T) in H s (T) and a simple approximation argument, we only need to prove Theorem 2.8 assuming u 0 ∈ H 3 (T). We thus obtain a unique solution of (2.1), defined in some nontrivial interval [0, T [, and such that u

∀ 1 ≤ β ≤ e+1 e-1 , (p + β∂ x p) b 2 u 2 + 3-b 2 u 2 x ≥ [(e + 1) -β(e -1)] δ b 2 u 2 . (2.
∈ C([0, T [, H 3 (T)) ∩ C 1 ([0, T [, H 2 (T))
. The starting point is the analysis of the flow map q(t, x) of (2.1)

(2.35) q t (t, x) = u(t, q(t, x)) x ∈ T, t ∈ [0, T * ), q(0, x) = x, x ∈ T. As u ∈ C 1 ([0, T [, H 2 (T))
, we can see that u and u x are continuous on [0, T [×T and x → u(t, x) is Lipschitz, uniformly with respect to t in any compact time interval in [0, T [. Then the flow map q(t, x) is well defined by (2.35) in the time interval [0, T [ and ). Differentiating (2.1) with respect to the x variable and applying the identity ∂ 2 x p * f = p * f -f , we get:

q ∈ C 1 ([0, T [×R, R
u tx + uu xx = b 2 u 2 - b -1 2 u 2 x -p * b 2 u 2 + 3 -b 2 u 2 x .
Let us introduce the two C 1 functions of the time variable depending on β. The constant β, will be chosen later on

f (t) = (-u x + βu) (t, q(t, x 0 )) and g(t) = -(u x + βu) (t, q(t, x 0 )).
Using (2.35) and differentiating with respect to t, we get

df dt (t) = [(-u tx -uu xx ) + β(u t + uu x )](t, q(t, x 0 )) = - b 2 u 2 + b -1 2 u 2 x + (p -β∂ x p) * b 2 u 2 + 3 -b 2 u 2 x (t, q(t, x 0 )),
and

dg dt (t) = [(-u tx -uu xx ) -β(u t + uu x )](t, q(t, x 0 )) = - b 2 u 2 + b -1 2 u 2 x + (p + β∂ x p) * b 2 u 2 + 3 -b 2 u 2 x (t, q(t, x 0 )).
Let us first consider b ∈]1, 3]. Recall that we work under the condition β b < ∞. By the definition of β b (2.11) we deduce that there exists β ≥ 0 such that

β 2 ≥ 2 b -1 b 2 -J(b, β) . (2.36)
Applying the convolution estimate of (2.2) and the fact that J(b, β) = J(b, -β), we get

df dt (t) ≥ b -1 2 u 2 x + J(b, -β) - b 2 u 2 (t, q(t, x 0 )) ≥ b -1 2 (u 2 x -β 2 u 2 ) (t, q(t, x 0 )) = b -1 2 [f (t)g(t)]
In the same way,

dg dt (t) ≥ b -1 2 u 2 x + J(b, β) - b 2 u 2 (t, q(t, x 0 )) ≥ b -1 2 (u 2 x -β 2 u 2 ) (t, q(t, x 0 )) = b -1 2 [f (t)g(t)].
The assumption u 0 (x 0 ) < -β b |u 0 (x 0 )| guarantees that we may choose β satisfying (2.36) with β -β b > 0 small enough so that

u 0 (x 0 ) < -β |u 0 (x 0 )| .
For such a choice of β we have f (0) > 0 and g(0) > 0.

We now make use of the following result:

Lemma 2.13 (See [START_REF] Brandolese | On permanent and breaking waves in hyperelastic rods and rings[END_REF]). Let 0 < T * ≤ ∞ and f, g ∈ C 1 ([0, T * [, R) be such that, for some constant c > 0 and all t ∈ [0, T * [,

df dt (t) ≥ cf (t)g(t) dg dt (t) ≥ cf (t)g(t).
If f (0) > 0 and g(0) > 0, then

T * ≤ 1 c f (0)g(0)
.

The blow-up of u then follows immediately from our previous estimates applying the above lemma.

estimates of β b

Theorem 2.8 is meaningful only if b ∈ (1, 3] is such that β b < ∞. We recall here that β b is defined by Eq. (2.11):

β b = inf β > 0 : β 2 + 2 |b -1| J(b, β) - b 2 ≥ 0 .
Next, we propose three lower bound estimates for the convolution term

(p ± β∂ x p) * b 2 u 2 + 3 -b 2 u 2 x ,
or -what is equivalent, owing to Proposition 2.2-three lower bound estimates for J(b, β)). Such estimates will allow us to determinate sufficient conditions on b ∈ [START_REF] Benjamin | Internal waves of permanent form in fluids of great depth[END_REF][START_REF] Brandolese | Blowup issues for a class of nonlinear dispersive wave equations[END_REF] in order to β b to be finite and will provide upper bounds for β b . Estimate 1 and Estimate 2 below are presented mainly for pedagogical purposes, as they are self-contained. But these two estimates will be later on improved by Estimate 3, which is more technical and deeply relies on a few involved computations made in [START_REF] Brandolese | On permanent and breaking waves in hyperelastic rods and rings[END_REF]. We point out however that Estimate 1 suffices to claim that Theorem 2.8 is not vacuous.

3.1. Estimate 1. Let 0 ≤ β ≤ e+1 e-1 and 1 < b ≤ 3. We start considering the obvious estimate

(p ± β∂ x p) * b 2 u 2 + 3 -b 2 u 2 x ≥ 0.
Thanks to definition (2.11), we see that a sufficient condition on b which entails β b < ∞, is the existence of a constant β satisfying

(3.1) b b -1 ≤ β ≤ e + 1 e -1 .
This holds when b ≥ (e+1) 2 4e ≡ α. In this case, the corresponding bound for β b is

(3.2) β b ≤ b b -1 < +∞, for (e+1) 2 4e ≤ b ≤ 3.
(See Figure 1a).

∃ 0 ≤ β ≤ 1 such that β 2 + 2 b -1 δ b - b 2 ≥ 0, (3.3) or ∃ 1 ≤ β ≤ e + 1 e -1 such that β 2 + 2 b -1 [(e + 1) -β(e -1)] δ b 2 - b 2 ≥ 0, (3.4)
where δ b is as (2.24). The study of the function b →

J(b, β) =      3-b 2 I b 3-b , β , if b = 3 3 2 inf 1 0 w(x) u 2 dx; u ∈ H 1 (0, 1), u(0) = u(1) = 1 , if b = 3.
where I(α, β) is as in [START_REF] Brandolese | On permanent and breaking waves in hyperelastic rods and rings[END_REF]. If b = 3, borrowing the computation made in [START_REF] Brandolese | On permanent and breaking waves in hyperelastic rods and rings[END_REF], we get

J b, e + 1 e -1 = 3 -b 2 I b 3 -b , e + 1 e -1 = 3 -b 4e (e + 1) 2 P υ(b) P υ(b) (cosh 1)
where

υ(b) = - 1 2 + 1 2 • 1 + 4 • b 3 -b ∈ {z ∈ C : (z) ≥ 0}.
and P υ(b) is Legendre function of the first kind, of the degree υ(b), arising when solving the Euler-Lagrange equation associated with the minimization problem of I(α, e+1 e-1 ). The reason for considering here the limit case β = e+1 e-1 is twofold: on one hand, in this case the weight function has a simpler expression, namely w(x) becomes in this case w(x) = p(x) + e+1 e-1 ∂ x p(x) = 2e (e-1) 2 sinh x, x ∈ (0, 1); this allow to reduce the Euler-Lagrange equation to a linear second order ordinary differential equation of Legendre type. See [START_REF] Brandolese | On permanent and breaking waves in hyperelastic rods and rings[END_REF] for more details. On the other hand, by Lemma 2.10, we have J(b, β) ≥ J b, e+1 e-1 for all 0 ≤ β ≤ e+1 e-1 . Now, for 0 ≤ β ≤ e+1 e-1 , we have x (x) dx.

β 2 + 2 b -1 J(b, β) - b 2 ≥ β 2 + 2 b -1 3 -b 4e (e + 1) 2 P υ(b) P υ(b) (cosh 1) - b 2 . ( 3
On the other hand, multiplying (3.9) by v and integrating with respect to the spatial variable, we get The above solution v of the minimization problem, depending on the parameters b and β, cannot be computed analytically, but it it can be computed numerically with the standard numerical schemes for linear ODEs, with an arbitrary good precision. This allow to compute numerically the above function J(b, β). This being done, a simple algorithm allows to compute numerically the quantity β b (with an arbirary good precision). Such numerical computations illustrate that in fact β b < +∞ for 1.0012 . . . ≤ b ≤ 3, which is (slightly !) better than the range 1.012 ≤ b ≤ 3 obtained via Estimate 3. The actual value of β b is actually slightly smaller than its upper bound computed in (3.7). See Figure 3 and 4. We summarize in the last picture all our previous estimates and numerical approximate of β b . In this plot we can see the different estimates that we have worked out (green curve first estimate, blue curve second estimate and red curve third estimates), as well as the numerical approach of β b .

(2. 32 ) p 2 1 p

 3221 (x) ≤ e p 1 (x), ∀x ∈ (0, 1), we get, for 1 ≤ β ≤ e+1 e-+ β∂ x p = (1 + β)p 1 -(β -1)p 2 , ≥ [(e + 1) -β(e -1)]p 1 . (2.33)

34 )

 34 Remark 2.12. If -1 ≤ b ≤ 3, then it follows by the preceding proposition that |β| ≤ 1, then J(b, β) ≥ δ b , and if 1 ≤ |β| ≤ e+1 e-1 then J(b, β) ≥ δ b 2 [(e + 1) -|β| (e -1)].

2 b- 1 b 2 - 3 . 3 .

 1233 δ b in the interval[START_REF] Benjamin | Internal waves of permanent form in fluids of great depth[END_REF][START_REF] Brandolese | Blowup issues for a class of nonlinear dispersive wave equations[END_REF] however reveals that condition(3.3) is satisfied only for b = 2. We have δ 2 = 1 2 and so β = 1. The corresponding estimate for β 2 is then β 2 ≤ 1. This situation corresponds to the Camassa-Holm equation. We thus recover the result in[START_REF] Brandolese | Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF]. (See Figure1b.) On the other hand, solving (3.4) is possible if and only if the largest real zero φ(b) of the quadratic polynomial β → P b (β) = β 2 + β δ b e+1 b-1 + δ b e+1 b-1 -b b-1 is inside the interval [1, e+1 e-1 ]. A simple computation shows that this is indeed the case when α ≤ b ≤ 3. Here α = (e+1) 2 4e is the same as in Estimate 1. For α ≤ b ≤ 3, now we get the bound (3.5) β b ≤ φ(b) < +∞, that considerably improves our earlier estimate (3.2). See Figure 1b Estimate 3. This part relies on the properties of J(b, β) which are described in Lemma 2.10 and the computations made in [4] Let I(α, β) as in [4, Section 2]. For b ∈ (1, 3], and |β| ≤ e+1 e-1 , the relation between I and J is the following:

( a )

 a The plot of the function b → b b-1 , providing the bound (3.2). The upper-bound estimate of β b given by Eq.(3.2), showing that Theorem 2.8 applies for b ∈ [α, 3], where α = (e+1) 2 4e (blue and gray region). (b) The function b → φ(b), providing the bound (3.5). The upper-bound estimates of β b given by Eq.(3.5) and the Theorem 2.8 are valid inside the interval [α, 3] (grey region).

Figure 1 .

 1 Figure 1. First and Second estimate of β b .

. 6 )- 1 b 2 - 3

 6123 Computing the Legendre function shows that the right hand-side of the above expression is nonnegative when γ ≤ b ≤ 3, with γ ≈ 1.012. See Figure 2. Therefore, in the range b ∈ [γ, 3] we have β b < +∞ (3.7) β b ≤ 2 b -b 4e (e + 1) 2 P υ(b) P υ(b) (cosh 1) , for γ ≤ b ≤ 3, and Theorem 2.8 applies in such range.

Figure 2 .- 1 b 2 - 3 . 4 .

 21234 Figure 2. The function b →

1 0( 3 1 0( 3 1 0parts 1 0( 3 - 1 0( 3 - 1 0[

 1313113131 -b)wv xx v dx + -b)w x vv x dx -bw(v 2 + v) dx = 0.Integrating by b)w x vv x dx, and using that v(0) = v(1) = 0, we get w dx = 1 and (3 -b)(wv xx + w x v x ) = bw(v + 1), we getJ(b, β) wv x ] x dx = 3 -b 2 (wv x )(1 -) -(wv x )(0 + ) .

Figure 3 .

 3 Figure 3. The plot of the function b → β b . This numerical approach of β b , allows us to say: if 3 ≥ b ≥ α 0 ≈ 1.0012, then the Theorem 2.8 is valid (gray region).

Figure 4 .

 4 Figure 4. In this plot we can see the different estimates that we have worked out (green curve first estimate, blue curve second estimate and red curve third estimates), as well as the numerical approach of β b .

  3.2. Estimate 2. Proposition 2.3 provides a better sufficient condition ensuring that β b < +∞. Namely:

Acknowledgement

Also, I would like to thank Escuela Politécnica Nacional del Ecuador (EPN) where most of this paper was written. I appreciate the comfortable and relaxing place it is. This work is supported by the Secretaría de Educación Superior, Ciencia, Tecnología e Innovación del Ecuador (SENESCYT).


Guido Ala

Maria Luisa

Di Silvestre

Elisa Francomano

Adele Tortorici

An Advanced Numerical Model in Solving Thin-Wire Integral Equations by Using Semi-Orthogonal Compactly Supported Spline Wavelets

In this paper, the semi-orthogonal compactly supported spline wavelets are used as basis functions for the efficient solution of the thin-wire electric field integral equation (EFIE) in frequency domain. The method of moments (MoM) is used via the Galerkin procedure. Conventional MoM directly applied to the EFIE, leads to dense matrix which often becomes computationally intractable when large-scale problems are approached. To overcome these difficulties, wavelets can be used as a basis set so obtaining the generation of a sparse matrix; this is due to the local supports and the vanishing moments properties of the wavelets. In the paper, this technique is applied to analyze electromagnetic transients in a lightning protection systems schematized as a thin-wire structure. The study is carried out in frequency domain; a discrete fast Fourier transform algorithm can be used to compute time profiles of the electromagnetic interesting quantities. The unknown longitudinal currents are expressed by using multiscale wavelet expansions. Thus, the thin-wire EFIE is converted into a matrix equation by the Galerkin method. Results for linear spline wavelets along with their comparison with conventional MoM that uses triangular basis functions and the point matching procedure are presented, for two case studies. Good agreement has been reached with a strong reduction of the computational complexity.

I. INTRODUCTION

A NALYSIS of electromagnetic (EM) transients can be car- ried out by employing a field approach in frequency domain, based on an appropriate integral equation. This approach is a powerful method for the analysis of EM antennas and scatterers. In particular, in the case of a lightning discharge that strikes a lightning protection system (LPS), thin-wire integral equations can be used. In fact, the whole structure can be simulated by means of a set of thin cylindrical conductors in which the lightning current flow. This current distribution can determine significant electromagnetic interference (EMI) with electric and electronic apparatus placed inside the structure to be protected; dangerous conditions for person can also arise. Moreover, parasitic EM coupling phenomena may cause: abnormal of the earth electrode. Dangerous transient step and touch voltages can arise on the ground surface due to transient currents injected into the soil by the earth electrode. The EM characterization of the LPS environment can be carried out through the realistic evaluation of the lightning current flowing in the various parts of the LPS. Then, from this knowledge, it is possible to correctly estimate electric and magnetic fields and their effects at any point inside the protected volume.

In order to study this electromagnetic compatibility (EMC) problem, integral equation method seems to be very appropriate [START_REF] Harrington | Field Computation by Moment Methods[END_REF]- [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF]. For small and medium scale problems, the discretization of the integral equation by the method of moments (MoM), enables to obtain efficient numerical solutions. On the other hand, for large-scale problems the numerical algorithms resolving dense matrix equations have heavy computational cost. In fact, in the conventional MoM the resolving matrix is dense due to the integral operator; thus the solution of such a system of linear equations needs a large memory requirement and a high computational time. To overcome these difficulties, the use of wavelet theory has been recently proposed in technical literature [START_REF] Goswami | On solving first-kind integral equations using wavelets on a bounded interval[END_REF]- [START_REF] Xiang | An effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations[END_REF].

Wavelets theory is a relatively new and an emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representations and segmentations, time-frequency analysis and fast algorithms for easy implementation [START_REF] Chui | Wavelets: A Mathematical Tool for Signal Analysis[END_REF]. Recently, the use of wavelets and wavelets-like basis functions in computational electromagnetics has received considerable attention whether for time-domain differential algorithms [START_REF] Zhu | Multiresolution time-domain analysis of plane-wave scattering from general three-dimensional surface and subsurface dielectrics targets[END_REF]- [START_REF]Scattering analysis by the multiresolution time-domain method using compactly supported wavelet systems[END_REF], and for the numerical analysis of integral equations [START_REF] Goswami | On solving first-kind integral equations using wavelets on a bounded interval[END_REF], [START_REF] Alpert | Wavelet-like bases for the fast solution of second-kind integral equations[END_REF]. In particular, by employing wavelets, the unknowns in the integral equation are expressed as a twofold summation of shifted and dilated forms of properly chosen basis functions. Moreover, the wavelets have localization properties in both the space and frequency domain and used as basis functions generate a strong de-correlation among the expansion coefficients and weaken the coupling effects in the integral equation. However, it has been underlined that, also with wavelets bases the resultant matrix is dense but, due to local supports and vanishing moment properties of wavelets, many of the matrix entries are very small compared to the largest ones. Consequently, these elements can be set to zero with an opportune threshold technique without significantly affecting the solution. signals in telecommunication cables; electric stresses dangerous for the insulation of electric components, disruptive discharges, danger to person, a significant voltage between any two points Thus, by using the multiresolution analysis (MRA) a sparse matrix equation can be obtained from an integral equation.

Most of the technical papers on wavelets in electromagnetics deal with orthogonal wavelets, which usually do not have closed form expressions. In [START_REF] Wang | Application of wavelets on the interval to the analysis of thin-wire antennas and scatterers[END_REF], the thin-wire electric field integral equation (EFIE) is solved by the hybrid wavelet expansion and boundary element method [START_REF] Wang | A hybrid wavelet expansion and boundary element analysis for multiconductor transmission lines in multilayered dielectric media[END_REF], using orthogonal wavelets on the interval [0,1]. The same method has been employed in [START_REF] Wang | Analysis of electromagnetic scattering from conducting bodies of revolution using orthogonal wavelet expansions[END_REF] to study the EM scattering from conducting bodies of revolution by means of the magnetic field integral equation.

In this paper, the semi-orthogonal (SO) compactly supported spline wavelets are used as basis functions for the efficient solution of the thin-wire EFIE in frequency domain. Very few references have been found in technical literature dealing with these kind of wavelets [START_REF] Goswami | On solving first-kind integral equations using wavelets on a bounded interval[END_REF] and their application to thin-wire integral equation. The MoM is used via Galerkin procedure that is, wavelets are employed as testing and weighting functions. This, in order to investigate EM transients as those observed in grounding systems due to switching operations or a lightning stroke, or in a LPS during its operation. In Fig. 1 a typical scenario of an earthing system integrated with a LPS in an electric power substation, is shown. A discrete fast Fourier transform (DFFT) algorithm can be employed to compute time profiles of the electromagnetic interesting quantities. The unknown longitudinal currents in each conductor of the structure are expressed by using multiscale wavelet expansions. Thus, the thin-wire EFIE is converted into a matrix equation by the Galerkin method.

The use of SO compactly supported spline wavelets is justified by their interesting properties [START_REF] Goswami | On solving first-kind integral equations using wavelets on a bounded interval[END_REF]. Among them the following can be explicitly cited: they satisfy all the properties on a bounded interval that are verified by the usual wavelets on the real line, but they do not determine the difficulty related to enforce boundary conditions explicitly, when applying such wavelets to problems in finite bounded domain; unlike most of the continuous orthogonal wavelets, the SO compactly supported spline wavelets have closed form expressions.

Results for linear spline wavelets along with their comparison with conventional MoM that uses piecewise linear basis functions and the point matching procedure are presented, for two case studies. Good agreement has been reached with a strong reduction of the computational complexity, the computer CPU time and the memory requirement.

II. WAVELETS AND MRA

This section is dedicated to briefly present the basic concepts of the wavelet theory [START_REF] Chui | An Introduction to Wavelets[END_REF]- [START_REF] Strang | Wavelets and Filter Banks[END_REF]. A natural framework for wavelet theory is the MRA that characterizes wavelets in a general way. The goal of MRA is to express a function at various levels of details, as the space of square integrable function on .

A multiresolution approximation of is characterized by a set of nested subspaces in [START_REF] Harrington | Field Computation by Moment Methods[END_REF] which satisfy the following properties:

(2)

(3

)
where is the set of integers. For each , exists a function referred to as scaling function such that (4) is a basis for . An approximation of a function at a resolution of , can be defined as the orthogonal projection of into , i.e., [START_REF] Ala | An advanced algorithm for transient analysis of grounding systems by Moments Method[END_REF] where represents the inner product of and . Since is a basis of , but not of , some information is being lost in the projection reported in [START_REF] Ala | An advanced algorithm for transient analysis of grounding systems by Moments Method[END_REF]. This lost information can be interpreted as the details in that are finer in resolution than . It is therefore necessary to describe a function in term of its approximation at the resolution and the remaining orthogonal terms containing local measures of the finer details. This is formally facilitated by the fact that exists an orthogonal complementary subspace of in : [START_REF] Xiong | Transient performance of substation grounding systems subjected to lightning and similar surge current[END_REF] as well as a function , referred to as wavelet, such that the set [START_REF] Dawalibi | Transient performance of substation structures and associated grounding systems[END_REF] is a basis for , whilst is a basis for . Therefore, by considering [START_REF] Grcev | Frequency dependent and transient characteristics of substation grounding systems[END_REF] as the orthogonal projection of a function into , from the equation ( 6) it follows that each function , can be written as follows:

(9)

By iterating this decomposition on an arbitrary number of levels, a description of in terms of its approximation at the resolution and its finer details can be written as [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF] The scaling function and the wavelet function satisfy the fundamental two-scale relations [START_REF] Chui | An Introduction to Wavelets[END_REF] (11) The sequences , are filters for the scaling function and for the wavelet function , respectively. In particular, for a good approximation and a data compression the following condition have to be satisfied: [START_REF] Wagner | A study of wavelets for the solution of electromagnetic integral equations[END_REF] where " " is the number of vanishing moments.

Let be the th-order cardinal B-spline function [15, p. 52] with otherwise [START_REF] Wang | Application of wavelets on the interval to the analysis of thin-wire antennas and scatterers[END_REF] and . It is well known that these compactly supported functions generate an MRA [START_REF] Chui | Wavelets: A Mathematical Tool for Signal Analysis[END_REF] with two scale relations [START_REF] Xiang | An effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations[END_REF] Then, the corresponding th-order cardinal SO compactly supported B-wavelet function is given by [15, p. 102] with ( 15)

III. WAVELET AND SCALING FUNCTIONS ON THE INTERVAL

Wavelet and scaling functions can be used to expand the unknowns of the integral equation, selected for modeling the electromagnetic problem. This integral equation can be expressed in a general way as [START_REF] Zhu | Multiresolution time-domain analysis of plane-wave scattering from general three-dimensional surface and subsurface dielectrics targets[END_REF] where is the unknown function, is the known kernel, is the known function, and is the domain of integration.

Wavelet and scaling functions are defined on the entire real line so that they could be outside of the integration domain. This behavior may require an explicit enforcement of the boundary conditions. In order to avoid this occurrence, SO compactly supported spline wavelets, constructed for the bounded interval [0,1], have been taken into account in this paper. These wavelets satisfy all the properties verified by the usual wavelets on the real line [START_REF] Chui | Wavelets: A Mathematical Tool for Signal Analysis[END_REF].

Let and be positive integers and [START_REF] Tentzeris | Multiresolution time-domain (MRTD) adaptive schemes using arbitrary resolutions of wavelets[END_REF] an equally-spaced knots sequence. The functions [START_REF] Dogaru | Application of Haar-wavelet-based multiresolution time-domain schemes to electromagnetic scattering problems[END_REF] and otherwise [START_REF]Scattering analysis by the multiresolution time-domain method using compactly supported wavelet systems[END_REF] are called cardinal B-spline functions of order for the knots sequence , and . For the sake of simplicity, the in- ; for each level the scaling functions of order " " can be defined as follows [START_REF] Goswami | On solving first-kind integral equations using wavelets on a bounded interval[END_REF], [START_REF] Chui | Wavelets: A Mathematical Tool for Signal Analysis[END_REF]: The scaling functions occupy segments and the wavelet functions occupy segments. There-fore the condition , must be satisfied in order to have at least one inner wavelet.

In the following, the scaling and wavelet functions used in the paper, for and , are reported:

(26) (27) (28)
(29)

(30) (31)
(32)

(33) (34)
Thus, the unknown function in ( 16) can be expanded in terms of the selected scaling and wavelet functions as follows:

(35)

By substituting this expression in an integral equation of the type ( 16) but with the domain of integration (36)

and by employing the Galerkin method, the following set of linear system of order is generated:

(37)

where

(38) (39) (40) (41)
where ; ; ; and . In the integrals (38)-(41) much smaller limits than [0,1] are involved in the computations, because of the finite support of the wavelet and scaling functions. The integrals involving wavelet functions are quite small, to the SO property of the wavelets; therefore, the entries with relevant magnitude are in the LL submatrix. In this way, a significant number of entries in the MoM wavelet matrix in (37) can be set to zero using an opportune threshold technique.

It is interesting to underline that in the conventional MoM all the entries of the matrices are comparable in magnitude. As a consequence, no threshold can be used to sparsify the system matrix.

IV. NUMERICAL SCHEME OF EMC PROBLEM

The "thin-wire" approximation [START_REF] Wang | Generalized Moment Methods in Electromagnetics[END_REF] is used in order to schematize a set of metallic straight cylindrical conductors embedded into a homogeneous medium with assigned specific electric parameters. In order to obtain an integral equation in the unknown longitudinal currents distribution, the start point is the wellknown relation that expresses the total electric field as the sum of the incident and scattered components [START_REF] Wang | Generalized Moment Methods in Electromagnetics[END_REF], in the frequency domain. The incident field can take into account, for example, of the influence of the lightning channel. The following statement in a point on the surface of the generic excited wire can be used as a boundary condition:

on ( 42
)
The left-hand side of (42) represents the tangential component of the total electric field, parallel to the axis of the excited wire; is the per-unit-length surface impedance of a cylindrical straight wire; is the longitudinal unknown current on the axis of the excited wire. The scattered field can be expressed by means of the retarded scalar and vector potentials (43)

The longitudinal currents and the surface charges of each wire of the structure, are the source of and , respectively; for a wire, the expression of and , are the following:

(44)
where, in a general rectangular coordinate system (O, X, Y, Z) (Fig. 2), and are the position vectors of the observation and of the source points, respectively; is along the axis of the exciting wire; is the magnetic permeability, is the conductivity, is the complex permittivity of the medium; is a unity vector along the axis of the exciting conductor;

is the longitudinal current on the axis of the exciting wire; is the scalar Green's function for an unbounded region as the wave number (45)

In the second equation of (44), the continuity equation with thin-wire approximation has been taken into account; this equation expresses the divergence of the current density vector as a function of the first time derivative ( , in time-harmonic variation) of the charge volume density. In the local cylindrical coordinate system ( , , of the exciting wire (Fig. 2), can be expressed as the vector sum of the axial and radial components (in this case the azimuthal component does not exist) (46) By introducing (43), (44), and (46) in (42), the following EFIE is obtained [START_REF] Goswami | On solving first-kind integral equations using wavelets on a bounded interval[END_REF]- [START_REF] Wang | Application of wavelets on the interval to the analysis of thin-wire antennas and scatterers[END_REF]: on (47) where schematizes the domain of integration (i.e., the LPS arrangement and the related complex images [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF]); are unit vectors associated with the oriented wires (Fig. 2) and " " is the inner product;

is the longitudinal current associated to the excited wire;

is the longitudinal current on the axis of the generic wire.

The two elementary components of in a point on the surface of the excited wire due to an elemental " " of the exciting wire, can easily be obtained by using the partial derivative with respect to the distance and referring to the local coordinates and (Fig. 2), as follows:

(48)

with " " as the local axial coordinate of the source point on the axis of the exciting wire (Fig. 2). By substituting (45), and (48) in (47), the following integral equation on the surface of the generic wire is obtained:

(49)
In solving (49), the unknown longitudinal current on each conductor of the thin-wire arrangement, has to be expressed by means of a convenient set of basis functions. In order to use the SO spline wavelets on the interval [0,1] as basis functions, the following domain transformations have to be made. Let us introduce a linear mapping between the local axial coordinate of the th wire and the domain, with so reporting each conductor length to the unit value. Moreover, let us introduce another mapping relation among the domains and the domain, with [START_REF] Wang | Application of wavelets on the interval to the analysis of thin-wire antennas and scatterers[END_REF]. The following relations hold:

(50)

with

, where is the number of wires, is the length of the th conductor, are equally spaced knots (Fig. 3) in the resulting domain. The mapping process scheme is reported in Fig. 3 for . The geometric quantities , , , can be also expressed as a function of the variables and ; these last are related to the positions of the source point on the exciting wire and to the field point on the excited wire, respectively. In such a way in (49), the transformed domain is completely introduced. The domain of integration is also reported to the transformed domain . In fact, schematizes the contribution of all the thin conductors to the field point of the selected excited wire. Thus, in conformity with the position of the source point on the axis " " of the exciting wire, the corresponding value in the domain belongs to the related subinterval of [0,1] (so as in Fig. 3).

Moreover, in conformity with the position of the field point on the surface of the excited conductor, the values of the geometric quantities , , , will change in accordance as a function of , also. The information related to the real mutual geometric position of the oriented wires of the structure is taken into account by means of the inner products and , which are computed successively between the exciting and excited conductors, depending on the values of and in the integrals of (49). The following further assumptions are made. For sake of simplicity, the incident field is supposed to be zero. On the other hand, the term in (49) enables us to easily take into account both aerial field and ground current in the case of both direct and indirect stroke on a LPS, as reported in [START_REF] Ala | Scattering effects of large earth electrodes in the EM lightning field[END_REF], [START_REF] Amoruso | Lightning channel's EM effects of large earth electrodes by moments method[END_REF]. Thus, in this paper, a lightning current that directly strikes the LPS structure is considered; this current can be schematized by an ideal current source generator. The Sommerfeld half-space problem is approximated by using modified image theory with mirror complex image sources [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF], [START_REF] Takashima | Calculation of complex field in conducting media[END_REF]; detailed description of such employed schematization is reported in [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF] to which the reader is invited to refer. Based on the previous considerations, (49) is so modified (51) Each of the three summations of the integrals in the left side of (51) extend the domain of integration over the entire bounded interval [0,1]. Equation ( 51) is of the form reported in (36) with different integral kernels. As reported in Section III, the unknown current , is expressed in terms of scaling and wavelet functions as follows:

(52) are chosen in this paper for the applications (see Section V). As reported in [START_REF] Goswami | On solving first-kind integral equations using wavelets on a bounded interval[END_REF], the highest level verify the condition that does not exceed , where is the minimum wavelength associated with the maximum work frequency. Because of their bandpass filter characteristics the wavelets extract successively lower and lower components of the unknown function with increasing values of the level parameter " ", while retains the coarsest approximation of the original function.

The Galerkin method is used to solve the integral equation (51). Thus, the functions of the expansion (52) are used as weighting functions also. Equation ( 51) is first rearranged by handling the integrals in the second an third summations of the left-hand side; these integrals are of the following type: where . In the first term of the right side of (56) the values of the current functions at the ends of the subinterval of [0,1], appear. These current values are those at the ends of the th conductor of the thin-wire structure. In this way, for each node of the real structure, it is possible to force the incident currents to satisfy the appropriate Kirchhoff's current law [START_REF] Ala | High frequency coupling among buried electrodes[END_REF], [START_REF] Ala | An advanced algorithm for transient analysis of grounding systems by Moments Method[END_REF], [START_REF] Ala | The method of moments for electromagnetic transients in grounding systems on distributed memory multiprocessors[END_REF], [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF].

By applying systematically (56) to all the integrals in the second and the third summation of (51), and by using the following identity:

(57) (51) is so modified on (58) Now, it is possible to solve the integral equation by the Galerkin method: by multiplying each side of (58) for the selected wavelet and scaling functions and integrating over the bounded interval [0,1], with respect to the external variable , with , (58) is so modified on

In (59) is the selected weighting function (i.e., scaling or wavelet . Based on the properties described in Section III, each of the integrals over the interval [0,1] is split into several integrals over subintervals depending on the small support of the functions. Moreover, a lot of the integrals become zero due to the SO and vanishing moments properties of the wavelet functions. In fact, the entries with significant magnitude are in the (LL) submatrix (38) which is of order not depending on the decomposition level ( as the order of the used cardinal B-spline functions, and the initial level) (Fig. 4).

Moreover, let be a small number so that , and the unknowns number, a thresholding operation on the matrix elements can be performed by choosing the following :

(60) as a threshold parameter, where are the MoM wavelet matrix entries [see (37)]. So operating, a large number of entries in the MoM wavelet matrix can be set to zero, and a reasonable correct solution is already performed by working with entries of the and submatrices as it will be shown in the next section. In this way, a strong increase of the efficiency, for both speed and memory, is reached. As already underlined, in the conventional MoM, all the entries of the matrices are comparable in magnitude. As a consequence, no threshold can be used to sparsify the system matrix. Moreover, the dimension of the MoM wavelet LL submatrix results independent either from the specific length of each conductor of the structure under consideration and from the wavelength associated to the maximum work frequency. On the contrary, in the conventional MoM a proper segmentation of each thin wire of the structure has to be chosen, depending either from the wire length and from the minimum wavelength in frequency domain.

V. APPLICATIONS AND MODEL VALIDATION Equation (59 is solved for each frequency of the discretized Fourier spectrum of the selected lightning current source; a DFFT algorithm can be then applied in order to obtain time profile of interesting electromagnetic quantities (i.e., currents distribution in the wire structure, electric and magnetic fields in the surrounding medium and on the ground surface [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF]). A double exponential wave form (1.2/50 s) has been considered for the lightning current time profile . The discretized Fourier spectrum of this function contains significant frequency up to few megahertz. In particular, in order to obtain a correct reconstruction by means of a DFFT algorithm, the maximum frequency value has to be set around some tens of megahertz: the value of 10 MHz has been selected so obtaining good results in the time reconstruction [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF]. In this way, a rectangular window function of amplitude in time domain s is used with a sampling time s, and samples. In the frequency domain, samples have to be used in order to satisfy the Nyquist theorem [START_REF] Brigham | The Fast Fourier Transform[END_REF]. Thus, (59) is solved for all the sampled frequency values [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF]. It is clear that, based on the consideration reported in Sections III and IV, the method proposed in this paper determines a strong reduction of the computer CPU time and memory requirement for a fixed problem, with respect to the conventional MoM solution.

In order to validate the proposed model, the following examples application in frequency domain are considered; computation results obtained with the cardinal B-spline functions of order , , and by working with the and submatrices of the MoM wavelet matrix, are compared with the conventional MoM solution that uses a direct approach with piecewise linear basis functions and point matching procedure [START_REF] Ala | High frequency coupling among buried electrodes[END_REF], [START_REF] Ala | An advanced algorithm for transient analysis of grounding systems by Moments Method[END_REF], [START_REF] Ala | The method of moments for electromagnetic transients in grounding systems on distributed memory multiprocessors[END_REF], [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF]. The details related to conventional MoM solution for LPS structures are reported in [START_REF] Ala | A simulation model for electromagnetic transients in lightning protection systems[END_REF].

At first, a single buried conductor in homogeneous medium, ten meters length, has been considered for two significant frequencies of the Fourier spectrum of the current source directly injected into one end (Fig. 5).

Moreover two separated wires, of 10-m length, buried in a homogeneous soil, have been considered. One of the wire is directly injected by a current source generator simulating a direct lightning stroke; the second wire is in the EM field of the exciting one (Fig. 6). Only one significant frequency of the Fourier spectrum of the source has been considered. In the case studies, each wire is partitioned in 50 segments in conventional MoM solution, and all the wires are supposed to be of copper with their radius equal to 5 mm.

As is shown in Figs. 789, a good agreement has been reached. In order to better explain the benefits of the performed computational model, the following quantities have been also considered:

(61) where , , and are the solution, computational time, and number of matrix entries with conventional MoM, respectively, whilst , , and are the solution, computational time, and number of matrix entries involved in the wavelet MoM process, i.e., and entries. entries hold back in the MoM wavelet matrix after having introduced an opportune threshold value : the range ; has been selected for the case studies. In Table I, for the case studies previous reported, the parameters , , and are shown in percentage.

Further improvements are expected with the use of cardinal B-spline functions of order and by increasing the scale level [START_REF] Goswami | On solving first-kind integral equations using wavelets on a bounded interval[END_REF].

VI. CONCLUSION

An advanced numerical model in solving thin-wire EFIE, in the frequency domain, by means of SO compactly supported spline wavelets, has been presented. The MoM is used via the Galerkin procedure. The model is applied to study EM transients in grounding systems or, in a more general way, in a LPS. A DFFT algorithm can be used in order to obtain the time profiles of the EM interesting quantities. By virtue of different mapping processes the thin-wire structure is transformed into the bounded interval [0,1]. In this way the SO spline wavelets defined on the interval [0,1] have been used. A sparse matrix equation is obtained from the thin-wire EFIE by using this technique. Thus operating, the traditional drawback of the integral equation methods, due to the dense matrices generates by the conventional approach, is overcome. Two case studies have been presented to illustrate the merits of the used approach.

Fig. 1 .

 1 Fig. 1. Typical scenario of EM transients in an LPS.

  terior B-spline functions, while the remaining , and are boundary B-spline functions, for the bounded interval . Since the boundary B-spline functions at 0 are symmetric reflections of those at , it is sufficient to construct only the first half functions by simply replacing with . By considering the interval , at any level , the discretization step is , and this generates number of segments in [0,1] with knots sequence (20) Let be the level for which

  of spline functions of order with knots sequence on . The is a proper subspace of the space on the finer knots sequence obtained by inserting a knot between and , , and . The and are bases for the and spaces, respectively. So, the orthogonal complementary subspace of relative to has order and a basis is formed by the functions , i.e., the B-wavelets with knots sequence relative to knots sequence . The relation[START_REF] Chui | Wavelets: A Mathematical Tool for Signal Analysis[END_REF] suggests the following formulations for the th-order SO compactly supported B-wavelet functions[15, p. 149] are boundary wavelets and inner wavelets in the boundary interval . Finally, by considering the level with , the B-wavelet functions in [0,1] can be expressed as follows[15, p. 184]:[START_REF] Strang | Wavelets and Filter Banks[END_REF]

Fig. 2 .

 2 Fig. 2. Geometric references for exciting and excited wires.

Fig. 3 .

 3 Fig. 3. Mapping process scheme among the wires domains and 2 [0; 1] for N = 12.

Fig. 4 .

 4 Fig. 4. MoM wavelet matrix for m = 2 and different values of the decomposition level j ; N as the unknowns number.

Fig. 5 .

 5 Fig.5. Case study 1: single buried conductor in homogeneous medium with a current source generator.

Fig. 6 .

 6 Fig.6. Case study 2: two separated wires buried in homogeneous soil: a current source generator excites one conductor; the second wire is in the EM field of the first one.

Fig. 7 .

 7 Fig. 7. Magnitude current profile for the case study 1 with frequency value 50 Hz. Results obtained with this paper (dotted line) conventional MoM (plain line).

Fig. 8 .

 8 Fig. 8. Magnitude current profile for case study 1 with frequency value 1 MHz. Results obtained with this paper (dotted line) conventional MoM (plain line).

Fig. 9 .

 9 Fig. 9. Magnitude current profile along the axis of the two parallel wires of case study 2, with frequency value 50 Hz. Results obtained with this paper (dotted line); conventional MoM (plain line).

TABLE I PRECISION

 I OF SOLUTION, SPEED OF SOLVER, AND MEMORY REQUIREMENT

FOR REPORTED CASE STUDIES


Marc Eisenstadt
email: m.eisenstadt@open.ac.uk

Jiří Komzák
email: j.komzak@open.ac.uk

Stefano A Cerri
email: cerri@lirmm.fr

Peer Conversations for e-Learning in the Grid

We take the view of 'learning as a conversational process', and argue that this can be extended to include the notions of peer-group interactions among students, tutors, and even artificial agents. This in turn lends itself to an approach to distributed conversation which builds upon modern Instant Messaging tools, and extends such tools to include what we call enhanced presence: a way of monitoring the availability of (and interacting with) fellow students and tutors at a distance. We describe a scenario being developed in the recently-started European Learning Grid Infrastructure 'ELeGI' project, deploying our own BuddySpace tools, and describe the novel contributions that this approach can bring to e-learning.

Conversational framework, e-learning, enhanced presence, grid.

INTRODUCTION: ELEGI

The European ELeGI project (European Learning Grid Infrastructure, www.elegi.org), started in February 2004, aims to provide an integrated framework for e-learning that builds upon practical and theoretical lessons learned over the past ten years, and harnesses the new promise of Grid technologies [START_REF] Allison | Human Learning as a Global Challenge: European Learning Grid Infrastructure[END_REF]. Our role in the ELeGI project is to elaborate, expand, and deliver a model of conversational processes and enhanced presence that we believe is central to the future of such e-learning frameworks.

Three activities have been influential in our work:

• It is apparent that many learning situations involve creative dialogues and conversations, and indeed strong theories of learning are built upon this concept •

The communicative aspect of the Internet is precisely one element that has proved exceptionally popular, as evidenced by the rapid growth of Instant Messaging and Peer-to-peer software •

The ability of web services to find and deploy intelligent agent software offers exciting promise for future intelligent applications

In this paper we provide the beginnings of one small part of our framework, namely the part concerned with enhanced presence, motivate it by reference to earlier influential theories, and illustrate our future directions by means of an evolving usage scenario.

A CONVERSATIONAL FRAMEWORK

One of the most influential theories in European Higher Education of the past 20 years is the Conversational Framework of Laurillard [START_REF] Laurillard | Rethinking University Teaching[END_REF], who argues that learning can be viewed as a series of teacher-learner conversations taking place at multiple levels of abstraction. As summarized in [START_REF] Laurillard | Affordances for Learning in a Non-Linear Narrative Medium[END_REF]:

At the most general level of description, the learning process is characterized as a 'conversation' between teacher and student (see Figure 1), operating on two levels, discursive and interactive, the two levels being linked by the twin processes of adaptation and reflection.

In fact, the framework is more powerful than the simple diagram shows, because multiple styles of interaction are possible, as well as iterations within any type of interaction. As Laurillard et al describe the framework:

Students' work on an interactive resource will take place at the interactive level, where students' adapt their actions in the light of their current conceptual understanding, and discuss and reflect on their practical work in order to develop their conceptual understanding at the discursive level. Similarly, the interactive resource, expressing the 'teacher's experiential environment' is an adaptive response by the teacher who chooses it in the light of discursive level conversations with students, and a sense of what they need to do in order to learn the topic. Again, the teacher's reflection on what the students do during the interaction will drive their further discussions at the discursive level.

FIGURE 1:

The conversational framework for the learning process (from [START_REF] Laurillard | Affordances for Learning in a Non-Linear Narrative Medium[END_REF])

As Britain and Liber argue [3], Laurillard's Conversational Framework serves as an excellent starting point for evaluating modern Virtual Learning Environments (VLE's). They argue that any VLE can be analyzed in terms of how well it supports discourse, whether it is adaptable, and how well it supports interaction and reflection -in other words the four key ingredients of the Conversational Framework (the extent to which an environment is discursive, adaptable, interactive, and reflective). In line with Laurillard's own description of the Conversational Framework, which emphasizes the possible iterations at each step, and multiple layers of abstraction possible, we have adapted Figure 1 by combining the analysis provided by Britain and Liber, and noted the scope for iterations, the involvement of agent software, and peer-to-peer interactions, all of which are presented in Figure 2. The original diagram of Britain and Liber shows the expansion of the four bi-directional arrows of figure 1 to illustrate the nature of the 'reflection' and 'adaptation' steps undertaken by the student and teacher (these are the up/down arrows), and also the articulation and action/feedback steps undertaken by the student and teacher (these are the left/right arrows).

Interactive level

Teacher's experiential environment

Student's conceptual representation reflection/ adaptation

Student's goal-oriented behaviour levels of abstraction); (ii) the 'Peer-to-peer' (P2P) multiple boxes in the lower right hand corner, to highlight the fact that peer group interactions (not only student-student but also teacher-teacher) are in fact fundamental to many learning scenarios, and (iii) the important caveat that any instance of Teacher T or Student S may also be viewed as an agent, and indeed may be either an artificial agent or a human agent, in an arbitrary mix (here labelled 'Teacher/agent = T/A' and 'Student/agent = S/A'. The relationships among the possible T/A and S/A dyads is explored further in [START_REF] Cerri | Dynamic Learning Agents and Enhanced Presence on the Grid[END_REF], whereas in the rest of this paper we concentrate on the peer group interactions.

How does the concept of 'Grid computing' fit into such a model? To answer this question, we draw a parallel between the essence of our approach and the initial reasons for conceiving and developing Grid technologies in the mid-1990's. The original observation was that facing a major increase of computational needs, for instance in the scientific community (for weather modelling and nuclear physics), there was a parallel increase of computational resources (machines and networks), for which many resources were proprietary and therefore not accessible directly nor on-demand. The original question was how to build a new infrastructure (the Grid) that would satisfy the computational needs in an on-demand fashion by exploiting the computational resources in a seamless way, not requiring 'physical possession' of the resource by the potential clients. That simple observation was based upon an optimization approach: previously idle machines would be used by those who would have needs, satisfying both (in fact, a win-win solution). Our 'conversational processes and enhanced presence' approach proposes, in exactly the same manner, to combine latent and explicit learning needs with potentially available teaching/learning resources able to satisfy those needs in a kind of 'human grid' -in this case offer and demand are combined, driven by the demand and enabled by the infrastructure. This is a first approximation to our proposed 'Open Grid Human Service Architecture' in the ELeGI project, where we study conversational processes among humans (mediated by machines) trying to offer human resources to learners in an on-demand (or 'as needed') manner.

ENHANCED PRESENCE AND MESSAGING WITH BUDDYSPACE

The notion of peer-group interactions, highlighted in the lower right corner of Figure 2, is critical to many learning interactions. Johnson and Johnson [START_REF] Johnson | Action research: Cooperative learning in the science classroom[END_REF] and numerous others have described the benefits that accrue from activities such as joint problem solving and related peer interactions that take place in the presence of other students. But in an e-learning situation, what exactly does it mean to be 'in the presence of other students'? The notion of 'presence' must therefore be extended, because the other students, and indeed the teacher or tutor, may not be physically present: instead they are only virtually present. Our belief is that simply being able to see the other person, although obviously useful, is neither necessary nor sufficient to impart a sense of presence: rather it is more of a mental state that depends on knowing the other person is available, even at a distance, and this can be de-coupled from 'visibility'. This mental state is crucial for what we called 'enhanced presence' (see the Step 5 in the scenario).

As we described in an earlier TelEduc03 paper [START_REF] Eisenstadt | Instant messaging + maps = powerful tools for distance learning[END_REF] and in [START_REF] Bachler | Collaboration in the Semantic Grid: a Basis for e-Learning[END_REF], 'enhanced presence' builds upon the ideas of 'peripheral awareness of others' made so popular by Instant Messaging, and extends it to include a variety of 'desktop radar' or 'dashboard' capabilities to facilitate collaborative interactions. Our research in this area has led to the development of BuddySpace, an Instant Messaging environment (based on the Jabber/XMPP protocol) with both client and server functionality extended to enhance presence awareness. Specifically, it introduces automatic roster ('buddy list') construction and intelligent service discovery on the server and the graphical visualization of people and their presence states on an image, geographical or conceptual map (as in Figure 3 later in the paper). This allows for multiple views of collaborative workgroups: the immediacy gives users a snapshot of a virtual organization, which is critical in modern learning organizations: We know from [START_REF] Whitelock | Perfect Presence: What does this mean for the design of virtual learning environments?[END_REF] that presence awareness increases emotional well-being, and from [START_REF] Nardi | Integrating communication and information through ContactMap[END_REF] that users benefit from knowing who else is around via presence and messaging tools. In a virtual classroom setting, the instant message capabilities of BuddySpace naturally provide a "backchannel" to the classroom, for example, conveying URLs of documents discussed or as a non-disrupting communication. For distributed activities involving videoconferencing, the presence of individuals gives an extra indication of co-location (especially if the videoconferencing technology is failing). The back-channel can also be used for meeting control tasks, such as queuing of speakers and voting on issues.

In an e-learning context, BuddySpace leverages the overwhelming power of social cohesiveness that can be brought about by knowledge of the presence and location of others in both real and virtual spaces, in the style argued persuasively by [START_REF] Rheingold | Smart Mobs: The Next Social Revolution[END_REF]. We know also from our own informal observations that this kind of presence awareness is used by students to locate resources, for quick exchange of information and to organize meetings either online or face-to face. Indeed we argue that Enhanced Presence is much more than just 'messaging' and 'maps'. In particular, we aim to provide tools that enable us to express the entire situated context of the learner, which is clearly a lot more than just 'location X' and 'online' or 'offline'. The learner's current state of mind, including goals, plans and intentions, must be understood, as well as the way this connects with ongoing activities and devices accessible to the learner. As these are made explicit, plausible inferences can be drawn about what the leaner wants and needs to know, and this gives us an important 'foot in the door' for addressing the problem of delivering the right knowledge to the right people in the right place at the right time. So far, this notion of 'right knowledge' has been nothing more than a Knowledge Management 'slogan', but our belief is that Enhanced Presence capabilities can make this dream a reality. For instance, an important precondition for 'the right knowledge' consists in having access to someone (a human) who eventually would possess it and would be available for delivering it as a service. Enhanced presence aims at answering exactly this question: 'Who out there somewhere can help me now, or has solved a similar or related problem?' Note that in this model, humans are of course potential service providers too, and therefore any 'service discovery tools' must also be capable of discovering relevant people.

A SCENARIO DEPLOYING ENHANCED PRESENCE FOR E-LEARNING

Below we describe an e-learning scenario, based on actual Open University BuddySpace experiences, involving groups of students enrolled in 3rd-level Spanish language courses at a number of universities across the UK.

They are asked to work in groups of 4 collaboratively to edit, design, and produce a Spanish-language web-based newsletter, containing items of interest to the Spanish-language community. Sub-tasks include deciding what the scope of the content should actually be (e.g. sports, news, culture) -this is a deliberately open-ended assignment.

Students work together to produce the newsletter, but must ultimately produce their own write-up for their own tutor. Key players in the scenario include the students, their tutors, and network administrators at several universities. The scenario comprises the following steps, elaborated in detail further below:

Step 1 (tutor mutual discovery, meet and plan)

Step 2 (recruit students)

Step 3 (set up signup and rosters)

Step 4 (BuddySpace client install and student mutual discovery)

Step 5 (collaborative knowledge construction)

Step 6 (student intra-group brainstorms)

Step 7 (service discovery and trigger services)

Step 8 (mentoring interactions)

Step 9 (project/assignment submission)

Below we expand the nine steps and illustrate the aspects of enhanced presence, conversation, and service discovery that are important for e-learning.

THE EXPANDED SCENARIO

Step 1 (Tutor mutual discovery, meet and plan)

The tutors are part of a UK-wide Spanish Language Universities Group. They 'meet' in a regularly-scheduled multi-user chat-room using BuddySpace, having first located each other through BuddySpace's 'semantic matchmaking' service, which identified them according to their user profiles as each (a) interested in CALL (Computer Assisted Language Learning), (b) experts in several relevant technologies, (c) fluent in or instructors in Spanish. A quick chat/brainstorm one evening in real-time results in a coherent project suggestion for their students.

Important benefits for e-learning:

• Automatically constructed profiles: Each institution/university can automatically fill in some of the fields in personal profile of each employee according to his/her job description and previous experience mentioned in her/his CV. More personal details can be entered by the tutors later.

•

Matchmaking: Work on a common project requires participating tutors to have several skills, but also interests. The matchmaking itself can be done by finding similarities between requirements and tutors' profiles.

•

Chat/brainstorm: During the short discussion the tutor can learn more about experience of others in teaching the particular part of the subject. They also negotiate the best formulation of the project: suggestions -> evaluation -> modifications…

Step 2 (recruit students)

The tutors inform their respective students of the proposed task, and ask students to register for a new BuddySpace group, to grant authorization for their thumbnail photos to be used, and for coarse-grained locations (and possibly other conceptual information) to be made visible to other group members.

Important benefits for e-learning:

• Opt-in: Students need to agree to take part. They also confirm what information can be used to populate their profiles.

•

Zones-of-trust: Access to all information (e.g. in profiles) can be set to different levels of detail for groups or individuals. Thus, for each course/project group, the visible profile can be different.

Step 3 (set up signup and rosters)

The tutors ask the network admin people to set up custom rosters for these groups of BuddySpace users. The servers synchronize with each other, so that students see all remote group members.

Important benefits for e-learning:

• Automatic rosters: Automatic generation of rosters (contact lists) is one of the ways to push customized contextual information to the students. It's an easy way how to learn who the fellow students are.

Step 4 (BuddySpace client install and student mutual discovery)

Students install/launch BuddySpace, inherit their automatically-generated roster group, and fill in a user profile in their client, specifying from among recognized interest areas that have been pulled form the server. They then seek out fellow students and self-form small groups/teams. Important benefits for e-learning:

• Custom profiles: Entering a profile can be thought of not only as 'filling in a database', but more significantly as providing a 'benign agent' with a collection of significant facts about the person. Thus, it can not only be just filling-in a form, but also more generic learning with definitions of new terms. Using machine learning, an agent can deduce even more information from user's behaviour (e.g. on-line patterns) and conversations with others (e.g. topics).

•

Matchmaking: Students can form groups in different ways -looking for fellows with similar interests, living in the same area, previous courses, and so on.

Step 5 (collaborative knowledge construction)

While studying the course materials, a student may be unclear about some principles or terms, or may want to discuss them with other students. The student can discuss this during a scheduled group session, or use the enhanced presence status to find other students who are studying the same materials or working on similar materials (e.g. using a trigger service described further in Step 7) and start an ad-hoc chat with them. The interaction with the others can take two forms: one of the students can take the role of the tutor and explain the principle (while learning to formulate ideas and express him/herself) while others comment and discuss whether that understanding is correct; alternatively, all students may come up with possible explanations and opinions, which help others to see different perspectives or understand the underlying processes and rules. In these cases, we believe that common or collaborative knowledge (understanding) is actually constructed during the chat.

Important benefits for e-learning:

• Enhanced presence: the entire context of the learner includes not only his/her state of mind (e.g. goals, plans and intentions), but also current and ongoing activities and devices accessible to the learner.

•

Ad-hoc chat: Chat can be triggered by the current needs of the student or state-change of some of his/her fellow students. The enhanced presence status helps to find a matching counterpart for this type of chat (somebody who is suitable, willing and ready to chat).

Step 6 (student intra-group brainstorms)

The students have formed into 12 groups of 3 and one group of 4, with tutor permission. Each group now begins the real task, starting to brainstorm about the issues. Most groups do this within BuddySpace multi user chat rooms, though some use telephone conference calls and other existing VLE tools, including our own FlashMeeting and Hexagon; the advantage of using BuddySpace is that it provides a layer of scaffolding, monitoring, and intelligent agent intervention unavailable with other approaches.

Important benefits for e-learning:

• Chat/brainstorming: Again, students can learn more about others -who is more willing or 'suitable' to do each task. Then they negotiate the best way to proceed: suggestions -> evaluation -> modifications -> voting…

•

Agent intervention: dialogue patterns can be detected and used for 'benign intervention' and assistance with service discovery.

Step 7 (service discovery and trigger services)

One group decides to create a blog style newsletter about (say) the forthcoming Spanish elections, using recommended VLE tools, but gets confused due to a lack of understanding of all the political acronyms such as 'PP'. Using BuddySpace they find a foreign language acronym-service, which expands acronyms immediately in a separate chat dialogue. Another valuable service they find is a 'trigger' service that will alert them, via instant (service) message directly to BuddySpace, of 'hot stories' pertinent to their election newsletter. They can bring these up in compact form via an RSS feed within BuddySpace, and launch full stories of interest with one click directly in a browser window.

Important benefits for e-learning:

• Services/resources discovery: Services and resources can be either GRID or Jabber-based, or both. Here we talk about the Jabber ones, although they can use similar description formats (from WSDL to IEEE LOM). The Jabber Publish/Subscribe protocol allows users to browse a tree hierarchy of published nodes and view their properties. The Jabber Service discovery protocol is similar.

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Use of services: Services generally can use their own protocol. It can be anything from natural language to filling of forms or even telephone-like "press 1 for…". The acronym service could provide a simple chat-like interface, answering queries such as "explain FAQ" and possibly allowing the entering of new acronyms.

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Trigger services: Trigger services allow users to define a set of conditions and actions to be performed when the conditions are fulfilled. The conditions can include presence of specific people, state of a device, newly published items (e.g. RSS feed or Pub/Sub) or other defined properties (e.g. location) in certain range (e.g. close to my location).

Triggers can send a message, change their own properties or behaviour, and so on. In the language of agents, a trigger service is an agent observing its environment and acting in specific situations. Again the conditions can be fuzzy or even automatically learned.

Step 8 (mentoring interactions)

During every 4th session one of the tutors 'drops into' the ongoing BuddySpace session, to observe and comment on the way things are going. Tutors also send out their own IM alerts with updates, thoughts, and any stop-press info, as these are deemed to be just-in-time and ephemeral.

Important benefits for e-learning:

• Chat/teaching: Tutors observing the behaviour of students can learn much about skills and problems of students during the task solving. They can immediately provide explanations and feedback -specific mentoring actions in the style of the STROBE model proposed by LIRMM would be embedded in the detailed expansion of this step.

Step 9 (project/assignment submission)

The students complete their newsletters, submit their assignments via a custom form no their respective e-learning sites and flash a BuddySpace alert around to their cohorts about two celebration parties to take place: one virtual and one real, at the geographical centre of their respective locations.

Important benefits for e-learning:

• Publishing: The assignments can be published using Jabber's own mechanism which also allows access permission control, in line with the 'zones of trust' described earlier. It's generally a way to provide information (e.g. assignment, location or profile) to certain set of people (subscribers).

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Trigger services: See above.

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Social chat: Celebrations!!! Learning in a social context could be promoted under this heading.

Although the above steps only depict a 'scenario' for which we are currently developing the suitable services to be delivered throughout the recently-started ELeGI project, all of the steps in the scenario are based on capabilities with which we have extensive experience at The Open University, including pilot studies already undertaken with hundreds of students in the Department of Languages [START_REF] Rossade | Audio-graphic-conferencing and Instant Messaging in language learning[END_REF], and more general derived from our practical experiences delivering distance learning to 200,000 students annually [START_REF] Daniel | Mega-Universities and Knowledge Media: Technology Strategies for Higher Education[END_REF]. We therefore believe it is a plausible route for delivering realistic next-generation services for our students.

CONCLUSIONS

'Conversation' is clearly an important ingredient of the learning process: indeed, even if you disagree with the underlying model put forward by Laurillard and her colleagues, it is still clear that conversations occur at many levels of abstraction during learning! We have argued that in light of the importance of conversation for learning, then modern communication media such as Instant Messaging are a natural and important focus for e-learning. But such media are not normally considered to be supportive of extended tutorial dialogues, and it is therefore important to emphasize that 'enhanced presence' is much more than just 'messaging': it is an environment for representing the mental states of all participants, and a vehicle for alerting peer groups at a distance in ways that facilitate many different styles of interaction.

We have described our motivation for investigating enhanced presence and messaging with tools such as BuddySpace, and presented a scenario which looks at a very coarse-grained view of an entire distance-learning project life-cycle. Indeed, this life-cycle (nine steps) represents a much more ambitious focus than typical shortterm e-learning dialogues. We have argued that there are numerous benefits for the learner, particular in the innovations that can be delivered through the deployment and discovery of services such as matchmaking and trigger alerts. Because our scenario is based on years of experience with hundreds of thousands of distance learning students at The Open University, we believe it is realistic, and look forward to pursuing our vision and delivering enhanced presence services within the ELeGI project.

FIGURE 2 :

 2 FIGURE 2:Extended conversational model, adapted from[START_REF] Britain | A framework for the pedagogical evaluation of virtual learning environments[END_REF]

FIGURE 3 :

 3 FIGURE 3: Automatically-generated map (in this case showing ELeGI users of BuddySpace), to illustrate how existing technology would be deployed in the Scenario. Green (darkest) dots depict currently-online ('available') users.

ACKNOWLEDGEMENTS

This work is supported by the European Community under the Information Society Technologies (IST) programme of the 6th Framework Programme for RTD -project ELeGI, contract IST-002205. This document does not represent the opinion of the European Community, and the European Community is not responsible for any use that might be made of data appearing therein.

We are grateful to our colleagues Martin Dzbor (KMi, Open University) for significant input into the concepts, design, and implementations underlying BuddySpace, and Joost Breuker, Clement Jonquet, and Philippe Lemoisson (LIRMM, Université de Montpellier) for significant input into the concepts, designs, and implementations underlying our work on conversation.