Daniel Richard-Molard

Laurence Lesage-Meessen

B Cahagnier

Effect of water-activity on mold growth and mycotoxin production

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Paul Colonna

Alain Buleon

Christiane Mercier

Hydrolysis of native wheat starch by an alpha-amylase from Bacillus subtilis

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K Latrach
email: khalid.latrach@math.univ-bpclermont.fr

B Lods

SPECTRAL ANALYSIS OF TRANSPORT EQUATIONS WITH BOUNCE-BACK BOUNDARY CONDITIONS

Keywords: Transport operator, bounce-back boudary conditions, transport semigroup, essential spectrum, compactness. AMS SUBJECT CLASSIFICATIONS (2000): 47D06, 47D05, 47N55, 35F05, 82C40

published or not. The documents may come

Spectral analysis of transport equations with bounce-back boundary conditions

Khalid Null Latrach, Bertrand Lods

INTRODUCTION

The spectral theory of transport equations with no-reentry boundary conditions (i.e. with zero incoming flux in the spatial domain) received a lot attention in the last decades (see, for example, the works [START_REF] Greiner | Spectral properties and asymptotic behavior of the linear transport equation[END_REF][START_REF] Jörgens | An aysmptotic expansion in the theory of neutron transport[END_REF][START_REF] Mokhtar-Kharroubi | Time asymptotic behaviour and compactness in neutron transport theory[END_REF][START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF][START_REF] Takac | A spectral mapping theorem for the exponential function in linear transport theory[END_REF][START_REF] Vidav | Existence and uniqueness of nonnegative eigenfunction of the Boltzmann operator[END_REF][START_REF] Vidav | Spectra of perturbed semigroups with applications to transport theory[END_REF][START_REF] Voigt | Positivity in time dependent transport theory[END_REF][START_REF] Voigt | Spectral properties of the neutron transport equation[END_REF][START_REF] Weis | A generalization of the Vidav-Jörgens perturbation theorem for semigroups and its application to transport theory[END_REF] and the references therein). The picture is fairly complete by now and almost optimal results have been obtained in [START_REF] Mokhtar-Kharroubi | Optimal spectral theory of the linear Boltzman equation[END_REF] for bounded spatial domains and in [START_REF] Mokhtar-Kharroubi | Critical spectrum and spectral mapping theorems in transport theory[END_REF] for unbounded domains.

When dealing with reentry boundary conditions (including periodic boundary conditions, specular reflections, diffuse reflections, generalized or mixed type boundary conditions), many progress have been made in the recent years in the understanding of the spectral features of one-dimensional models [START_REF] Chabi | On singular monoenergetic transport equation in slab geometry[END_REF][START_REF] Dehici | Spectral properties and time asymptotic behaviour of linear transport equations in slab geometry[END_REF][START_REF] Latrach | Regularity and time asymptotic behaviour of solutions of transport equations[END_REF][START_REF] Latrach | Weak spectral mapping theorems for C0-groups associated to transport equations in slab geometry[END_REF][START_REF] Latrach | Spectral properties and regularity of solutions to transport equations in slab geometry[END_REF][START_REF] Lods | Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions[END_REF][START_REF] Lods | On the spectrum of mono-energetic absorption operator with Maxwell boundary conditions. A unified treatment[END_REF]. However, to our knowledge, for higher dimensions, only few partial results are available in the literature [START_REF] Chen | Linear transport equations with specular reflection boundary condition[END_REF][START_REF] Frosali | Transport equations with boundary conditions of reverse reflection type[END_REF][START_REF] Makin | Spectrum of a stationary uniform-speed transport equation with reflection-type boundary conditions[END_REF][START_REF] Palczewski | Spectral properties of the spaces nonhomogeneous linearized Boltzmann operator[END_REF], dealing in particular with very peculiar shapes of the spatial domain. Our paper deals with the following two problems concerning multidimensional transport equations with bounce-back (reverse) boundary conditions in convex bounded domains:

(1) The spectral analysis of the streaming operator subjected to bounce-back boundary conditions and the explicit expression of the streaming semigroup. [START_REF] Greiner | Spectral properties and asymptotic behavior of the linear transport equation[END_REF] The compactness of the difference of the (perturbed) transport semigroup and the streaming semigroup.

To be more precise, we are concerned with the following initial-boundary-value problem in L p -spaces (1 p < ∞)

∂ψ ∂t (x, v, t) = -v • ∇ x ψ(x, v, t) -Σ(x, v)ψ(x, v, t) + R N κ(x, v, v ′ )ψ(x, v ′ , t)dv ′
= T ψ(x, v, t) + Kψ(x, v, t), (x, v) ∈ D × R N , t > 0;

(1.1a)

with bounce-back boundary conditions:

ψ |Γ -(x, v, t) = γψ |Γ + (x, -v, t), (x, v) ∈ Γ -, t > 0; (1.1b)
and the initial condition

ψ(x, v, 0) = ψ 0 (x, v) ∈ L p (D × R N ). (1.1c)
Here D is a smooth convex open subset of R N (N 1), γ is a real constant belonging to (0, 1) and Γ ∓ represent the incoming and outgoing parts of the boundary of the phase space (see Section 2 for details). The collision frequency Σ(•, •) ∈ L ∞ (D ×R N ) is a non-negative function. The scattering kernel κ(•, •, •) is nonnegative and defines the linear operator K called the collision operator which is assumed to be bounded on

L p (D × R N ) (1 p < +∞). The operator T appearing in (1.1a) is called the streaming operator while T + K denotes the (full) transport operator. It is well-known [25] that, since γ < 1, T generates a C 0 -semigroup of contractions (U (t)) t 0 in L p (D × R N ) (streaming semigroup) and, since K is bounded, T + K is also the generator of a C 0 -semigroup (V (t)) t 0 in L p (D × R N ) (transport semigroup).
Our work is displayed into two parts, referring to the above points (1) and (2). First, we present a fine description of the spectrum σ(T ) of the streaming operator in L p (D × R N ), 1 p < ∞. Moreover, we derive the explicit expression of the streaming semigroup (U (t)) t 0 for the particular case of a space-homogenous collision frequency. Second, we prove that the difference

R 1 (t) = V (t) -U (t) is compact in L p (D × R N ) (1 < p < ∞)
for any t 0 under natural assumptions on the scattering operator K. The interest of such a compactness result lies in the fact that it implies that the streaming semigroup and the transport semigroup possess the same essential spectrum (see [START_REF] Schechter | Principles of Functional Analysis[END_REF] for a precise definition). In particular, their essential types coincide. This shows that the part of the spectrum of the transport semigroup outside the spectral disc of the streaming semigroup consists of, at most, eigenvalues with finite algebraic mutiplicities. Assuming the existence of such eigenvalues, the transport semigroup can be decomposed into two parts: the first containing the time development of finitely many eigenmodes, the second being of faster decay.

Although the well-posedness of the problem (1.1) is a known fact [START_REF] Arlotti | Boundary conditions for streaming operator in a bounded convex domain[END_REF][START_REF] Beals | Abstract time-dependent transport equations[END_REF][START_REF] Frosali | Transport equations with boundary conditions of reverse reflection type[END_REF][START_REF] Lods | Semigroup generation properties of streaming operators with non-contractive boundary conditions[END_REF], the description of the spectrum of the streaming operator and the analytic expression of its semigroup seem to be new. Let us also notice that besides the interesting consequences of the compactness of R 1 (t) on the behavior for large times of the solution of the problem (1.1), it is an interesting result in itself. Actually, it is the first time that the compactness of the first order remainder term of the Dyson-Phillips expansion of the transport operator with reentry boundary conditions is discussed in higher dimensions. For the one-dimensional case we refer to the work [START_REF] Lods | Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions[END_REF] while the compactness of R 1 (t) in the case of non-reentry boundary condition was established in [START_REF] Mokhtar-Kharroubi | Optimal spectral theory of the linear Boltzman equation[END_REF].

As in [START_REF] Lods | Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions[END_REF], the mathematical analysis is based upon a recent result owing to M. Sbihi [START_REF] Sbihi | A resolvent appraoch to the stability of essential and critical spectra of perturbed semigroups on Hilbert spaces with applications to transport theory[END_REF] (see also Section 4) valid for Hilbert spaces. Actually, under some natural assumptions on the collision operator, we prove, via approximation arguments, the compactness of R 1 (t) on L 2 (D × R N , dx ⊗ dv). The result is then extended to L p (D × R N , dx ⊗ dv) with p ∈ (1, 2) ∪ (2, ∞) by an interpolation argument. Unfortunately, the limiting case p = 1 is not covered by our analysis and requires certainly another approach. Notice that the compactness of R 1 (t) for p = 1 is also an open problem in the one-dimensional case [START_REF] Lods | Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions[END_REF] and for no-reentry boundary condition [START_REF] Mokhtar-Kharroubi | Optimal spectral theory of the linear Boltzman equation[END_REF].

The outline of this work is as follows. In Section 2 we introduce the functional setting of the problem and fix the different notations and facts needed in the sequel. Section 3 is devoted to the spectral analysis of the streaming operator with bounce-back boundary conditions and to the analytic expression of the streaming semigroup. The compactness of the first order remainder term of the Dyson-Phillips expansion is the topic of Section 4.

PRELIMINARY RESULTS

For the definitions of the different spectral notions used throughout this paper we refer, for example, to the book [START_REF] Schechter | Principles of Functional Analysis[END_REF]. If X is a Banach space, B(X) will denote the set of all bounded linear operators on X.

Let D be a smooth bounded open subset of R N . We define the partial Sobolev space

W p = {ψ ∈ X p ; v • ∇ x ψ ∈ X p } where X p = L p (D × R N , dx ⊗ dv) (1 p < ∞).
Let us denote by Γ -(respectively Γ + ) the incoming (resp. outgoing) part of the boundary of the phase space

D × R N Γ ± = (x, v) ∈ ∂D × R N ; ±v • n(x) 0
where n(x) stands for the outward normal unit at x ∈ ∂D. Suitable L p -spaces for the traces on Γ ± are defined as L p ± = L p Γ ± ; |v • n(x)|dγ(x) ⊗ dv , dγ(•) being the Lebesgue measure on ∂D. For any ψ ∈ W p , one can define the traces ψ |Γ ± on Γ ± , however these traces do not belong to L p ± but to a certain weighted space (see [START_REF] Beals | Abstract time-dependent transport equations[END_REF][START_REF] Cessenat | Théorèmes de traces Lp pour des espaces de fonctions de la neutronique[END_REF][START_REF] Cessenat | Théorèmes de traces pour des espaces de fonctions de la neutronique[END_REF]). For this reason, one defines

W p = ψ ∈ W p ; ψ |Γ ± ∈ L p ± .
In all the sequel, we shall assume that Σ(•, •) is a measurable non-negative function on D × R N that fulfills the following.

Assumption 2.1. The collision frequency Σ(•, •) is an even function of the velocity, i.e. for any

(x, v) ∈ D × R N , Σ(x, v) = Σ(x, -v).
Let us define the absorption operator with bounce-back boundary conditions:

T : D(T ) ⊂ X p -→ X p ϕ -→ T ϕ(x, v) := -v • ∇ x ϕ(x, v) -Σ(x, v)ϕ(x, v),
with domain

D(T ) := ψ ∈ W p such that ψ |Γ -(x, v) = γ ψ |Γ + (x, -v)
where 0 < γ < 1. We recall (see e.g. [START_REF] Beals | Abstract time-dependent transport equations[END_REF][START_REF] Frosali | Transport equations with boundary conditions of reverse reflection type[END_REF]) that T is a generator of a non-negative C 0semigroup of contractions

(U (t)) t 0 in X p (1 p < ∞) Remark 2.2.
Notice that our analysis also applies to the more general case γ 1 provided the associated transport operator T generates a C 0 -semigroup in X p . Practical conditions on the geometry of D ensuring the latter to hold are given in [START_REF] Lods | Semigroup generation properties of streaming operators with non-contractive boundary conditions[END_REF].

Definition 2.3. For any (x, v) ∈ D × R N , define t ± (x, v) = sup{ t > 0 ; x ± sv ∈ D, ∀ 0 < s < t } = inf{ s > 0 ; x ± sv / ∈ D}.
For the sake of convenience, we will set

τ (x, v) := t -(x, v) + t + (x, v) for any (x, v) ∈ D × R N . Let us define ϑ(x, v) = t -(x,v) -t + (x,v) Σ(x -sv, v)ds, (x, v) ∈ D × R N .
One proves easily the following thanks to Assumption 2.1.

Lemma 2.4. For any (x, v) ∈ D × R N , ϑ(x, v) = ϑ(x, -v) = τ (x,v) 0 Σ(x + t -(x, v)v -sv, v)ds.
In particular, for any

(x, v) ∈ Γ - ϑ(x, v) = t + (x,v) 0 Σ(x + sv, v)ds.
Now let us investigate the resolvent of T . For any λ ∈ C such that Reλ > 0, let us define

M λ ∈ B(L p -, L p + ) by M λ u(x, v) = u(x -τ (x, v)v, v) exp - τ (x,v) 0 λ + Σ(x -sv, v)ds , (x, v) ∈ Γ + ,
and let B λ ∈ B(L p -, X p ) be given by

B λ u(x, v) = u(x -t -(x, v)v, v) exp - t -(x,v) 0 λ + Σ(x -sv, v)ds , (x, v) ∈ D .
In the same way, let G λ ∈ B(X p , L p + ) be given as

G λ ϕ(x, v) = τ (x,v) 0 ϕ(x -sv, v) exp - s 0 λ + Σ(x -tv, v)dt ds, (x, v) ∈ Γ + ;
and C λ ∈ B(X p ) be defined as

C λ ϕ(x, v) = t(x,v) 0 ϕ(x -tv, v) exp t 0 λ + Σ(x -sv, v)ds dt, (x, v) ∈ D .
It is not difficult to show the following in the spirit of [START_REF] Latrach | Spectral analysis and generation results for streaming operators with multiplying boundary conditions[END_REF].

Proposition 2.5. Let 0 < γ < 1 be fixed and let H ∈ B(L p + , L p -) be defined by:

H(φ + )(x, v) = γφ + (x, -v) for any (x, v) ∈ Γ -. (2.1) If λ ∈ C is such that 1 ∈ ̺(M λ H), then λ ∈ ̺(T ) with (λ -T ) -1 = B λ H(I -M λ H) -1 G λ + C λ . (2.2)
In particular, if there is

λ 0 ∈ R such that r σ (M λ H) < 1 ∀ Reλ > λ 0 , then {λ ∈ C ; Reλ > λ 0 } ⊂ ̺(T )
and the resolvent of T is given by (2.2).

STUDY OF THE STREAMING OPERATOR AND SEMIGROUP

We shall focus in this section on the streaming operator associated with bounce-back boundary conditions and the associated semigroup.

3.1. Description of the spectrum of T . To discuss the spectrum of T , we provide a more precise description of the inverse operator of (I -M λ H) ∈ B(L p + ). Precisely, let us define the measurable function

m λ (x, v) = γ exp - τ (x,v) 0 λ + Σ(x -sv, v)ds , (x, v) ∈ Γ + .
Before stating our first result we recall that the essential range of the measurable function

m λ (•, •), R ess (m λ ), is the set u ∈ C : {(x, v) ∈ Γ + ; |m λ (x, v) -u| < ε} = 0 ∀ε > 0
where |A| denotes the Lebesgue measure of the set A. Then, one has the following:

Proposition 3.1. Let λ ∈ C be such that 1 / ∈ R ess (m λ ). Then, (I -M λ H) ∈ B(L p + ) is invertible with inverse given by (I -M λ H) -1 ψ (x, v) = (1-m 2 λ (x, v)) -1 [(I + M λ H)ψ] (x, v), ∀ ψ ∈ L p + , (x, v) ∈ Γ + . Proof.
Let us fix λ ∈ C and consider the equation:

ψ -(M λ H)ψ = g, (3.1)
where g ∈ L p (Γ + ) as well as the unknown function ψ. From (3.1), one sees that

ψ(x, v) -m λ (x, v)ψ(x -τ (x, v)v, -v) = g(x, v), (x, v) ∈ Γ + .
For any fixed (x, v) ∈ Γ + , one has

g(x -τ (x, v)v, -v) = ψ(x -τ (x, v)v, -v) -m λ (x -τ (x, v)v, -v)× × ψ(x -τ (x, v)v + τ (x -τ (x, v)v, -v)v, +v). Now, one sees easily that m λ (x -τ (x, v)v, -v) = m λ (x, v) while ψ(x -τ (x, v)v + τ (x -τ (x, v)v, -v)v, +v) = ψ(x, v).
Therefore, one has

g(x -τ (x, v)v, -v) = ψ(x -τ (x, v)v, -v) -m λ (x, v)ψ(x, v) so that [M λ Hg] (x, v) = m λ (x, v)g(x -τ (x, v)v, -v) = m λ (x, v)ψ(x -τ (x, v)v, -v) -m 2 λ (x, v)ψ(x, v). Since m λ (x, v)ψ(x -τ (x, v)v, -v) = [M λ Hψ] (x, v), one obtains from (3.1) that g(x, v) + [M λ Hg] (x, v) = 1 -m 2 λ (x, v) ψ(x, v), (x, v) ∈ Γ + .
This leads to an explicit expression of the solution to (3.1):

ψ(x, v) = (1 -m 2 λ (x, v)) -1 [(I + M λ H) g] (x, v). Defining R λ g(x, v) = (1 -m 2 λ (x, v)) -1 [(I + M λ H) g] (x, v) it is not difficult to see that 1 / ∈ R ess (m λ ) =⇒ R λ ∈ B(L p + ), and the above calculations show that R λ = (I -M λ H) -1 .
The precise picture of the spectrum of T is given by the following, which is in the spirit of [START_REF] Lods | On the spectrum of mono-energetic absorption operator with Maxwell boundary conditions. A unified treatment[END_REF] Theorem 3.2. For any k ∈ Z, let us define

F k (x, v) = log γ -ϑ(x, v) τ (x, v) -i 2kπ τ (x, v) , ∀(x, v) ∈ D × R N . Then, σ(T ) = k∈Z R ess (F k )
where R ess (F k ) stands for the essential range of F k .

Proof. Let us begin with the inclusion ⊃. Given λ ∈ R ess (F k ) (k ∈ Z). Let ε > 0 and define

Λ ε := {(x, v) ∈ D × R N ; |λ -F k (x, v)| ε}.
By the definition of R ess (F k ), |Λ ε | = 0 for any ε > 0. For any integer n ∈ N, define

B n = {(x, v) ∈ D × R N , τ (x, v) 1/n}. Since τ (x, v) 0 for a. e. (x, v) ∈ D × R N , one has Λ ε = n B n ∩ Λ ε so that, for any ε > 0, there exists n(ε) ∈ N such that |B n(ε) ∩ Λ ε | = 0. Define A ε := B n(ε) ∩ Λ ε , ε > 0 so that |A ε | = 0 for any ε > 0, while ess -inf τ (x, v) ; (x, v) ∈ A ε 1/n(ε) > 0, for any ε > 0. (3.2)
Then, let χ ε stand for the characteristic function of the measurable set A ε (ε > 0). One sees that

χ ε (x, v) = χ ε (x, -v), and
χ ε (x + tv, v) = χ ε (x, v)
for t > 0 small enough. Now, for any ε > 0, one can define

ϕ ε (x, v) = χ ε (x, v) exp - t -(x,v) 0 F k (x, v) + Σ(x -sv, v) ds , (x, v) ∈ D × R N . One sees that ϕ ε ∈ L ∞ (D × R N ) for any ε > 0, since ess -sup (x,v)∈Aε exp - t -(x,v) 0 F k (x, v) + Σ(x -sv, v) ds < ∞ by virtue of (3.2). Moreover, since t -(x, v) = 0 for any (x, v) ∈ Γ -, one sees that ϕ ε|Γ -= χ ε .
Given (x, v) ∈ Γ -and ε > 0, one has

ϕ ε (x, -v) = χ ε (x, -v) exp - t -(x,-v) 0 F k (x, -v) + Σ(x + sv, -v) ds . Since t -(x, -v) = t + (x, v) = τ (x, v) and F k (x, -v) = F k (x, v) (see Lemma 2.4
), one has

ϕ ε (x, -v) = χ ε (x, v) exp -τ (x, v)F k (x, v) - t + (x,v) 0 Σ(x + sv, -v) ds . Now, one checks that -τ (x, v)F k (x, v) = -log γ + ϑ(x, v) + 2ikπ = -log γ + 2ikπ + t + (x,v) 0 Σ(x + sv, v)ds ∀(x, v) ∈ Γ -,
where we used again Lemma 2.4. Thus, one sees that, for any (x, v) ∈ Γ -,

ϕ ε (x, -v) = 1 γ ϕ ε (x, v)
which exactly means that ϕ ε fulfils the boundary conditions (2.1). Finally, it is easy to see that ϕ ε ∈ D(T ). Define now the net (ψ ε ) ε by

ψ ε = ϕ ε / ϕ ε , ε > 0 (3.3)
and let g ε = (λ -T )ψ ε , i. e.

g ε (x, v) = (λ + Σ(x, v)) ψ ε (x, v) + v • ∇ x ψ ε (x, v).
Using the fact, for any (x, v) ∈ D × R N , and for any t > 0 small enough, ϑ(x + tv, v) = ϑ(x, v) (note that the same occurs for F k ) one can prove that

g ε (x, v) = λ -F k (x, v) ψ ε (x, v), (x, v) ∈ D × R N , ε > 0 and so g ε ess -sup (x,v)∈Aε |λ -F k (x, v)| ψ ε ε.
This, together with (3.3) achieves to show that (ψ ε ) ε>0 is a singular net of T so that λ ∈ σ(T ).

The closedness of the spectrum ensures that k∈Z R ess (F k ) ⊂ σ(T ).

Let us prove now the converse inclusion. Assume that λ / ∈ R ess (F k ) for any k ∈ Z. Then, for any k ∈ Z, there exists

β k > 0 such that |λ -F k (x, v)| β k a.e. (x, v) ∈ D × R N , i.e. log γ -ϑ(x, v) -2ikπ τ (x, v) -λ β k a.e. (x, v) ∈ D × R N .
Then,

| log γ -ϑ(x, v) -2ikπ -λ τ (x, v)| τ (x, v) β k a. e. (x, v) ∈ D × R N .
This means that, for any integer n 0, there exists c n > 0 such that

log γ -ϑ(x, v) -λ τ (x, v) 2π n ± i c n τ (x, v) a. e. (x, v) ∈ D × R N , n 1,
and

| log γ -ϑ(x, v) -λ τ (x, v)| c 0 τ (x, v) a. e. (x, v) ∈ D × R N .
Arguing as in [START_REF] Lods | On the spectrum of mono-energetic absorption operator with Maxwell boundary conditions. A unified treatment[END_REF], one can choose M > 0 such that

|log γ -ϑ(x, v) -λτ (x, v)| M a. e. (x, v) ∈ D × R N ,
and

|exp {log γ -ϑ(x, v) -λτ (x, v)} -1| C 2 N n=1 c 2 n τ (x, v) 2 a. e. (x, v) ∈ D × R N ,
where

C = ess -inf (x,v)∈D×R N exp 1 2 (log γ -ϑ(x, v) -λτ (x, v)) .
Moreover, one can easily see that

lim inf τ (x,v)→0 | exp{log γ -ϑ(x, v) -λτ (x, v)} -1| |1 -γ| > 0.
In particular, ess -inf

(x,v)∈D×R N | exp{log γ -ϑ(x, v) -λτ (x, v)} -1| > 0.
This proves that 1 / ∈ R ess (m λ ). From Proposition 3.1, one gets that (I -M λ H) is invertible and λ ∈ ̺(T ).

3.2.

Explicit expression of the semigroup (U (t)) t 0 . We derive in this section the explicit expression of the semigroup (U (t)) t 0 generated by the streaming operator T associated to the bounce-back boundary conditions H given by (2.1). For simplicity, we shall restrict ourselves to the case of a homogeneous collision frequency:

Σ(x, v) = Σ(v), ∀(x, v) ∈ D × R N .
Theorem 3.3. The C 0 -semigroup (U (t)) t 0 in X p generated by T is given by

U (t) = ∞ n=0 U n (t), ∀t 0,
where, for any fixed t 0,

[U 0 (t)ϕ] (x, v) = ϕ(x -tv, v) exp(-Σ(v)t)χ {t<t -(x,v)} , ϕ ∈ X p , (x, v) ∈ D × R N while, for any n 0 [U 2n+2 (t)ϕ] (x, v) = γ 2n+2 exp(-Σ(v)t)χ I 2n+1 (x,v) (t)ϕ x -tv + (2n + 2)τ (x, v)v; v , and
[U 2n+1 (t)ϕ] (x, v) = γ 2n+1 exp(-Σ(v)t)χ I 2n (x,v) (t)ϕ x+tv-2t -(x, v)v-2nτ (x, v)v, -v
for any ϕ ∈ X p , and any (x, v) ∈ D × R N , with

I k (x, v) = [kτ (x, v) + t -(x, v); (k + 1)τ (x, v) + t -(x, v)],
for any k ∈ N.

Proof. The proof is based upon the representation of the resolvent (2.2) and the use of the uniqueness of the Laplace transform. Precisely, let λ > 0 be fixed. According to Proposition 3.1,

(I -M λ H) -1 ψ (x, v) = (1 -m 2 λ (x, v)) -1 [(I + M λ H)ψ] (x, v) = ∞ n=0 γ 2n exp(-2n(λ + Σ(v))τ (x, v)) [(I + M λ H)ψ] (x, v),
for any nonnegative ψ ∈ L p + , i.e.

(I -M λ H) -1 ψ (x, v) = ∞ n=0 γ 2n exp(-2n(λ + Σ(v))τ (x, v))ψ(x, v)+ ∞ n=0 γ 2n+1 exp(-(2n + 1)(λ + Σ(v))τ (x, v))ψ(x -τ (x, v)v, -v), (x, v) ∈ Γ + .
It is then easy to see that, for any fixed ϕ ∈ X p :

B λ H(I -M λ H) -1 G λ ϕ = ∞ n=0 J n (λ)ϕ (3.4)
where, for any n 0,

[J 2n+1 (λ)ϕ] (x, v) = γ 2n+1 exp {-2n(λ + Σ(v))τ (x, v)} exp {-(λ + Σ(v))t -(x, v)} × × τ (x,v) 0 ϕ(x -t -(x, v)v + sv, -v) exp {-(λ + Σ(v))s} ds, and
J 2n (λ)ϕ(x, v) = γ 2n+2 exp {-(2n + 1)(λ + Σ(v))τ (x, v)} exp {-(λ + Σ(v))t -(x, v)} × × τ (x,v) 0 ϕ(x -t -(x, v)v + τ (x, v)v -sv, v) exp {-(λ + Σ(v))s} ds. For fixed (x, v) ∈ D × R N , performing the change of variable t = 2nτ (x, v) + t -(x, v) + s, dt = t ∈ I 2n+1 (x, v)
in the above expression of J 2n+1 (λ) leads easily to

[J 2n+1 (λ)ϕ] (x, v) = ∞ 0 exp(-λt) [U 2n+1 (t)ϕ] (x, v)dt.
In the same way, the change of variable

t = (2n + 1)τ (x, v) + t -(x, v) + s, dt = ds, t ∈ I 2n+2 (x, v)
in the above expression of J 2n+2 (λ) allows to prove that

[J 2n+2 (λ)ϕ] (x, v) = ∞ 0 exp(-λt) [U 2n+2 (t)ϕ ] (x, v) dt, ∀n 0.
Finally, it is easily seen that

[C λ ϕ] (x, v) = ∞ 0 exp(-λt) [U 0 (t)ϕ ] (x, v) dt.
Therefore,

(λ -T ) -1 ϕ = ∞ n=0 ∞ 0 exp(-λt)U n (t)ϕ dt
for any ϕ ∈ X p for which the series converges. Moreover, since T generates a C 0 -semigroup (U (t)) t 0 in X p , one also has

(λ -T ) -1 ϕ = ∞ 0 exp(-λt)U (t)ϕ dt.
From the uniqueness of the Laplace transform, this yields

U (t)ϕ = ∞ n=0 U n (t)ϕ
for any nonnegative ϕ ∈ X p and, since all the operators involved are clearly nonnegative and the positive cone of X p is generating, the result holds for arbitrary ϕ ∈ X p .

SPECTRAL ANALYSIS OF THE PERTURBED SEMIGROUP

We investigate now the spectral properties of the full semigroup governing the problem (1.1). Let us define the collision operator K by

Kϕ(x, v) = R N κ(x, v, w)ϕ(x, w) dw where the kernel κ(•, •, •) is nonnegative over D × R N × R N . We shall assume here that K is a bounded operator, K ∈ B(X p ), 1
p < ∞, so that by the standard bounded perturbation theory, the operator (T + K, D(T )) generates a C 0 -semigroup (V (t)) t 0 given by the following Dyson-Phillips expansion series:

V (t) = ∞ j=0 V j (t) where V 0 (t) = U (t), V j (t) = t 0 U (t -s)KV j-1 (s)ds, ( j 1)
.

As indicated in the Introduction, our analysis does not cover the case of transport equation in L 1 -spaces, so we shall assume in all this section that

1 < p < ∞.
Throughout the sequel, we shall assume that K is a regular operator in the following sense: Definition 4.1. An operator K ∈ B(X p ) (1 < p < ∞) is said to be regular if K can be approximated in the operator norm by operators of the form:

ϕ ∈ X p -→ i∈I α i (x)β i (v) R N θ i (w)ϕ(x, w)dw ∈ X p (4.1)
where

I is finite, α i ∈ L ∞ (D), β i ∈ L p (R N , dv) and θ i ∈ L q (R N , dv), 1/p + 1/q = 1. Remark 4.2. Since 1 < p < ∞, one notes that the set C c (R N ) of continuous functions with compact support in R N is dense in L q (R N , dv) as well as in L p (R N , dv) (1/p + 1/q = 1).
Consequently, one may assume in the above definition that β i (•) and θ i (•) are continuous functions with compact supports in R N .

We prove in this section the following compactness result, generalizing known ones for 1Dtransport problems [START_REF] Lods | Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions[END_REF] Theorem 4.3. Assume 1 < p < ∞. If K ∈ B(X p ) is a regular operator, then the difference V (t) -U (t) is compact for any t 0. As a consequence, σ ess (V (t)) = σ ess (U (t)) for any t 0.

Remark 4.4. Notice that the compactness of the difference V (t) -U (t) for any t 0 implies that of (λ -T -K) -1 -(λ -T ) -1 for sufficiently large λ. In particular,

σ ess (T + K) = σ ess (T ).
This result was already obtained in [START_REF] Latrach | Compactness results for transport equations and applications[END_REF] and is valid for more general reentry boundary conditions.

The rest of the paper is devoted to the proof of the above Theorem. We shall adopt the so-called resolvent approach which allows to infer the compactness of

R 1 (t) = V (t) -U (t), t 0
from properties of the resolvent (λ-T ) -1 and K only. The basis of our approach is a fundamental result owing to M. Sbihi [1, Theorem 2.2, Corollary 2.1] which, applied to our case, asserts that, for p = 2, if T is dissipative and there exists α > w(U ) (w(U ) denoting the type of the semigroup (U (t)) t 0 ) such that

α + iβ -T ) -1 K(α + iβ -T ) -1 is compact for all β ∈ R (4.2)
and lim β→∞

( K * (α + iβ -T ) -1 K + K(α + iβ -T ) -1 K * ) = 0, (4.3)
then R 1 (t) = V (t) -U (t) is compact on X 2 for all t 0. Notice that here, the streaming operator T is dissipative on X p , p ∈ (1, ∞), in particular for p = 2. Moreover, the compactness assumption (4.2) follows from Theorem 3.1 in [START_REF] Latrach | Compactness results for transport equations and applications[END_REF] and holds true for more general boundary conditions. Therefore, we have only to check that (4.3) holds true provided K is a regular collision operator.

Though M. Sbihi's result is a purely Hilbertian one, it has already been noticed in [START_REF] Lods | Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions[END_REF] that it can be applied successfully to neutron transport problems in L p -spaces for any 1 < p < ∞. Actually, since K is regular and R 1 (t) depends continuously on K ∈ B(X p ), one may assume that K is of the form (4.1) where, according to Remark 4.2, the functions β i and θ i are continuous with compact supports in R N . In this case, K is bounded in any B(X r ) and, by an interpolation argument already used in [START_REF] Lods | Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions[END_REF], we may restrict ourselves to prove the compactness of R 1 (t) in X 2 . Moreover, using a domination argument as in [START_REF] Lods | Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions[END_REF], there is no loss of generality in proving the compactness of R 1 (t) in the special case

Σ(v) = σ > 0, γ = 1
Now, since K is given by (4.1), by linearity, Eq. (4.3), and consequently Theorem 4.3, follow from the following Lemma 4.5. Let β j , θ j be continuous functions with compact support in R N and α j ∈ L ∞ (D), j = 1, 2. Then, there is some α > -σ such that

lim |β|→∞ K 1 (α + iβ -T ) -1 K 2 = 0
where

K j ϕ(x, v) = α j (x)β j (v) R N θ j (w)ϕ(x, w)dw, j = 1, 2; ϕ ∈ X 2 .
Proof. Let α > -σ be fixed. According to Eq. (2.2)

(λ -T ) -1 = B λ H(I -M λ H) -1 G λ + C λ .
Moreover, it is well-known from [START_REF] Mokhtar-Kharroubi | Time asymptotic behaviour and compactness in neutron transport theory[END_REF] that

lim |β|→∞ K 1 C α+iβ K 2 = 0, ∀α > -σ. (4.4)
since C λ is the resolvent of the transport operator with no-reentry boundary conditions. Therefore, one has to prove that

lim |Imλ|→∞ K 1 B λ H(I -M λ H) -1 G λ K 2 = 0 where Reλ > -σ.
According to Eq. (3.4), it suffices to establish that

lim |β|→∞ K 1 J n (α + iβ)K 2 = 0, ∀n ∈ N, α > -σ.
Let us prove the result for J 2n+1 (α + iβ), n ∈ N. Let λ = α + iβ, α > -σ and let n ∈ N be fixed. Technical calculations show that

K 1 J 2n+1 (λ)K 2 ϕ = A 3 A 2 (λ)A 1 ϕ
where

A 1 : ϕ ∈ L 2 (D × R N ) → A 1 ϕ(x) = α 2 (x) R N ϕ(x, w)θ 2 (w)dw ∈ L 2 (D), A 3 : ψ ∈ L 2 (D) → A 3 ψ(x, v) = α 1 (x)β 1 (v)ψ(x) ∈ L 2 (D × R N ) and A 2 (λ) : L 2 (D) → L 2 (D) is given by A 2 (λ)ϕ(x) = R N exp -2n(λ + σ)τ (x, v ′ ) -(λ + σ)t -(x, v ′ ) β 2 (-v ′ )θ 1 (v ′ )dv ′ τ (x,v ′ ) 0 exp(-(λ + σ)s)ϕ(x -t -(x, v ′ )v ′ + sv ′ )ds.
Therefore, it is sufficient to prove that

lim |β|→∞ A 2 (α + iβ) B(L 2 (D)) = 0, α > -σ. (4.5)
To do so, we adopt the approach of [START_REF] Mokhtar-Kharroubi | Time asymptotic behaviour and compactness in neutron transport theory[END_REF] and [START_REF] Sbihi | A resolvent appraoch to the stability of essential and critical spectra of perturbed semigroups on Hilbert spaces with applications to transport theory[END_REF]. Precisely, setting µ = λ + σ and h

(v) = θ 1 (v)β(-v), v ∈ R N , the change of variable s → t = s -τ (x, v ′ ) leads to A 2 (λ)ϕ(x) = R N h(v ′ ) exp{-2nµτ (x, v ′ )}dv ′ t + (x,v ′ ) -t -(x,v ′ ) ϕ(x + tv ′ ) exp(-µt)dt. Since D is convex, given (x, v ′ ) ∈ D × R N , t ∈ (-t -(x, v ′ ), t + (x, v ′ )) ⇐⇒ y = x + tv ′ ∈ D
The change of variable y = x + tv ′ shows that A 2 (λ) is an integral operator :

A 2 (λ)ϕ(x) = D κ(λ, x, y)ϕ(y)dy where κ(λ, x, y) = R h y -x t exp -µt -2nµτ x, y -x t dt t N = R h y -x t exp -µt -2nµ |t| |x| τ x + z, - x |x| dt t N ,
where we used the know property τ (x, v s ) = |s|τ (x, v) for any (x, v) ∈ D × R N and any s ∈ R. Notice that the very rough estimate

A 2 (λ) B(L 2 (D)) D×D |κ(λ, x, y)| 2 dxdy 1/2
apparently does not lead to (4.5). We have to estimate the norm of A 2 (λ) more carefully. With respect to [START_REF] Mokhtar-Kharroubi | Time asymptotic behaviour and compactness in neutron transport theory[END_REF], one of the difficulty in estimating A 2 (λ) is that A 2 (λ) is not a convolution operator because of the dependence in x of τ (x, •). To overcome this difficulty, we follow the approach of [START_REF] Sbihi | A resolvent appraoch to the stability of essential and critical spectra of perturbed semigroups on Hilbert spaces with applications to transport theory[END_REF]. Precisely, set

N λ (x, z) = R h - x t exp -µt -2nµ |t| |x| τ x + z, - x |x| dt t N ,
where (x, z) ∈ D × D with x + z ∈ D. Let us point out that, from assumption, there is no loss of generality assuming that there exist two constants a, b > 0 such that Supp (h) ⊂ v ∈ R N ; a |v| b .

In this case, in the above integral, one sees that t ∈ R is such that Using the fact that, for any fixed x ∈ D, the mapping z → τ x + z, -x |x| is bounded, this can be done as in [START_REF] Sbihi | A resolvent appraoch to the stability of essential and critical spectra of perturbed semigroups on Hilbert spaces with applications to transport theory[END_REF] thanks to the Riemann-Lebesgue's Lemma. This achieves to prove that Clearly, this result may be also derived from Theorem 4.3 using the fact that the spectral mapping theorem holds true for the point spectrum.

  |t| |x|/a. This means that the above integral over R reduces actually to an integral over -d a , d a where d is the diameter of D. Then, κ(λ, x, y) = N λ (x -y, y), for any (x, y) ∈ D × D and, setting G λ (x) = sup z∈D-x |N λ (x, z)| , x ∈ D one has |A 2 (λ)ϕ(x)| D G λ (x -y)|ϕ(y)|dy, ∀x ∈ D, ϕ ∈ L 2 (D).

Consequently, A 2

 2 (λ) B(L 2 (D)) D G λ (x)dx.To prove(4.5), one has then to show thatlim |β|→∞ D G α+iβ (x)dx = 0, ∀α > -σ.First, one sees that for any (x, z) ∈ D × D with x + z ∈ D and any λ = α + iβ, one has|N λ (x, z)| R h -x t exp -(α + σ)t dt t Nwhere we used the fact that τ (•, •) 0. Then,D sup λ=α+iβ |G λ (x)|dx R exp -(α + σ)t dt t N D hexp -(α + σ)t dt R N |h(v)|dv < ∞,where we performed the change of variables v = x t in the x integral. Therefore, from the dominated convergence theorem, it suffices to prove that lim |β|→∞ G α+iβ (x) = 0, a. e. x ∈ D.

lim |β|→∞ K 1 JK 1 JRemark 4 . 6 .

 1146 2n+1 (α + iβ)K 2 = 0, ∀n ∈ N, α > -σ, ∀n ∈ N.One proves in the same way thatlim |β|→∞ 2n (α + iβ)K 2 = 0, ∀n ∈ N, α > -σ, ∀n ∈ Nand, combined with (4.4) and (3.4), yield the result. Let us observe that Lemma 4.5 allows to describe the asymptotic spectrum of the transport operator T + K. Indeed, combining Lemma 4.5 with the compactness of (λ -T ) -1 K [32] and [4, Lemma 1.1] we infer that i) σ(T + K) ∩ {λ ∈ C : Reλ > -λ * } consists of, at most, isolated eigenvalues with finite algebraic multiplicity; ii) for any η > 0, the set σ(T + K) ∩ {λ ∈ C : Reλ > -λ * + η} is finite or empty.


Copyright

Uniaxial compression as a model for studying the shaping of food powders F.


Shunsuke Nakata
email: nakata.shunsuke.dpe@tufs.ac.jp

Yohann Meynadier°
email: yohann.meynadier@univ-provence.fr

Final accent and lengthening in French

Keywords:

An experiment was carried out to examine the possibility of word-final F0 rise without lengthening in French. The results show that, although a variability is observed depending on speaker, syllable structure and syllable number, the final rise can be unaccompanied by vowel lengthening in the phrasemedian final syllable of a noun, in difference to the phrasefinal syllable which is lengthened to signal the prosodic boundary. The duration of infra-syllabic components is also investigated.

Introduction

French word-final accent has traditionally been described as an "accent of duration" which is essentially characterized by lengthening of the final full vowel, also accompanied by a tonal movement [1]. Although the potential location of this final accent is determined morphemically, its domain is a group of words which forms a certain sense unit, and a word can be "deaccented" when it does not occupy the final position of a prosodic group [4].

It is observed that the grouping by the final accent has a hierarchical structure, the intonational level and another inferior level which marks the components of an intonational group. These two levels have been described as Major / Minor Intoneme [1][6], Intonational Phrase / Accentual Phrase [4] [START_REF] References Delattre | L'accent final en français: accent d'intensité, accent de hauteur, accent de durée[END_REF], or Intonational Phrase / Prosodic Phrase [2] and the two levels are distinguished by the significant difference in the lengthening of the group-final vowel.

One of the divergences in the literature concerns the prosodic structure inside the inferior level, namely the phonological status of an F0 rise realized without vowel lengthening. Jun & Fougeron [5] include this phenomenon as Hi in their phonological unit of French intonation described as /LHiLH*/, where H* is a high tone associated to a lengthened syllable. While the second L tone collaborate with H* to mark the right boundary as a group-final accent, the first L tone serves, together with Hi, to indicate the left boundary of a prosodic group (Accentual Phrase). Hi is realized on "one of the initial syllables of the first content word of the AP". The default pattern /LHiLH*/ has variant forms where some of the components are truncated depending on various factors as in example (1) b. In the model of Di Cristo [2], the F0 rise without lengthening can occur on the word-final as well as the word-initial position and the two realizations share the same prosodic properties. The F0 rise forms the elementary rhythmic unit (Tonal Unit) <LH>, which reflects the basic metric structure of the language at a level inferior than the boundary operation by the final accent, as in (2). Final accents without lengthening indicate the realization of the <LH> Tonal Unit and not a boundary of certain higher domain. In (2)b, a likely F0 rise on the initial syllable Pré-no longer indicates the location of Tonal Unit but marks the right boundary of a Prosodic Phrase. The latter is distinguished from the former by the global concave contour throughout the phrase as well as a likely short pause [2]. As in (2)c, F0 rises without lengthening are not necessarily realized on initial syllable as a marking of the initial boundary. Jun & Fougeron observe that an F0 rise can appear on final syllable of a functional word before the first lexical word in AP [START_REF] References Delattre | L'accent final en français: accent d'intensité, accent de hauteur, accent de durée[END_REF] or on medial syllables of a exceptionally long polymorphemic word [6]. Di Cristo's modelling in (2)c differs from these observations in that it also includes the final syllable of a content word as the location of F0 rise without lengthening.

The present study aims to examine whether the F0 rise without lengthening can be realized on the word-final position as Di Cristo's model predicts, in which case it is supposed to reflect the basic metric structure inside a lowest prosodic group, or its occurrence is rather confined to the word-initial position to indicate the right boundary of a prosodic group. An experiment was carried out to test the hypothesis of a lengthening-less accent in French produced in the content word-final position but not at the right boundary of prosodic or intonational phrase.

Methods

Materials

Three types of sentences are chosen to examine the duration of the target syllable in three different positions as in (3):

(3 a) J'aime le carnaval musical de Saint-Denis.

(I like the music carnival of Saint-Denis.) b) J'adore la cavalcade infernale des Rapetout.

(I like the infernal cavalcade of the Beagle Boys.) c) Le grand carnaval commence mardi prochain.

(The big carnival begins next Tuesday.)

In (3a), the target syllable is the final vowel of a noun in <noun+adjective> type noun phrase as a verb compliment. In (3b), it is situated in the middle of a noun in the same type of phrase. In (3c), it is the final vowel of a noun phrase which serves as the subject of the sentence. The target syllables at these three positions realize different prosodic properties and also differ in their phonological status. We expect that (i) the target syllable in (3a The onset consonants and the nuclear vowels are not totally controlled, but the coda is always [l]. The phoneme number of the whole sentence and of the <article+noun+adjective> portion is equalized for each set of (3a)-(3c) to attenuate the possible sentence-length effect. However, the interspeaker variation of 1-3 phonemes is observed depending on the realization or not of final schwa or liaison. In total, 24 sentence-types are analyzed.

Subjects and Procedure

4 speakers, 1 male (SP1 from Tour) and 3 females (SP2 and SP3 from Provence, and SP4 from Loraine) participated in the experiment. None of them had a strong regional accent.

The sentences are randomized and the participants read the corpus 12 times. 10 cases other than the shortest and the longest ones are retained for analysis. In total, 320 cases for (3b) and (3c) and 319 for (3a) are analyzed, 1 case of (3a) with a pause being excluded. The recordings are digitized at 16kHz and the target syllables are labeled phonemically by hand on PRAAT. Two parameters are examined: F0 and duration. The whole sentence duration is also measured to check that the speech rate remains constant. For F0, the tonal patterns of <article+noun+adjective> part are labeled using L, Hi, H1, and H2. H1 is assigned to the target syllable when it accompanies F0 rise or peak. H2 is labeled to the adjective-final syllable when it accompanies an F0 rise. Hi indicates the initial rise or peak on the first syllable(s) of nouns or adjectives. The syllables where the pitch is low or going down are represented by L. Figure 1 shows an example of tonal pattern [LLH1LLH2] where both the target syllable on noun and the final syllable of adjective have an F0 rise with no initial rise. For sentences (3a), only the cases where the target syllable is labeled H1, i.e. the ones which matched our hypothesis, were retained for further duration analysis. That is, the vowel duration of target syllables with H1 in (3a) is measured and compared statistically with the vowel duration of target syllables in (3b) and (3c). All the target syllables in (3b) have no F0 rise, and on the contrary, all of those in (3c) are realized with the expected F0 rise. In this paper we don't treat the tonal difference between perceptual "rise" and "peak" which will be examined in our further studies.

Results

Tonal pattern

On the target syllables in (3a), the expected F0 rise is realized on 121 of 319 cases, i.e. 37.9% of all the realizations by 4 speakers. The interspeaker variation is observed for the frequency of H1: 32.9 % (SP1), 68.8% (SP2), 20.0% (SP3), and 30.0% (SP4). Its frequency varies also by syllable number of the noun : while the nouns of 2 and 3 syllables accompany an F0 rise on final syllable for more than half cases (67.1% and 53.7% respectively), those of 1 an 4 syllables realized less final F0 rises (16.3% and 15.0% respectively). Table 2 represents the distribution of tonal patterns for the cases with H1 (final F0 rise on the noun) in (3a). The tonal patterns can be categorized by the occurrence of Hi on the noun or/and adjective. Table 1 shows that the recurrence of L and H (LH1LH2 or LHiLH1LH2) is preferred to the collision of H tone (HiH1 or H1Hi), being realized on 83.5% of all the cases. Hi appears mainly on the adjective (20/23 cases), indicating the beginning of the 2 nd element of the noun phrase. Hi on the 1 st element appears only on the noun of more than 2 syllables. We also labeled the tonal patterns of the cases without H1 (62,1% of 319 cases) which are excluded from the duration analysis. Table 3 represents the observed tonal patterns by noun syllable number. The noun-final L tone is indicated separately from the next L on the beginning of the adjective. ), it appears more frequently on the nouns with less number of syllables (1 or 2) where the sequence of L does not exceed 3 syllables. Generally, it can be said that the speakers prefer the recurrence of the elemental rhythmic pattern 'LH' where H was realized as either initial Hi or final H1.

Duration of target syllables

Here we examine the durational properties of the noun-final H1 tones (from (3a)) in comparison with phrase-final tones (from (3b)) as well as with word-median unaccented syllables (from (3c)). We expect that it does not accompany lengthening as the boundary tones do. Therefore, only the cases with H1 accent (37.9% of (3a)) will be treated in the further analysis.

Speech rate control

Preliminarily, the possible inter-and intra-speaker variation of speech rate is examined. Table 4 shows the average speech rate (ms per phoneme) by speaker and by sentence type. As the table shows, the inter-and intra-speaker variation of average speech rate does not exceed 5 ms. It was also observed that the maximum difference throughout the corpus does not exceed 11 ms. We can conclude that the speech rate is closely similar between speakers or sentences, variation being negligible in our corpus.

General tendency

ANOVA was carried out with 4 factors: Prosodic Position of the target syllable ((3a)-( 3c)), Speakers (1-4), Syllable Number of the noun (1-4) and Syllable Structure (open or closed). The results of the 4-way ANOVA show that, at a 5% significant level, all the factors and interactions are significant (p<0.0001 except p<0.008 for Syllable Structure) except the Speaker-Syllable Number interaction (p = 0.604). We examine below the effect of the Position factor and its interactions with three other factors, by means of Tamhane and of Fischer's PLSD post-hoc tests (with a significant level at 5%).

Main effect of the Prosodic Position

Henceforth, the positions (3a), (3b) and (3c) of target vowels are called respectively WfA (Word-final Accent), UNA (unaccented) and PfA (Phrase-Final Accent). The post-hoc tests on vowel duration show that vowels in WfA position are not significantly lengthened compared with those in UNA position (p=0.499), PfA differing significantly (p<0.0001) from the two other positions (Figure 2). In other words, a word-final F0 rise can appear without lengthening. ), but their difference is near 30 ms for the open syllables while it is less than 15 ms for the closed syllables (Figure 3). The open and closed syllables also differ from each other in that WfA is not distinguished significantly from UNA for the open syllables (p=0.997) while it is for the closed syllables (p<0.0001). However, the difference between WfA and UNA remains very small, that is less than 10 ms. The investigation of standardized duration (by z-transform, see [7]) of the sub-syllabic components enhances an important interaction of the syllable structure with the prosodic positions (Figure 4). The distribution of lengthening among the syllabic components seems to be modified by the [l] syllable-closing.

1,

(

1)UNA (2)WfA (3)PfA Noyau Coda Attaque -1,25 -1 -,75 -,5 -,25 0 ,25 ,5 ,75 1 1,25 1,5 (1)

Figure 4 Average standardized duration (y axis) of onset, nucleus and coda by positions for closed and open syllables

Thus in an unexpected way, at prosodic boundary (PfA), vowel lengthening is reduced and the coda duration does not increase at all. Instead, a compensatory lengthening of the onset consonant seems to take part in syllable-final lengthening. Concerning WfA, the onset duration is affected by syllable structure: with regard to the unaccented position, it is significantly reduced in open syllables while significantly increased in closed ones (p<0.0001 for both). Nevertheless, both of these opposite differences do not exceed on average about 10 ms. Besides, regarding the onset and nucleus duration for closed syllables, WfA might be considered as an intermediate level between UNA and PfA. These results do not match the patterns of sub-syllabic duration drawn by previous studies on accentual and boundary lengthening in French [8]. Furthermore, they are largely weakened when the interaction of the other factors are considered. The vowel duration also varies by the syllable number of the noun (Figure 5). The post-hoc tests show that the difference between WfA and PfA is always significant and generally longer than 20 ms (p<0.0001). The increase of syllable number of the noun from 1 to 4 does not seem to affect differently the durational property related to these three prosodic positions: WfA and PfA remain two different phonological categories. WfA is not statistically distinguished from UNA, except with the three syllable nouns. However, we cannot explain this only difference. Finally, we also observed a speaker variation where the general vowel duration of SP1 is significantly different from those of SP2 and SP4 (p<0.0001). Figure 6 shows that the general inter-speaker variation is mainly due to that of PfA, for which the same significant differences as the general variation are observed. There is no significant inter-speaker difference for vowel duration in WfA. As for UNA, a significant difference is observed only between SP1 and SP4 (p<0.0001). The non-significance of inter-speaker difference for WfA seems to indicate that F0 rise without lengthening is a prosodic property of this position. In contrast, the important variability at PfA suggests that speakers may realize different kinds of prosodic boundary, i.e. either Prosodic Phrase or Intonational Phrase [2], or may use differently tonal and temporal features to signal the same boundary level: SP1, for example, uses lengthening much less than the other speakers.

Discussion and Conclusion

Overall, the results seem to attest the occurrence of final F0 rise without lengthening on content words, which could be called a 'final rhythmic accent' in the sense that it does not demarcate a prosodic boundary but rather serves to form the elemental rhythmic pattern inside the boundaries. No strong interactions of the syllable number of accented words or of the speaker variability on vowel duration are observed which importantly weaken this main result. Nevertheless, complex crossing effects of the syllable structure on sub-syllabic durations blur this conclusion. Further investigations on subsyllabic temporal pattern seems be needed to test if WfA could be distinguished from PfA, UNA and/or word-initial accent, as previous studies let us envisage. Besides, it is observed in our corpus that the recurrence of 'LH' is preferred and that the final F0 rise, in concurrence with the initial F0 rise, participated in the realization of this recurring pattern. This final F0 rise without lengthening differs from both Hi and H* in /LHiLH*/ or medial H by Jun & Fougeron [4][5]. According to Jun & Fougeron [START_REF] References Delattre | L'accent final en français: accent d'intensité, accent de hauteur, accent de durée[END_REF], Hi occurs on 'some initial (i.e. 1 st or 2 nd , and rarely the 3 rd )' syllable of the AP-initial content word and marks the initial boundary of an AP together with the preceding L tone. However, it is difficult to consider the final F0 rise as a delayed Hi which happens to fall on the final syllable when the target syllable is no longer at the initial position of an AP, as in the cases with the nouns of 3 or 4 syllables. Furthermore, the co-occurrence of the initial and final rise before the groupfinal accent are observed ([LHiLH1LH2] and [LHiLH1HiLH2] in Table 2), even if they are rare cases. The final F0 rise without lengthening observed in our corpus also differs from the medial H [START_REF] References Delattre | L'accent final en français: accent d'intensité, accent de hauteur, accent de durée[END_REF] in that it was realized at the final syllables of content words. Di Cristo's model [2], which sees the final F0 rise without lengthening as a part of the Tonal Unit, would seem to be better able to integrate this low accent-level. However, it is to be verified if the initial and final F0 rise without lengthening share the same prosodic property as [2] predicts. This point and the possible distinctive temporal infra-syllabic patterns of initial vs final accent inside AP will be the focus of further investigations with the aim of discussing about a hypothetical lowest prosodic constituent (i.e. TU) inside the Prosodic Phrase [2] or Accentual Phrase [4] [START_REF] References Delattre | L'accent final en français: accent d'intensité, accent de hauteur, accent de durée[END_REF] in French.

  a. les dé)claration} b. {Président de la République} c. [La jolie) voiture) italienne} qu'il lui avait offert} pour son anniverSAIRE}] ([ ]: Intonational Phrase, {}: Prosodic Phrase, ): Tonal Unit)

Figure 1 :

 1 Figure 1: Example of the tonal pattern [LLH1LLH2] on the noun phrase "le carnival musical (the music carnival)"

Figure 2 :

 2 Figure 2: Average vowel duration of three positions (error bar: 95% confidence intervals) 3.2.4. Interactions of Position with the other factors The closed syllables do not have exactly the same behavior as the open ones. WfA and PfA are significantly different with the open syllables (p<0.0001) as well as with the closed ones (p<0.0001), but their difference is near 30 ms for the open syllables while it is less than 15 ms for the closed syllables (Figure3). The open and closed syllables also differ from each other in that WfA is not distinguished significantly from UNA for the open syllables (p=0.997) while it is for the closed syllables (p<0.0001). However, the difference between WfA and UNA remains very small, that is less than 10 ms.

Figure 3 :

 3 Figure 3: Average vowel duration of three positions by syllable structure

Figure 5 :

 5 Figure 5: Average vowel duration of three positions by syllable number of the noun (Note that the shorter vowel for two- syllable nouns in all positions is due to the intrinsic shorter duration of [i] used in only this context.)

Figure 6 :

 6 Figure 6: Average vowel duration of three positions by 4 speakers

Table 1 :

 1 (3b) has no F0 rise nor lengthening, being unaccented position; (iii) the target syllable in (3c) has an F0 rise accompanied by lengthening, realizing a demarcating Prosodic Phrase accent which marks a boundary of this larger prosodic group.Besides the position in the sentence, the syllable number of the noun and the structure of the target syllable (open / closed) are controlled as follows: Target syllables by syllable number of the noun and by the syllable structure

	), which accompanies an F0 rise without
	lengthening at the final position of a content word (noun),
	would serve to indicate a smaller rhythmic unit inside the
	group, i.e. a non-demarcating Tonal Unit final accent; (ii) the

Table 2 :

 2 Distribution

of tonal patterns on <noun + adj.> with final F0 rise on noun (number in brackets: percentage; contiguous Ls are indicated by a single L)

without Hi on Adj. with Hi on Adj. total

	without Hi on N	LH1LH2	98 (81.0)	LH1HiLH2	19 (15.7)	117 (96.7)
	with Hi on N	LHiLH1LH2	3 (2.5)	LHiLH1HiLH2	1 (0.8)	4 (3.3)
	total		101 (83.5)		20 (16.5)	121 (100)

Table 3 :

 3 Distribution of tonal patterns on <noun + adj.> without final F0 rise on noun

	Tonal pattern Art.+N Adj.	1	syllable number of N 2 3	4	total
	L	LH2	13	16	5	8	42
	LHiL	LH2			6		6
	L	HiLH2	43		3	25	71
	L	HiLH2	8	9		15	32
	L	HiH2	3				3
	L	LHiL		1			1
	LHiL	HiLH2			23	20	43
	total		67	26	37	68	198

Table 4 :

 4 Average

		sentence(3a)	sentence(3b)	sentence(3c)	Average
	SP1	68.8	70.4	66.4	68.5
	SP2	66.6	68.0	66.0	66.9
	SP3	68.6	70.6	67.1	68.8
	SP4	68.5	70.2	68.2	68.9

speech rate by speaker and by sentence type (ms per phoneme)


Guy Della Valle

A Kozlowski

Paul Colonna

J Tayeb

Christiane Mercier

Energy balance during twin screw extrusion cooking of maize starch

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Claude Broussolle

Pierre Daucé

Jean-Claude C Poupa

P Dauce

Tentative regional projections building up scenarios for agriculture in Brittany

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The research presented hereafter has consisted in exploring, from the current 1980 situation ta the year 2000, several possible scenarios for agriculture and food industries in Brittany. The building up of the scenarios is not exclusively based on a totally pre-set, formal and standardized method ; the characteristic of prospective studies is ta combine factual approach and the use of mathematical means, to try and grasp the complexity of reality together wilh avoiding the risks of subjectivity. In the present case, the proposed proceedings, partly new as to the method in use, find their originality in their application to a problem of regional economics.

Taking into account the involvement of Brittany in world markets, the most apropriate method of working out coherent scenarios consists in starting from national and international environment, which forms the surrounding structure which conditions the functioning and evolution of regional agricultural economics. lt has hence appeared necessary to take position within the scopes of reflexion which have been set up at various geographic levels, so as not to lose contact with perspectives revealed within other systems . That is what was done, with reference mainly to the prospective studies by O.E.C.D. (Interfuturs 1979) and, in France, by the General Planning Commission (Commissariat Général au Plan, LESOURNE, 1980). ln all, four wide scopes of evolution ha ve been selected to cont emplate the possible future issues for Breton agriculture ; they have lead to the four following scenarios A. Slow growth: in a conte xt of economic depression and uncertainty, an assist ed and dominated agriculture.

8. Liberalism and important growth: bounda r y-free liberalism competing agriculture.

C. Breakaway economics : protectionism, strong exchange limitations, national self-sufficiency, privileged agriculture. X This article develo ps the major methodological aspects of a prospecti ve st udy , the complete results of which can be found in "Scenarios pour l'agriculture Bretonne, hy C. Broussolle, P. Dauce, P. Houée, et C. Mouchet, INRA, Rennes, Déc. 1983, 245 p. D. Another development : a new international balance, new values, an independant agriculture.

To explain the method in use, and bring out its logic, we have taken as an example the slow growth scenario (A) and we shall present the methological sequence which was applied toit.

I. Structural analysis of the Breton agricultural and food production system

The analysis of the system formed by the economics of a region leads to the inventory of its specific variable factors and to reveal their interconnections.

Identification of the main variable factors.

Fifty-seven variable factors have been selected. Twenty-eight are related to agriculture in itself, fifteen to the food production industries, and fourteen to environment. That choice is the result of a dual necessity : -establishing a list of variable factors as complete as possible in order to identify the key factors, -distinguishing between internal factors: those related to agriculture and industry, and external factors : those related to environment.

Those variable factors, displayed in lines and columns in a double entry table, form the matrix of a structural analysis, the filling in of which is carried out :

. either in columns, marking the influence of each factor on all the others : 1 if there is influence, 0 if there is none . . or along lines, showing which factor influences which.

The division between the variable factors of agriculture, food production industries and environment leads to the structuration of the matrix into blacks. That matrix may be presented as an oriented graph. Bearing in mind that the degree of dependence or materiality of the variable factors is linked to the number of paths (as meant in the graph th eory) to or from, it is possible, from that matrix, common to all scenarios, to reveal the direct and indirect chains of influence of the variable factors upon one another. Also, the division of the variable factors into blacks permits to class hierarchically the factors in each of them according to their influence on those of other blacks. Under those conditions, the problem which is posed is that of counting up or enumerating all paths of any length joining one peak of the graph to another. Several methods are available to salve that problem. {. [START_REF] Derniame | Problèmes du cheminement dans les graphes. Collection monographies d'informatique AFC~T[END_REF], [START_REF] Gondran (m | Graphes et algorithmes[END_REF]), (H[LARY, PEDRONO, 1983) Sorne methods account for all paths, including the loops ; such is the case for that consisting in bringing up the "Boolean" matrix to several powers, successively (MICMAC method). Others only account for e]ementary paths, i.e. those on]y passing once on one peak of the graph. We have used that latter method with a simple recursive algorithm.

N being the number of peaks, the length of an elementary path is inevitably smaller than n and we have at the most 2 to the power of n-1 paths of n-1 length. The total number of elementary paths of any length is hence calculated.

Within the scenarios study and inasmuch as the problem was not so much the sorting of the relations than that of the variable factors according to the number of paths joining each one of them, we have limited ourselves to the length 3 paths.

Direct effects

The direct effect of a variable factor is measured through the number of length 1 paths going from that factor . The most important driving factors are shown in table 1. We find:

. The degree of regional specialisation,

The price maintenance in animal production Ability to innovate within agriculture and food production industries Energy cost The number of specialised farms • Etc . .•.

On the contrary, the most de pendant factors, those to which lead the largest number of length 1 paths are:

The proportion of intermediate consumption in the final agricultural production, Gross operating incarne, The amount of final agricultural production, . The amount of annual investments, • The amount of imported production factors, Etc ....

Those results can be presented in a "motoriality/dependence" chart divided into four sectors :

Sector 1 : "autonomous" variable factors, little motorial and little dependent.

Sector 2 "resuJting" variable factors, little motorial and very dependent.

Sector 3 "relay" variable factors, very motorial and very dependent. Sector 4 : "explanatory" variable factors, very motorial and little dependent.

The sector 1 factors, hardly linked ta the system, cannot be controlling factors. Those of sector 2 have an evolution which can be explained by the "explanatory" factors and the "relay" factors. The sector 3 factors are intermediate ones which amplify or reduce the effect of the motorial factors of sector 4.

Indirect effects

The motorial variable factors with the greatest indirect effect are the following (table 1) :

Animal production price maintainance. There are 15,392 paths with a length of <3 going from that factor to the other variables. Energy cost (14,666 paths) The degree of regional specialisation (14,619 paths) Ability ta innovate in the agriculture and in the food production industries ( 14,051 paths) Gross National Product increase rate (12,198 paths) etc .... One can observe that some variable factors with Iimited direct effects rnay, on the contrary, ha ve greater indirect effects. Such is the case, for instance, wi th the "dispersion in food production industries" factor, going from 32nd in the former ranking ta the 21st in the latter.

Division of variable factors into blacks according ta indirect

effects.

ln addition ta the aforesaid rankings, one can also grade hierarchically the factors of a sub-system or black, according ta the influence they exercise on or receive from factors in olher blacks. ~---------------------------------------------------------------------- The general economic situation which is likely ta prevail over the period covered by the study, can be characterized by a number of events. Faced with an uncertain future, the prospective method consists in selecting several combinations of different events which formas many scenarios.

---------------------------------------------------------------------- E E FPI E A A E E A E FPl FPI FPl A A A A E Degree of
A A A A E A A A A A A A FPI A A A
21. Building up the scenarios the mathematical approach.

Within a given scenario, certain events can be considered as given facts; they form, in a way, the rigid frame of the scenario: such is the case, for instance, with unemployment and inflation in the slow growth scenario. Others, on the contrary, are hazardous : the accrued solidarity between E.E.C. countries, a dynamic policy of the Food Production Industries, etc ... lt would then be for a group of experts ta evaluate the conditional or simple probabilities for those various events to occur.

The provided empirical probabilites are not necessarily coherent, in the sense of the probability theory. The proposed method allows for the calculation of "corrected" probabilities . From those, a script is made for a system of relations linking together the probabilities of occurence of various possible situations or sub-scenarios, and one calculates, for each sub-scenario, an interval of variation for the probabilit y of occurence under the system of constraints, as expressed by those relations. The sub-scenario for which the upper limits of the variation interval are the highest, are considered as the most likely. The mathematical approach is as follows

Ta a list of Boolean factor X • if e i happens, 0 if value of vector X. n events (e1, ••• , e1, ••• en) is associated a (xi, •• •, x1, • • •, x 0 ), the component x1equalling 1 not. A sub~scenario is characterized by one single r • 2n sub-scenarios are counted up.
Experts provide simple probabilities of occurence of events (marked p(ei) •) and conditional probabilities of or.curence of an event e1 , know1ng that e j has occured (marked p(e1/e j) ) or has not occured (marked p(e 1 /ë:j)). To be coherent, s uch empirical probabilities must veri fy Lhe following relations :

Û,( (1)

p(e 1 /ej)•p(ej). p(ej/ei)p(ei)=p(ei !!_ej)

(2)

p(e1/ej)•p(ej) + p(e1/ej) p(ej) • p(e1) (3)
The corrected probabilities (marked p• ) are calculated through minimising the function: under constraints (1 ) (2) and(3) To the r scenarios 5 k, characterized by vectors Xk, are associated probabilities markedn k Hence the function f (i,j ,k) equal to 1 i f (ei €.

Sk) and (e j €. Sk), 0 i f not. We can then write• n(n+1) const raints of the type : The calculation of the upper limit of the variation interval of the nk probability is done Lhrough maximizing Lhat probability under the aforesaid constraint. The use of the simplex method requires the calculation of an initial solution. The problem is solved through adding C C = n(n+1) + 1 artificial variables, the initial solution being Lhat 2 which cancels out the total of the artificiel variable factors, but at the same lime allowing for the constraints of the basic problem. One can then apply r times the simplex algorythm to the r probabilities n k• Inasmuch as the selected time limit is rather far off, the progression which is being attempted, from the initial situation to the final prospect, can prove more and more difficult the further we get from the present. The division of the period in sub-periods permits us to account for possible mutations and facilitates analysis. Hence it is necessary to establish an impact matrix to try and measure the influence of events having occured or not during the first period, on the probability of events likely to occur during the second one.

Three situations are then possible 1) the event ej of the first period has no infl uence on the e vent e1 of the second one. They are independent :

Pi "" p(1/ j) (7)
2 ) The event e j has an amplify ing effect on the event e i p1< p(i/j) ~8)

3) The event e j has an inhibiting effect on the event e1

Pi >p(i/j)

(9) 2, l >t r . . If i,J
is the impact ration of the event ej of the first period on the event ei of the second one, when the event ej is part of the most probable scenario selected for the first period end if n*1 is the number of events ej. occuring during that period, thë'°rlthe expression of the probability of events during the second period will be as follows:

1) if there is an amplifying effect of ej on e1

i:

2, i:~( j r . . p(i/j)=p. + (1-p.) l., J • ' (10) l. l. M 2 n 1
2 ) If there is an inhibitino effect of ej on ei i: 2, lx j r . . p(i/j) = p. + p. 2 nx lt will be noted that c~anging from likelihood es timates to probabilities can be interpreted in a Bayesian way. Pa rt icularl y , the adjustment of the probabilities of one period to the other according to the occurence of events, illustrates an adaptation process into the realms of chance.

Scenarios and sub-scenarios

The building up of scenario A will illustrate our move, bearing in mind that we are in the following general situation: within a world environment of economic crisis and monetary instability, we do not expect any fundamental change in the French economy. The rhythm of growth was eut short in 1974 and has ne ver recovered s i nce. Asto agriculture, it has slipped into a slow de cli ne, aggravated by the deep international changes.

The certain events which have been selected to characterize that scenario are the following -poor economic growth -domestic market for food products near saturation -external market for agricultural products under poor development -uncertainty of necessary suppli es for agriculture -inflation rate remaining high -high unemployment rate -energy cost increase -slight decr ease in pri ces for agricultural products To those eight events, which are part of the slow growth scenario, other events are added, which do not have the same degree of c~~tainty, as they are, to some extent, the resulting factors of powerplays between the active participants, hence lheir occurence is Jess certain. 8 ln that second category, the six forecast events, marked e1, e2, e3, e4 0 e5 0 e6 are the following:

e1
an increased solidarity between the E.E.C. countries e2 a tendency to encourage regional specialization e3 a dynamlc policy of the food production industries e4

governrnent withdrawal e$ balance of prices in favor of animal productions e6 launching of major technological innovations

The sub-scenarios Simple and conditional probabilities have been attributed to the above six events. The comhination of those events with optimisation of the prospective function have allowed for the classification of the subscenarios according to their highest probahility of occurence , on the one hand for the 1980-1990 sub-period and, on the other had for the 1990-2000 sub-period.

221. The scenarios of the 1980-1990 sub-period.

The Ai• A2 , and A3 scenar ios form the tendential core of evolution. there is ~ore than one chance out of two that one of the three fits into the 1990 situation.

1. The A1 sub-scenario =(e1, e2. e3, ëi;, es,eo)has a probability of 20 %. (The non-occurence of an event e1 is marked e1)• lt is an average scenario which can be used as reference and helps to reveal other possible e volutions. lt is also an indicator for freedom of action.

2. The sub-scenario A2 • (ëi, probability of 16,$ %.

es, e 6 ) has a Except for the Go vernment not withdrawing, everything is negative. It is a pessimistic scenario ; we get deeper into the crisis.

The sub-scenaric-A3 • (e1,

probability of 16 %. This scenario is more optimistic than the former one because it foresees an increased solidarity between the [.E.C. countries, a dynamic policy on the part of Food Production Industries and major technological innovations. lt is a productivity prone scenario which, in the middle term, may help us out of the crisis, if we can work out a concerted program of growth through technology.

The sub-scenarios of the 1990-2000 period

We take the hypothesis that in 1990, one of the three previous scenarios will have taken place. On such basis, and with the same events, new sub-scenarios are built up to outline the possible pictures of Breton agricullure of the 1980-1990 period. They are actualised through building up three impact matrixes which measure the influence of the first per iod events on those of the second.

23. The sensitivity of sub-scenarios ta the variations in events probability.

When reading the obtained results, one may wonder about the stability of the sub-scenarios, i.e. about their sensi tivity to the variation of probabilities that have been attributed to the events.

Calculations provide an answer to that question.

The probabilities of three events have been changed : solidarity between the E.E.C. countries, the withdrawal of the Government, the balance of prices favourable to animal productions. We then examine what are the consequences of that, in terms of occurence probability for the various sub-scenarios. For instance, the results collected after changing the first event probabilities are as follows :

Two possibilities have been considered. The first one is an increase in the solidarity probability; that probability increases from .4 to .5; the second one decrease s , falling to .3. We obser ve that when there is an increase -the probability of lhe pessimistic suh-scenario diminishes (15 % against 16.5 %) -the probabilit y of the optimistic sub-scenario increases highly ( 21 % against 16 %) lf there is a decrease: the probability of the pessimi s tic sub-scenario increases (18,5 % a ga inst 16,5 %) -the probability of the opti mistic sub-scenario decreases very much, from 16 % to 5 %. That sub-scenario does not appear any more in the tendenlial core.

Importance and strategy of the acti ve participants

Within the framework delimited by the characteristic events of each scenario or sub-scenario, the future of the . region is not tot~lly defined. History and observation have taught us that we must also take into account the action of economic agents and social groups in contact. Taking into account the active participants ' performance is one of peculiarities of the scenarios methods presented herein.

The table of active participants

After a debate between experts, a list of active participants is composed; it gathers together those which are the most likely to intervene in the agricultural change of the region. It features truly economic categories {farrns, cornpanies, etc .• ) as well as administrative or professional institutions (unions, chambers of commerce, trade, agriculture and industry .. ) and currents of thought or opinion ( political parties). The econornic dimension is predominant, but the other factors {social and cultural ) are also taken into consideration. Sorne groups are essentially local, while others, with a national scope , are herein considered for their regional implications.

From the sarne experts' judgement, a "weight" from O to 4 is affected to each sector ; that weight is equivalent to the relative importance it is granted within the considered scenario. It is a first approach to the role of each economic or social force in contact.

Within the dornestic and international environrnent as defined in each scenario, the active participants aim to orientate the evolutions in accordance with their best interests or their principles, and agreements and disagreernents will arise between them, whether they defe nd simi l ar positions or not. To reveal the alliances and conflicts which are susceptible to occur henceforth, a series of potential tendencies are listed, which are active in the scenario (e.g. productivism, Government withdrawal, European solidarity ..•. ) and each active participant is placed in relation to those tendencies, through a numerical grading. The chosen notation, the result of a debate between experts in the agricultural economics of the region, expresses neutrality or indifference {O), strong or very strong hostility ( -1 and -2 ) or, on the contrary an active or very active support (+1 and +2 ) (Table 2).

Hierarchical classification of the active participants

The objective is to work out a typology for the active participants, in order to disclose opposed or combined strategies. Because of the variable balancing, the individual, as meant in the factual analysis, is not an active participant, but the amount of power which is materialised. Lerman's method for hierarchical classification, which is being used, uses the algorithrn of likelihood of the link [START_REF] Lerman | La classification caractéristiques d'une méthodologie données[END_REF] which allows for the grouping of the active par ticipants in a decreasing proximity order.

The algorithm looks f or a classification which, with a limited number of classes, would form the "best" global synthesis, i.e. the "best" typology of strategies. The global statistics, marked I: , expresses the classification quality at a given level, the firsl level indicating the part for which each individuel forms a class, and the last one the single class which gathers togethet all individuals. Hence, for scenario A 11 Table 2 . Weighl and behaviour of active participant~ '

• A) "' .:cenar10

. . •2

• 1 •2 -l •2 • l 0 -2 Î .. or'-forct- J • 2 -1 •2 • l • 2 •2 •2 -2 12. MQO,,,rn1sf'd follfll 1 lt fa r-s •llP'I o ff. qround c,0 11t 11 nanl 1 !. fllcektn1st'd r...,11~ •ai-s w1 tt'I J •l -2 0 0 .. , •2 •2 -1 C'a l l lt a "° aranl" <1~1n•nt l -l -2 -2 -2 0 .. , •2 • 1
1.:.. ~l l 1t"ulturaJ fa m s ,,, !.,,.;bu t.l VIC't' 91'1d p lU l -ll "'f" f • r-.s 2

:-1 -1 • 1 -1 -l 0 0 •l 16. Aqr , c:1.1J tur-a l ~J o,er • ù 11. ~st.0c 1at 1~tfar111s 1 • J -1 -l -1 •l • J • 2 0 11 . .. 10Cl l f' -Sut" CQlftPan 14'!t •llh rrÇJO"la ) ] •I 0 0 •2 •l •l • 1 • 1 22. Hv ll1Dutposr c-OOPC'tall ... l'!, "° J • ] -l •l •l •l •2 •1 0 11. Sctec:1ahsec ('OOOrrn i-l's 3 •2 -1 -1 •2 •2 •2 •2 -1 20. ,at , ona t f I r"'s l • 2 •1 • 1 •2 •2 -1 • I -2 H. H-Jl l 1nal1 ortal r I rms 3 +2 •2 • l • 2 •2 -2 -1 -2 z~. ~~lili I nf'l • OTlo ) +2 •2 0 •2 (2 -1 •l -:.i 11 \.r,,)-ar;r 1cu llur11 ) Kl 1 , t 1nh.Jb1tants 1 •o 0 •l 0 0 0 0 • I ... '>t-co,"l(jar~ res10~"l& anc pt'-1\s. ,o,wrs 1 -1 -_, +I -1 -1 0 0 •2 . ,. 1 JSl'll"G ana Nr Il ) lllf-a('\ I\ lt\ 1 0 -1 0 0 •1 •2 • 2 0 .z. \ t("onoa r ~ U C' tOr 2 •I 0 •I • 2 •2 0 0 -1 -'• lour J.S"• 2 -2 -1 • 1 -1 0 0 0 12 --. ~t "'t"r trrl i•t \ S.f'CIOrS 2 0 • l 0 0 • l 0 0 0 _ ,;. l ! r e l , on , r ~ ar.o ao\anc ro t • e-nnoJoq> 2 • 2 -1 • l • 2 •2 •I • I -1 ". ""il JOrl h t raOl"'•uf'l1on5 r ,~t ~-r,Jil J •I -2 • 1 -1 • l • 2 +1 -1
.. , ;.ip'l(Hl l , l radf'-vr-10ns

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• ~rs~•Hel"I ,na lra1 n1nq 1nst 1tut ions 2

• I -1

0 • I •2 • I .. 1 0 • •- ... J.,, .. r-;,roff"<S:s 1onal org;r,u1t1ons 2 •l -1 +I • l • 2 •2 •2 -2 (,~. { ti.ar.'IC>t l S Of C~f't"t'f' l'ld 2 +2 •2 f'•;l l o~ir r ~ • as soc, a t ions • 2 • 2 •2 -1 0 -2 " '-Or lo lP'r'-a nô ~molo\.-f'S un10f'l1 1 • l •1 -1 0 0 u ., -• l -,;. . '-On-a qr 1c• •t,.,,.~1 t>ank , n.o 5tttor 1 •2 +2 •1 •2 • 2 -2 •I -2 ,,. (o"s~rs orqar,1sal 1ori5 1 -1 -1 +1 -1 0 -1 -1 0 ,,. locïtl ;;md t f'q1onal po,,ot"r5 ~ •l -1 -1 •1 +1 •1 •2 -1 81. lnt • ! Jec tuai ttJwPrll("nls ] -1 0 +l -1 •1 0 0 ... 2 81. Cvlturtl lr'IOw~nts 1 -2 -1 0 -1 -l +I 0 .. 2 8l. R• l1q1ov,; a,o..,~t • J 0 -1 -1 0 • 1 • 2 0 •1 60. Cc-ol0q1sl 1!10\IPIM'n t 1 l -2 -2 0 -2 -1 0 -2 •2 e~. Rrq10f'lit l nl IIIQw~ls 0 • 86. PrPSS , ""'di8 l • 1 0 •I • J +2 +I • I -1 8 7. R1Qht -1hq po l 1t1eaJ p•rlte-5 l •2 0 •I +I •2 •1 0 -1 ee. ltfl • 1"Q 001 11 1.eal (:lar tie-i. 1 • 1 -2 0 -l • 1 +I •l . ) -• ... Lot"• J 11nd rl"q1ona l il!Ô"'\& nis\ r•t 1oni ] •l -1 0 +1 • 2 • l . ] • l 92. r ~"t r.i l •dfll1n1 s t r.it , an 2 0 •2 •2 •l -) -1 •I • 1 • l. l.l .C' . ) .. 1 •l +I • 2 •2 •2 • 1 -1 ... l nt ern.iil •O"'al orqan1s al10n-Lr.0 • •-•--• -•• 0 -
with balanci~a out of the active participants, the classification related to level 29(E =26.2) is "better" than that of level 31 ( E • 21.6) .. That level 29 classification can be represented as a 5 branch tree (diagram 1).

Local statistics are also given, the value of which increases when a class in the process of being formed is completed; it decreases afterwards. The peak is recorded as an indication of the level of class completion, and the knot of the associated tree is called "significant" (Lerman, 1981 b). lt must be ment ioned also that each active participant is, in that method, given a dispersion index ; the active participants with the highest index are those which frame best the group in which they belong. In the scenario A example , five classes of active parti cipants, whose behaviours are relatively similar, have been selected. They are the following (in brackets are the most significant categories in each class) Group 1 -Group II -Group III Group IV -Group V "Liberal-producti vist" (multinational firm s) 11 Manager of the system" (Services and development organisations) "Regional dynamics" (Specialised coperatives ) "Protection of family-farming" (Minority uni ons, cattle/arable farms) "New rurals'' (cultura l and ecologistical movements)

"Objective" alliances are hence formed within the various classes they enlighten the action of economical agents and they show how their influence can be enha nced by the synergies crea ted to defend neighbouring positions. At the same time, appurtenance lo different sub-groups may foreshadow opposition or even deadlocked situations between groups with divergent interests.

It is the confrontation between the group of active participants' strategies, the consequences of the selected events and the dynamics of each major cultural factor which lead to the construction of scenarios as scheduled.

4. Framework of the scenarios and sub-scenarios. Connection between events, active participants and variable factors.

The matching of variable factors, events and classes of actors is achieved through the constitution of a double entr y table in which the major variable factors are set in columns, the events and active participants along lines (Table 3, as an exemple for the tendencial subscenario A1) . The selected variable factors are, excepted a few, those which have appeared as the most important active factors in t~ê structural analysis matrix.

•ne.

... ,_...., iÇ"79' ,;

• Signifi cant knots .....

•••" ••-l.lM -• • - t . n1
.....

•• 111 !-f.!.:!!~ 1 '•"' ••-..... .... , ...... ••N• 1 .... , • •.n, 1
.. , .. ..... :: ':' ;; ~ ,., > i:l !: .., !2 :,: .., .., .., .

~ '< 1J • ..... • '° '< • ..., , .,. ;r (tJ DJ r
.

1 • • 1 .. 1 . ' 1 ' ' •

Relation between events and variable factors

All the events forming the sub-scenario are taken into account (certain events and chance events, the occurence of which defines the subscenario), and we investigate what influence they are likely ta exert on the structural factors. Three modes only are selected:

A positive influence : the occuring event tends ta increase the value of the factor under consideration; -a negative influence: the occurence of the event causes a decrease of the factor under consideration -a neutral influence, when the event only exerts an action considered as negligible on the evolution of the factors.

A matrix of correspondence is hence made up, with one part common ta all the variants of a scenario (certain events) and a part specific to each sub-scenario (chance events). the summing-up of the positive and negative incitations received by each variable factor is an indication of the general orientation it tends to take in the interested sub-scenario. The interpretation of that "general tendency" vector is ta be handled \vi th care, for the implicit hypothesis is made, that the weight of each negative or positive influence is equivalent. But it is a means of differenciating scenarios and sub-scenarios and of ascertaining the situation of the groups of active participants. 42. Relation between active participants and variable factors, and possibilities for the groups of active participants.

Ta carry out their projects, the various categories of active participants will act on the factors they control. Whether they choose ta privilege or diminish their importance,+ and -marks will be attributed ta the corresponding factors. Jn table 3 it is specified, for instance, that "liberaJ-productivists" are in faveur of an increase in integrated production, or in the size of Food Production Companies, and that the "new rurals", on the contrary, are opposed la integrati~n or farm specialisation.

That second reflexion guide also provides indications on the degree of coherence between the active participants' aspirations and the general characteristics of the scenario ; their coherence can effectively be very uneven, according ta the groups, between the variable factors on which they aim ta act and the general orientat ions which came out of the occurence of the events. The measuring of such gaps is done through comparing the target-factor of each active participants'group with the general tendency of the sub-scenario. The wider the gap, the more difficulties will the interested group encounter in carrying out its projects.

Framing of the relations between events/active participants/factors

Beyond the simple comparison between the privileged actions of the classes of active participants and the general tendencies resulting from the impact of events, a factorial analysis of theconnections permits one outline the frame of the table of the influences exerted by events and active participants on the variable factors . It is hence necessary, in order to do that, to change the symbolic coding (+, blank or-) into a logical (or binary) coding. The variable factors are those of the influences table and the observations are composed of certain and chance events and categories of active participants.

Here will be presented the results obtained from the first two charts. According to diagram 2 (Axes 1 and 2), it appears clearly that the first axis contrasts on one side the "depressive" events and factors, i.e. those related to a regression of agricultural and food regional activity, on the other, the events and growth factors associated with the rtsreton Model". The second axis, for its part, distributes variable factors, active participants and events according to their position in relation to Government intervention.

On the whole, the space delimited by the first two axes can be divided into three large zones -A recession zone, where are gathered the assured events of scenari o A ( latent crisis scenario), two chance events of sub-scenario A1 (absence of major innovations, lack of dynamics on the part of Food Production Industries) and the variable factors, agricultural or not, corresponding to a very limited economical growth in its whole as in regional agriculture. No group of active participants is present in that zone.

-A zone of important growth, with Government support, where are gathered such variable factors as the increase in agricultural population, of the number of farms, of agricultural revenues or of s upport to animal productions. The event "Government withdrawal" can be found there, together with the group of family farming defenders.

-A zone of important growth in a liberal context, meeting the desires of two classes of active participants : "Liberal-productivists" and "regional dynamics" ; the corresponding variables are those related ta the development of the highly specialised Breton agricultural model.

-Axis 3 introduces another part contrasting a purely agricultural pole with an economic a nd FPI pole. One can then define within that second factorial chart, four sub-assemblies composed respectively of agricultural criteria for growth or recession, and environmental features linked to the crisis continuance or a re-start of national production.

Crowth pol e l!>

Diagram 2

Func tional anal ys i s of co r respondancesa -Re l a t ion between events -ac t ive partici pants-va r iable factors Ax i s 2 l nter vent i oni s t J pol e St rong growth with gove r nment s timulation Those are additional elements of thought illustrated in a peculiar application but which, in their coherence, tend ta confirm the methodological validity of thal approach, which consists i n the building up of a matrix of correspondence between events/active participants/factors ; they make up all the malerials to wor k out possi ble sketches for agriculture and Food Production Industries in t he region, for the 2000's.

CONCLUSION

Considering the insufficiencies and limitations of the classical method for economic forecasting, even more obvious within the curr ent period of changes and uncertainties, the prospective approach seems to be the most appropriate to answer questions about the evolution of a fastevolving regional, agricultural system.

The result of prospective study, if they do not permit one to foresee the future, provide elements for those who will have to make t he future : farmers, of course, but also planning experts, Food Production Companies, regional communities, etc .•. The advantage of scenarios is to display all the factors which drive regional agricultural economics (starting situation, role of active participants, influence of external events, not forgetting technical progress) and to assess the connection between all those elements in order to offer a coherent pict ure, global and operational for the 2000's, following the selected hypotheses, and to evaluate the degree of liberty enjoyed by the various categories of active participants, interested parties in the regional development, ta carry out their projects.

One should note that the scenarios melhod has, when experienced, certain limitations. Apart from the fact that the process implies the listing of events and the evaluation of their probabilily, one cannot multiply the variable factors, for one must be able to understand the results. To the crossing of chance events within the economic space, ar e superimposed possible crossings of chance events intime, and it then becomes difficult lo situate the scenarios in relation to one another. As a malter of fact, they could be extended by the building up of economic models. It is however difficult, at the present lime, to f ind examples for such a symbiosis. One can easily undersland, nevertheless, the possibility of building up one model for each scenario, the latter guaranteeing the former its validity through a combination of possible and coherent hypotheses on the explanatory variable factors.

  Min ~ . (( p ( e . / e . )

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~ 1

 1 Peid-workere r • r~e _ _ _ _ _ -1..._..i....:=.:.

  .. .. f!: ! ! "' % "' ~ "' ~ ~ M i q "

•

  -group s of ac t ive par tici pants *-agr icu ltur al va r iab lJ + wi t h a tendency Lo factors r l ~e •,foorl 1ndustry varia-• wlth no tendency • ble factors ! -~i th a tendency to ' environment al variab e

Table 1 .

 1 Classification of variable factors

		Motorial factors (active)
	Factor		Direct	Indirect
	type	Factor headings	Effects	Effects

.

' . . . .. This article develops the major methodological aspects of this prospective study, The proposed proceedings, partly new as to the method in use, find their originality in their application to a problem of regional economics,


Yves Lespinasse

Yongxiang Zhang

Vitro In

Androgenesis

In vitro androgenesis in cultivated apple (Malus x domestica Borkh)

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.