Thai Ha-Huy

Subjective Expected Utility on Arbitrary State Spaces *

Keywords: Savage theorem, sure-thing principle, Independence axiom, expected utility representation. JEL Classification: D11, D90

1 Introduction

Motivation and result

After seven decades since the publication of "The foundation of Statistics," [START_REF] Savage | The foundation of statistics[END_REF]) and scholars continue to admire this tour de force, calling it a "crowning glory of decision theory". 1 Based on a list of seven axioms, this legendary contribution reconciled the ideas of de [START_REF] De Finetti | La prévision: ses lois logiques, ses sources subjectives[END_REF] about subjective probability and those of von [START_REF] Von Neumann | Theory of Games and Economic Behavior[END_REF] about expected utility behavior. The proof of Savage does not rely on the mathematical structures of de Finetti or von Neumann -Morgenstein, which are crucial for using the separate theorem in convex analysis.

Besides the famous and perhaps most discussed Savage axiom, the sure-thing principle, the structure of Savage's world also relies on "technical axioms" P 6 and P 7 which ensure a continuum nature on the set of states. This continuum property allows Savage to divide the set of states by two equal parts and, then by four, eight, sixteen and so on. We can therefore measure every event by passing through the limit a sequence of events with probabilities in the form k 2 n , with 0 ≤ k ≤ 2 n .2 Naturally, efforts have been done to extend the result of Savage to configurations that encompass the possibility of atoms, events that can not be divided into smaller non-null ones. It is not surprising that we must make a trade-off between the richness of the set of states and the set of outcomes. The existence of atoms is problematic.

The seminal contribution of [START_REF] Gul | Savage's Theorem with a Finite Number of States[END_REF] proposes an idea initiated by [START_REF] Ramsey | Truth and probability[END_REF], assuming the existence of a pair event, an event that the decision maker believes has the same chance to occur as its complementary one. This choice is clearly intuitive: if a subjective probability exists, the probability of this event should be 1 2 . [START_REF] Gul | Savage's Theorem with a Finite Number of States[END_REF] followed the approach by [START_REF] Ramsey | Truth and probability[END_REF] rather than Savage and defines the probability of an event through a utility function. Instead of the sure-thing principle, [START_REF] Gul | Savage's Theorem with a Finite Number of States[END_REF] assumed an Independence property, which states that combining two acts with a third one using a non-null event does not alter the comparison between them.

This article is a studiy of Savage's theorem in a configuration where the continuum property of the set of states may not be satisfied. We proposed a weaker condition used in the literature with three considerations. First, there is no restriction on the set of states. It may be finite or infinite, contain atoms or not, countable or continuum. Second, we present a weakened version of Independence axiom, requiring its verification with one special non-null event. Third, we do not assume the existence of a pair event.

The main idea of our approach is as follows. Given every two outcomes x and y, using the event in Independence axiom, we construct an outcome that can be considered their midpoint, or equivalently, an outcome that represents the an equiprobability of 1 2 to x and y. Following the construction midpoints between x, y and their midpoint, and so on, we can find the equivalent outcome of every distribution that attributes a probability of the form k 2 n to x and 1 -k 2 n to y. For a given event, a lottery that wins x if that event occurs and y if not, is equivalent to the limit of a sequence of midpoints. Hence, the probability of that event can be determined. We can then prove the existence of an expected utility representation of the comparison between acts.

Related literature

For a finite number of states, [START_REF] Kraft | Intuitive Probability on Finite Sets[END_REF] and [START_REF] Scott | Measurement strutures and linear inequalities[END_REF] considered cancellation as a necessary and sufficient condition for the existence of a probability measure. [START_REF] Kraft | Intuitive Probability on Finite Sets[END_REF] also gave a counterexample to prove that the additivity is not strong enough for positive answer when the number of states is bigger than or equal to 5.

For an infinite number of states, [START_REF] Chateauneuf | Archimedean qualitative probabilities[END_REF] and [START_REF] Chateauneuf | On the existence of a probability measure compatible with a total preoder on a boolean algebra[END_REF] considered the problem under the Archimedean property and prove that this condition is sufficient for the establishment of a probability measure. The curious readers can refer to the excellent reviews of [START_REF] Fishburn | The axioms of subjective probability[END_REF][START_REF] Fishburn | Foundation of decision analysis: along the way[END_REF] and [START_REF] Mackenzie | A foundation for probabilistic beliefs with or without atoms[END_REF].

The article by [START_REF] Gilboa | Maximin expected utility with non-unique prior[END_REF], a "second Big Bang after [START_REF] Savage | The foundation of statistics[END_REF],"3 opens a huge body of literature for the configurations where Savage's famous sure-thing principle is not satisfied, and paves the way for numerous contemporaneous developments of the ambiguity literature. For a survey, see [START_REF] Etner | Decision theory under ambiguity[END_REF].

To the best of our knowledge, the idea of establishing a midpoint without passing through a pair event appeared first in [START_REF] Ghirardato | A subjective spin on roulette wheels[END_REF] and is in application after by [START_REF] Ghirardato | A general theory of subjective mixture[END_REF]. In these two contributions, the authors used an essential event to determine a midpoint of two given outcomes, which they called a preference average. Applying this idea to an order that represents biseparable references, the midpoints serve to construct probability mixtures of acts. They can therefore enjoy a rich structure similarly to [START_REF] Anscombe | A definition of subjective probability[END_REF] and [START_REF] Gilboa | Maximin expected utility with non-unique prior[END_REF].

Without relying on the continuum nature of the set of states neither of the set of outcomes, [START_REF] Mackenzie | A foundation for probabilistic beliefs with or without atoms[END_REF][START_REF] Mackenzie | On atoms and event richness in Savage's model[END_REF] studied different atoms swarming properties to establish probabilistic beliefs. His results did not require the usual conditions such as no atoms, cancellation, or solvability. In our opinion, atoms swarming conditions may be considered a beautiful bridge linking the configurations with discrete and continuum states.

Adding a continuity property may allow us to change some conditions' statements from "for every" to "exists," as shown in the weakened Independence property of this article. An interesting discussion about this situation may be found in the article by [START_REF] Segal | ∀ or ∃?[END_REF].

Organization

The article is organized as follows. Section 2 presents the fundamental notions. Section 3 describes the establishement of the sujective probability and expected utility behavior. All proofs are given in the Appendix.

Fundamentals

Let S be the set of states and A an algebra of events on S. The set S can be a continuum, discrete, atomless, or hybrid type which contains continuum subsets and atoms. For each event A ∈ A, let A c be its complementary event:

A c = S \ A.
The set of outcome X is endowed with a topology τ . Denote by F 0 the set of measurable finite-value acts from S with algebra A to X.

F 0 = {f : S → X such that f is measusable and f (S) is finite} .
For a measurable partition A 1 , A 2 , . . . , A n of S, outcomes x 1 , x 2 , . . . , x n in X, denote by x 1,A 1 x 2,A 2 . . . x n,An the act that obtains outcome x k for s ∈ A k . For example,

x A y A c denotes the act that takes value x if s ∈ A and value y otherwise. In the same spirit, for every acts f and g, f A g A c denotes the act that is equal to f (s) on A and g(s) on A c . Let P 0 be the set of finite support probability distributions on X. For p 1 , p 2 , . . . , p n belonging to the interval [0, 1] such that n k=1 p k = 1 and x 1 , x 2 , . . . , x n ∈ X, let (p 1 : x 1 , p 2 : x 2 , . . . , p n : x n ) be the random distribution on X which takes value x k with probability p k .

The comparison between acts is defined by an order . The strict and equivalent comparisons are denoted by and ∼, respectively. Before establishing some fundamental properties on order , let us define some notations. An event A is called a null-event if for every outcomes x and y, act h, we have

x A h A c ∼ y A h A c .
The outcomes set X can be considered the set of constant acts and a subset of F 0 .

For a given sequence of outcomes {x n } ∞ n=0 , we use the notion convergence in to an outcome x, in the sense that for every y x z, there exists N such that for n ≥ N , y x n z. We use this notion to distinguish the convergence with respect to the topology τ of X. Throughout this work, when we say that some sequence converges to some outcome, we always refer to the convergence in .

Axiom F1. Fundamental i) Completeness, transivitity, monotonicity and non-triviality For every f, g ∈ F 0 , either f g or g f . If f g and g h, then f h. For every non-null even A

x y if and only if

x A g A c y A g A c .
There exist z, z ∈ X such that z z.

ii) Weak comparative probability For any A, B ∈ A and x y, x y ,

x A y A c x B y B c if and only if x A y A c x B y B c .
iii) Continuity For any x ∈ X, the sets {y ∈ X such that y x} and {y ∈ X such that x y} are closed with respect to the τ -topology. Moreover, the space (X, τ ) is connected and separable.4

Conditions (i) and (ii) are standard in literature and are presented in the contribution by [START_REF] Savage | The foundation of statistics[END_REF]. Condition (iii) assumes a continuum property on the set of outcomes.

Thanks to Weak comparative probability property, we can define an order on the set of events as follows, A B if there exists x y such that

x A y A c x B y B c .
The comparison using does not depend on the choice of outcomes x and y.5

3 Expected utility representation 3.1 Independence axiom and the determination of the midpoint

The first purpose of this section is to provide a mixture of acts that replaces the lack of linear structure being in use in [START_REF] Anscombe | A definition of subjective probability[END_REF], and [START_REF] Gilboa | Maximin expected utility with non-unique prior[END_REF].

Definition 1. For every acts f, g ∈ F 0 , a non-null event H, define the mixture of f and g through H an act f satisfying: for any s ∈ S,

f (s) ∼ f (s) H h(s) H c .
We denote by Hf + H c g a mixing act of f and g through H. One must avoid confusing Hf + H c g with f H g H c . The former can be considered as a convex combination act of f and g with weighted parameters defined using the event set H, while the latter is an act equal to f on H and to g on H c . From now on, we always assume axioms Fundamental and Independence on the order .

Axiom A1. Independence There exists an event H such that both H and H c are non-null and for all acts f, g, h ∈ F 0 ,

f g if and only if Hf + H c h Hg + H c h.
Mixing f and g with h using event H does not change the comparison between f and g. The intuition is clear once a probability measure µ on the set of events is established. The mixing act between f and g through H is the act µ(H)f + µ(H c )g that corresponds each state s to a lottery that gives f (s) with probability µ(H) and g(s) with the probability 1 -µ(H). The independence axiom condition therefore

requires that f g if and only if µ(H)f + µ(H c )h µ(H)h + µ(H c )h. A more
detailed intuition is presented in [START_REF] Gul | Savage's Theorem with a Finite Number of States[END_REF]. It is worth noting that here we require Independence for only one event H, instead of every non-null event.

The relation between the independence axiom and the sure-thing principle is an important question. In [START_REF] Gul | Savage's Theorem with a Finite Number of States[END_REF], where the states space S is finite, independence implies the sure-thing principle. Proposition 1 states the same conclusion for a general set of states S.

Proposition 1. The sure-thing principle is satisfied. For every non-null event A, acts f, g, h, h,

f A h A c g A h A c if and only if f A hA c g A hA c .
Under Independence, using event H, we can determine the midpoint of two outcomes x and y, which can be considered an equivalent outcome of uniform distribution. This allows us to overcome the lack of a pair event, as proposed in [START_REF] Ramsey | Truth and probability[END_REF], and used by [START_REF] Gul | Savage's Theorem with a Finite Number of States[END_REF] in a configuration where the set S is finite.

Lemma 1. For events x and y, define sequences {x n } ∞ n=0 and {y n } ∞ n=0 as follows. Let x 0 = x and y 0 = y. For every n ≥ 0, let

x n+1 ∼ x n,H y n,H c and y n+1 ∼ y n,H x n,H c .
Then both {x n } ∞ n=0 and {y n } ∞ n=0 converge to an outcome z that we denote as

z ∼ 1 2 x ⊕ 1 2 y.
Let us present some intuition for the definition of the midpoint and the result of Lemma 1. Assume for an instance that a subjective probability has been established.

Let p be the probability of event H. For every n, outcomes x n and y n are equivalent correspondingly to distributions (p n : x, (1 -p n ) : y) and ((1 -q n ) : x, q n : y), with p 0 = 1, q 0 = 0 and for n ≥ 0,

p n+1 = p × p n + (1 -p) × q n , q n+1 = (1 -p) × p n + p × q n .
Assume that x y and H H c . By induction, x n y n , p n ≥ q n and p n + q n = 1 for every n. Since {p n } ∞ n=0 is decreasing and {q n } ∞ n=0 is increasing, they converge respectively to p * and q * and p * ≥ q * . This implies p * = q * = 1 2 . Hence, the sequences {x n } ∞ n=0 and {y n } ∞ n=0 have the same limit, that is an equivalent outcome of distribution 1 2 : x, 1 2 : y and this outcome can be considered a midpoint of x and y. The cases H c H or y x are similar.

We can therefore work with distributions of the form 1 2 : x, 1 2 : y without relying on a pair event set. Following [START_REF] Debreu | Mathematical economics: Twenty papers of Gerard Debreu[END_REF], 6 Theorem 1, chapter 9, as [START_REF] Gul | Savage's Theorem with a Finite Number of States[END_REF], we obtain a utility function u that represents the comparison between these special distributions.

Lemma 2. There exists a utility function u such that for every x, y, x , y ∈ X,

1 2 x ⊕ 1 2 y 1 2 x ⊕ 1 2 y if and only if 1 2 u(x) + u(y) ≥ 1 2 u(x ) + u(y ) .
The function u is unique up to a strictly increasing transformation.

Expected utility representation

We use midpoint z as an equivalent outcome of distribution 1 2 : x, 1 2 : y . The construction of equivalence of distributions on the set {x, y} is intuitive. Using 1 2 x ⊕ 1 2 z, we obtain an equivalent outcome of the distribution 3 4 : x, 1 4 : y . Similarly, 1 2 z ⊕ 1 2 y represents 1 4 : x, 3 4 : y , and so on. Continuing with this line of reasoning, we can create equivalent representations of every distribution of the form k 2 n : x, 2 n -k 2 n : y , for 0 ≤ k ≤ 2 n . Taking the limits when n converges to infinity, we construct the representation of every distribution which takes at most two values: p : x, (1 -p) : y , with x, y ∈ X and 0 ≤ p ≤ 1. Moreover, if z is equivalent to p : x, (1 -p) : y , then

u(z) = pu(x) + (1 -p)u(

y).

Proposition 2. There exists a unique probability measure µ and a utility function u such that for every events A, B and outcomes x, y, x , y ,

x A y A c x B y B c is equivalent to µ(A)u(x) + 1 -µ(A) u(y) ≥ µ(B)u(x ) + 1 -µ(B) u(y ).
Once the subjective probability has been established, an act f = x 1,A 1 x 2,A 2 . . . x n,An , with a partition A 1 , A 2 , . . . , A n S, can be considered an equivalence of the distribution (µ(A 1 ) : x 1 , µ(A 2 ) : x 2 , . . . , µ(A n ) : x n ). Therefore, the comparison of distributions can be realized through equivalent outcomes. We obtain Savages's theorem in a new context without relying on the continuum nature of the set of states.

Theorem 1. There exists unique finitely additive probability measure µ and unique utility function u (up to a strictly increasing affine transformation) such that for every acts f and g belonging to F 0 :

f g if and only if S u (f (s)) µ(ds) ≥ S u (g(s)) µ(ds).

Appendix

Proof of Proposition 1

Consider a non-null event A. Assume that for acts f, g, h, we have

f A h A c g A h A c .
We want to prove that for every h ∈ F 0 , f A hA c g A hA c . First, we prove the following claim: if there is some ĥ ∈ F 0 such that for every

s ∈ S, h(s) ∼ h(s) H ĥ(s) H c , then f A h A c g A h A c if and only if f A hA c g A hA c .
This claim is a direct consequence of the following two:

i) f A h A c g A h A c if and only if f A hA c (Hg + H c f ) A hA c . ii) f A hA c g A hA c if and only if f A hA c (Hg + H c f ) A hA c .

First, we prove (i). Indeed, from Independence axiom, f

A h A c g A h A c if and only if H (f A h A c ) + H c f A ĥA c H (g A h A c ) + H c f A ĥA c ,
which is equivalent to Now, consider (ii). Using once again the Independence axiom, f A hA g A hA c if and only if

f A hA c (Hg + H c f ) A hA c .
H f A hA c + H c f A hA c H g A hA c + H c f A hA c ,
which is equivalent to

f A hA c (Hg + H c f ) A hA c . Hence f A h A c g A h A c if and only if f A hA c g A hA c
. The claim is proven.

Fix outcome x such that z x z. We will prove that

f A h A c g A h A c if and only if f A x A c g A x A c .
Indeed, consider the sequence of acts h 0 , h 1 , . . . , h n , . . . ∈ F 0 defined as

h 0 = h, h n+1 = Hh n + H c x for n ≥ 0.
Using the same arguments as the case h = Hh + H c ĥ, we have

f A h A c g A h A c is equivalent to f A h 1 A c g A h 1 A c , which is equivalent to f A h 2 A c g A h 2
A c and so on. By induction, for every n, f

A h A c g A h A c is equivalent to f A h n A c g A h n A c .
The sequence of acts {h n } ∞ n=0 converges to x, in the sense that for every y x z, there exists N such that for n ≥ N , s ∈ S, y h n (s) z.

Let n be sufficiently big,

h n (s) H z H c x h n (s)z H c ,
for every s ∈ S. This implies the existence of h * ∈ F 0 such that for every s ∈ S,

h n (s) H h * (s) H c ∼ x.

This is equivalent to Hh

n + H c h * ∼ x. Hence f A h n A c g A h n A c is equivalent to f A x A c g A x A c .
Applying the same arguments for h, we get f A hA c g A hA c if and

only if f A x A c g A x A c .
The sure-thing principle is proven.

Proof of Lemma 1

Consider the case H H c and x y. Then we have

x 0 x 1 . . . x n . . . y n y n-1 . . . y 1 y 0 .

Let Z be the set of outcomes z such that x n z for every n ≥ 0. This set is closed with respect to the topology τ of X. Let Z c be the close envelope of the open set

Z c . The connectivity of X implies that Z ∩ Z c = ∅. Fix z ∈ Z ∩ Z c
. This outcome is found on the boundaries of Z and Z c . We will prove that z can be considered as an infimum of the sequence {x n } ∞ n=0 , in the sense that for every outcome y z, there exists n such that y

x n z. Indeed, the contrary implies that z belongs to the interior of the set Z: a contradiction. Using the same argument, we prove the existence of an outcome z that can be considered a supremum of the sequence

{y n } ∞ n=0 .
To finish the proof, we must prove z ∼ z . Assume the contrary, z z . Then H and x y, we use the same arguments, with the only observation that the sequences {x 2n } ∞ n=0 and {y 2n+1 } ∞ n=0 are decreasing, the sequences {x 2n+1 } ∞ n=0 and {y 2n } ∞ n=0 are increasing. They converge all to a midpoint of x and y. The case y

z z H z H c z .
x now becomes trivial. It is obvious that 1 2 x ⊕ 1 2 y ∼ 1 2 y ⊕ 1 2 x for every x and y.

Proof of Lemma 2

The proof of Lemma 2 is long and will be presented in two steps.

(1) Step 1: We prove that for every outcomes x, y, x , y , we have

1 2 1 2 x ⊕ 1 2 y ⊕ 1 2 1 2 x ⊕ 1 2 y ∼ 1 2 1 2 x ⊕ 1 2 x ⊕ 1 2 1 2 y ⊕ 1 2 y . (1)
(2) Step 2: We prove that for every outcomes

x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , if 1 2 x 2 ⊕ 1 2 y 1 1 2 x 1 ⊕ 1 2 y 2 and 1 2 x 3 ⊕ 1 2 y 2 1 2 x 2 ⊕ 1 2 y 3 , (2)
then 1 2 x 3 ⊕ 1 2 y 1 1 2 x 1 ⊕ 1 2 y 3 .
The proof of Step 1 requires two preparation statements. First, consider x, y, x , y , z, z such that

z ∼ x H y H c and z ∼ x H y H c .
Fix w, v, t ∈ X satisfying:

w ∼ x H x H c , v ∼ y H y H c , t ∼ w H v H c . Then we have t ∼ z H z H c .
Indeed, by Independence axiom, the mixture of acts x H y H c and x H y H c using H is equivalent to the mixture between z and z using H. We obtain:

t ∼ w H v H c ∼ H(x H y H c ) + H c (x H y H c ) ∼ Hz + H c z ∼ z H z H c .
Second statement: consider x, y, x , y , z, z such that

z ∼ 1 2 x ⊕ 1 2 y and z ∼ 1 2 x ⊕ 1 2 y .
Fix w, v, t ∈ X satisfying:

w ∼ x H x H c , v ∼ y H y H c , t ∼ 1 2 w ⊕ 1 2 v.
Then we have t ∼ z H z H c .

Let x 0 = x, x 0 = x , y 0 = y, y 0 = y, w 0 = w, and v 0 = v. Consider sequences Let sequence {t n } ∞ n=0 be defined as

{x n } ∞ n=0 , {x n } ∞ n=0 , {y n } ∞ n=0 , {y n } ∞ n=0 , {w n } ∞ n=0 ,
t n = w n,H v n,H c ,
for every n.

Using the first statement, by induction, we have, for every n,

w n ∼ x n,H x n,H c , v n ∼ y n,H y n,H c .
Let n goes to infinity, we have x n and y n converge to the midpoint z of x and y, x n and y n to the midpoint z of x and y , and w n and v n to the midpoint t of w and v.

Hence, t ∼ z H z H c .

We finish Step 1 in the proof. Consider x, y, x , y , z, z such that

z ∼ 1 2 x ⊕ 1 2 y and z ∼ 1 2 x ⊕ 1 2 y .
Fix w, v, t ∈ X satisfying:

w ∼ 1 2 x ⊕ 1 2 x , v ∼ 1 2 y ⊕ 1 2 y , t ∼ 1 2 w ⊕ 1 2 v.
We will prove that t ∼ 1 2 z ⊕ 1 2 z . Let w 0 = x, w 0 = x , v 0 = y, v 0 = y , z 0 = z, z 0 = z . Consider sequences

{w n } ∞ n=0 , {v n } ∞ n=0 , {z n } ∞ n=0 , {z n } ∞ n=0
being defined as:

w n+1 ∼ w n,H w n,H c and w n+1 ∼ w n,H w n,H c , v n+1 ∼ v n,H v n,H c and v n+1 ∼ v n,H v n,H c , z n+1 ∼ z n,H z n,H c and z n+1 ∼ z n,H z n,H c . Let t n ∼ 1 2 w n ⊕ 1 2 v n .
Since w n and v n converge respectively to w and v, t n converges to t ∼ 1 2 w ⊕ 1 2 v. By the second statement and induction, we have

t n ∼ z n,H z n,H c .
Since z n and z n converge to a midpoint of z and z , t n converges to t ∼ 1 2 z ⊕ 1 2 z . The claim in Step 1 is proven. Now, we begin Step 2. Assume (2). Consider the case where we can "decrease" y 1 and "increase" y 3 such that the comparisons in (2) become "equivalent". Precisely, assume the existence of outcomes y 1 and y 3 such that y 1 y 1 and y 3 y 3 , satisfying

1 2 x 2 ⊕ 1 2 y 1 ∼ 1 2 x 1 ⊕ 1 2 y 2 and 1 2 x 3 ⊕ 1 2 y 2 ∼ 1 2 x 2 ⊕ 1 2 y 3 . Then 1 2 1 2 x 3 ⊕ 1 2 y 2 ⊕ 1 2 1 2 x 2 ⊕ 1 2 y 1 ∼ 1 2 1 2 x 1 ⊕ 1 2 y 2 ⊕ 1 2 1 2 x 2 ⊕ 1 2 y 3 .
By (1), we obtain 1 2

1 2 x 3 ⊕ 1 2 y 1 ⊕ 1 2 1 2 x 2 ⊕ 1 2 y 2 ∼ 1 2 1 2 x 1 ⊕ 1 2 y 3 ⊕ 1 2 1 2 x 2 ⊕ 1 2 y 2 .
Since two sides of this equivalence relation share the common term 1

2 x 2 ⊕ 1 2 y 2 , 1 2 x 3 ⊕ 1 2 y 1 ∼ 1 2 x 1 ⊕ 1 2 y 3 .
Recall that y 1 y 1 and y 3 y 3 . We obtain

1 2 x 3 ⊕ 1 2 y 1 1 2 x 1 ⊕ 1 2 y 3 . (3)
Consider the general case with (2). Fix x such that z x z. For every outcome w ∈ {x 1 , x 2 , x 3 , y 1 , y 2 , y 3 }, let the sequence {w(n)} ∞ n=0 be defined as: w(0) = w, and for every n ≥ 0,

w(n + 1) = 1 2 w(n) ⊕ 1 2 x.

Proof of Proposition 2

Consider a construction of the following sequence z k,2 n , with n ≥ 0 and 0 ≤ k ≤ 2 n .

For n = 1, fix outcomes z 0,2 , z 1,2 and z 2,2 as:

z 0,2 = y, z 1,2 ∼ 1 2 x ⊕ 1 2 y, z 2,2 = x.
For n ≥ 1, 0 ≤ k ≤ 2 n+1 , fix the elements z k,2 n+1 ∈ X as:

z k,2 n+1 = z k ,2 n if k = 2k , with 0 ≤ k ≤ 2 n , z k,2 n+1 ∼ 1 2 z 2k ,2 n ⊕ 1 2 z 2k +1,2 n if k = 2k + 1, with 0 ≤ k ≤ 2 n -1.
The following assertions are intuitive and can be proven by induction. Assume that x y.

a) For every 0 ≤ k ≤ n, x z 2 n -1,2 n . . . z k+1,2 n z k,2 n . . . z 1,2 n y. b) For every 0 ≤ k ≤ n, u z k,2 n = k 2 n u(x) + 1 - k 2 n u(y).
c) For any event A ∈ A and n, there exists unique k n (A) such that:

z kn(A)+1,2 n x A y A c z kn(A),2 n .
Fix an event A ∈ A and outcomes x y, consider the sequence

{(k n (A), 2 n )} ∞ n=0 such that for every n, z kn(A)+1,2 n x A y A c z kn(A),2 n .
Observe that the sequence kn(A)

2 n n≥0
is increasing in respect to n. We may define the probability measure of A using outcomes x and y as

µ x,y (A) = lim n→∞ k n (A) 2 n .
However, the sequence {(k n (A), 2 n )} ∞ n=0 and the limit may depend on the choice of x and y. Under the Independence axiom, we can discard this possibility and prove that the value of µ x,y (A) is independent with respect to the choice of x and y. Now, we begin the main part of the proof of this Proposition. First, we prove the independence of µ x,y (A) with respect to the choice of x, y.

Fix any x * , y * ∈ X such that x * y * . Consider an event A ∈ A. Let p = µ x * ,y * (A), with the same construction of µ x * ,y * , using the sequences of z k,2 n .

First, we prove that under any other choice of x ∼ z k,2 n and y ∼ z k ,2 n with k > k : µ x,y (A) = µ x * ,y * (A).

The proof will be given by induction. Consider first the case n = 1. Consider z 1,2 and z 0,2 . Let x = z 1,2 , and y = z 0,2 = y * . Let p = µ x * ,y * (A) and p = µ x ,y (A). Since y = y * , using the first statement in the proof of Lemma 2, we get

x A y A c ∼ 1 2 (x A y A c ) ⊕ 1 2 y * .
Taking (4) minus (5), we obtain u(x) -u(y) µ A ∪ B = u(x) -u(y) µ(A) + µ(B) .

Hence, µ A ∪ B = µ(A) + µ(B).

Proof of Theorem 1

By Proposition 2, there exists unique probability measure µ and a utility function such that for every events A, B ∈ A, outcomes x, y, x , y ∈ X, the comparison Suppose that the assertion of the theorem is true for the acts which take almost n-1 different values, with some n ≥ 2. We will prove that it is verified for n different values.

For a partition {A k } n k=1 of S, let f = x 1,A 1 x 2,A 2 . . . x n,An , with {A k } n k=1 a partition of S. Fix a constant outcome v ∈ X. For 1 ≤ k ≤ n, let p k = µ(A k ).

We will prove that f v if and only if

n k=1 p k u(x k ) ≥ u(v).
Fix w ∈ X such that

x 1,A 1 x 2,A 2 x 3,A 3 . . . x n,An ∼ x 1,A 1 w ∪ n k=2 A k .
By the sure-thing principle property, replacing x 1 by x 2 , this implies

x 2,A 1 x 2,A 2 . . . x n,An ∼ x 2,A 1 w ∪ n k=2 A k ,
which is equivalent to p 1 u(x 2 ) + p 2 u(x 2 ) + p 3 u(x 3 ) + . . . + p n u(x n ) = p 1 u(x 2 ) + (p 2 + p 3 + . . . + p n )u(w).

Hence

u(w) = 1 n k=2 p k n k=2 p k u(x k ).
This allows us to deduce the value of f :

u x 1,A 1 w ∪ n k=2 A k = p 1 u(x 1 ) + n k=2 p k u(w) = n k=1 p k u(x k ).
We have f v if and only if

x 1,A 1 w ∪ n k=2 A k v, which is equivalent to n k=1 p k u(x k ) ≥ u(v).
For every acts f = x 1,A 1 x 2,A 2 . . . The proof is completed.

  Recall that,u(x A y A c ) = pu(x * ) + (1 -p)u(y * ), u(x A y A c ) = p u(x ) + (1 -p )u(y )

x

  A y A c x B y B c is equivalent to: µ(A)u(x) + µ(A c )u(y) ≥ µ(B)u(x ) + µ(B c )u(y ).

  x n,An and g = y 1,B 1 y 2,B 2 . . . y m,Bm , by considering v such that v ∼ g, one has f g if and only if n k=1 µ(A k )u(x k ) ≥ m k=1 µ(B k )u(y k ).

  The continuous nature of the set of outcomes X ensures the , with a direct consequence that z x n,H y n,H c ∼ x n+1 : a contradiction. Hence, both {x n } ∞ n=0 and {y n } ∞ n=0 converge to z. In the case H c

	existence of y	z and z	y such that z	y H y H c	z . For n sufficiently big,
	y x n z z	y n y

  {v n } ∞ n=0 being defined as:x n+1 ∼ x n,H y n,H c and y n+1 ∼ y n,H x n,H c , x n+1 ∼ x n,H y n,H c and y n+1 ∼ y n,H x n,H c , w n+1 ∼ w n,H v n,H c and v n+1 ∼ v n,H w n,H c .

See Kreps (1988).

The proof of Savage is long and complicated. For a more simple proof and better explainations, seeGilboa (2009), chapter 10, and[START_REF] Abdellaoui | Savage for dummies and experts[END_REF][START_REF] Abdellaoui | Savage for dummies and experts[END_REF].

See Karni et al. (2022).

We can not split X into two disjoint closed subsets, and there exists a countable and dense subset of X.

See Gilboa (2009), Chapter 10.

This is a posthumous publication of the article "Topological methods in cardinal utility theory" by Gerard Debreu, Mathematical Methods in the Social Sciences, 1959.

to sends many thanks to Xianguy Qu for helpful comments.

He acknowledges the Labex MME-DII (ANR-11-LBX

This implies u(x

Hence p = p , or µ x ,y (A) = µ x,y (A).

For the case of the choice x and z 1,2 , we use the same arguments. The conclustion is immediate for x = x * , y = y * . Now assume that the assertion is true for any number n. We will prove that it is also true for n + 1. Consider any 0 ≤ k ≤ k ≤ 2 n+1 . By the construction of the sequence {z k,2 n } ∞ n=0 , there exist x, x , y, y ∈ z k,2 n 2 n k=0 such that x y, x y and

Applying the first step in the proof of Lemma 2, the equivalence t ∼ 1 2 w ⊕ 1 2 v is satisfied. Hence

This implies

Consider now any x, y such that x * x y y * . Let z kn,2 n ∞ n=0 and z k n ,2 n ∞ n=0 be sequences that converge correspondingly to x and y. Then

which is equivalent to

For any x y and x y , fix x * and y * such that x * x, x and y, y y * , we have

Hence the choice of value µ(A) does not depend on the choice of x, y.

Now, we prove that µ is a finitely countable probability. We will prove that for every events A and B such that

x A y B∪C , there exists outcome w such that

Applying the sure-thing principe by replacing x by y on the event A, we get

From x A∪B y C ∼ x A w B∪C and x B y A∪C ∼ y A w B∪C we get (5)


Michele Quattromini
email: michele.quattromini@poliba.it

Michele Alessandro Bucci

Stefania Cherubini
email: stefania.cherubini@poliba.it

Onofrio Semeraro
email: onofrio.semeraro@universite-paris-saclay.fr

ENHANCING DATA-ASSIMILATION IN CFD USING GRAPH NEURAL NETWORKS A PREPRINT

We present a novel machine learning approach for data assimilation applied in fluid mechanics, based on adjoint-optimization augmented by Graph Neural Networks (GNNs) models. We consider as baseline the Reynolds-Averaged Navier-Stokes (RANS) equations, where the unknown is the meanflow and a closure model based on the Reynolds-stress tensor is required for correctly computing the solution. An end-to-end process is cast; first, we train a GNN model for the closure term. Second, the GNN model is introduced in the training process of data assimilation, where the RANS equations act as a physics constraint for a consistent prediction. We obtain our results using direct numerical simulations based on a Finite Element Method (FEM) solver; a two-fold interface between the GNN model and the solver allows the GNN's predictions to be incorporated into postprocessing steps of the FEM analysis. The proposed scheme provides an excellent reconstruction of the meanflow without any features selection; preliminary results show promising generalization properties over unseen flow configurations.

Introduction

Computational Fluid Dynamics (CFD) is a cornerstone for the simulation and analysis of fluid flows across a multitude of scientific and engineering fields. However, the computational complexity associated with solving the governing equations is a persistent challenge, as it requires high-performance computing resources and elaborate numerical techniques. The combination of Machine Learning (ML) techniques and CFD numerical solvers is emerging as a possible solution to overcome these limits (see reviews by [START_REF] Duraisamy | Turbulence modeling in the age of data[END_REF] and [START_REF] Brunton | Machine learning for fluid mechanics[END_REF]). In this paper, we consider a data-assimilation problem augmented by a Graph Neural Network (GNN) model. In particular, we aim at computing the meanflow based on known measurements. Standard approaches rely on dataassimilation via adjoint-based optimization [START_REF] Foures | A data-assimilation method for Reynolds-averaged Navier-Stokes-driven mean flow reconstruction[END_REF]. Here, we introduce an end-to-end assimilation process, that includes a GNN model capable of predicting the closure term for the RANS equations, namely the Reynolds stress, following the works by [START_REF] Donon | Deep Statistical Solvers[END_REF] and [START_REF] Ströfer | End-to-end differentiable learning of turbulence models from indirect observations[END_REF]. The choice of GNN models is justified by the strong interest for this architecture in the fluid mechanics community [START_REF] Lino | Current and emerging deep-learning methods for the simulation of fluid dynamics[END_REF]. Indeed, GNN allows to address some of the limitations often found when integrating Deep Neural Networks (DNN) and numerical solvers for physics. First, data hungriness is often a limiting factor when training a DNN, as from an industrial point of view the availability of data can be limited. GNN are known to be more data-parsimonious [START_REF] Hamilton | Graph representation learning[END_REF].

Figure 1:

Unstructured mesh around a 2D cylindrical body.

Second, some DNN architectures combined with numerical solvers lack of flexibility when considering CFD simulations in complex geometries. An example is given by standard Convolutional Neural Networks (CNN), often used as approximators of closure terms, known to face challenges with unstructured meshes. Although very recent works address the latter limitation in CNN [START_REF] Coscia | A continuous convolutional trainable filter for modelling unstructured data[END_REF], successful works are already available relying on GNN, for instance, for the up-scaling of low-resolution simulations [START_REF] Belbute-Peres | Combining differentiable PDE solvers and graph neural networks for fluid flow prediction[END_REF].

In the following, we briefly introduce our data-assimilation scheme and consider the dynamics of the wake past a cylinder at low Reynolds number, Re = 120, as a test case. The flow exhibits the von Karman Street instability and is simulated by means of Direct Numerical Simulations (DNS) using a Finite Element Method (FEM) solver. The GNN model is pre-trained on a dataset composed by a single pair of meanflow (input) and Reynolds stress (output) of the wake past cylinder bluff body on unstructured mesh at Re = 120. Then, the training iteration proceeds by enforcing the RANS equations in the post-training loop. We show that such a model improves the learning path by effectively assimilating data under the constraint provided by the RANS equations.

Methodology 2.1 Baseline equations and numerical simulations

We consider incompressible, two-dimensional (2D) fluid flows developing past bluff bodies. We focus on timeaveraged quantities and on second order-statistics, the Reynolds stress. To this end, we introduce the Reynolds decomposition u(x,t) = u(x) + u ′ (x,t),

(1) where u = (u, v) T is the meanflow, u ′ the fluctuation field, and x = (x, y) T is the vector composed by the streamwise direction x and the wall-normal direction y. Formally, any unsteady flow can be described using this decomposition, whether we consider a laminar case or turbulent one [START_REF] Foures | A data-assimilation method for Reynolds-averaged Navier-Stokes-driven mean flow reconstruction[END_REF]. Plugging Eq. ( 1) in the Navier-Stokes equations, after time-averaging we get

u • ∇u + ∇p - 1 Re ∇ 2 u = f (2a) ∇ • u = 0, ( 2b
)
where p is the average pressure field. The Reynolds number is defined as Re = U ∞ D/ν, with the reference velocity chosen as the free stream velocity U ∞ at the inlet, D the reference length of the flow and ν the kinematic viscosity. Mathematically, the resulting system provides the Reynolds-averaged Navier-Stokes Equations (RANS) and the forcing f is the closure term or Reynolds stress. This term can be modeled or -when data are available -directly computed as f = -u ′ • ∇u ′ .

(3) In this article, the numerical solution of the Eq. 2 is performed using a FEM solver based on FEniCS [START_REF] Alnaes | The FEniCS project version 1.5[END_REF]. Temporal discretization is implemented using a second-order Backward Differentiation Formula (BDF). FEM solutions are computed on unstructured mesh, thus allowing for a great versatility in handling complex geometries (see Fig. 1). The computational domain is discretized using 13283 nodes and has dimensions [L x , L y ] = [27, 10].

Data assimilation problem

A common problem often found in fluid mechanics is to reconstruct a flow field or a meanflow starting from local measurements. The problem can be cast as an optimization process of data-assimilation, where the Eq. 2 can be used as baseline and the forcing term f is the control parameter to be determined. The cost function to be minimized can be written as

J( û) = 1 2 ||(u -û)|| 2 , ( 4
)
where u is a known field and û is the solution of the RANS equations based on f. The variable f is the iterative solution of an adjoint-based optimization, based on the procedure introduced in the paper by [START_REF] Foures | A data-assimilation method for Reynolds-averaged Navier-Stokes-driven mean flow reconstruction[END_REF] and approximating the term f. The solution is iteratively updated until convergence using the gradient ∂ J /∂ f, which corresponds to the adjoint state. Here, this process is augmented through an additional GNN model that provides the closure term.

Augmented training process

The training process is inspired by the Deep Statistical Solver framework introduced by [START_REF] Donon | Deep Statistical Solvers[END_REF], and the data assimilation scheme by [START_REF] Ströfer | End-to-end differentiable learning of turbulence models from indirect observations[END_REF], both developed as end-to-end learning cycles. Our algorithm is sketched in Fig. 2, where the forward step path and the backward step are represented; θ θ θ are the GNN's trainable parameters. Following the Stochastic Gradient Descent (SGD) method and employing the gradients chain-rule, the gradient of the cost function J with respect to θ θ θ is

∂ J ∂ θ θ θ = ∂ J ∂ û • ∂ û ∂ f • ∂ f ∂ θ θ θ = ∂ J ∂ f • ∂ f ∂ θ θ θ . ( 5
)
The novelty of this paper lies in the combination of these gradients from diverse computational strategies, achieved through the use of our GNN-FEM interface. In particular, the term ∂ J /∂ f comes from the adjoint formulation of the RANS equations and is computed using the FEM simulation; ∂ f/∂ θ θ θ is obtained from the Automatic Differentiation (AD) package provided by the Torch python library. To compute the chain-rule as stated in Eq. 5, these terms from different approaches need to be accumulated and back-propagated all together through the entire GNN model. This operation is made possible by the two-folded interface that can convert a FEM discretized vectorial field from FEniCS in a numerical tensor to be used by Torch and vice-versa, without any loss of information. In the following we will denote the data assimilation scheme as DA-GNN.

Figure 2: End-to-end training loop; u is the GNN's input meanflow; θ θ θ are the GNN's trainable parameter; f is the GNN's predicted forcing; J( û) is the cost function as expressed in Eq. 4.

∂ J /∂ û ∂ û/∂ f ∂ f/∂ θ θ θ

GNN architecture description

A custom GNN architecture has been designed to operate on the unstructured meshes. In the initial phase of the forward step, each node within the mesh is assigned with an embedded state, denoted as h i for each node i. The state is initialized to a zero vector. The architecture employs two dedicated Multi-Layer Perceptrons (MLPs) to perform the message-passing mechanism [START_REF] Hamilton | Graph representation learning[END_REF], one for the in-going messages to the node and one for the out-going information from the node itself. These MLPs extract and integrate the spatial flow information from the neighboring nodes and, together with the embedded state of the node itself of the current iteration, they are processed via a third MLP to provide an update of the embedded state at each node.

The operation reads as

h (k) i = h (k-1) i + αΨ (k) h (k-1) i , {u, Re}, φ φ φ (k) →,i , φ φ φ (k) ←,i , φ φ φ (k) ⟳,i , (6)
where φ φ φ

(k) →,i = 1 N d ∑ N d j=1 φ φ φ (k)
j,i is the message that the node i sends to its N d neighbours, averaged over the number of neighbours; φ φ φ

(k) ←,i = 1 N d ∑ N d j=1 φ φ φ (k)
i, j is the message that the node i receives from its N d neighbours, averaged over the number of neighbours; φ φ φ

(k) ⟳,i = φ φ φ (k)
i,i the message that the node i sends to itself to avoid loss of information as the process advances. Ψ (k) is a generic differentiable operator that can be approximated also in this case by MLP; α is a relaxation coefficient that allows to scale each update of the embedded states with the previous one during the message passing loop. At the end of the process, the last embedded state is projected back to a physical state using a decoder, a fully connected MLP, to obtain the closure term prediction of the GNN. We adopted an automatic hyper-parameters landscape explorer, the python library Optuna, for tuning the 5 hyper-parameters of the GNN model: the dimension of the embedded state on each node; the number k of GNN layers; the relaxation coefficient α used for the updates; the learning rate and γ, a coefficient used to weight the contribution to the cost function by each layer of the GNN.

Results

We consider as a test case the reconstruction of the meanflow of a cylinder wake at Re = 120. In Fig. 3, the training loss curve is shown. The training process consists of two phases; the first 1,000 epochs are based on a supervised learning of the forcing terms. This initial phase is necessary to provide a starting forcing term for the subsequent phase that allows the RANS solver to converge, addressing FEM numerical stability problems. Therefore, the cost function is designed to minimize the difference between the predicted forcing terms and the actual ones trained over a dictionary of synthetic data obtained for different bluff body shapes [START_REF] Quattromini | Operator learning of rans equations: a graph neural network closure model[END_REF]. Then, we move to the full end-to-end training process for another 2,500 epochs, where the cost function J in Eq. 4 evolves, comparing the predicted meanflow û with ground truth from DNS. The global training loss shows that the cost function is consistently decreasing, assuring Training: Cylinder case Generalization: Random shape case

Conclusions

In this paper, we introduced a novel algorithm of data assimilation combining the versatility of GNN with a FEM solver. An interface between the GNN model and the FEM solver enables the communication between the two computational environments for a robust back-propagation of gradients during the training phase. The resulting dataassimilation scheme provides an appropriate reconstruction of the meanflow without any features selection, relying on a pre-trained GNN model and keeping physical coherence of the prediction through the constraint provided by the RANS equations. We are currently moving to corrupted or incomplete meanflow reconstruction and refining the technique for enhancing the generalization properties over unseen cases, including different geometries and flow parameters.

Figure 3 :

 3 Figure 3: Training loss curve based on the cost function in Eq. 4. The blue line (pre-training) and the orange line (post-training) are the moving average of the mean curve from 5 different GNN models, with different random seeds. The grey area shows the standard deviation around the mean curve.

Figure 4 :

 4 Figure 4: Comparison of the streamwise component between DNS results and DA-GNN predictions; (a, e) u from DNS; (c, g) û from DA-GNN; (b, f ) f from DNS; (d, h) f from DA-GNN;

Acknowledgements. The Ph.D. fellowship of M. Quattromini is supported by the Italian Ministry of University. A part of the research was funded by the grant PRIN2017-LUBRI-SMOOTH of the Italian Ministry of Research and ANR-21-REASON from the French Agency for National Research.


Ana Paixão

Maria Gabriela

Maria Gabriela Llansol

João Barrento

Maria Etelvina Santos

Na Assírio

Alvim

Keywords: portuguesa, existe antes uma forma cultural portuguesa: a fronteira, o estar na fronteira

come

Os textos de Maria Gabriela Llansol revisitam identidades e reinventam fronteiras espaciotemporais, linguísticas, do cânone literário e do «corp'a'screver» : «Cada cidade, cada Lugar, convergem para um ponto situado no corpo que imagina» (Llansol 2015: 301). A obra fulgurante de Llansol delineia leituras límpidas da Europa, de um Portugal omnipresente na escrita e omnidistante pela escrita: «Ir a Portugal. Quando digo "ir a Portugal", digo ir à sua vastidão histórica. Há dias fui como [que] projectada deste Brabante sobre um outro espaço onde nasci, ou seja, onde aprendi a falar a língua com que escrevo. Mais longe. Mais longe» (Llansol 2010: 124-125) 3 .

Palavras-chave : Maria Gabriela Llansol; espaços-tempos de escrita; exílio.

____________ Artigo

Désormais l'espace en lui-même et le temps en lui-même sont destinés à s'évanouir comme des ombres, et seule pourra prétendre à une existence indépendante une espèce d'union de l'un et de l'autre (Minkowski in Hoffmann 1975 : 100).

Em 1908, o físico Hermann Minkovski foi o primeiro a teorizar a interdependência espaciotemporal e as dobras do espaço, noções fundamentais para a relatividade einsteiniana, e que Maria Gabriela Llansol utiliza como fundamento do seu processo poiético, iniciado no béguinage de Bruges, logo após a sua chegada à Bélgica, em 1965. Dobras do espaço que acompanharão o «corp'a'screver» llansoliano ao longo de toda a sua criação, e que correspondem neste texto a três etapas: 1. O béguinage de Bruges e a «terra aliena», 2. HerbaiSintra, 3. O espaço do «Logos de ser só».

O béguinage de Bruges e a «terra aliena»

Em 1440, o pintor flamengo Rogier van der Wyden fez um retrato de uma beguina (van der Weyden 1440) dando rosto a um movimento feminino que havia começado cerca de duzentos e cinquenta anos antes na Flandres. As beguinas eram de todas as classes sociais -aristocratas, viúvas e filhas de cavaleiros, campesinas ou burguesas -e de todas as idades, a partir dos 14 anos. Dedicavam-se a atividades caritativas e gozavam de independência económica em comunidades de caráter religioso, que duraram até ao século XX, e onde a regra monástica não era imposta. Paralelamente as beguinas praticavam o debate teológico, eram professoras, artistas e intelectuais e exímias escritoras místicas4 . Na Bélgica há mais de 100 béguinages identificados como património protegido pela UNESCO.

Maria Gabriela Llansol visitou o béguinage de Bruges, logo após 1965, data da sua chegada à Flandres, onde se exilou com o marido, Augusto Joaquim, desertor da guerra colonial. De formação católica e nutrindo grande interesse pelos escritores esotéricos como S. João da Cruz ou Ana de Peñalosa, a visita da autora ao béguinage assumirá um caráter iniciático. A atividade profissional que exercerá na Bélgica será disso um exemplo através da criação de um projeto de inovação pedagógica5 : Carta a João Décio: Eu vivo na Bélgica há quinze anos; nem sempre em Jodoigne, que é uma cidadezinha do Brabante, a cinquenta quilómetros de Bruxelas; dantes vivia em Lovaina, onde meu marido e eu fundámos, com alguns belgas e portugueses, uma escola que era sobretudo uma maneira de viver e de conhecer. Durou oito anos, depois evoluiu para uma comunidade de produção e ensino, e há um ano e meio desapareceu. Ficou uma pequena empresa de produção de móveis e distribuição de géneros alimentares, onde trabalho. É uma actividade com perspectivas ecológicas (Llansol 2013: 205).

A visita ao béguinage trará sobretudo mudanças do ponto de vista da produção literária. As beguinas surgem como figuras centrais de obras da primeira e da segunda trilogias llansolianas6 . O béguinage será fundamental na poiética de Gabriela pela criação de um procedimento de escrita que Llansol designa por «cena fulgor»:

Estava eu de visita ao béguinage de Bruges quando, de súbito, tive a sensação estranha de que vários níveis de realidade ali aprofundavam a sua raiz, coexistindo sem nenhuma intervenção do tempo. Havia as mulheres beguinas, ao lado dos portugueses descobridores de novos mundos […]. Geograficamente, era a encruzilhada do espiritual, num sítio ainda vazio, em que eu perguntava a mim própria […] "O que se passou aqui? O que é que aqui, no que se passou continua a passar?" (Llansol 1994a: 126-127).

A «cena fulgor» consiste numa fusão espaciotemporal, na abolição de fronteiras geográficas e de época: as beguinas, os descobridores, e outras presenças, fundem-se numa «encruzilhada do espiritual». A especificidade da escrita llansoliana baseia-se precisamente nesta localização na fronteira, no «estar na fronteira» que Boaventura Sousa Santos associa a muita da criação artística em Portugal (Sousa Santos 2002: 134). Não se trata apenas de uma permeabilidade de influências exteriores, mas sobretudo de uma forma de singularidade. A escrita llansoliana é fronteiriça em vários domínios, retomando o conceito de dobrar espaços:

É estranho: nem Aossê está em Lisboa, nem Lisboa está dentro de Portugal. Nem a Alemanha contém Lisboa e Leipzig, nem Herbais se situa na curva verde da Bélgica. Cada cidade, cada Lugar, convergem para um ponto situado no corpo que imagina e no Logos que atentamente se perde (Llansol 2015: 301).

À abolição de fronteiras espaciotemporais junta-se ainda a reunião de personagens de períodos históricos distintos, como Aossê (Pessoa) e Bach em Lisboaleipzig, aos quais se acrescentam ainda nesta obra Espinosa ou Höderlin e, noutros textos, Eckhart, Camões, Copérnico, Suso, Bruno, Isabel de Portugal, Nietzsche, Müntzer, entre outros, em «encontros inesperados do diverso», como a autora os designa.

A confluência de tempos, espaços, de personalidades da cultura europeia converge para o «corpo que imagina», o lugar que Gabriela designará por «corp'a'screver» (Llansol, 1977: 9-10). A interioridade da escrita e o processo criativo são frequentemente compartilhados com o leitor ao longo dos 24 livros que a autora publicou e nas mais de 35.000 páginas dos seus Livros de horas 7 . A escrita llansoliana encontra impregnada-se de contaminações, de fusões e de transfusões de géneros que mantêm o registo autoficcional como uma das linhas condutoras. Os cadernos dos Livros de horas aparentam-se a diários íntimos, sendo na realidade aquilo que Michel Tournier designa por «diários extimos» 8 , recolocando novamente a problemática da fronteira, neste caso a da exteriorização do «corp'a'screver» llansoliano.

O posicionamento na fronteira é o ponto de partida do processo poiético da autora pela permanente disrupção espaciotemporal, pela junção de personalidades improváveis, pela mescla e fusão de géneros literários e pela exteriorização do «corp'a'screver». Quando regressa a Portugal após 20 anos de exílio belga, Gabriela questiona-se «De que modo se canta em terra alheia?» (Llansol 1977: 87), reafirmando a não pertença a nenhum desses espaços, como veremos no ponto seguinte. Todos os lugares são assim «terra aliena» e o único território seu é o da escrita.

HerbaiSintra

Eu quero ser Herbais -disse-me Sintra -, eu quero ser para ti, ao menos, um pouco do que foi Herbaisdisse-me Sintra, a serra, a vila, e a possível cidade (Llansol 2015: 717).

O «corp'a'screver» posicionar-se-á em geografias distintas ao longo dos anos da escrita. O primeiro livro de Gabriela será publicado em Lisboa em 1962 [START_REF] Llansol | Os Pregos na Erva[END_REF], cidade onde escreverá ainda um segundo 9 , antes da sua partida para a Flandres em 1965. Um texto decadente será um primeiro livro de transição: começou a ser escrito em Lisboa em 1964 e foi terminado em Lovaina 1968, sendo dedicado à ascese S. João Da Cruz10 .

O caminho da Flandres 11

Após os vinte anos de exílio belga, Gabriela descreve desta forma a sua chegada à Flandres e as motivações que a guiaram:

Não tinha mais de trinta anos; acabara de me casar e vinha, sozinha, ter com o Augusto que, umas semanas antes, se recusara a participar na guerra colonial, e desertado; deixava um livro publicado e outro, embora já concluído, ainda inédito. […] Vim com muito temor e, também, com uma enorme sede de liberdade, de novo, de atingir o âmago do ser (Llansol 1994a: 125).

O «corp'a'screver» de Gabriela passará por lugares diversos como Lovaina, Jodoigne e Herbais, que aparecem referidos nos seus cadernos ou nas epígrafes das obras. Nesse mesmo texto de balanço dos vinte anos passados na Bélgica, diz a autora:

As particularidades, múltiplas e incontáveis, de um país conservador, percorrido por pessoas livres e trabalhadoras que, embora achando muito original a imaginação das gentes vindas de outras terras, não podiam deixar de marcá-las com os indícios de uma espécie de bom senso degenerado. Não queriam acreditar no que viam. Nessa atitude havia, sem dúvida, muita generosidade, mas também uma muralha intransponível. Uma real dificuldade de partir para o vasto mundo.

Reconheço que sempre vivi nesse país sem tentar mergulhar nele, e torná-lo meu; mas como poderia ser diferente, se eu própria me afastava daquele em que tinha nascido (Llansol 1994a: 126).

Os lugares belgas que cruzamos nos livros de Gabriela são fundamentalmente os da sua escrita, sobretudo de refúgio para a sua produção e até performance literária, como Herbais:

Se um dia existisse um lugar -um circo -para a representação do conteúdo do livro -e seu exterior, esse lugar seria Herbais. Herbais-sobre-Ática (Llansol 1996: 164).

Herbais converte-se na residência de escrita da autora, simultaneamente lugar de asilo e de cativeiro, como o refere na capa de um dos seus cadernos. Ao mesmo tempo, depois de regressar a Portugal, considerará Herbais como o espaço de criação privilegiado:

____________ Eu quero ser Herbais -disse-me Sintra -, eu quero ser para ti, ao menos, um pouco do que foi Herbais -disse-me Sintra, a serra, a vila, e a possível cidade_____________ Mas és prosa -respondi-lhe, sem querer. À minha volta, fala-se a mesma língua, ou seja, vou supondo que falam a mesma língua. E em Herbais, sozinha, minha língua materna recém-nascida é que existia (Llansol 2015: 717).

Herbais era esse território predileto, onde a autora se podia isolar com a sua língua, numa relação essencial da produção llansoliana, como veremos adiante. As geografias belgas surgem como referências frequentes nos textos de Gabriela, neutros cenários de fundo da sua escrita, paisagens de telas flamengas, apenas nomes de lugares que se dobram no espaço e se conjugam com outros: Lovaina (1965-75), Jodoigne (1975-80) e Herbais (1980-85). (Llansol 1994a: 125).

[…] aquele em que tinha nascido

Que Portugal se espera em Portugal? (Sena 1984: 270-271) Portugal surge traçado na obra de Llansol de forma mais presente e nítida do que a Flandres. É o país da infância, das fotografias de Alpedrinha e dos primeiros cadernos de orações. É o país da ditadura e da censura:

Ninguém conseguirá ter uma pálida imagem da densidade do ar que, lá em baixo, se respirava, no exíguo cubículo fechado das nossas vidas. Eu procurava evadir-me a escrever. A maior parte das situações que eu apontava eram-me totalmente desconhecidas. Tinha-as "vivido" no cinema, e nos romances americanos (Llansol 1994a: 125).

Portugal é ainda o país do medo: dos medos coletivos e individuais, dos ligados à religião nos primeiros cadernos de infância, dos medos associados à escrita, à falta de editor, à morte. Gabriela participa das reflexões de José Gil, a propósito desse traço identitário nacional: «O medo herda-se. Porque interiorizado, mais inconsciente do que consciente, acaba por fazer parte do "carácter dos portugueses"» (Gil 2005: 78-79). Não apenas do consciente ou inconsciente coletivo, assim como do medo individual. No Livro das comunidades, são identificados três desses medos:

Há três coisas que metem medo […] A primeira é a mutação. […] A segunda é a tradição […]. A terceira é um corp'a'screver. Só os que passam por lá, sabem o que isso é. E que isso justamente a

ninguém interessa (Llansol 1977: 9-10).

O Inquérito às quatro confidências termina com um diálogo em que a autora é interpelada por um oculto a propósito da mesma temática:

-Gabriela! -Sim! -Ver-nos-emos face a face, daqui a milhões de anos.

-Sim! -Faça a sua parte! Sem medo, sem medo, sem medo (Llansol 1996: 184).

Ou, em junho de 1998, num dos diários, identifica-se ainda outra necessidade correlacionada:

Livro a escrever: A Caixa de Música da Leitura ________________ contra o medo da morte (Llansol 2015: 708).

A principal reflexão em torno da temática do medo associada à identidade portuguesa, surgirá em (Llansol 1994b: 154-155).

As imagens de Portugal e da integração europeia apresentadas neste texto mantêm-se de uma atualidade intocável, sobretudo neste momento de pandemia em que «[…] deixou de haver no mundo muro que separe os que têm medo dos que não têm medo» [START_REF] Couto | texto proferido na Conferência do Estoril[END_REF]. No conjunto da obra de Maria Gabriela Llansol surgem de forma caleidoscópica retratos lúcidos sobre as perspetivas socioeconómicas de um país dependente. Dependente de um passado colonial desaparecido, dependente de um futuro limitado ao progresso económico, ou dependente de financiamentos atribuídos por uma gestão europeia, ela própria subsidiária da plutocracia, distante e «futurenta».

Além do quadro mais geral do país na sua correlação com a Europa, é possível ainda encontrar na obra de Gabriela panorâmicas mais aproximadas sobre outras problemáticas, como a do estado da edição em Portugal, por exemplo. Apesar de ter publicado a maioria dos seus textos em português e em Lisboa manteve uma relação com os editores nacionais feita de incertezas: 27 de Abril de 1979 É uma época em que quase não escrevo; não tenho editor; não sei onde conduzir o livro (Llansol 2013: 71).

Simultaneamente mantém uma ligação com a crítica literária nacional:

de Janeiro de 1980

Chegaram páginas literárias de Portugal, esse mundo ainda me afecta, ou será que só simplesmente me faz reflectir sobre as linhas com que a literatura portuguesa, com os seus críticos, se cose? (Llansol 2013: 187).

A autora ocupará sempre um lugar à parte no cânone, com uma obra considerada singular no panorama literário português: é-lhe atribuído o rótulo «esotérica de serviço» 12 que lamentará numa carta endereçada a Eduardo Prado Coelho. Postumamente outros rótulos lhe serão atribuídos 13 e a 12 Carta de 1999 endereçada a Eduardo Prado Coelho: «Nas experiências que fiz, no meio literário, não encontrei qualquer caminho para o meu texto. A falta de humanidade é de regra, ninguém sabe porque escreve, a minha obra é apresentada como um meteoro desconhecido, nunca ninguém pediu a quem sabe um pouco mais sobre ela para a apresentar, eu própria estou, desde sempre, catalogada como inacessível, nunca encontrei ninguém que estivesse minimamente interessado em escutar os problemas que me punha a obra que nascia de mim e à qual eu sempre estive firmemente disposta a dar corpo. […] Mas o mais grave é a situação em que sempre fui colocada. Eu que sou frontal, gosto de conversar e rir, adoro facécias, antes mesmo de abrir a boca, já a minha palavra estava inscrita numa rede de total incompreensão. […] No quadro da literatura 'nacional', serei sempre a esotérica de serviço» (in Lúcia Crespo 2017). 13 Eduardo Lourenço dirá: «Provavelmente, a Gabriela Llansol será -penso eu -o próximo grande mito literário português» [START_REF] Lourenço | Revista Ler-Livros[END_REF]. António Guerreiro escreveu aquando da sua morte: «O século XX português, no domínio da literatura, é inquestionavelmente pessoano; um dia também será llansoliano. Não se trata de medir grandezas, mas de reconhecer o que há de incomensurável na obra de Maria Gabriela Llansol, pelo modo como inventou um género literário (para o qual não servem as categorias de romance nem de ficção, e em rigor se esbatem as fronteiras entre prosa e poesia), criou a sua própria tradição e projectou-se fora das constelações literárias», [START_REF] Guerreiro | [END_REF]. Ainda durante a vida da autora, Maria Silvina Rodrigues escreveu Teoria da Des-crítica procurará outros prismas de leitura da obra llansoliana. Ainda em vida, a crítica literária portuguesa atribuiu-lhe vários prémios 14 , a partir de 1985, data do regresso da autora a Portugal. Os novos refúgios de escrita serão em Colares (1985Colares ( -1994)), na «Casa da Saudação» e em Tokialai (em basco "abrigo", "refúgio"). A casa de Sintra (1994Sintra ( -2008) ) será o último território de Gabriela, lugar onde esteve instalado o Espaço Llansol, entre o ano do desaparecimento da autora e 2017. O Espaço Llansol 15 , associação que tem procurado manter viva a obra da autora, encontra-se neste momento em Lisboa, em Campo de Ourique a poucos metros da casa onde Gabriela nasceu 16 , mantendo os objetos, a biblioteca da autora e os indícios do «corp'a'screver» na sua ligação com a palavra.

3) O espaço do "Logos de ser só"

Colares, 1990 __________ eu nasci em 1931, no decurso da leitura silenciosa de um poema (Llansol 2000: 23).

Maria Gabriela Llansol refere a propósito da sua infância e adolescência que: «Na Casa não se administrava bem a justiça da língua» [START_REF] Barrento | Depoimento sobre Maria Gabriela Llansol, in Alagamares, associação cultural[END_REF], nessa Casa que era simultaneamente também a sua Casa-país. Dessa busca incessante pela «justiça da língua», surgirá Témia, «A rapariga que temia a impostura da língua», personagem que nasce da relação intersticial e mística entre a autora e o seu idioma:

[…] não possuía do passado senão uma língua de que nada, nem ninguém, conseguiriam separar-me. E, hoje, sei que essa língua se tinha tornado o meu único ponto firme -a minha âncora: o meu real; o nó de certeza do meu corpo com o mundo. O meu órgão de convicção, se assim vos aprouver chamarlhe (Llansol 1994a: 126).

A produção da obra llansoliana foi sempre concretizada no mesmo idioma: «[…] eu perguntava a mim própria em português, em português e não em qualquer outra língua» (Llansol 1994a: 126). Também as «cenas fulgor» llansolianas se exprimem na mesma língua, numa poiesis de busca incessante:

«VI -O que […] eu procurava sem o saber, era o logos, a que mais tarde chamei cena fulgor -o logos do lugar; da paisagem; da relação; a fonte oculta da vibração e da alegria, em que uma cena -uma morada de imagens -, dobrando o espaço e reunindo diversos tempos, procura manifestar-se» (Llansol 1994a: 128).

Após a «cena fulgor» do béguinage de Bruges, deu-se ainda outro encontro fundamental para a consciencialização linguística de Llansol: «Foi aí que encontrei um pequenino ser dominicano […] possessão: ensaio sobre textos de Maria Gabriela

  Mas o nome que eu, Pessoa, lhe dou é o medo da insignificância do pequeno excluído. O medo de ser pobre […] os caminhos da prata e do mando já não passam por aqui, que isto foi desafectado. Nestes carris vazios, e quase integralmente cobertos de areia, muito se tem imaginado, Baruch. Também eu imaginei. Imaginado como obrigar as rotas do mundo a passar (já não digo por aqui) mas por aqui perto. O facto é que essas rotas fogem dessas paragens por nem sequer saberem o que viriam procurar aqui. Talvez que pudessem vir à procura dessa imaginação de não-saber em que esta gente é fértil, se não fosse o medo. Mas há o medo, Baruch

	Lisboaleipzig 2, num diálogo entre Aossê (Pessoa) e Baruch (Spinoza) acerca da adesão de Portugal à
	Comunidade Económica Europeia:

Na Lisboa que vim encontrar agora, a crença é o progresso que, na realidade, é o dinheiro a correr mais ligeiro e abundante e que, na mais-realidade ainda, é o nome do nada-de-prata que a Europa pluthocrata, telemathica e futurenta deu a alguns aqui. Agora vem e, amanhã, como sempre, deixa de vir.

  Llansol, Lisboa: Black Sun Editores, 1988. Em torno da obra de Gabriela têm trabalhado dois filólogos incontornáveis -João Barrento e Maria Etelvina Santos -, assim como outros críticos e ensaístas como Eduardo Pitta, Pedro Eiras, Maria Graciete Besse, entre outros. 14 O Prémio D. Dinis da Casa de Mateus, 1985, Um Falcão no Punho; o Prémio Inasset, em 1986, Contos do Mal Errante; Prémio da Crítica da Associação Portuguesa de Críticos Literários e o Grande Prémio do Romance e Novela da Associação Portuguesa de Escritores, 1991, Um beijo dado mais tarde; Grande Prémio do Romance e Novela da Associação Portuguesa de Escritores, 2007, Amigo e Amiga: curso de silêncio de 2004. 15 A criação do Espaço Llansol foi precedida pelo Grupo de Estudos Llansolianos (GELL) na Universidade Nova de Lisboa, por João Barrento, numa iniciativa impulsionada pelo marido e companheiro de escrita da autora, Augusto Joaquim. O espaço Llansol tem incitado atividades em torno da obra da autora, como a exposição «Sobreimpressões. Maria Gabriela Llansol: uma visão da Europa», em 2011, no Centro cultural de Belém, e em 2017, produziu o Ciclo Autor do Festival Silêncio dedicado a Gabriela. 16 Maria Gabriela Llansol nasceu no número 8 da Rua Saraiva de Carvalho, e o Espaço Llansol encontra-se atualmente no número 11 da mesma rua de Campo de Ourique.

Bastará pensar em Marie D'Oignies, Mechthild de Magderburg, Beatrijs of Nazareth, Jadewijch e Marguerite Porete (Swan 2016).

Maria Gabriela e Augusto Joaquim criaram uma escola de inovação pedagógica (Ecole de la Rue de Namur) e posteriormente surgiu La Maison, integrada numa cooperativa de produção e de ensino, a Ferme Jacob, junto a Louvain-la-Neuve.

Como na Casa de Julho e Agosto (1983) e Causa Amante (1984).

Os livros O Estorvo(Lisboa, 1961(Lisboa, -1963) ) e Um Texto Decadente(Lisboa, 1964 -Lovaina, 1968) serão publicados com o primeiro livro escrito na Bélgica -E que não escrevia(Lovaina, 1968(Lovaina, -1971) -, numa compilação intitulada Depois de Os Pregos na Erva[START_REF] Llansol | Depois de Os Pregos na Erva[END_REF].

Expressão do livro Na Casa de Julho e Agosto[START_REF] Llansol | Na Casa de Julho e Agosto, Geografia de Rebeldes III[END_REF].


II.1 -Session d'étude du matériel lithique dans le cadre de la publication du niveau IV0 -année 2019

Pierre BODU, Grégory DEBOUT

I -Bref rappel sur les travaux achevés et ceux toujours en cours

Sans y revenir de manière développée, dans la mesure où nous avons déjà eu l'occasion de le faire dans les deux précédents rapports (années 2017 et 2018), rappelons en premier lieu ce qui, au titre des activités et analyses préalables à la mise en oeuvre de la rédaction finale du manuscrit monographique, est aujourd'hui achevé et ce qui nécessite encore quelques mois de travail pour y parvenir.

 ce qui est achevé :

-l'inventaire total informatisé de l'industrie lithique et la création des divers tableaux de décompte qui agrémenteront les futurs textes de la monographie ;

-les tentatives de raccords et remontages des objets en matériau allochtone et le travail de restitution par DAO (photographie de quelques remontages et raccords choisis ; plan des liaisons obtenues à partir des éléments raccordés et/ou remontés à l'échelle du campement).

 ce qui est en cours :

-les tentatives de raccords et remontages des objets en matériau local et le travail de restitution par DAO (photographie des ensembles remontés de manière significative ; plan de répartition des éléments en silex concernés). Une description détaillée de cette activité est présentée en partie 2 du présent chapitre (cf. infra) ;

-le dessin technique des objets (outils, armatures et nucléus notamment) les plus représentatifs provenant de chacune des unités du campement IV0. Pour l'heure, nous disposons d'une panoplie de dessins relativement complète concernant l'industrie des unités V97 et M 103 (cf ; rapport année 2018). Une seconde session de dessins, toujours réalisés par N. Delacroix, est en cours. Elle concerne cette fois-ci l'unité D 110 ;

-l'analyse tracéologique d'une sélection d'objets visant, d'une part, à enrichir les discussions relatives à la taphonomie du niveau archéologique et, d'autre part, à documenter les aspects fonctionnels des pièces esquillées, dont la présence en grand nombre constitue l'une des réelles originalités de ce niveau IV0. La sélection d'objets réalisée par nos soins a été confiée à Mickaël Baillet qui, de son côté, n'a réellement pu entreprendre les premières analyses qu'à partir de novembre ;

-la réalisation des plans de répartition des différents types d'objets retouchés (outillage, armatures) et des nucléus. Nous avions initié ces travaux dès 2018. La base de données constituant l'inventaire du matériel lithique avait alors été formatée selon un modèle déterminé afin de la relier aux relevés topographiques (x, y, z) et au plan des découvertes du

* Quelques observations à partir des remontages :

Le tableau ci-dessous (tabl. 1) synthétise les remontages réalisés entre 2017 et 2019 sur le matériel lithique taillé des unités D110, M103, V97. Les 40% restant de silex de l'unité M103 proviennent donc pour l'essentiel de débitages effectués à l'extérieur de ce locus. Si quelques lames et plus rares lamelles de bonne venue retouchées ou non ont été apportées (au moins 12 fragments de lames et lamelles présentent soit une bonne régularité, soit un talon en éperon ou les deux ensemble), on compte également un nucléus à éclats laminaires, quelques éclats corticaux ou non corticaux plutôt fins sans véritable intérêt économique, soit des supports qu'à priori on transporte rarement.

Ce regroupement plutôt atypique apparaît ainsi en raison de la faible quantité de vestiges lithiques retrouvés sur place, ce qui permet d'identifier des pièces « étrangères » même si elles sont de prime abord peu intéressants en termes fonctionnels. Le nucléus M105.4, jusqu'à présent isolé de toute pièce de débitage a été remonté d'ailleurs lors de cette session de juillet 2019 avec un proximo-mésial d'éclat de l'unité D110 (pièce I109.97). Ainsi ce bloc (rognon ?) aurait été débité (tout ou partie) en D110, le nucléus étant ensuite déposé en M105 en bord est extrême de l'unité M103 (fig. 16). Parmi les objets un peu particuliers de l'unité M103, il y a deux éclats, l'un court semi-cortical (M102.20) l'autre non cortical plus allongé (L103.43 + M104.12 + L103.32) qui remontent directement l'un sur l'autre et qui ne sont associés à aucun autre silex de l'unité (fig. 18). Etant donné la préparation soigneusement facettée d'un des talons de ces éclats, on comprend qu'il s'agit d'un débitage de qualité qui provient peut-être d'une mise en forme sophistiquée d'un bloc de bonne qualité (crête arrière ?). Mais ce qui étonne c'est le fait qu'ils soient seuls, qu'ils remontent ensemble et qu'aucun autre nucléus ou éclat leur correspondant ne soient présents en M103. Ainsi au gré des remontages et de l'observation de similitudes techniques, le locus M103 semble tisser des liens étroits avec l'unité proche D110, il pourrait même en être une émanation, une aire annexe à la fois technique si l'on en juge par les quelques débitages et outils présents sur place mais aussi peut-être quelque chose de symbolique au regard des restes humains et des vestiges osseux animaux sélectionnés qui s'y trouvent (scapulas de renne et mandibules de chevaux).

* En D110, peu de débitages très sophistiqués :

L'ensemble du matériel lithique de D110 a été à nouveau étudié afin de chercher les relations potentielles avec M103. A cette occasion des tentatives de remontages et raccords ont été réalisées avec M103 certes mais aussi avec V97 et partiellement avec T125. Il a été ainsi possible d'établir une autre relation formelle entre des pièces de D110 (F115.40 et F114.9, H110.123, cf ci-dessus) et une lamelle semi corticale découverte en V97 (U98.397). Ces 4 objets sont rapprochés du très beau débitage réalisé en T125 sur le nucléus Y123.5 dont beaucoup d'éléments se trouvent dans le mètre carré U 120 mais cette relation reste à confirmer (fig. 17).

Mais il existe véritablement peu d'indices de bons débitages laminaires ou lamellaires réalisés directement en D110, au centre ou en périphérie. La plupart des nucléus observés, il y en a 44 en silex secondaire dans l'unité D110, ont été peu productifs (fig. 19). Ces savoir-faire moyens ou médiocres se marquent par des ratés d'exploitation, des insistances inutiles, des productions peu standardisées de supports laminaires peu réguliers et plutôt courts. Cela s'accompagne du fait qu'en D110, en tout cas dans le coeur de l'unité, il n'existe pas ou très peu de gros amas de débitage comprenant de grands éclats corticaux ou semi corticaux. Les seuls éléments de ce gabarit et avec plus ou moins de cortex se trouvent en fait en périphérie sud-est de l'unité D110, dans ce que l'on peut interpréter comme une zone de rejet (pierres brûlées, silex) (X-A/111-117). La présence de blocs de silex à peine entamés comme celui présenté ci-dessous (fig. 23) pose par ailleurs la question des modalités de collecte de la matière première qui apparemment ne répondaient pas uniquement aux besoins en supports mais pouvaient s'apparenter aussi à la constitution de réserves de blocs pour utilisation futures. De la même façon l'apport de blocs volumineux mais fortement faillés, à l'instar de celui présenté ci-dessous découvert dans l'unité T125 et dont la mauvaise qualité était pourtant perceptible au premier test (fig. 24), ne semble pas être un comportement économique très stratégique et pose donc la question de la diversité des acteurs de ces collectes mais peut-être aussi du rôle que l'on assignait à certains de ces blocs (blocs à débiter, enclumes, poids pour caler, pierres de foyer ?). Un autre nucléus à lamelles T98.41, qui remonte avec un certain nombre de ses tablettes de réaménagement de plan de frappe évoque un même traitement spatial séquencé (fig. 33).

Malgré une matière première assez aisément identifiable, il nous a été en effet impossible de retrouver en V97 l'ensemble des éclats corticaux, semi corticaux et acorticaux de mise en forme de ce beau volume, ce qui signifie vraisemblablement que le nucléus a été apporté en V97 déjà mis en forme et peut-être en partie exploité pour une production de lames. Cette hypothèse sera à vérifier en 2020 par des tests de remontage notamment avec l'unité T125, car nous n'avons pas identifié ce type de silex en D110 et évidemment pas en M103.

* De la qualité des exploitations du silex dans le niveau IV0 :

On a donc pu constater qu'il existe une forte composante de blocs de silex de qualité moyenne à médiocre dans toutes les unités, qui ont conduit à des exploitations soit vite avortées en raison de la fracturation accidentelle des blocs, soit courtes en raison du faible potentiel des blocs, soit ratées en raison du faible niveau de technicité des tailleurs, tout ceci pouvant évidemment se combiner. A l'opposé, il existe quelques chaines opératoires de très bon niveau en T125, mais elles semblent assez peu présentes en V98 et encore moins en D110. Ainsi en T125 on compte 18 nucléus dont la production en lames et surtout en lamelles est de bon niveau tant qualitatif que quantitatif. En D110 il y en a seulement 2 ou 3 et en V98, il existe 5 à 6 exploitations productives. En M103 seul un débitage lamellaire a été productif ! Sur l'ensemble des blocs débités sur le niveau IV0 (environ 200) on compte donc seulement une trentaine de productions de bonne qualité ce qui est peu. On reste étonné par la collecte abondante et majoritaire de blocs de matière première qui ne pouvaient pas assurer une production en lames et en lamelles suffisante pour les besoins du campement. A cela a sans doute répondu l'apport de lames et dans une moindre mesure de lamelles en silex allochtone, puisque le niveau IV0 est celui qui présente le plus fort taux de silex tertiaire dans l'ensemble des niveaux magdaléniens de Pincevent.

III -Programme d'études pour 2020

Lors de la campagne à Pincevent en juillet 2020, nous avons prévu de poursuivre les remontages sur l'ensemble du matériel lithique en insistant notamment sur les unités V97 et D110. Nous serons néanmoins amenés à prendre en compte le matériel lithique de l'unité T125 étant donné que c'est d'elle que provient vraisemblablement un bon nombre de supports laminaires utilisés dans les autres locus. Nous consacrerons deux semaines à ce travail de remontages, assorti de la réalisation des listings des ensembles remontés et des photographies de ces ensembles.

L'étude tracéologique se poursuivra en 2020 avec de nouveaux questionnements et des compléments d'étude par rapport à ce qui a été réalisé à compter de fin 2019 par Mickael Baillet. Nous privilégierons ainsi les objets qui ont été pas ou peu étudié en cette fin d'année, en particulier les macrolithes présents dans chacune des unités, ce qui nécessitera aussi de consulter les blocs de pierre n'ayant pas servi dans les activités de combustion (pierres de foyer) et dont certains pourraient aussi avoir été utilisés en macrolithes, on pense ainsi au travail des bois de cervidé ou à la fracturation des ossements pour récupération de la moelle.

Enfin nous envisageons de compléter les dessins du matériel lithique dont deux séries ont déjà été réalisées par Nelson Delacroix. Pour l'instant plus d'une soixantaine d'objets ont été dessinés, nous allons prochainement confier une trentaine d'autres silex à N. Delacroix. Ces dessins actuels seront complétés dans la monographie d'une partie de ceux qui avaient été dessinés dans la précédente publication concernant le niveau IV0 mais aussi de clichés de l'industrie lithique notamment des remontages.

Référence bibliographique

Julien M., Karlin C. (2014) -Un automne à Pincevent. Le campement magdalénien du niveau IV20. Mémoire 57, Société Préhistorique Française, 639 p.

  niveau IV0 par l'intermédiaire d'un logiciel de SIG. Les plans de distribution obtenus n'ont au final pas donné satisfaction. Par cette méthode, seuls 73% des objets concernés se retrouvaient projetés sur les plans, et ce pour différentes raisons. Il a donc été décidé en 2019 de changer de méthode (projection par m² et non plus par x et y, mais toujours via un logiciel SIG, ce qui permet toujours d'interroger la base de données initiale pour extraire des plans analytiques). Par l'intermédiaire de celle-ci, il est dorénavant possible de projeter 94 % des objets concernés (outils, armatures et nucléus), ce qui est d'un point de vue statistique est beaucoup plus acceptable. Les plans obtenus s'avèrent ainsi beaucoup plus proches de réalité archéologique. Le travail d'exportation et de reprise par DAO n'étant pas encore achevé (cf. exemple ci-dessous), nous présenterons quelques exemples de ces plans définitifs dans le prochain rapport.Exemple de plan obtenu pour la catégorie des grattoirs (sortie brute du logiciel SIG) par la méthode mise en oeuvre en 2019. Les points jaunes représentent les effectifs par m². En bleu : la distribution totale des outils et des armatures du campement IV0, obtenue par la méthode de densité du noyau (Kernel Estimation Density).

Fig. 1 .Fig. 2 .

 12 Fig. 1. Session de remontages à Pincevent en juillet 2019 (P. Bodu ; CAP)

Fig. 3 .Fig. 4 .

 34 Fig. 3. Un remontage entre l'unité V97 et l'unité D110. La lame, C111.54 (unité D110, à gauche), provenant du débitage du nucléus Y97.83 (unité V97, à droite) a été débité dans l'unité V97 où se trouve l'essentiel de sa production (ex de la lame T98.211, unité V97, au centre ; .P. Bodu ; CAP)

Fig. 5 .Fig. 6 .

 56 Fig. 5. Le nucléus/macrolithe sur casson gélifracté de M103 (N101.2 ; N.Delacroix)

Fig. 7 .

 7 Fig. 7. Quelques outils sur lames, lamelle ou éclats laminaires de M103, pour l'essentiel non débités dans cette unité (P. Bodu ; CAP)

Fig. 8 .

 8 Fig. 8. Le nucléus N101.2 de l'unité M103 et le nucléus C112.89 de l'unité D110 (P.Bodu ; CAP)

Fig. 10 .

 10 Fig. 10. Le nucléus D113.49 de l'unité D110 : nucléus ? outil ? les deux ? (P.Bodu ; CAP)

Fig. 13 .Fig. 14 .Fig. 15 .

 131415 Fig. 13. Trois vues de talons ocrés appartenant au débitage du bloc L103.23, unité M103, niveau IV0 (P. Bodu ; CAP)

Fig. 16 .Fig. 17 .

 1617 Fig. 16. Un remontage entre les unités M103 (nucléus M105.4 en bas) et D110 (éclat I109.97en haut ; P. Bodu ; CAP)

Fig. 18 .

 18 Fig. 18. Remontage de l'éclat L103.43 + M104.12 + L103.32 et de l'éclat M102.20 (P. Bodu ; CAP)

Fig. 19 .Fig. 20 .Fig. 21 .

 192021 Fig. 19. Tri de quelques nucléus moyennement à peu productifs de différentes unités du niveau IV0 (P. Bodu ; CAP) La grande majorité témoignent de savoir-faire moyens à médiocres en tout cas lors de leur phase d'abandon, les volumes et les formes des blocs initiaux sont peu compatibles avec de bonnes exploitations laminaires et assez fréquemment la matière première est de piètre qualité (blocs faillés, gélifractés) (fig. 20). Cette constatation vaut aussi bien pour D110 que pour V97 (fig. 21) mais aussi pour T125 (fig. 22).

Fig. 23 .

 23 Fig. 23. Bloc de silex (N113.2) à peine ébauché de D110 (P. Bodu ; CAP)

Fig. 24 .

 24 Fig. 24.Grand bloc de silex de l'unité T125 (R125.112 : au moins 30 cm de longueur) fracturé accidentellement lors de la percussion en raison de la présence de nombreuses failles (P. Bodu ; CAP)

Fig. 25 .Fig. 26 .Fig. 27 .

 252627 Fig. 25. Bloc de silex W96.39 de l'unité V97 fracturé accidentellement lors de la percussion en raison de la présence de nombreuses failles (P. Bodu ; CAP)

Fig. 28 .

 28 Fig. 28. Bloc de silex de grande taille de l'unité T125 (G125.1) de qualité suffisante pour un débitage laminaire mais qui a été simplement débité en éclats épais (P. Bodu ; CAP)

Fig. 29 .

 29 Fig. 29. Grand nucléus de silex V130.23 de l'unité T125 (P. Bodu ; CAP) à gauche, accompagné de grands éclats corticaux de mise en forme du volume ; à droite, le même nucléus avec de grands éclats de mise en forme de la partie dorsale remontant

Fig. 30 .

 30 Fig. 30. Un autre exemple d'un débitage parfaitement maîtrisé (F131.4) de l'unité T125 ayant produit de grandes lames (P. Bodu ; CAP)

Fig. 31 .

 31 Fig. 31. Plusieurs des grandes lames (entières ou fragments) en silex local de l'unité V97 empruntées à une autre unité (Dessins : N.Delacroix, clichés : P. Bodu ; CAP)

Fig. 32 .

 32 Fig. 32. Le nucléus T98.185 : remontage de rares tablettes et d'un éclat laminaire débité dans l'unité V97 (P. Bodu ; CAP)

Fig. 33 .Fig. 34 .

 3334 Fig. 33. Le nucléus T98.41 et les tablettes de réaménagement de plan de frappe remontées (P. Bodu ; CAP)

3, fig. 4). Ainsi

  été ainsi remontés sur les 3 unités. La grande majorité associe entre 2 et 5 silex (96 soit 77,4%), 12 associent entre 6 et 10 objets (9,6%), et 8 entre 11 et 15 objets (6,4%). Les remontages plus conséquents, au-delà de 15 objets et jusqu'à une trentaine sont beaucoup plus rares (6,4%) 1 . Si les plus petits ensembles remontés apportent relativement peu d'informations en termes spatial et économique, mais ce n'est pas tout le temps vrai (cf. lien par remontage entre M103 et D110 qui se fait entre deux silex uniquement), les plus grands permettent de relier indiscutablement l'ensemble des unités entre elles, que ce soit V97 avec D110, D110 avec T125 mais aussi plus récemment V97 avec T125 et M103 avec D110 (fig. la contemporanéité de l'ensemble des unités du IV0 est démontrée et confirmée même si le long temps d'occupation de ce niveau par les Magdaléniens a probablement occasionné des décalages temporels légers entre l'installation des différentes unités et leur utilisation/réutilisation. La circulation « à double sens » de différents supports laminaires mais aussi d'éclats et de nucléus plaide, quoiqu'il en soit, pour une réelle synchronie de l'ensemble des locus du niveau IV0, témoignant d'une complémentarité entre eux dont il reste à définir les modalités.

	124 ensembles ont
		2-5
		objet	6 à 10	11 à 15	16 à 20	21 à 25	26 à 30	Sup à 30
	remontages	s	objets	objets	objets	objets	objets	objets	Total
	D110	67	7	5			1		80
	M103	1	1					1	3
	V97	28	4	3	3	1	1	1	41
	Total	96	12	8	3	1	2	2	124
					59

Tabl. 1. Remontages réalisés entre 2017 et 2019 sur le matériel lithique taillé des unités D110, M103, V97.

  Malgré cela, si une trentaine de blocs donnent entre 20 et 30 supports utilisables soit entre 600 et 900 lames et/ou lamelles au maximum, c'est insuffisant pour expliquer la présence de près de 2700 outils et armatures en silex local sur le niveau IV0. Cet écart quantitatif entre produits potentiellement obtenus sur les nucléus disponibles et les outils-armatures en silex local peut s'expliquer par la forte fragmentation de l'outillage du fond commun (plusieurs outils sur un même support ?) mais aussi par le fait que nous n'avons peut-être pas retrouvé tous les nucléus débités sur le niveau IV0, certains étant vraisemblablement hors de la zone fouillée ou emportés à l'extérieur du campement. Comme nous avons mené des vérifications au sud-centre et sud-ouest, à l'ouest et au nord du niveau IV0, les seules possibilités d'extension du campement se trouvent à l'est ou au sud-est soit sous le parking actuel ou sous le musée.

II -Bilan 2019 sur le travail de raccord et remontage des objets en matériau localCette année, deux semaines ont été consacrées à une session de raccords et remontages sur le matériel lithique du niveau IV0 (figs. 1 et 2). Depuis plusieurs années et au gré des campagnes de fouille, nous avions déjà pu réaliser des raccords systématiques sur le silex tertiaire mais aussi un certain nombre de remontages d'éléments taillés en silex local notamment ceux provenant des unités T125 et M103. La méthode avait été en revanche moins appliquée aux unités D110 et V97, cette dernière ayant été fouillée plus récemment que les autres. Depuis le début de la triennale consacrée à la monographie du niveau IV0, nous avons certes entamé ce travail de remontage mais la présence des stagiaires en 2017 notamment, a ralenti le processus en raison du temps qu'il fallait consacrer à leur encadrement et à leur formation. Cette année, nous avons donc choisi de ne pas impliquer d'étudiants dans ce travail, ce qui nous a permis de nous consacrer pleinement et pendant deux semaines à cette activité.

Certains appariements associent jusqu'à une quarantaine de silex mais les relations entre l'ensemble de ces objets doivent être confirmées


Ana Paixão

Un Programme Conçu Par Le Professeur

Manuela De Sá

Patrícia Modesto

Sara Afonso

Manuel Brandão

" Récital Les temps de la langue portugaise " dans le

teaching and research institutions in France or abroad, or from public or private research centers.

.

RECITAL -LES TEMPS DE LA LANGUE. TEMOIGNAGES

Manuela de Sá (Professeur à l'École de musique du Conservatoire national de Lisbonne) Nous présentons ici, sous forme de témoignages, les réflexions et les choix qui nous ont permis d'aboutir au Récital Les temps historiques de la langue portugaise qui a clos la première journée (18/10/19) du Colloque de Commémoration du Centenaire de l'Enseignement du Portugais dans l'Université Française -Sorbonne. Le premier est celui de Manuela de Sá, le second celui de Ana Paixão.

Choix musical et poétique : questions préalables

La conception d'un programme pour voix et piano focalisé sur l'illustration sonore de la langue portugaise, à travers l'association de textes poétiques et de compositions musicales, nous a amenées à nous poser les questions préalables qui suivent :

quelles périodes littéraires ? quelles caractéristiques phonétiques ? quels textes ? quels compositeurs ? quelles voix ?

Recherche et méthodes : poésie, musique, chant, périodes historiques

La recherche a porté sur des textes poétiques appartenant à une large période sous-divisée en deux grandes parties :

de la poésie médiévale des troubadours à l'époque des Grandes Découvertes (XV e -XVI e siècles) ; de la poésie du XIX e au XX e siècles. Sachant que le Brésil a joué un rôle important au XIX e siècle, comme prolongement de l'Empire, avec le transfert du siège de la Cour du Royaume de Lisbonne à Rio de Janeiro, nous avons jugé nécessaire d'intégrer dans notre choix poétique et musical des créations qui ont vu le jour sur le sol brésilien.

Un problème s'est alors posé quant au portugais des différentes époques historiques. Comment chanter les paroles poétiques ? Avec les sonorités du portugais européen actuel ou en respectant les sonorités des différentes époques ? Quel que fût le choix, les défis étaient de taille.

Le premier choix (sonorités du portugais actuel), posait le problème des sonorités plus consonantiques que vocaliques, compte tenu de l'évolution de la langue portugaise, dans sa variante européenne. Or, pour les chanteurs lyriques, cette diminution de sons vocaliques pose des problèmes quant à l'ajustement de résonateurs, pour qu'ils puissent atteindre des paliers adéquats d'énergie acoustique et ainsi réussir une bonne émission de la voix.

Nous rappelons un extrait très éclairant d'une étude du linguiste brésilien Celso Cunha: «O português era uma língua de base vocálica e assim continua na modalidade brasileira. Há cerca de dois séculos começou o português europeu a seguir outra deriva, ou seja, a fortalecer as consoantes e a obscurecer as vogais átonas»1 .

Le deuxième choixcertainement le plus intéressant pour le public -demandait la participation d'un expert en linguistique diachronique, d'une part, tout en posant des problèmes de cohérence, pour ce qui est de la relation musique / texte poétique, d'autre part.

En d'autres termes, du point de vue linguistique et poétique, la prononciation des poèmes avec la prononciation de l'époque s'avérait cohérente, mais, pour ce qui est du versant musical, des anachronismes étaient évidents, quant à l'instrument (le piano n'existait pas du temps des troubadours, ni du temps des découvertes) et quant à la musique (les compositeurs ne sont pas contemporains des poètes).

Ceci dit, nous avons assumé ces anachronismes et nous avons fait le choix de respecter les caractéristiques phonétiques de la langue portugaise, selon la période des textes poétiques. Nous avons voulu rendre vivant, le temps fugitif du concert, le parler des différents « temps de la langue », le parler des gens qui circulaient dans les rues à différentes époques historiques.

Bref, le concert ainsi conçu se voulait une performance vivante de voix et piano, par la voix de poètes, évoquant l'histoire de la langue portugaise et de ceux qui la parlaient, à différents moments de l'Histoire.

Le choix du répertoire

Tout d'abord, il convient de souligner l'apport fondamental du pianiste José Manuel Brandãoque nous remercionspour la sélection, selon des critères de maturité dans la création musicale, du point de vue mélodique et harmonique, mais aussi selon le critère de l'ajustement de la prosodie à la structure musicale.

Nous avons alors fait une première sélection de textes poétiques et de compositions musicales. À ce stade, l'expertise en linguistique historique de Esperança Cardeiraque nous remercions égalementde façon à reconstituer des traits phonétiques et prosodiques des textes de différentes époques, a été essentielle.

Nous reprenons un extrait d'une étude de Esperança Cardeira très éclairant :

Em rigor, quando traçamos a história do português referimo-nos apenas à história da língua escrita já que as fontes em que nos fundamentamos são documentos escritos. Não dispondo de uma milagrosa máquina de viajar no tempo nunca poderemos ouvir falar D. Dinis para sabermos como se pronunciava o português na sua época. Ainda assim, a Linguística Histórica tem feito um bom uso de maus dados, decifrando as pistas que a escrita oferece para a compreensão da oralidade. É esse trabalho quase detectivesco do estudo dos textos antigos, aliado à observação das variedades actuais da língua, que nos permite esboçar uma história da pronúncia do português desde a sua fase arcaica até aos nossos dias 2 . Un autre versant, non moins important et nécessaire pour atteindre l'objectif final, était celui de l'adéquation de chaque chanson aux caractéristiques du timbre de la voix 3 , des techniques et de la personnalité des chanteuses lyriques. Nous avons ainsi procédé à la distribution du programme, gardant toujours en vue les différents versants, d'abord par essais partiels, puis par essais de la totalité de chaque oeuvre du programme.

  phonétique des textes nous a permis de garantir que les particularités spécifiques à chaque époque étaient représentées. Ce critère a guidé notre choix final. Le public devrait pouvoir percevoir les différences et rendre son oreille sensible aux caractéristiques phonétiques de chaque période historique représentée. Nous tenons à souligner l'apport indispensable de Esperança Cardeiraet sa patiencepour que la rigueur phonétique soit respectée.

	Nous avons pu arriver au choix suivant :
	Les auteurs et les textes
	-Lourenço Jogral, Cantar d'amigo
	-Xohan Zorro, Em Lisboa sobre lo mar
	-Dom Denis, Levantou-se a velida
	-Dom Sancho I, A Ribeirinha
	-Luís de Camões, Se Helena apartar ; O céu, a terra o vento sossegado ; Descalça vai para a
	fonte ; Na fonte está Leonor
	-José Gomes Ferreira, Aquela que eu nunca vi
	-Almeida Garrett, Barca bela
	-Tolentino de Mendonça, Olham por dentro ; Tudo é sombra que passa
	-Ruth Valadares Corrêa, Tarde, uma nuvem rósea…
	Les compositeurs (suivant l'ordre du programme)
	-Cláudio Carneiro
	-Fernando Lopes Graça
	-Manuel Ivo Cruz
	-Joly Braga Santos
	-Croner de Vasconcelos
	-Alfredo Teixeira
	-Heitor Villa-Lobos
	-José Francisco Leal
	L'étude

Celso Cunha, Língua portuguesa e realidade brasileira, Rio de Janeiro, Tempo Brasileiro, 1986, p.