Adrien Deloro

Éric Jaligot

Lie rank in groups of finite Morley rank with solvable local subgroups

Keywords: 2000 Mathematics Subject Classification: Primary 20F11, Secondary 20G07, 03C60 Cherlin-Zilber Conjecture, Morley Rank, Bender Uniqueness Theorem

We prove a general dichotomy theorem for groups of finite Morley rank with solvable local subgroups and of Prüfer p-rank at least 2, leading either to some p-strong embedding, or to the Prüfer p-rank being exactly 2.

Introduction

The "size" of a simple algebraic group over an algebraically closed field can be captured by several means. One can measure its Zariski dimension, but one can also consider its Lie rank, which is the Zariski dimension of its maximal algebraic tori. For instance, it is often straightforward to argue by induction on the Zariski dimension, as is typically the case with solvable groups. On the other hand the Lie rank is sometimes necessary in classification problems: it leads to the notions of thin/quasi-thin/generic groups which are essential in the Classification of the Finite Simple Groups, as well as in algebraic groups. Now the only quasi-simple algebraic groups of Zariski dimension 3 are of the form PSL 2 or SL 2 ; they also are the only quasi-simple algebraic groups of Lie rank 1. On the other hand PSL 2 and SL 2 are the only "small" quasi-simple algebraic groups from a purely group-theoretic point of view, namely a "local solvability" condition: the normalizer of each infinite solvable subgroup remains solvable. Hence an algebraic group with the latter property must have small Lie rank. In the present paper we give a precise meaning to, and prove, such a statement in a much more general context.

Our framework will be that of groups of finite Morley rank, for the global theory of which we refer to [START_REF] Borovik | Groups of finite Morley rank[END_REF] or [START_REF] Altınel | Simple groups of finite Morley rank[END_REF]. Briefly put, groups of finite Morley rank are groups equipped with a rudimentary notion of dimension on their first-order definable subsets, called the Morley rank for historical reasons in model theory. Since the Morley rank satisfies basic axioms reminiscent of the Zariski dimension of algebraic varieties over algebraically closed fields, groups of finite Morley rank generalize algebraic groups over algebraically closed fields. Conversely, a major question which has stirred a huge body of work is the Cherlin-Zilber algebraicity conjecture, which postulates that infinite simple groups of finite Morley rank are in fact isomorphic to algebraic groups over algebraically closed fields. The algebraicity conjecture holds true at least of groups containing an infinite elementary abelian 2-group [START_REF] Altınel | Simple groups of finite Morley rank[END_REF]; so far, the proof is part of the Borovik program for groups with involutions, and based on ideas modelled on the Classification of the Finite Simple Groups. On the other hand there are potential configurations of simple groups of finite Morley without involutions for which the Borovik program is helpless. Here we may refer to the configurations of simple "bad" groups of Morley rank 3 discovered in [START_REF] Cherlin | Groups of small Morley rank[END_REF], or more generally to the "full Frobenius" groups of finite Morley rank studied in [START_REF] Jaligot | Full Frobenius groups of finite Morley rank and the Feit-Thompson theorem[END_REF].

In the context of groups of finite Morley, the "local solvability" condition mentioned above is equivalent to the following.

Definition. A group of finite Morley rank is * -locally •

• solvable if N • (A) is solvable for each nontrivial abelian connected definable subgroup A.

We now relate our group-theoretic notion of smallness to an abstract version of the Lie rank as follows. Since there is a priori no satisfactory first-order analogue of the notion of an algebraic torus, we shall deal with certain torsion subgroups throughout. Although basic matters such as the conjugacy and structure of Sylow p-subgroups of a group of finite Morley rank are not settled in general, enough is known about abelian divisible p-subgroups, which are called p-tori. The maximal ones are conjugate [START_REF] Cherlin | Good tori in groups of finite Morley rank[END_REF], and direct powers of a finite number of copies of the Prüfer p-group Z p ∞ [START_REF] Borovik | Tores et p-groupes[END_REF]. The latter number is called the Prüfer p-rank of the ambient group. This will, quite naturally, be our analogue of the Lie rank. Indeed, it can easily be seen that in a quasi-simple algebraic group the Lie rank and the Prüfer p-rank agree: maximal tori of SL 2 or PSL 2 are of dimension 1 and of Prüfer p-rank 1 for any prime p different from the characteristic of the ground field. We note that in our more abstract context we then have a notion of the Lie rank for each prime p such that the ambient group contains a non-trivial divisible abelian p-subgroup.

If S is an abelian p-group for some prime p and n is a natural number, then we denote by Ω n (S) the subgroup of S generated by all elements of order p n . In technical terms that we will define shortly, our main theorem takes the following form.

Main Theorem. Let G be a connected nonsolvable * -locally • -solvable group of finite Morley rank of Prüfer p-rank at least 2 for some prime p, and fix a maximal p-torus S of G. Assume that every proper definable connected subgroup containing S is solvable, that elements of S of order p are not exceptional, and let

B = C • (s) | s ∈ Ω 1 (S) \ {1} .
Then:

(1) either B < G, in which case B is a Borel subgroup of G; and if in addition

S is a Sylow p-subgroup of N N (B) (S), N (B) is p-strongly embedded in G,
(2) or B = G, in which case S, or equivalently G, has Prüfer p-rank 2.

According to the algebraicity conjecture for simple groups of finite Morley rank, or rather a consequence of it, the Prüfer p-rank of a connected nonsolvable * -locally • -solvable group of finite Morley rank should be 1. Hence our Main Theorem deals with configurations which are not actually known to exist, but the dichotomy it gives severely limits possibilities in both cases. It is obvious in the second case, and in the first case it suffices to recall from [START_REF] Deloro | Groups of finite Morley rank with solvable local subgroups[END_REF] that a definable subgroup M of a group G is called p-strongly embedded if it contains non-trivial p-elements and M ∩ M g contains none for any g in G \ M . This mimics a similar notion in the theory of finite groups which had been crucial with p = 2 in the Classification of the Finite Simple Groups. In any case we note that for groups of finite Morley rank, and typically for p = 2, the "bad" or "full Frobenius" groups mentioned above fit in case (1) of our Main Theorem.

The present paper is actually part of a series which aims at classifying configurations of nonsolvable * -locally • solvable groups of finite Morley rank. For much more details on such groups, including a few historical remarks, we refer to the preliminary article [START_REF] Deloro | Groups of finite Morley rank with solvable local subgroups[END_REF] of our series. We simply recall that this classification started in [START_REF] Cherlin | Tame minimal simple groups of finite Morley rank[END_REF] in the case of minimal connected simple groups: these are the infinite simple groups of finite Morley rank all of whose proper definable connected subgroups are solvable (as in PSL 2 ). After a series of generalizations of the original classification of [START_REF] Cherlin | Tame minimal simple groups of finite Morley rank[END_REF] it was realized that much of the theory of minimal connected simple groups transfers readily to * -locally • -solvable groups, which are defined as the * -locally • -solvable ones but where the main condition that N • (A) is solvable is required for any nontrivial abelian subgroup A. This shift from minimal connected simple groups to * -locally • -solvable groups corresponds, in finite group theory, to a shift from the minimal simple groups studied for the Feit-Thompson (Odd Order) Theorem to the N -groups classified later by Thompson. Of the two main variations on the notion of "local solvability" from [START_REF] Deloro | Groups of finite Morley rank with solvable local subgroups[END_REF] we shall work here with the most general * -local Since [START_REF] Cherlin | Tame minimal simple groups of finite Morley rank[END_REF] it seems unrealistic to hope for a complete reduction of *locally •

• solvable groups to the algebraic ones, even assuming the presence of involutions in order to proceed to a much sharper analysis. However a reduction to a very small number of configurations will be obtained in [START_REF] Deloro | Groups of finite Morley rank with solvable local subgroups and of odd type[END_REF], along the lines of [START_REF] Cherlin | Tame minimal simple groups of finite Morley rank[END_REF] and subsequent papers. This can be seen as part of the Borovik program for classifying simple groups with involutions, in the utterly critical case of minimal configurations. We simply note that the present part of the analysis does not depend on p and can be reached by very general means (and some of them, such as signalizer functors, could be borrowed from finite group theory).

We also note that our Main Theorem generalizes the dichotomy represented by Sections 6 and 7 of [START_REF] Cherlin | Tame minimal simple groups of finite Morley rank[END_REF], concerning p = 2 in minimal connected simple groups also satisfying a simplifying "tameness" assumption. Sections 6 (resp. 7) there corresponds, in Prüfer 2-rank at least 2, to C • G (Ω 1 (S)) not being (resp. being) a Borel subgroup, two cases corresponding respectively to our cases (2) and (1) here. Then came [START_REF] Burdges | Minimal connected simple groups of finite Morley rank with strongly embedded subgroups[END_REF] where our corresponding case (1) was shown not to exist, still for minimal connected simple groups and for p = 2, but without the "tameness" assumption. In [START_REF] Deloro | Groups of finite Morley rank with solvable local subgroups and of odd type[END_REF] we will reach essentially the same conclusions in the much more general context of * -locally •

• -solvable groups, applying the general dichotomy of the present paper with p = 2. In particular we will bound by 2, in full generality, the Prüfer 2-rank of a nonsolvable *locally •

• solvable group. But in any case, case (2) of our Main Theorem still stands around, even with p = 2 and in the context of tame minimal connected simple groups of [START_REF] Cherlin | Tame minimal simple groups of finite Morley rank[END_REF] where the configuration is described with high precision.

We collect some raw material in §2 and the proof of our Main Theorem takes place in §3. In §4 we will make more comments on the difficulty to deal with the configuration arising in case (2); we will also give a form of our Main Theorem more directly applicable in [START_REF] Deloro | Groups of finite Morley rank with solvable local subgroups and of odd type[END_REF] (essentially explaining how to deal which the two extra assumptions that every proper definable connected subgroup containing S is solvable and that elements of S of order p are not exceptional).

Preliminaries

For general reference on groups of finite Morley rank we refer to [START_REF] Borovik | Groups of finite Morley rank[END_REF] or [START_REF] Altınel | Simple groups of finite Morley rank[END_REF], and for more specific facts about * -locally •

• solvable groups of finite Morley rank we refer to [START_REF] Deloro | Groups of finite Morley rank with solvable local subgroups[END_REF]. We simply recall that groups of finite Morley rank satisfy the descending chain condition on definable subgroups, and that any group G of finite Morley rank has a connected component, i.e., a smallest (normal) definable subgroup of finite index, denoted by G • .

Unipotence theory

A delicate point in our proof is the use of the abstract unipotence theory for groups of finite Morley rank. We follow the general treatment of [START_REF] Frécon | Model theory with applications to algebra and analysis[END_REF] and [DJ11a, §2.1], and for the sake of self-containment we shall try and put unipotence theory in a nutshell. P denotes the set of prime numbers.

Definition. A unipotence parameter is a pair p = (p, r) ∈ ({∞} ∪ P) × (N ∪ {∞}) with p < ∞ if and only if r = ∞. In addition, p is called the characteristic of p and r is called the unipotence degree of p.

Such a notion enables a parallel treatment to two theories which had been considered distinct, namely p-unipotence theory (p a prime), and Burdges' more recent notion of null characteristic graded unipotence which originated in [START_REF] Burdges | A signalizer functor theorem for groups of finite Morley rank[END_REF]. For p a prime, a p-unipotent subgroup is by definition a nilpotent definable connected p-group of bounded exponent, and for p = ∞ a definition is given in terms of generation by certain definable connected abelian subgroups with torsion-free quotients of rank r [FJ08, §2.3]. In the extreme case where (p, r) = (∞, 0) one makes use of so-called decent tori [START_REF] Cherlin | Good tori in groups of finite Morley rank[END_REF]. All this gives rise, for any group G of finite Morley rank, to the group U p(G) as the group generated by subgroups as above. This group is always definable and connected, and for p a prime

U (p,∞) (G) is simply denoted by U p (G).
Imposing nilpotence, it leads to a general notion of p-subgroup for any unipotence parameter p, and now imposing maximality with respect to inclusion, it provides a notion of Sylow p-subgroup [FJ08, §2.4], with properties much similar to those of Sylow p-subgroups of finite groups.

In groups of finite Morley rank, the Fitting subgroup, i.e., the group generated by all normal nilpotent subgroups, is always definable and nilpotent, and in particular the unique maximal normal nilpotent subgroup. If one lets d p (H) be the maximal r such that H contains a non-trivial (p, r)-unipotent subgroup [DJ11a, Definition 2.5], then the interest for the unipotence theory comes from the following in connected solvable groups.

Fact 2.1. [DJ11a, Fact 2.8] Let H be a connected solvable group of finite Morley rank and p = (p, r) a unipotence parameter with r > 0. Assume d p (H) ≤ r. Then U p(H ) ≤ F • (H), and in particular U p(H ) is nilpotent.

Uniqueness Theorem in * -locally • • -solvable groups

The main result about * -locally • • -solvable groups is a Uniqueness Theorem analogous to a similar result of Bender about minimal simple groups in finite group theory. We give here the most general statement as proved in [START_REF] Deloro | Groups of finite Morley rank with solvable local subgroups[END_REF]. We will not use it as such, but the proof of our Main Theorem will be based on an elaborate version of this Uniqueness Theorem. • -solvable group of finite Morley rank, p = (p, r) a unipotence parameter with r > 0,

and U a Sylow p-subgroup of G. Assume that U 1 is a nontrivial definable p- subgroup of U containing a nonempty (possibly trivial) subset X of G such that d p (C • (X)) ≤ r. Then U is the unique Sylow p-subgroup of G containing U 1 , and in particular N (U 1 ) ≤ N (U ).
A Borel subgroup is a maximal definable connected solvable subgroup. Fact 2.2 has the following consequence on Borel subgroups of * -locally

(G) = r. Let B be a Borel subgroup of G such that d p (B) = r. Then U p(B) is a Sylow p- subgroup of G, and if U 1 is a nontrivial definable p-subgroup of B, then U p(B) is the unique Sylow p-subgroup of G containing U 1 , N (U 1 ) ≤ N (U p(B)) = N (B),
and B is the unique Borel subgroup of G containing U 1 .

Torsion

The results we shall use about torsion are rather elementary. We first give the general decomposition of nilpotent groups of finite Morley rank, in terms of both classical Sylow p-subgroups and Sylow p-subgroups. (1) G is the central product of its Sylow p-subgroups and its Sylow (∞, r)subgroups, which are all divisible for p of the form (∞, r) [FJ08, p. 16].

(2) If G is connected, then G is the central product of its Sylow p-subgroups.

Throughout the rest of this subsection, p denotes a prime number. For an arbitrary subgroup S of a group of finite Morley rank, one defines the (generalized) connected component of S by S • = H • (S) ∩ S, where H(S) is the smallest definable subgroup of G containing S. (Of course, H(S) exists by descending chain condition on definable subgroups.) It is easily checked that S • has finite index in S.

Fact 2.5. [BN94, Corollary 6.20] Let p be a prime and S a p-subgroup of a solvable group of finite Morley rank, or more generally a locally finite p-subgroup of any group of finite Morley rank. Then:

(1) S • is a central product of a p-torus and a p-unipotent subgroup.

(2) If S is infinite and has bounded exponent, then Z(S) contains infinitely many elements of order p.

We also recall that Sylow p-subgroups are conjugate in solvable groups of finite Morley rank [BN94, Theorem 9.35]. Furthermore they are connected in connected solvable groups of finite Morley rank by [BN94, Theorem 9.29]; in this case a Sylow p-subgroup S satisfies S = S • and has a decomposition as in Fact 2.5 (1), a point that will be frequently used below.

If G is a group of finite Morley rank, we denote by O p (H) the largest normal definable connected subgroup without p-torsion. It exists by ascending chain condition on definable connected subgroups and the following elementary property of lifting of torsion valid in groups of finite Morley rank (or more generally in groups with the descending chain condition on definable subgroups):

Fact 2.6. [ABC08, Lemma 2.18] Let G be a group of finite Morley rank, N a normal definable subgroup of G, and x an element of G such that x has finite order n modulo N . Then the coset xN contains an element of finite order, involving the same prime divisors as n.

The following will be useful when dealing with p-strongly embedded subgroups.

Lemma Let

F = F • (H). As O p (H) = 1, O p (F ) = 1
as well, and U q (H) = 1 for any prime q different from p. By assumption U p (H) = 1 also, and F is divisible by Fact 2.4. As F is torsion-free, by [BN94, Theorem 2.9] or Fact 2.4 and [DJ11a, Corollary 2.2], it must be trivial by assumption. (In connected groups of finite Morley rank, derived subgroups are definable and connected by a well-known corollary of Zilber's indecomposability theorem.) Hence F is divisible abelian.

To conclude it suffices to show that F is central in H, as then H is nilpotent, hence equal to F , and hence divisible abelian, as desired. Let h be any element of H; we want to show that [h, F ] = 1. Notice that that map f → [h, f ], from F to F , is a definable group homomorphism. As the torsion subgroup of F is central in H (by [DJ11a, Fact 2.7 (1)], or using [DJ11a, Fact 2.1]), it is contained in the kernel of the previous map and Fact 2.6 shows that the image of the previous map, i.e., [h, F ], is torsion-free. Hence [h, F ] ≤ O p (F ) = 1, as desired.

Generation by centralizers

In the present subsection we prove miscellaneous lemmas concerning generation by centralizers.

Lemma 2.8. Let p be a unipotence parameter and q a prime number. Let H be a p-group of finite Morley rank without elements of order q, and assume K is a definable solvable q-group of automorphisms of H of bounded exponent. Then C H (K) is a definable p-subgroup of H.

Proof. By descending chain condition on centralizers, C H (K) is the centralizer of a finitely generated subgroup of K, and by local finiteness of the latter we may assume K finite. In particular C H (K) is connected by [START_REF] Burdges | A signalizer functor theorem for groups of finite Morley rank[END_REF]Fact 3.4].

When p = (∞, 0), H is a good torus in the sense of [START_REF] Cherlin | Good tori in groups of finite Morley rank[END_REF], and in particular (∞, 0)-homogeneous in the sense of [FJ08, Lemma 2.17], and the connected subgroup C H (K) is also a good torus. Otherwise, C H (K) is also a p-group, by [Bur04, Lemma 3.6] when the unipotence parameter is finite, or by [FJ08, Lemma 2.17-c] when the characteristic is finite.

Fact 2.9. [Bur04, Fact 3.7] Let H be a solvable group of finite Morley rank without elements of order p for some prime p. Let E be a finite elementary abelian p-group acting definably on H. Then:

H = C H (E 0 ) | E 0 ≤ E, [E : E 0 ] = p .
We recall that a Carter subgroup of a group of finite Morley is, by definition, a definable connected nilpotent subgroup of finite index in its normalizer.

Lemma 2.10. Let H be a connected solvable group of finite Morley rank such that U p (H) = 1 for some prime p. Suppose that H contains an elementary abelian p-group E of order p 2 . Then:

H = C • H (E 0 ) | E 0 is a cyclic
O p (H) = C • O p (H) (E 0 ) | E 0 is a cyclic subgroup of order p of E .
But the generation by the full centralizers is given by Fact 2.9, and these centralizers are connected by [Bur04, Fact 3.4].

A subgroup is called p-toral, or just toral, if it is contained in a p-torus of the ambient group.

Lemma 2.11. Let H be a connected solvable group of finite Morley rank with an elementary abelian p-toral subgroup E of order p 2 for some prime p. Then:

H = C • H (E 0 ) | E 0 is a cyclic subgroup of order p of E .
Proof. For a connected nilpotent group of finite Morley rank L, we define the "complement" C p (L) of U p (L), namely the product of all factors of L as in Fact 2.4 (2), except U p (L). Now if H is any connected solvable group of finite Morley rank and Q a Carter subgroup of H, then H = QF • (H) by [FJ08, Corollaire 3.13], and H is the product of the definable connected subgroup C p (Q)C p (F • (H)) with the normal definable connected subgroup U p (H), and the first factor has trivial p-unipotent subgroups.

In our particular case, E is by torality contained in a p-torus, and the latter is contained in a Carter subgroup Q of H by [FJ08, Théorème 3.3]. By [BN94, Theorem 9.29] and Fact 2.5, E centralizes the normal definable connected subgroup U p (H), so it suffices to show the generation by the connected components of centralizers in C p (Q)C p (F • (H)). But this follows from Lemma 2.10.

Proof of our theorem

We now turn to proving our Main Theorem which we restate.

Main Theorem. Let G be a connected nonsolvable * -locally •

• -solvable group of finite Morley rank of Prüfer p-rank at least 2 for some prime p, and fix a maximal p-torus S of G. Assume that every proper definable connected subgroup containing S is solvable, that elements of S of order p are not exceptional, and let

B = C • (s) | s ∈ Ω 1 (S) \ {1} .
Then:

(1) either B < G, in which case B is a Borel subgroup of G; and if in addition

S is a Sylow p-subgroup of N N (B) (S), N (B) is p-strongly embedded in G, (2)

Getting rid of (1)

Assume first

(1) B < G.

By assumption B ≤ B 1 for some Borel subgroup B 1 of G. As S ≤ B ≤ B 1 , Lemma 2.11 implies that B = B 1 , and thus B is a Borel subgroup of G.

In particular, Sylow p-subgroups of B are conjugate, as in any solvable group of finite Morley rank.

We now make the new assumption that S is a Sylow p-subgroup of N N (B) (S).

Lemma 3.1. U p (C(s)) = 1 for every element s of order p of S.

Proof. It suffices to apply Fact 2.5 in the connected solvable group B, again with the fact that Sylow p-subgroups of connected solvable groups of finite Morley rank are connected.

We claim that M = N (B) is p-strongly embedded in G in this case by using a "black hole" principle (a term going back to Harada) similar to the one used in [BCJ07, §2.2], and already contained in [CJ04, Lemma 7.3]. We note that Lemma 3.1 implies that S is a Sylow p-subgroup of B indeed, and of M as well, as M = BN N (B) (S) by a Frattini Argument. In particular M/B has trivial Sylow p-subgroups by lifting of torsion, Fact 2.6.

Assume that M ∩M g contains an element s of order p for some g in G. Notice that s is actually in B ∩ B g , and p-toral. By connectedness and conjugacy of Sylow p-subgroups in connected solvable groups, the definition of B implies that C • (s ) ≤ B for any element s of order p of B. Similarly, C • (s ) ≤ B g whenever s has order p and is in B g . By conjugacy in B we may assume s in S, and Ω 1 (S) ≤ S ≤ C • (s) ≤ B ∩ B g . By Lemma 2.10 or 2.11 applied in B and in B g we get

B g = C • (s) | s ∈ Ω 1 (S) \ {1} = B. Thus g normalizes B,
and is in M .

Hence M = N (B) is p-strongly embedded in G under the extra assumption adopted here, and this proves clause (1) of the Main Theorem.

Case (2); p-elements

We now pass to the second case (2) B = G.

We will eventually show that clause (2) of our Main Theorem holds by reworking the begining of Section 6 of [START_REF] Cherlin | Tame minimal simple groups of finite Morley rank[END_REF]. We first put aside p-unipotent subgroups.

Lemma 3.2. Any Borel subgroup containing a toral element of order p has trivial p-unipotent subgroups.

Proof. Assuming the contrary, we may assume after conjugacy of decent tori [START_REF] Cherlin | Good tori in groups of finite Morley rank[END_REF] In other words, nontrivial p-toral elements commute with no nontrivial punipotent subgroups. This can be stated more carefully as follows.

Corollary 3.3. Any connected solvable subgroup which is s -invariant for some p-toral element s of order p has trivial p-unipotent subgroups. Proof. Otherwise s would normalize a nontrivial p-unipotent subgroup, and by Fact 2.5 it would centralize a nontrivial p-unipotent subgroup.

Our assumption (2) on B yields similarly a property antisymmetric to the black hole principle implied by assumption (1). Let E denote the elementary abelian p-group Ω 1 (S).

Lemma 3.4. Let E 1 be a subgroup of E of order at least p 2 . Then for any proper definable connected subgroup L there exists an element s of order

p of E 1 such that C • (s) L. In particular G = C • (s) | s ∈ E 1 \ {1} .
Proof. Assume on the contrary C • (s) ≤ L for any element s of order p of E 1 .

We claim that C • (t) ≤ L for any element t of order p of E. In fact, as

E 1 ≤ S ≤ C • (t), C • (t)
is by Lemma 2.10 generated by its subgroups of the form C • (t, s), with s of order p in E 1 . As these groups are all contained in L by assumption, our claim follows.

Hence we have B ≤ L < G. But under our current assumption B = G, and this is a contradiction.

Our last claim follows immediately.

Corollary 3.5. There exists an element s of order Lemma 3.6. There exists an element s of order p of E such that

p of E such that C • (S) < C • (s). Proof. C • (S) is S-local •
d(O p (C • (s))) ≥ 1.
Proof. Assume the contrary, and let s be an arbitrary element of order p of E.

By our assumption that d(O

p (C • (s))) ≤ 0, O p (C • (s)) is trivial or a good torus by [DJ11a, Lemma 2.6], and central in C • (s) by [DJ11a, Fact 2.7 (1)]. Notice that U p (C • (s)) = 1 by Lemma 3.2. As C • (s)/O p (C • (s)) is abelian by Lemma 2.7, C • (s) is nilpotent. Now S is central in C • (s) by Fact 2.4 (2). In particular C • (S) = C • (s)
, and this holds for any element s of order p of E. We get a contradiction to Corollary 3.5.

It follows in particular from Lemma 3.6 that there exist definable connected subgroups L containing C • (s) for some element s of order p of E and such that O p (L) is not a good torus. Choose then a unipotence parameter q = (q, r) different from (∞, 0) such that r is maximal in the set of all d q (O p (L)), where L varies in the set of all definable connected solvable subgroups with the above property.

Notice that there might exist several such maximal unipotence parameters q, maybe one for q = ∞ and several ones for q prime, except for q = p by Corollary 3.3.

It will also be shortly and clearly visible below that the notion of maximality for q is the same when L varies in two smaller subsets of all definable connected solvable subgroups containing C • (s) for some s of order p of E: the set of Borel subgroups with this property on the one hand, and exactly the finite set of subgroups of the form C • (s) on the other. Lemma 3.7. Let L be any definable connected solvable subgroup containing C • (s) for some element s of order p of E. Then U q (O p (L)) is a normal definable connected nilpotent subgroup of L.

Proof. As O p (L) is normal in L, it suffices to show that its definably characteristic subgroup U q (O p (L)) is nilpotent. But the latter is in F (O p (L)) by Fact 2.1 and the maximality of r.

Corollary 3.8. Let L be any definable connected solvable subgroup containing C • (s) for some element s of order p of E. Then any definable q-subgroup of L without elements of order p is in U q (F (O p (L))).

Proof. Let U be such a subgroup. As U p (L) = 1 by Lemma 3.2, L/O p (L) is (divisible) abelian by Lemma 2.7, and thus U ≤ O p (L), and U ≤ U q (O p (L)). Now it suffices to apply the normality and the nilpotence of the latter.

A Uniqueness Theorem via elementary p-groups

We now prove a version of the Uniqueness Theorem (Fact 2.2) with a combined action, more precisely where the assumption on unipotence degrees of centralizers is replaced by an assumption of invariance by a sufficiently "large" p-toral subgroup. For this purpose we first note the following. Lemma 3.9. Let E 1 be a subgroup of order at least p 2 of E, and H a definable connected solvable E 1 -invariant subgroup. Then d q (O p (H)) ≤ r.

Proof. Assume toward a contradiction r > r, where r denotes d q (O p (H)). In this case r is necessarily finite, and

q = ∞. By Fact 2.1, U (∞,r ) (O p (H)) ≤ F • (O p (H))
, and this nontrivial definable (∞, r )-subgroup is E 1 -invariant. Fact 2.9 gives an element s of order p in E 1 such that

C U (∞,r ) (O p (H)) (s) = 1.
But the latter is an (∞, r )-group by Lemma 2.8. Now considering the definable connected solvable subgroup C • (s) gives a contradiction to the maximality of r, as C • (s)/O p (C • (s)) is (divisible) abelian as usual and the centralizer above is connected without elements of order p, and thus contained in O p (C • (s)).

As already mentioned around the definition of maximal parameters q (after Lemma 3.6), the same argument shows that r is also exactly the maximum of the d q (O p (L)) different from 0, with L varying in the set of Borel subgroups containing C • (s) for some element s of order p of E (instead of all definable connected solvable subgroups L with the same property), and similarly with L varying in the set of subgroups C • (s) for some element s of order p of E.

We now prove our version of the Uniqueness Theorem, Fact 2.2, specific to the configuration considered here.

Theorem 3.10. Let E 1 be a subgroup of order at least p 2 of E. Then any E 1 -invariant nontrivial definable q-subgroup without elements of order p is contained in a unique maximal such.

Proof. Let U 1 be the q-subgroup under consideration. Let U be a maximal E 1 -invariant definable q-subgroup without elements of order p containing U 1 .

Assume V is another such subgroup, distinct from U , and chosen so as to maximize the rank of

U 2 = U q (U ∩ V ). As 1 < U 1 ≤ U 2 , the subgroup U 2 is nontrivial. As U 2 is nilpotent, N := N • (U 2 ) is solvable by * -local •
• solvability of G. Note that U 2 < U , as otherwise U = U 2 ≤ V and U = V by maximality of U . Similarly U 2 < V , as otherwise V = U 2 ≤ U and V = U by maximality of V . In particular, by normalizer condition [FJ08, Proposition 2.8], U 2 < U q (N U (U 2 )) and U 2 < U q (N V (U 2 )).

We claim that d q (O p (N )) = r. Actually d q (O p (N )) ≤ r by Lemma 3.9, and as O p (N ) contains U 2 which is nontrivial and of unipotence degree r in characteristic q we get d q (O p (N )) = r.

By Fact 2.1 and the fact that r ≥ 1 we get U q (O p (N )) ≤ F • (O p (N )). In particular U q (O p (N )) is nilpotent, and contained in a maximal definable E 1 -invariant q-subgroup without elements of order p, say Γ. Notice that N , being E 1 -invariant, satisfies U p (N ) = 1, and N/O p (N ) is abelian as usual. Now U 1 ≤ U 2 < U q (N U (U 2 )) ≤ Γ, so our maximality assumption implies that Γ = U . In particular

U q (N V (U 2 )) ≤ Γ = U . But then U 2 < U q (N V (U 2 )) ≤ U q (U ∩ V ) = U 2 , a contradiction.
Corollary 3.11. Let E 1 be a subgroup of order at least p 2 of E.

(1) If U 1 is a nontrivial E 1 -invariant definable q-subgroup without elements of order p, then U 1 is contained in a unique maximal E 1 -invariant definable connected solvable subgroup B. Furthermore U q (O p (B)) is the unique maximal E 1 -invariant definable q-subgroup without elements of order p containing U 1 , and, for any element s of order p of E 1 with a nontrivial centralizer in U 1 , C • (s) ≤ B and B is a Borel subgroup of G.

(

) U q (O p (C • (E 1 ))) is trivial. 2
Proof. (1). Assume B 1 and B 2 are two maximal E 1 -invariant definable connected solvable subgroups containing U 1 . We have U p (B 1 ) = U p (B 2 ) = 1. Hence B 1 and B 2 are both abelian modulo their O p subgroups.

Let U = U q (O p (B 1 ∩ B 2 )). This group contains U 1 and is in particular nontrivial, and is E 1 -invariant, as well as U q (O p (B 1 )) and U q (O p (B 2 )). Now all these three subgroups are contained in a (unique) common maximal E 1 -invariant definable q-subgroup without elements of order p by the Uniqueness Theorem 3.10, say Ũ . Notice that B 1 = N • (U q (O p (B 1 ))) and B 2 = N • (U q (O p (B 2 ))) by maximality of B 1 and B 2 . Now applying the normalizer condition [FJ08, Proposition 2.8] in the subgroup Ũ without elements of order p yields easily U q (O p (B 1 )) = Ũ = U q (O p (B 2 )). Taking their common connected normalizers,

B 1 = B 2 .
Our next claim follows from the same argument.

For the last claim, we note that there exists an element s in E 1 of order p such that C U1 (s) is nontrivial. By Lemma 2.8 the latter is a q-group, and of course it is E 1 -invariant. So the preceding uniqueness applies to C U1 (s), and as C U1 (s) ≤ U 1 ≤ B we get that B is the unique maximal E 1 -invariant definable connected solvable subgroup containing C U1 (s). But C U1 (s) ≤ C • (s) ≤ B s for some Borel subgroup B s and E 1 ≤ B s , so B s satisfies the same conditions as B, so B s ≤ B and B = B s is a Borel subgroup of G.

(2). Suppose toward a contradiction

U := U q (O p (C • (E 1 ))) nontrivial. It is of course E 1 -invariant. Recall that Q is a fixed Carter subgroup of G containing the maximal p-torus S. As Q ≤ C • (E 1 ), Q normalizes the subgroup U . Now for any element s of order p in E 1 we have U Q ≤ C • (s).
As E 1 ≤ Q, any Borel subgroup containing U Q is E 1 -invariant, and by the first point there is a unique Borel subgroup containing U Q. Now C • (s) is necessarily contained in this unique Borel subgroup containing U Q, and this holds for any element s of order p of E 1 . We get a contradiction to Lemma 3.4.

We note that the proof of the second point in Corollary 3.11 actually shows that any definable connected subgroup containing E 1 and U 1 for some nontrivial E 1 -invariant definable q-subgroup U 1 without elements of order p is contained in a unique Borel subgroup of G. Furthermore with the notation of Corollary 3.11 (1) we have in any case N (U 1 ) ∩ N (E 1 ) ≤ N (U q (O p (B))) = N (B).

Bounding the Prüfer rank via "signalizer functors"

There are two possible ways to prove that the Prüfer p-rank is 2 at this stage. One may use the Uniqueness Theorem 3.10 provided by the * -local •

• solvability of the ambient group, or use the general signalizer functor theory, which gives similar consequences in more general contexts. We now explain how to use the signalizer functor theory to get the bound on the Prüfer p-rank, but we will rather continue the analysis with the Uniqueness Theorem 3.10 which is closer in spirit to [CJ04, Lemma 6.1] and our original proof. It also gives much more information in the specific context under consideration, including when the Prüfer p-rank is 2, while the general signalizer functor theory just provides the bound.

Then, still assuming E of order at least p 3 , there exists a subgroup E 2 of E of order at least p 2 and disjoint from s . By Lemma 2.10,

C • (s) = C C • (s) (t) | t ∈ E 2 \ {1} .
But now if t is in E 2 as in the above equality, then E 1 := s, t has order p 2 as E 2 is disjoint from s , hence

C C • (s) (t) ≤ C(s, t) ≤ N ( s, t ) = N (E 1 ) ≤ N (θ(E)),
and this shows that C • (s) ≤ N • (θ(E)). This is a contradiction, and as our only extra assumption was that the Prüfer p-rank was at least 3, it must be 2.

Bounding the Prüfer rank: Uniqueness Methods

Anyway, we can get the bound similarly, by using more directly the Uniqueness Theorem 3.10 here instead of the axiomatized signalizer functor machinery. Actually the proof below is the core of the proof of the Nilpotent Signalizer Functor Theorem, and the Uniqueness Theorem here shortcuts the passage to a quotient for the induction in the general case (see [START_REF] Burdges | A signalizer functor theorem for groups of finite Morley rank[END_REF]). Theorem 3.13. S has Prüfer p-rank 2.

Proof. Assume towards a contradiction E has order at least p 3 .

We then claim that there exists a unique maximal nontrivial E-invariant definable q-subgroup without elements of order p. Let U 1 and U 2 be two such subgroups. Then by Lemma 2.8 and Fact 2.9 C U1 (E 1 ) and C U2 (E 2 ) are nontrivial q-subgroups for some subgroups E 1 and E 2 of E, each of index p in E. Assuming |E| ≥ p 3 then gives an element s of order p in E 1 ∩ E 2 . Now C U1 (s) and C U2 (s) are nontrivial q-subgroups by Lemma 2.8. Clearly both are Einvariant, as E centralizes s, and included in U q (O p (C • (s))) as usual, which is also E-invariant. Now the Uniqueness Theorem 3.10 gives U 1 = U 2 , as desired.

Hence there is a unique maximal E-invariant definable q-subgroup without elements of order p, say "θ(E)" in the notation of the signalizer functor theory. For the same reasons as mentioned above, Lemma 2.8 and Fact 2.9, it is nontrivial. Now by Lemma 2.8 and Fact 2.9 again, C θ(E) (E 1 ) is a nontrivial definable qsubgroup of θ(E) for some subgroup E 1 of E of index p. As U p (C • (E 1 )) = 1, the quotient C • (E 1 )/O p (C • (E 1 )) is abelian as usual, and the definable connected subgroup C θ(E) (E 1 ) is in O p (C • (E 1 )), and in U q (O p (C • (E 1 ))).

But as |E| ≥ p 3 , |E 1 | ≥ p 2 , and we get a contradiction to Corollary 3.11 (2). This proves clause (2) of the Main Theorem and completes its proof.

Afterword

We can also record informally some information gained along the proof of case (2) of our Main Theorem, which can be compared to [CJ04, 6.1-6.6]. We let G and S be as in case (2) of the Main Theorem and Q be a Carter subgroup of G containing S. Then Q is contained in at least two distinct Borel subgroups of G by Lemma 3.4, and in particular Q is divisible abelian by Fact 2.3 and [DJ11a, Proposition 4.46]. Now there are unipotence parameters q = (∞, 0) as in the proof of case (2) of the Main Theorem (maybe one for q = ∞, several for q prime, but none for q = p by Lemma 3.2). All the results of the above analysis apply, now with |Ω 1 (S)| = p 2 necessarily.

By Corollary 3.11, U q (O p (C • (Ω 1 (S)))) = 1.

As Ω 1 (S) has order p 2 , it contains in particular p 2 -1 p -1 = p + 1 pairwise noncollinear elements. It follows that there are at most p + 1 nontrivial subgroups of the form U q (O p (C • (s))) for some nontrivial element s of order p of S, and at most p + 1 Borel subgroups B containing Q (actually Ω 1 (S)-invariant suffices as noticed after Corollary 3.11) and such that U q (O p (B)) = 1. By Corollary 3.11, any such Borel subgroup would contain C • (s) for any element s of order p of S having a nontrivial centralizer in U q (O p (B)), and Ω 1 (S) has a trivial centralizer in U q (O p (B)).

The following corollary to the Main Theorem will be of crucial use in [START_REF] Deloro | Groups of finite Morley rank with solvable local subgroups and of odd type[END_REF] to get a bound on Prüfer 2-ranks. Then:

(1) either B < K, in which case B is a Borel subgroup of K; and if in addition S is a Sylow p-subgroup of N N K (B) (S), then N K (B) is p-strongly embedded in K,

(2) or B = K, in which case S, as well as S, has Prüfer p-rank 2.

Proof. It suffices to apply our Main Theorem in K. We note that S and S have the same Prüfer p-rank, as X is finite by [DJ11a, Lemma 3.18].

Cases (1) and (2) of the Main Theorem and Corollary 4.1 correspond respectively to Sections 7 and 6 of [START_REF] Cherlin | Tame minimal simple groups of finite Morley rank[END_REF] in presence of divisible torsion.

For p = 2 case (1) will entirely disappear in [START_REF] Deloro | Groups of finite Morley rank with solvable local subgroups and of odd type[END_REF] by an argument similar to the one used in [BCJ07, Case I] for minimal connected simple groups.

Fact 2

 2 .2. [DJ11a, Uniqueness Theorem 4.1] Let G be a * -locally •

  Fact 2.4. [DJ11a, Fact 2.3] Let G be a nilpotent group of finite Morley rank.

  or B = G, in which case S has Prüfer p-rank 2. Here begins the proof. Let M = N (B). As B is definable (and connected) by Zilber's generation lemma, M is definable as well. As B contains a generous Carter subgroup Q of G containing S by [DJ11a, Fact 3.32], the conjugacy of generous Carter subgroups of [Jal06] and a Frattini argument give M = N (B) ⊆ BN (Q), and as Q is almost selfnormalizing B = M • . (With the same notation, this holds of course for an arbitrary p-torus S in an arbitrary group G of finite Morley rank.)

  , and thus solvable by * -local • • solvability of G and [DJ11a, Lemma 3.4]. As C • (S) ≤ C • (s) for any element s of order p of S, it suffices to apply Lemma 3.4. Recall from [DJ11a, Definition 2.5] that for any group G of finite Morley rank, d(G) denotes the maximum of d ∞ (G) and of max p∈P (d p (G)), i.e., the infinite symbol if G contains a nontrivial p-unipotent subgroup, and otherwise the maximal r in N such that G contains a nontrivial (∞, r)-subgroup. This number exists and belongs to N {∞} when G is infinite [DJ11a, Lemma 2.6].

Corollary 4. 1 .

 1 Let G be a connected nonsolvable * -locally • • -solvable group of finite Morley rank and of Prüfer p-rank at least 2 for some prime p, and fix a maximal p-torus S of G. Let X be a maximal exceptional (finite) subgroup of S (as in [DJ11a, Lemma 3.29]), H = C • (X)/X, K a minimal definable connected nonsolvable subgroup of H containing S, and letB = C • K (s) | s ∈ Ω 1 (S) \ {1} .

  • • -solvable groups of finite Morley rank. Fact 2.3. [DJ11a, Corollary 4.4] Let G be a * -locally • • -solvable group of finite Morley rank, p = (p, r) a unipotence parameter with r > 0 such that d p

  2.7. (Compare with [CJ04, Lemma 3.2]) Let H be a connected solvable group of finite Morley rank such that U p (H) = 1. Then H/O p (H) is divisible abelian. Proof. Dividing by O p (H), we may assume it is trivial and we want to show that H is divisible abelian.

For s a nontrivial element of E we let θ(s) = U q (O p (C(s))).

If t is another nontrivial element of E, then it normalizes the connected nilpotent q-group without p-elements of order θ(s), and by Lemmas 2.8 and 2.7, C θ(s) (t) ≤ U q (O p (C(t))) = θ(t). Hence one has the two following properties:

(1) θ(s) g = θ(s g ) for any s in E \ {1} and any g in G.

(2) θ(s) ∩ C G (t) ≤ θ(t) for any s and t in E \ {1}.

In the parlance of finite group theory one says that θ is an E-signalizer functor on G. In groups of finite Morley rank one says that θ is a connected nilpotent Esignalizer functor, as any θ(s) is connected (by definition) and nilpotent, which follows from Corollary 3.8. When E 1 is a subgroup of E one defines

In groups of finite Morley rank there is no "Solvable Signalizer Functor Theorem" available as in the finite case [Asc93, Chapter 15] (see [START_REF] Goldschmidt | 2-signalizer functors on finite groups[END_REF][START_REF] Goldschmidt | Solvable signalizer functors on finite groups[END_REF][START_REF] Glauberman | On solvable signalizer functors in finite groups[END_REF][START_REF] Bender | Goldschmidt's 2-signalizer functor theorem[END_REF] for the history of the finite case). However Borovik imported from finite group theory a "Nilpotent Signalizer Functor Theorem" for groups of finite Morley rank [START_REF] Borovik | Simple locally finite groups of finite Morley rank and odd type[END_REF] [BN94, Theorem B.30], stated as follows in [Bur04, Theorem A.2] (and which suffices by the unipotence theory of [START_REF] Burdges | A signalizer functor theorem for groups of finite Morley rank[END_REF] for which it has been designed originally). Fact 3.12. (Nilpotent Signalizer Functor Theorem) Let G be a group of finite Morley rank, p a prime, and E ≤ G a finite elementary abelian p-group of order at least p 3 . Let θ be a connected nilpotent E-signalizer functor. Then θ(E) is nilpotent. Furthermore θ(E) = O p (θ(E)) and θ(s) = C θ(E) (s) for any s in E \ {1}.

(In the finite group theory terminology one says that θ is complete when it satisfies the two properties of the last statement.)

In our situation one thus has, assuming toward a contradiction the Prüfer prank to be at least 3, that θ(E) is nilpotent. Notice that the definable connected subgroup θ(E) is nontrivial, as θ(s) is nontrivial at least for some s by Lemma 2.8 and Fact 2.9. In particular N • (θ(E)) is solvable by * -local •

• solvability of G. From this point on one can use arguments formally identical to those of [Bor95, §6.2-6.3] used there for dealing with "proper 2-generated cores".

If E 1 and E 2 are two subgroups of E of order at least p 2 , then for any s in

In particular θ(E) = θ(E 1 ) for any subgroup E 1 of E of order at least p 2 . Now if g in G normalizes such a subgroup E 1 , then θ(E) g = θ(E 1 ) g = θ(E g 1 ) = θ(E 1 ) = θ(E) and thus g ∈ N (θ(E)). Take now as in Lemma 3.4 an element s of order p in E such that C • (s) N • (θ(E)).


D D Doan

E Ramasso

V Placet

S Zhang

L Boubakar

N Zerhouni

An unsupervised pattern recognition approach for AE data originating from fatigue tests on polymer-composite materials

Keywords: organic-matrix composites, acoustic emission clustering, fatigue datasets, noise reduction, sequential feature selection

This work investigates acoustic emission generated during tension fatigue tests carried out on a carbon fiber reinforced polymer (CFRP) composite specimen. Since fatigue data processing, especially noise reduction remains an important challenge in AE data analysis, a Mahalanobis distance-based noise modeling has been proposed in the present work to tackle this problem. A Davies-Bouldin-index-based sequential feature selection has been implemented for fast dimensionality reduction. A classifier offline-learned from quasi-static data is then used to classify the processed data to different AE sources with the possibility to dynamically accommodate with unseen ones. With an efficient proposed noise removal and automatic separation of AE events, this pattern discovery procedure provides an insight into fatigue damage development in composites in presence of millions of AE events.

Introduction

AE testing has become a recognized nondestructive test (NDT) method, commonly used to detect and locate defects in mechanically loaded structures and components. AE can provide comprehensive information on the origination of a discontinuity (flaw) in a stressed component and also pertaining to the development of this flaw as the component is subjected to continuous or repetitive load [1]. Moreover, the method has been developed and applied in numerous structural components, such as steam pipes and pressure vessels, and in the research areas of rocks, composite materials and metals [2].

Acoustic emissions (AE) are stress waves produced by the sudden internal stress redistribution of the materials caused by the changes within the structure [START_REF] Huang | Using acoustic emission in fatigue and fracture materials research[END_REF]. For polymer-composite materials, these changes are mainly due to crack initiation and growth, crack opening and closure, fiber breakage and fiber-matrix debonding. The use of AE for structural health monitoring has been investigated several decades ago with the objective to predict material failure [START_REF] Graham | Characterization of acoustic emission signals and application to composite structures monitoring[END_REF][START_REF] Murthy | Application of pattern recognition concepts to acoustic emission signals analysis[END_REF][START_REF] Otsu | Pattern recognition analysis of acoustic emission from unidirectional carbon fiber-epoxy composites by using autoregressive modeling[END_REF].

With a huge noisy amount of data originating from fatigue loading tests, a major challenge in the use of AE technique is to associate each signal to a specific AE source related to noise or to a damage mechanism. This analysis is a non-trivial task for two main reasons. First, AE signals are complex so that it has to be characterized by multiple relevant features. Second, there is generally no a priori knowledge of the acoustic signatures of damage events which are generally scattered due to the high variability of the properties of composite materials [START_REF] Pineau | Subsampling and homogenization to investigate variability of composite material mechanical properties[END_REF].

In the literature, dealing with the challenge of massive data due to high sensitivity of AE sensors and to long-term fatigue loading experiments, several processing approaches have been proposed [START_REF] Henry | Suivi de l'endommagement en fatigue d'un composite a renfort interlock carbone/epoxy par emission acoustique[END_REF][START_REF] Yu | Quantification of fatigue cracking in CT specimens with passive and active piezoelectric sensing[END_REF][START_REF] Calabrese | Noise removal by cluster analysis after long time AE corrosion monitoring of steel reinforcement in concrete[END_REF]. In [START_REF] Henry | Suivi de l'endommagement en fatigue d'un composite a renfort interlock carbone/epoxy par emission acoustique[END_REF], it is considered that only signals with amplitude higher than 70 dB or recorded above 80% of peak load contain information related to damage mechanisms. In [START_REF] Yu | Quantification of fatigue cracking in CT specimens with passive and active piezoelectric sensing[END_REF], "friction emission" tests in which the maximum cyclic load was decreased to a level that was insufficient to generate crack growth were performed to understand the AE signal characteristics arising from hydraulics, machine start/stop and slippage. All of the AE events at this lower peak load were therefore assumed to be due to friction emission. Emission having the characteristics of friction emission was then filtered. A more complex denoising process developed by [START_REF] Calabrese | Noise removal by cluster analysis after long time AE corrosion monitoring of steel reinforcement in concrete[END_REF] that combines Principal Component Analysis (PCA) and K-means and several validation techniques was presented to be able to classify more than 60% of the detected signals as noise during long time corrosion monitoring of a pre-damaged post tensioned concrete beam.

High dimensional feature space reduction is a remaining challenge to statistical processing and classification of AE data. In the literature, many approaches for AE data processing [1,[START_REF] Godin | Clustering of ae signals collected during tensile tests on unidirectional glass/polyester composite using supervised and unsupervised classifiers[END_REF] rely on the Principal Component Analysis (PCA). The PCA takes a set of features calculated from AE signals, such time-frequency features, and generates a set of articifial variables made of a linear combination of the input features depicting the largest variance.

Other approaches [START_REF] Gutkin | On ae for failure investigation in CFRP: pattern recognition and peak frequency analyses[END_REF][START_REF] Moevus | Analysis of damage mechanisms and associated ae in two SiCf/[SiBC] composites exhibiting different tensile behaviours. Part II: unsupervised ae data clustering[END_REF][START_REF] Sause | Pattern recognition approach to identify natural clusters of ae signals[END_REF] rely on a specific subset of features such as energy, rise time, duration, amplitude [START_REF] Gutkin | On ae for failure investigation in CFRP: pattern recognition and peak frequency analyses[END_REF] or have reduced the dimension of the feature space by using complete link hierarchical clustering in order to merge the correlated features into groups [START_REF] Moevus | Analysis of damage mechanisms and associated ae in two SiCf/[SiBC] composites exhibiting different tensile behaviours. Part II: unsupervised ae data clustering[END_REF]. Those apply a greedy approach that generates all possible feature combinations and then selects the one which optimizes a given criterion [START_REF] Sause | Pattern recognition approach to identify natural clusters of ae signals[END_REF][START_REF] Halkidi | On clustering validation techniques[END_REF]. The goal of the criterion is generally to evaluate the quality of the partition provided by the clustering. It can be noticed that the PCA and the K-means clustering method are theoretically related to each other as shown in [START_REF] Ding | K-means clustering via principal component analysis[END_REF]. An alternative approach to Euclidean distance-based clustering methods was proposed [START_REF] Placet | Online segmentation of ae data streams for detection of damages in composite structures in unconstrained environments[END_REF] and based on the Gustafson-Kessel algorithm (GK) [START_REF] Gustafson | Fuzzy clustering with a fuzzy covariance matrix[END_REF]. It makes use of a modified Mahalanobis distance for each cluster which is iteratively adapted to fit ellipse-shaped clusters. The use of hyper-ellipses instead of hyper-spheres is more appropriate for AE clustering in presence of low density and high scattering. In the GK algorithm, the covariance between each pair of features is estimated so that possible redundancy or complementarity between features can be taken into account. The Mahalanobis distance has also been shown to be robust to outliers in statistical analysis [START_REF] Peel | Robust mixture modelling using the t distribution[END_REF].

The processing of large AE datasets, in particular originating from fatigue, requires to develop efficient methods in terms of memory and time consumption. Some approaches have been proposed which are able to work online (or real-time), that means that clusters parameters are updated without iterative procedure but as new data arrive. As underlined in the GKbased method proposed in [START_REF] Placet | Online segmentation of ae data streams for detection of damages in composite structures in unconstrained environments[END_REF] and in the Kmeans-based method developed in [START_REF] Pomponi | A real-time approach to acoustic emission clustering[END_REF], external AE sources (corresponding to noise) may have an important influence on the clusters' updating. In this paper we propose a methodology to estimate efficiently the partition of AE data obtained in fatigue loading in presence of noise sources. The methodology also includes an automated sequential feature selection based on the GK algorithm and relying on quasistatic (QS) tests. The clusters obtained are then adapted to be applied on large fatigue tests. The next section is dedicated to presentation of the proposed methodology.

Unsupervised pattern recognition

The flow chart of the methodology is shown on Fig. 1.

[Figure 1 about here.]

AE fatigue data pre-processing

All acoustic emissions even originating from outside the area of interest bounded by the sensors were taken into account (no spatial filtering). Thus a pre-processing step of such AE data is highly important and requires adapted filtering methods [START_REF] Barat | Detection of AE signals against background friction[END_REF].

Signal screening

Continuous background noise due to hydraulic flows is essentially eliminated from the AE signal by a floating signal threshold, which is adjusted at a 40 dB level. This threshold makes it possible to loose signals originated from friction. Optimal denoising, for instance using wavelets [START_REF] Donoho | De-noising by soft-thresholding[END_REF], would be necessary if those signals are important for the monitoring.

Noise model-based filtering

Typical field and environmental noise such as electromagnetic interference (EMI), fretting, mechanical or hydraulic vibration encountered in real applications generate extraneous noise detected by the broadband and high sensitive AE sensors. Assuming that this AE activity is not due to damages, a noise model is built using a multivariate statistical test based on the Mahalanobis distance as used in novelty detection [START_REF] Farrar | Unsupervised learning novelty detection[END_REF][START_REF] Serir | E2GKpro: An evidential evolving multi-modeling approach for system behavior prediction with applications[END_REF]. For that, the AE hits recorded before the loading phase are considered as representative of the AE hits corresponding to external AE sources (such as noise). The statistical mean (center of the noise model) and covariance of those samples define an ellipsoid in the feature space, and its boundary is estimated as the average of the Mahalanobis distances between each sample and the center. An AE hit recorded during loading is then considered as noise if it falls within the boundary of the ellipsoid.

Sequential selection algorithm of AE features

An automated technique is presented to detect relevant feature subsets for clustering of AE events. In contrast to feature reduction procedures (for example based on correlation dendrogram [1]) or exhaustive search of global optimal feature combinations [START_REF] Sause | Pattern recognition approach to identify natural clusters of ae signals[END_REF], the principle of the approach is to combine gradually each feature from an available feature space with an initial feature subset [START_REF] Guerif | Unsupervised variable selection: when random rankings sound as irrelevancy[END_REF]. The feature selection is achieved by minimizing the value of Davies and Bouldin (DB) index [START_REF] Davies | A cluster separation measure[END_REF] defined by:

DB = 1 c c i=1 max i =j d i + d j D ij (1)
where c is number of clusters, d i and d j are the average within-class distances of clusters i and j respectively, and D ij denotes the distance between the two clusters i and j. This clustering validity index has been used by several authors in order to select optimal cluster number [START_REF] Moevus | Analysis of damage mechanisms and associated ae in two SiCf/[SiBC] composites exhibiting different tensile behaviours. Part II: unsupervised ae data clustering[END_REF] or to evaluate feature subset partition [START_REF] Sause | Pattern recognition approach to identify natural clusters of ae signals[END_REF]. The lower is its value, the better is the compactness and the separability within the partition. Figure 2 shows the diagram of the proposed algorithm based on a feature filtering approach [START_REF] John | Irrelevant features and the subset selection problem[END_REF].

[Figure 2 about here.]

Considering an initial subset of features S (empty by default), the algorithm takes each of the available features from F to update S. This subset is then partitioned by the GK clustering algorithm. At the k th iteration, a feature f l ∈ F is added to the current subset of features S k , and the corresponding DB index DB l of the partition obtained by the GK algorithm is computed.

The computation of the DB index makes use of the Mahalanobis-like distance defined in the GK algorithm [START_REF] Gustafson | Fuzzy clustering with a fuzzy covariance matrix[END_REF] to estimate the distance between AE hits and cluster centers and finally obtain the estimate of the average within-class distances used in Eq. 1 (d i and d j ).

The subset of features S k+1 for the next iteration is given by S k ∪ f l * with l * = arg min l DB l and the partition is then evaluated by the DB criterion.

The feature that minimizes the value of DB index is selected and transfered from F to S. At each iteration, the procedure generates |F | new subsets since each new subset contains the features from S plus a new one taken from the remaining features in F . The algorithm stops when no new subsets can improve the DB criterion.

For each iteration k, an improvement rate IR(k) is calculated as follows:

IR(k) = DB(S k ) -DB(S k-1 ) DB(S k-1 ) (2)
where DB(S k ) and DB(S k-1 ) represent the value of the DB-index of the best feature selection for the k th and (k-1) th iteration respectively. The sign of IR indicates if the DB criterion is improved (negative) or not (positive). For the last iteration k last (for which

IR(k last ) > 0), if IR(k last ) < min k<k last |IR(k)|
then the feature with the best DB-index is added to S to establish the final selected feature set.

AE source clustering

Quasi-static (QS) tests are first applied to obtain a relatively low amount of data compared to fatigue and by supposing that damage sources in QS tests are mostly similar to fatigue. The GK algorithm is thus applied to estimate the parameters of a given set of k clusters on AE originated from QS tests. To cope with possibly additional AE sources that can occur during fatigue [START_REF] Baere | Assessment of the interlaminar behaviour of a carbon fabric reinforced thermoplastic lap shear specimen under quasi-static and tension-tension fatigue loading[END_REF], an additional k + 1 th cluster is estimated based on fatigue data to include all feature vectors located "far" from the previous k clusters. For that, the boundary of each cluster characterized on QS tests is estimated by the average of the Mahalanobis-like distance (used in GK) [START_REF] Serir | E2GKpro: An evidential evolving multi-modeling approach for system behavior prediction with applications[END_REF]. A feature vector obtained during fatigue belongs to the k + 1 th cluster if its distance to nearest cluster is above the corresponding radius. This adaptation of clusters is supposed to take into account one (or more) AE sources that is (or are) not present in quasi-static tests (e.g. noise due to repeated tensile loading, acoustic waves related to cumulated damage ...).

Experiments

Composite split disks were considered subjected to cyclic fatigue loading up to failure determined when a complete break of the specimen was observed in the hoop direction. The specimens were cyclically tested under a tensile/tensile sinusoidal loading with constant amplitude and frequency of 5

Hz and under constant stress ratio R = 0.1 at room temperature. Quasi-static tests were preliminarily conducted on five different specimens with a constant loading rate of 0.3 kN.s -1 . The static failure stress was equal to 1520 ± 165

MPa. The tests were performed according to ASTM D2290 "Apparent hoop tensile strength of plastic or reinforced plastic pipe by split disk method".

Rings were produced by cutting and machining filament-wound carbon fiber reinforced epoxy tubular structures intended for the manufacturing of flywheel rotors with a (90 • ) 6 lay-up configuration. The transient elastic waves

were recorded during test at the material surface using a multi-channels data acquisition system from EPA (Euro Physical Acoustics) corporation (MIS-TRAS Group). The system is made up of miniature piezoelectric sensors (micro-80) with a range of resonance of 250 -325 kHz, preamplifiers with a gain of 40dB and a 20 -1000 kHz filter, a PCI card with a sampling rate of 1MHz and the AEWin software. Two AE sensors were coupled on the specimen faces using silicon grease. The experimental set-up is shown in Fig. 3.

[Figure 3 about here.]

The calibration of the system was performed after installation of the transducers on the specimen and before each test using a pencil lead break procedure. A part of the ambient noise was filtered using a threshold of 40dB.

The acquisition parameters: PDT (Peak Definition Time) = 60 µsec; HDT (Hit Definition Time) = 120 µsec and HLT (Hit Lock Time) = 300 µsec were optimized for this specific experimental configuration to extract transient signals. The optimization of these time-driven parameters was performed using the standard pencil-lead breakage proposed by Hsu and Nielsen [START_REF] Hsu | Characterization and calibration of acoustic emission sensors[END_REF]. Many features such as absolute energy, counts, hits, amplitude, duration, frequency centroid were calculated from recorded waves.

Results and discussion

According to different percentages of the ultimate tensile stress determined in the tensile test (1520 MPa), the S-N curve was obtained as illustrated in Figure 4.

[Figure 4 Four datasets were considered denoted as A1 (quasi-static test) and A2, A3 and A4 (fatigue tests for 90%, 80% and 70% of the ultimate tensile strength respectively). A brief description of the obtained datasets is summarised in Table 1.

[Table 1 about here.]

Noise reduction

According to the scenario of the quasi-static test A1 (Fig. 5(a)), around the time-instant t1, the actuator was pressurized and the stress was applied only at t2. The noise modeling phase (Section 1.1.2) has been made from AE data recorded before t1, i.e. while the specimen was let in its environment without any mechanical loading. Noise during loading is then filtered by this model.

[Figure 5 about here.] [Figure 6 about here.]

The application of the noise model to fatigue dataset A3 made of 1,682,434 AE hits led to a similar separation between noise and denoised data (Fig. 7(a)). In spite of 93% of AE hits recorded associated to "noise" (Fig. 7(c)), this highest population represents negligible AE cumulated energy level in comparison with that of denoised data (Fig. 7(b)).

[Figure 7 about here.]

Feature selection

Many energy-based approaches of damage characterization or identification have been studied since AE energy provides a good correlation with damage mechanisms. Thus, in this work, absolute energy (Fig. 8) is used to initialize the subset of relevant features. As the number of clusters is unknown, 3 cases were addressed to check the stability of the selection algorithm by considering 4, 5 and 6 clusters.

The selection algorithm was applied on the quasi-static dataset A1 with 4 clusters. At the first iteration, given the absolute energy feature, the optimal DB index is given by the combination with the amplitude feature (Fig. 8(a)).

At the second iteration, the best score was obtained by the combination with the MARSE energy (Fig. 8 The denoised and selected feature subset obtained previously is now used to identify the clusters in quasi-static dataset A1 using the GK clustering algorithm. Four well-separated clusters with different sizes and shapes have been obtained in the duration-amplitude space (Fig. 9(a)). After projection onto the amplitude dimension, four distinct distributions can be obtained, among which three are located above 75 dB. These distributions have been often used to identify AE sources [START_REF] Graham | Characteristics of ae signals from composites[END_REF][START_REF] Short | Amplitude distribution ae signatures of unidirectional fibre composite hybrid materials[END_REF].

Ono and Gallego [2] recently underlined a misconception that fiber fracture always produces high-energy event, and that still persists to this day.

For the considered material, the damage process involves fiber tow breakage.

If the breakage of an elementary fiber (7µm diameter) can cause the release of low energy transient, the breakage of fiber tows including hundreds or thousands of elementary fibers (up to 12,000 in the considered material) are likely to induce highly energetic signals.

As a complementarity view, the temporal evolution of the logarithm of the Cumulated Sum of Cluster Appearance (logCSCA) [START_REF] Placet | Online segmentation of ae data streams for detection of damages in composite structures in unconstrained environments[END_REF] has been depicted in Fig. 9(c) (for each cluster) together with the cumulated energy and the load.

When an AE hit (emitted after the activation of an AE source) is associated to a given cluster at a given time, the corresponding logCSCA curve depicts a step. When several consecutive steps appear in a short time period, this visualisation allows to point out that the activity of the corresponding AE source is particularly sustained which may be related to propagations of cracks [START_REF] Sornette | Statistical physics of rupture in heterogeneous media[END_REF].

In the sequence shown in Fig. 9(c), the first cluster is activated at the very beginning before applying the load. Despite the number of AE hits in this cluster is important (73% at the end), the cumulated energy of AE hits in this cluster is the lowest one among all clusters (Fig. 9(e)). These observations are coherent with the activation of an AE source related to mechanical and hydraulic emission such as vibration and friction between the specimen and the half-cylinders.

Both cluster 2 and 3 start early when the actuator has been pressurized.

The main activity of cluster 2 occurs after a certain level of load (Fig. 9(c))

and the cumulated energy of AE hits in this cluster (Fig. 9(e)) as well as the amplitudes (> 95 dB, Fig. 9(a)) are the highest ones compared to all other clusters. The number of AE hits in this cluster is particularly important at the end of the test, as expected with the ruine of the specimen induced by a cascade of fiber tow breakage. This cluster is thus related to the activity of highly energetic sources, in particular carbon fiber tow breakage.

The low cumulated energy in cluster 3 as well as the amplitudes around 75 and 90 dB make this cluster related to minor damage (probably matrix micro-cracks).

The partition also emphasizes an important cascade of AE hits at t ≈ 352 s during which the activity of cluster 3 increases importantly and this increase is synchronised with both the appearance of cluster 4 and a high activity of cluster 3. The load level at this time, the mean value of amplitudes in cluster 4 (around 95 dB, Fig. 9(a)) and the level of the cumulated energy in this cluster (around 13% of the total cumulated energy, Fig. 9(e)) make this cluster related to macro-cracking and interface failures starting around the specimen's notches and propagating gradually in the hoop direction.

Sequence of AE hits in a fatigue test

Afterwards, the model estimated on A1's AE hits is used to infer the partition on the fatigue dataset A3. Direct application of the model generates overlapping zones between clusters in the duration-amplitude space of A3 (Fig. 9(b)). We can observe a similar distribution of clusters in this feature space compared to A1 (Fig. 9(a)). However, we can also observe clusters overlap, particularly important between clusters 2 and 3. As a consequence, the projection onto the amplitude axis would not give distinct distributions as for the quasi-static test. This phenomenon finds its origins in the fact that, compared to quasi-static tests, additional mechanisms can play a role during fatigue such as the temperature [START_REF] Gregory | Constituent and composite quasi-static and fatigue fracture experiments[END_REF] or the cycling which implies crack opening/closing initially not observed during QS tests [START_REF] Baere | Assessment of the interlaminar behaviour of a carbon fabric reinforced thermoplastic lap shear specimen under quasi-static and tension-tension fatigue loading[END_REF]. Therefore, it was expected to find out that a pattern recognition model learned from a quasi-static test and simply applied on a fatigue test may present a limited generalization capability. Based on the assumption that a new AE source is activated during fatigue and which has not been observed in quasi-static tests, the proposed methodology (Section 1.3) includes the creation of new cluster to cope with this problem. The result is a new segmentation with less overlapping between clusters as shown in Fig. 9(d).

[Figure 9 about here.]

The comparison of partitions with the previous quasi-static test yield similar conclusions concerning the possible damage scenario. The main difference holds in the position of the new cluster, which has been automatically found from AE hits. Indeed, the cluster 3 identified as the friction and possibly micro-cracking in the quasi-static test (Fig. 9 when the AE sources are activated. On dataset A3, it can be observed in Fig. 10(a) to 10(f) that during 20%-fatigue-life of the specimen, many AE hits appear, related to all AE sources. This phenomenon is well known as the accommodation phase [START_REF] Degallaix | Low cycle fatigue of a duplex stainless steel alloyed with nitrogen[END_REF] which generally appears at the first stage of materials undergoing fatigue testing and may lead to partial fractures.

Indeed, AE hits with high energy (from AE sources 2 and 4) are activated during this phase (and during failure). After this stage, the clusters' activities globally slow down for a while (stabilization phase). Beyond 65% of the fatigue life, an important number of highly energetic AE hits occur up to the ruine of the specimen (from AE sources 2 and 4). It is interesting to notice a repetitive phenomena that takes place all along the test and represented by the activation of AE sources 3a and 3b: The latter is mainly activated in loading phases while the former occurs in unloading phase (Fig. 11(b)). As for the quasi-static test, the latter may correspond to internal frictions and interfaces fretting as observed in previous papers [START_REF] Cheng | Internal friction behavior of unidirectional carbon/carbon composites after different fatigue cycles[END_REF]. It can also be observed that the AE hits originated from these clusters occur between 5-7 kN at the beginning of the tests and between 3-4 kN at the end. AE source 3a is much more activated than AE source 3b between 20% and 50%, then the activity of 3b substantially increases until the ruine. This increasing is followed by the activation of AE source 5 that is particularly active between 70% and 90%, just before the ruine. Therefore, as expected, the fatigue plays a role on the loading level required to activate some sources and the chronology of activation may give insights to the understanding of damage mechanisms.

Sequence of AE hits in two other fatigue tests

The complexity of damage mechanisms involved during fatigue is illustrated in this section. For that, two other specimens denoted as A2 and A4, corresponding to 90% and 70% of the tensile strength respectively, are considered. The behavior of A2 is similar to the previous specimen A3 as depicted in Fig. 12(a). The activity of the AE hits including high energy and high duration signals is rather high (relatively to the remaining AE hits) at the very beginning of loading and increases again at about 60% of the specimen life, as for A3. Although AE hits generated by AE source 3a are more scattered than in the previous test, overlaps between clusters related to this source and to AE source 3b have also been detected by the proposed algorithm. Table 2 summarises the clusters assigned to each AE source according to the previous observations.

[Table 2 about here.]

Rather different than the previous tests, the partition obtained on dataset A4 at 70% of the ultimate static strength is depicted in Fig. 12(b). The initial (accommodation) phase occurs within the first cycles as for the previous loading levels, but it is then followed by a silence of most of AE sources.

Only AE source 1 is activated (representing possible external sources which has been filtered out) and a few highly energetic AE hits occur (such as fiber tow breakage). Then, at 20% of the fatigue life, a progressive activation of all AE sources can be observed. In the load band 2 -10 kN, only cluster 1 is activated but this band is gradually reduced with respect to the number of cycles to reach 4 -7 kN when approaching the end-of-life. The progressive and continuous reduction of the band beyond which clusters are activated can be of interest for predicting the remaining lifetime of the composite if confirmed on other specimens and lower loading levels. It can also be noticed that more AE hits related to AE sources 2 and 4 (i.e. with the highest energy) can be found compared to the two previous specimens. Therefore, the failure process of specimen A4 tested at 70% of the ultimate tensile strength is more gradual and more related to the progressive weakness of the material during the repeated stress until the ruine.

[Figure 12 The first results on three real fatigue tests of thermoset ring-shaped CFRP involving until 10 millions AE hits demonstrate that the proposed methodology allows to identify some relevant clusters. Of particular interest:

• Four main phases have been identified: Accommodation with many AE hits with the highest energy and amplitude (0-20% of the lifetime), a slowdown of AE activity (20-50%), a resumption of the AE activity (50-85%) and a failure progress up to the final failure (85-100%). The fatigue at 70% of the ultimate strength depicts a particular pattern during the degradation involving an envelop which gradually reduces until the ruine.

• Two clusters detected by the adaptation phase occur at similar loading levels. A modification of their kinetics with report to the cumulated loading lets suppose that those two clusters can be due to damage. It is also interesting to emphasize that the level required to activate the AE sources related to those two clusters depicts a slight and progressive decreasing together with the degradation of the material until the ruine.

The visualization of clusters in the amplitude-duration feature, the logarithm of the cumulated AE hits and energy in each cluster as well as the the positioning of clusters onto the loading level have allowed to connect some clusters to possible AE sources. In order to validate the identification of AE sources observed, complementary non-destructive techniques and insitu measurements is under study on more specimens. The application of the proposed methodology is currently investigated on thermoplastic CFRP composites and compared to finite element models [START_REF] Sause | Finite element modelling of cracks as acoustic emission sources[END_REF]. Finally, the proposed methodology is under improvement for robust AE-based prognostics of composite structures.

  about here.] Nine samples were used to generate the S-N curve. This was a good compromise between the six specimens recommended by ASTM D-3479 for preliminary and exploratory test campaign and the twelve specimens required for research and development on testing of components and structures. The results presented in this work are part of a wider study including the generation of S-N curves of different types of composites (with different carbon fibers) and with different lay-up configurations. The main goal is to select a composite of choice for the application concerned, namely rotors of flywheels.

Figure 5 (

 5 Figure 5(b) and 5(c) represent dataset A1 (made of 52,832 AE hits) in the duration-amplitude space segmented into three populations: noise before and during loading in Figure 5(b), and denoised data after application of noise model in Figure5(c). The two first populations (noise) possess the same characteristics, the same location and the same scattering. This observation is justified by the graphic of AE cumulated energy in Fig.6(a). Indeed, the level of AE cumulated energy of noise before and during loading is negligible and the total energy is conserved within denoised data while the latter occupies only 12% of the whole dataset in terms of quantity (Fig.6(b)).

  (b)). No more improvement of the DB index is made at the next iteration, so the algorithm is stopped by selecting the subset made of absolute energy, amplitude and MARSE energy. The same selection result was obtained with 5 and 6 clusters. In what follows, 4 clusters are used as initial number of AE sources. [Figure 8 about here.] 3.3. AE source classification 3.3.1. Sequence of AE hits in the quasi-static case

  Figure 11 represents the positioning of clusters onto the load level for the fatigue dataset A3. This figure enables one to visualise the load level

  about here.]ConclusionAn unsupervised pattern recognition approach for AE data originating from fatigue tests on polymer-composite materials has been presented to tackle different existing challenges of AE analysis and damage detection: 1) data pre-processing, especially noise reduction; 2) automatic and fast feature selection; 3) clustering of massive data from fatigue tests with cluster adaptation. The methodology relies on the estimation of clusters during static tests. Its application to big fatigue data based on the adaptation phase allows to add a new cluster to cope with new AE sources. The assignment of a cluster to a AE hit is not iterative and only requires to find the closest cluster by using a Mahalanobis-like distance that allows to cope with data scattering. The processing of a fatigue dataset is made faster than iterative procedures which requires to load a dataset and to perform interative optimization on large matrices.

Figure 1 :FigureFigure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :RFigure 8 :Figure 9 :

 13456789 Figure 1: Unsupervised damage detection methodology

Figure 10 :Figure 11 :Figure 12 :

 101112 Figure 10: Classified AE events during cyclic loading of specimen A3 (80%): (a) All sources during the whole test; (b)-(f) Individual AE source 36

Acknowledgement

This work has been carried out in the framework of the Laboratory of Excellence ACTION through the program "Investments for the future" managed by the National Agency for Research (references ANR-11-LABX-01-01).

The authors thank the reviewers for the relevant and fruitful comments.

[1] A. A. Anastassopoulos, T. P. Philippidis, Clustering methodology for the evaluation of ae from composites, J. of Acoust. Emission 13 (1-2) (1995) 11-22.

[2] K. Ono, A. Gallego, Research and applications of ae on advanced composites, Journal of Acoustic Emission.

List of Figures

Fiber-related damage (rupture of tows, pull-out) 3a

Friction-related source due to fatigue crack closure under cyclic loading 3b

Matrix-related damage (micro/macro cracking, splitting) 4

Interface-related damage (fiber/matrix)