H A Minh

Lam Morales

Marcel Abtracts

Binomial extensions of Simplicial ideals and reduction number

In this article, we define a class of binomial ideals associated to a simplicial complex. This class of ideals appears in the presentation of fiber cones of codimension 2 lattice ideals [HM], and in the work of Barile and Morales [BM2], [BM3], [BM4]. We compute the reduction number of Binomial extensions of Simplicial ideals. This extends all the previous results in this area.

Introduction

According to the classification resulting from the successive contributions by Del Pezzo, Bertini, and Xambó (see [EG] for literature), the equidimensional algebraic subsets X ⊂ P n of minimal degree which are connected in codimension one are of three types: quadric hypersurfaces, the cone over the Veronese surface in P 5 , and unions X = ∪ n i=0 X i of scrolls embedded in linear subspaces such that for all i = 1, . . . , n -1, we have:

(X 1 ∪ . . . ∪ X i ) ∩ X i+1 = Span(X 1 ∪ . . . ∪ X i ) ∩ Span(X i+1 ). ( * )
Homologically, varieties of minimal degree were characterized by Eisenbud -Goto [EG] (see Theorem 2.1).

Under the algebraic point of view, the condition ( * ) was considered at first in [START_REF] Barile | On certain algebras of reduction number one[END_REF], and later in [START_REF] Barile | On unions of scrolls along linear spaces[END_REF], the authors give a complete constructive characterization of the ideals defining varieties of unions of scrolls satisfying the above condition ( * ).

Later Eisenbud-Green-Hulek-Popescu [EGHP], define a linearly joined sequence of varieties, as an union of varieties satisfying the condition ( * ). They prove that an algebraic set X ⊂ P r is 2-regular if, and only if, X = X 1 ∪ . . . X n , with X 1 , . . . , X n is a sequence of varieties of minimal degree.

Recall that the homogeneous coordinate ring of a scroll is of the type A = S/I 2 (M ), where S is the polynomial ring S = k[T i,j | 1 ≤ i ≤ l and 1 ≤ j ≤ s i + 1], and I 2 (M ) is the ideal generated by 2 × 2 minors of the matrix M = (M 1 | M 2 | . . . | M l ), with each M u is the generic catalecticant matrix M u = T u,1 T u,2 . . . T u,su-1 T u,su T u,2 T u,3 . . . T u,su T u,su+1 .

We call an ideal of type I 2 (M ) a scroll ideal.

1

In this article, we define a class of binomial ideals associated to a simplicial complex. This class of ideals appears in the presentation of fiber cones of codimension 2 lattice ideals [HM], and in the work of Barile and Morales [BM2], [START_REF] Barile | On unions of scrolls along linear spaces[END_REF], [START_REF] Barile | On the equations defining minimal varieties[END_REF]. Let △ be a simplicial complex over a set of vertices V △ = {x 1 , x 2 , . . . , x n }. We will call proper facet a facet F l with a star of some edges belonging only to F l (called also proper edges).

To each proper facet of △, we associate a set of points Y (l) (which can be empty), and a scroll ideal I l of variables in Y (l) and in vertex set of these proper edges. The new simplicial complex obtained from △ and the sets Y (l) is called an extension complex, and denoted by △.

The binomial extension of a simplicial ideal B △ associated to △ is defined to be the one generated by all I l and the Stanley-Reisner ideal of △.

The aim of this article is to prove that binomial extension of simplicial ideal is a good generalization of Stanley-Reisner theory to the case of binomial ideals.

In the first section, we will define the class of binomial extension of simplicial ideals, we will give the prime decomposition. From this, we deduce that our class of ideals, in fact, defines an union of scrolls along linear spaces.

In Section 2, we study the reduction number of binomial extension of simplicial ideals. Our aim is to extend the results of Barile and Morales : to describe explicitly the reduction ideals through the complexes. In [START_REF] Barile | On Stanley-Reisner Rings of Reduction Number One[END_REF], they described a class of squarefree monomial ideals whose reduction number is 1 by coloring the graph of a simplicial complex. In [START_REF] Barile | On certain algebras of reduction number one[END_REF], Barile and Morales considered a class of binomial ideals, which indeed are particular cases of binomial extension of simplicial ideals B △ where G △ is a generalized d-tree and each vertex belongs to at most two extension facets. They proved that this class of ideals is of reduction number 1, and an explicit expression of the reduction is given.

In the case of binomial extension of simplicial ideals, we have the following theorem: Theorem 0.1. If the graph associated to △ admits a good (d + 1)-coloration, and in addition, for each proper facet F the origin of the star of proper edges belongs only to F , then the ring K[x, y]/B △ has reduction number 1.

In this case, the reduced graph associated to △ admits also a good (d + 1)-coloration, and

(g 1 , g 2 , . . . , g d+1 )m △ + B △ = m 2 △ , where m △ = (x, y) is the irrelevant ideal of K[x, y], and g i is the sum of all variables with color i. Theorem 0.2. Let G △ be a generalized d-tree. Then we can find a good (d + 1)coloration for the reduced graph associated to △, such that

(g 1 , g 2 , . . . , g d+1 )m △ + B △ = m 2 △ ,
where m △ = (x, y) is the irrelevant maximal ideal of the polynomial ring K[x, y], and g i is the sum of all variables with color i.

1 Simplicial ideals and binomial extension of simplicial ideals binomial extension of simplicial ideals is an extension of Stanley-Reisner monomial ideals. It associates an ideal to a simplicial complex and a family of ideals indexed by a set of its facets. In this article, we will consider some particular cases, which define in fact an union of scrolls, and study some properties of these binomial extension of simplicial ideals.

First of all, let us recall some definitions.

A simplicial complex △ over a vertex set V △ = {x 1 , x 2 , . . . , x n } is a collection of subsets of V △ with the property that:

• For all i, the set {x i } is in △, • If F ∈ △ and G ⊂ F, then G ∈ △.
An element of a simplicial complex △ is called a face of △. The dimension of a face F of △, denoted by dim F, is defined to be |F | -1, where |F | denotes the number of vertices in F. The dimension of △, denoted by dim △, is defined to be the maximal dimension of a face in △. The maximal faces of △ under inclusion are called facets of △.

Let us remark that by taking all faces of dimension 0 and 1 of △, i.e. all vertices and edges, we associate to △ a simple graph G △ . An arbitrary facet F of △ becomes a completed subgraph of G △ , so called a |F|-clique of G △ .

Notation 1.1. We denote by (E) (resp. K[E]) the ideal generated by E (resp. the polynomial ring with the variables in E). And for a facet F , let F • denote the set of the points of F which belong to no other facets of △. To a set of vertices V △ = {x 1 , x 2 , . . . , x n }, one associate a ring of polynomials R = K[x 1 , x 2 , . . . , x n ] (here, by abuse of notation, we use the x i 's to denote both the vertices in V △ and the variables in the polynomial ring). It is known that to a simplicial complex △ on this vertex set, there is an associated ideal, called the Stanley -Reisner ideal, defined as follows

I △ = (x i1 x i2 • • • x ir | i 1 < i 2 < • • • < i r , {x i1 , x i2 , . . . , x ir } / ∈ △) .
This ideal is generated by monomials. Now, we introduce the definition of a binomial extension of a simplicial complex.

Definition 1.2. A facet F l of dimension d l of △ is said to be proper if F contains k l edges x (l) 0 , x (l) i1 , x (l) 0 , x (l) i2 , . . . , x (l) 0 , x (l) i k l
which belong uniquely to F . If it is the case, these edges are called the proper edges of F .

To each proper edge x (l) 0 , x (l) ij of a facet F l of △, we associate a set of points Y (l) j (which can be empty). The simplex which is the product of Y (l) := ∪Y (l) j and F l is called the extension of F l by Y (l) . By abuse of notation, we use F l to denote this new facet (without any confusion). Let us remark that this extension can be trivial for some facets of △. Let △ be the complex which facets are all the extension facets of △. We call this complex an extension complex of △.

Let △ be an extension complex constructed by △ and a set of points. We associate to △ a polynomial ring R := K[V △ ] = K[x, y]. We will denote by x Definition 1.3. To each non trivial extension facet F l of △ we associate the prime ideals I l , J l , where

• I l = 0 if Y (l) = ∅, • if Y (l) := {y (l)
ij m } = ∅, the I l is the ideal generated by the 2 × 2 minors of the matrix:

M l :=   x (l) 0 y (l) i11 . . . y (l) i1ji 1 y (l) i11 y (l) i12 . . . x (l) i1 y (l) i21 . . . y (l) i2ji 2 y (l) i22 . . . x (l) i2 . . . . . . y (l) i k l 1 . . . y (l) i k l ji k l y (l) i k l 2 . . . x (l) i k l   . J l = I l , (V △ \ F l ) ⊂ K[V △ ]
, where V △ \ F l denote the set of vertices of △ which are not in F l .

The binomial extension of simplicial ideal

B △ ⊂ K[V △ ] is defined by B △ =   F l facet of △ I l , I △   ,
where I △ is the Stanley-Reisner ideal associated to the simplicial complex △.

The couple △(B △ ) := (△, B △ ) is called a binomial extension of △.

It is well-known that the Stanley-Reisner ideal of a simplicial complex admits a decomposition into prime ideals corresponding to the facets of the complex (each ideal is generated by the variables which are not in the correspondent facet). We will prove the same property for the ideal B △ .

We have the Primary decomposition of B △ in the ring K[V △ ]: Proposition 1.4.

B △ = F l facet of △ J l .
Before proving the proposition, let us remark that

• x (l) 0 x (l) is ∈ (V △ \ F i ) for all proper edge x (l) 0 , x (l) is of F l (s = 1, k l ), and all facet F i of △, F i = F l , since either x (l) 0 ∈ V △ \ F i or x (l) is ∈ V △ \ F i . • (F • l ) ⊂ (V △ \ F i ), for all facet F i of △, F i = F l .
It implies that for all facet F i of △, F i = F l we have

I l ⊂ (V △ \ F i ) ( * )
Proof: First of all, recall a well-known fact that if I, J are disjoint sets of variables and I = (I) , J = (J) then I ∩ J = (pq | p ∈ I, q ∈ J ) .

We will prove the proposition by induction on the number m of facets of △. The case m = 1 is trivial. If m > 1, then we denote by F m the m th facet. Denote by △ ′ the complex constructed by m -1 facets of △, and denote by B △ ′ the binomial ideal associated to △ ′ , and J ′ i the prime ideal associated to the i th facet of △ ′ . Remark that

J i = (J ′ i , x | x ∈ F • m ) for all (i = 1, 2, . . . , m -1). By induction, we have B △ ′ = J ′ i . We have that J m = I m , V △ \ F m , and
B △ ′ =   1≤k≤m-1 I k , I △ ′   , and B △ =   1≤k≤m I k + I m , I △   .
For the Stanley-Reisner ideal I △ , it is known that:

I △ ′ , F • m V △ \ F m = I △ . (α)
In addition, since

I k ⊂ V △ \ F m for all k = m, we have: 1≤k≤m-1 I k ⊂ V △ \ F m . (β) Moreover, since I m ⊂ V △ \ F m ′ for all facet F m ′ = F m in △,
we have:

I m ⊂ m ′ =m V △ \ F m ′ = I △ ′ , F • m . (γ)
It implies that:

B △ ⊂ B △ ′ , F • m I m , V △ \ F m .
Now, we will prove the other inclusion.

If r ∈ B △ ′ , F • m I m , V △ \ F m , then r = u + v = p + q, where u ∈ 1≤k≤m-1 I k , v ∈ I △ ′ , F • m , p ∈ I m , and q ∈ V △ \ F m . Due to (β) and (γ), we have v -p = q -u ∈ V △ \ F m ∩ I △ ′ , F • m (α) = I △ . Hence, r = u + (v -p) + p ∈ 1≤k≤m-1 I k + I △ + I m = B △ .
From this it follows that:

B △ = B △ ′ , F • m I m , V △ \ F m .
From the induction hypotheses, we deduce that

B △ = m-1 i=1 J ′ i , F • m J m = m-1 i=1 (J ′ i , F • m ) J m = m-1 i=1 J i J m .
The proposition is proved. 2

Remark 1.5. For all facet F l of △, the ideal J l is prime and the ring

K[V △ ]/J l is of dimension 1 + d l , where d l is the dimension of F l .
We deduce from that a corollary on the dimension of B △ as follows:

Corollary 1.6. dim(K[V △ ]/B △ ) = 1 + dim(△).

Reduction number one

First, we recall a theorem of Eisenbud-Goto [EG]:

Theorem 2.1. Let R be a reduced graded ring, defining an algebraic projective variety.

Then we have:

(1) R is Cohen-Macaulay and e(R) = 1 + codim R, where e(R) is the multiplicity of R;

⇒ (2) R admits a 2-linear resolution; ⇒ (3) r(R) = 1; ⇒ (4) e(R) ≤ 1 + codim R
Moreover, if R is Cohen-Macaulay, then the above implications are equivalences.

Definition 2.2. A generalized d-tree on a set of vertices V is a graph defined recursively by the following properties:

(a) A complete graph on d + 1 elements of V is a generalized d-tree.

(b) Let G be a graph on the set V . Assume that there exists a vertex v ∈ V such that:

1. The restriction G ′ of G on V ′ = V \ {v} is a generalized d-tree,
2. There is a subset V ′′ ⊂ V ′ with 1 ≤ j ≤ d vertexes such that the restriction of G on V ′′ is a complete graph, and 3. G is the graph generated by G ′ and the complete graph on V ′′ ∪ {v}.

The vertex v as above is called a extremal.

If j = d, then we say that G is a d-tree.

Remark 2.3. Let ∆(G) the "clique complex" of G, i.e. the simplicial complex whose vertices are the ones of G and the facets are the simplexes with support on the complete subgraphs of G. M. Morales associate to ∆(G) a graph H(G) whose vertices are the facets of ∆(G), and an edge of H(G) links two vertices such that the intersection of their associated facets is non-empty. He proved that G is a generalized d-tree if and only if H(G) is a tree.

The following theorems are proved by Fröberg [Fr]:

Theorem 2.4.

2. ∆ = ∆(G △ ).
Definition 2.6. Let R be the polynomial ring of n variables on the field K. Let I ⊂ R be a homogeneous graded ideal under the standard graduation and d = dim R/I. A set of linear forms {g 1 , g 2 , . . . ,

g d } is a reduction of R/I, if (g 1 , g 2 , . . . , g d )m ρ = m ρ+1 ( mod I)
where m is the irrelevant maximal ideal of R.

The smallest number ρ for all the possible reductions is called the reduction number of R/I.

In [START_REF] Barile | On Stanley-Reisner Rings of Reduction Number One[END_REF], Barile and Morales described a class of square-free monomial ideals whose reduction number is 1. First of all, we recall some definitions.

Definition 2.7. A (d+ 1)-coloration of a graph G is a partition of the vertex set V G into d + 1 subsets, which are called "class of colors", such that two neighbors in G belong to different classes of colors. For each vertex x ∈ G, we denote by C(x) the class containing x.

A (d + 1)-coloration of G is good if every cycle of G is colored by at least three colors.
Remark that this definition is considered only in the case where d ≥ 2.

Let us recall that by taking all faces of dimension 0 and 1 of a simplicial complex △, i.e. all vertices and edges, we associate to △ a simple graph G △ .

Proposition 2.8. [BM1, Theorem 1.1] Let △ be a simplicial complex of dimension d. Denote by R △ the associated Stanley-Reisner ring. Assume that G △ admits a good

(d + 1)-coloration. Let C 1 , C 2 , . .

. , C d+1 sign the classes of colors and for each

i = 1, 2, . . . d + 1, put g i = xj∈Ci x j .
Then g 1 , g 2 , . . . , g d+1 is a system of parameters of R △ . In particular, the reduction number of R △ is 1.

In [START_REF] Barile | On certain algebras of reduction number one[END_REF], Barile and Morales considered a class of binomial ideals defining an union of scrolls, which indeed are binomial extension of simplicial ideals B △ where G △ is a generalized d-tree and each vertex belongs to at most two extension facets. They proved that this class of ideals is of reduction number 1, and an explicit expression of the reduction is given.

In this section, we will extend these results to the binomial extension of simplicial ideals.

Notation 2.9.

Let F l = x (l) 0 , x (l) 1 , . . . , x (l) i1 , . . . , , x (l) i k l , ..., x (
(l) 0 , x (l) i1 , . . . , x (l) 0 , x (l) i k l
, and we denote

I l = x (l) 0 , x (l) i1 , . . . , , x (l) i k l .
For the binomial extension of simplicial ideals, it is not necessary to color all G △ . In fact, for each extension facet F l , it is sufficient to color the extremal points in each bloc of the associated matrix M l . The graph obtained from these points is defined as follows:

Definition 2.10. The reduced graph, denoted by G (△,B) , is given by:

• The vertex set V e G (△,B)
consists of the points of △ and the points y (l) ij 1 (with j ≥ 2) for all extension facet F l of △.

• The set of edges is

E e G (△,B) := F l ∈△s.tY (l) =∅ E △ \ { x (l) 0 , x (l) ij } k l j=2 ∪ { x (l) 0 , y (l) ij 1 , x (l)
i2 , y

(l) ij 1 } k l j=2 .
Example 2.11. Consider the following binomial extension with the binomial ideal associated to the facet F = [a, b, c, d] extended by x, y, z is generated by 2 × 2 minors of the matrix

M = a x x b y c z d .
Then, the reduced graph associated to this extension complex is as in the figure.

Lemma 2.12. Let B M be the ideal generated by all the 2 × 2 minors of the matrix M M := x 0 y 1,1 . . . y 1,j1 and m ′ = {x 0 , ..., x k } ∪ Y be the set of variables in the matrix M , then for any two distinct variables x, y ∈ m ′ with y ∈ Y the product xy is equivalent modulo B M to one of the following monomials:

1. x 1 x 0 , 2. x m p, p ∈ Y n 3. qy n,1 , q ∈ Y m , n ≥ 2, 4. x 0 y 1,j , 5. x 0 y n,1 , Where 1 ≤ m ≤ n ≤ k
Proof. We can assume that x is a variable in the m-block of M , and y is a variable in the n-block of M , with m ≤ n. We have the following cases:

1. 1 ≤ m < n ≤ k (a) x = x m ,
In this case we have the monomials

x m p, with m ≥ 1, p ∈ Y n , (b)
x any and y = y n,1 , In this case we have the monomials x 0 y n,1 , qy n,1 , x m y n,1 with q ∈ Y m , m ≥ 1.

(c) x = y m,u , y = y n,v , or the case x = y m,u , y = x n . We only consider the first case, the proof of the second case is similar.

xy = y m,u y n,v = (y m,u y n,v -y m,u+1 y n,v-1 ) + y m,u+1 y n,v-1 = y m,u+1 y n,v-1 mod B M = ... = x m p mod B M , p ∈ Y n qy n,1 mod B M , q ∈ Y n . 2. m = n > 1 (a) x = y m,1 .
In this case we have the monomials y m,1 p, y m,1 x m with p ∈ Y m .

(b) y = x m . In this case we have the monomials px m with p ∈ Y m .

(c) x = y m,u , y = y m,v , with u < v. In this case we have

xy = y m,u y m,v = (y m,u y m,v -y m,u-1 y m,v+1 ) + y m,u-1 y m,v+1 = y m,u-1 y m,v+1 mod B M = ... = x m p mod B M , p ∈ Y m qy m,1 mod B M , q ∈ Y m . 3. m = n = 1 (a) x = x 0 .
In this case we have the monomials x 0 p with p ∈ Y 1 .

(b) x = x 1 . In this case we have the monomials px 1 with p ∈ Y 1 .

(c) x = y 1,u , y = y 1,v .

xy = y 1,u y 1,v = (y 1,u y 1,v -y 1,u-1 y 1,v+1 ) + y 1,u-1 y 1,v+1 = y 1,u-1 y 1,v+1 mod B M = ... =      x 1 p mod B M , p ∈ Y 1 x 1 p mod B M , p ∈ Y 1 x 0 x 1 mod B M , p ∈ Y 1 .
Definition 2.13. We will say that G (△,B) admits a binomial-coloration if G (△,B) admits a (d + 1)-coloration C such that for every facet F l :

1. C(x (l) 0 ) ∩ F l = {x (l) 0 , x (l) 2 }, 2. C(y (l) j1 ) ∩ F l = {y (l) j1 , x (l)
j+1 } pour tout (j = 2, k l -1),

C(y

(l) k,1 ) ∩ F l = {y (l) k,1 }, 4. For all the other vertices x ∈ G (△,B) ∩ F l , C(x) ∩ F l = {x}.
Proposition 2.14. Suppose that G (△,B) admits a binomial-coloration C, we set

g i = x∈ e Ci
x, G := (g 1 , g 2 , . . . , g d+1 ).

Consider a facet F l with a nonzero associated scroll matrix.

1. If x (l) 0 y (l) n,1 ∈ Gm + B ∆ for any 2 ≤ n ≤ k l , then xy ∈ Gm △ + B △
for any variables x = y ∈ F l , excepts for the products x

(l) 0 x (l) 1 , y (l) 1,u y (l) 1,v , x (l)
0 y (l) , for y (l) ∈ F l but not appearing in M l , and x (l) y (l) , for x (l) , y (l) ∈ F l ∩ △ \ {x

(l) 0 }. 2. If x (l) 0 ∈ F • l , then xy ∈ Gm △ + B △
for any variables x = y ∈ F l , excepts for the products x (l) y (l) , for x (l) , y (l) ∈ F l ∩ △ \ {x

(l) 0 }.
Proof.-We call this facet F , we also delete all scripts l from the variables defining vertices in F and the associated matrix M . from now on, we will denote by ≡ the equivalence relation introduced by Gm + B ∆ . We have the following:

Remark 2.15.

1. If x, y are two distinct element in F such that C(x) ∩ F = {x}, and y belongs only to the facet F , (i.e. y ∈ F • ), then xy ≡ 0, since we can write xy = g x y -z∈C(x),z =x zy, but y belongs only to the facet F and z ∈ F , so zy ≡ 0, hence xy ≡ 0.

2. If x, y are two distinct element in F such that C(x) ∩ F = {x, x ′ }, and y ∈ F • then xy ≡ x ′ y.

We have the following cases:

1. x doesn't appears in the matrix M and y ∈ Y , 2. x, y appear in the matrix M , but one of them belongs to Y , 3. x = x 0 , and y = x m , for some 1 ≤ m ≤ k Now we consider each case:

1. If x doesn't appears in the matrix M and y ∈ Y , then C(x)∩F = {x}, the Remark 2.15 applies so xy ≡ 0.

2. If x, y appear in the matrix M , but one of them belongs to Y .

By applying the Lemma 2.12 the monomial xy is equivalent modulo Gm + B ∆ to one of the following monomials:

(a) For 1 ≤ m ≤ n ≤ k, x m p, p ∈ Y n , or qy n,1 , q ∈ Y m , n ≥ 2,
a-1) The remark 2.15 applies to the monomial x 1 p, p ∈ Y n , n ≥ 1, so it belongs to Gm + B ∆ . a-2) If we are in the first case then by hypothesis x 0 y n,1 ≡ 0. If we are in the second case the following argument is also true when q = x 0 . Consider the case qy n,1 , q ∈ Y 1 , n ≥ 2.

If n = k, and q ∈ Y 1 the remark 2.15 applies, so qy k,1 ≡ 0 for any q ∈ Y ∪ {x 0 }, q = y k,1 . So we may assume that 2 ≤ n < k. In this case C(y n,1 ) ∩ F = {y n,1 , x n+1 }, by applying the remark 2.15 we have:

qy n,1 ≡ -qx n+1 ,
so by using the binomial relations in the matrix M we have

qy n,1 ≡ -qx n+1 ≡ x 1 p ′ , p ′ ∈ Y n+1 q ′ y n+1,1 , q ′ ∈ Y 1 .
Since the case x 1 p, p ∈ Y n , n ≥ 1 was considered in the item b-1), after a finite number of steps we have either qy n,1 ≡ 0 or qy n,1 ≡ q ′ y k,1 , but it was proved before that q ′ y k,1 ≡ 0, so qy n,1 ≡ 0. a-3) We now consider the monomial x 2 p, with p ∈ Y n , n ≥ 2. Since C(x 2 )∩ F = {x 0 , x 2 }, by applying the remark 2.15 we have

x 2 p ≡ -x 0 p.
By using the binomial relations in the matrix M we have

x 0 p ≡      x 0 y n,1 x 1 p ′ , p ′ ∈ Y n qy n,1 , q ∈ Y 1
, So this case is done by taking care of the previous cases. a-4) We consider the monomials x m p, for 2 < m ≤ n ≤ k, and the monomi-

als qy n,1 with q ∈ Y m , 2 ≤ m < n ≤ k. Since C(x m ) ∩ F ) = {x m , y m-1,1 }
by applying the remark 2.15 we have

x m p ≡ -y m-1,1 p.
By the proof of the Lemma 2.12, case 1.c, we have

y m-1,1 p ≡ x m-1 p p ∈ Y n qy n,1 q ∈ Y m-1 .
We have either

x m p ≡ x m-1 p ≡ ... ≡ x 2 p, or x m p ≡ qy n,1 , for some q ∈ Y m , 2 ≤ m < n ≤ k.
So it should be enough to consider the monomial qy n,1 with q ∈ Y m , 2 ≤ m < n ≤ k. The case n = k was considered in b-3). So we may assume 2 ≤ m < n < k. By applying the remark 2.15 we have

qy n,1 ≡ -qx n+1 ,
and by using the binomial relations in

M qx n+1 ≡ x m p p ∈ Y n+1 q ′ y n+1,1 q ′ ∈ Y m .
So after a finite number of steps we will have:

• Either qy n,1 ≡ x 2 p, p ∈ Y n , n ≥ 3, yet considered in b-1) or • qy n,1 ≡ q ′ y k,1 , q ′ ∈ Y m , yet considered in b-2).
(b) If x, y belongs to the first block of M then we have to consider the following monomials x 1 y 1,j , x 0 y 1,j , x 0 x 1 or y 1,u y 1,v . For the monomial x 0 y 1,j , by applying the remark 2.15 we have

x 0 y 1,j ≡ -x 2 y 1,j ,
which is equivalent modulo the binomial relations in the matrix M either to the monomial x 1 p, with p ∈ Y 2 , or to the monomial qy 2,1 , with q ∈ Y 1 . The monomial x 1 p, with p ∈ Y 2 , was yet considered in the first item, and the the monomial qy 2,1 , with q ∈ Y 1 was considered in the second item.

The monomial x 1 y 1,j , was consider before. By the Lemma 2.12 the monomial y 1,u y 1,v is equivalent modulo B ∆ to one of the monomials x 1 y 1,j , x 0 y 1,j , x 0 x 1 . If x 0 ∈ F • then the remark 2.15 applies to x 1 x 0 so we have x 1 x 0 ≡ 0. So this subcase is done.

3. We consider the monomial x 0 x m , where x m appears in the matrix M , m > 1. By the proof of the Lemma 2.12, 1.c, for any 1 < m, we have either x 0 x m ≡ x 1 p, with p ∈ Y m , or x 0 x m ≡ qy m,1 , with q ∈ Y 1 . Both monomials were considered in the previous items and we have seen that they belong to Gm + B ∆ . The proposition is proved.

2 Proposition 2.16. Suppose that G (△,B) admits a binomial-coloration C. Let C 1 , C 2 , . . . , C d+1
the classes of colors of the vertices of G (△,B) , we set

g i = x∈ e Ci
x, G := (g 1 , g 2 , . . . , g d+1 ).

1. Suppose that this coloration on the graph G ′ , with set of edges

E e G ′ := E e G (△,B) ∩E G∆ , is a good d + 1-coloration C
, that is every cycle has more than three colors. Let m △ = (x, y) be the maximal ideal of the polynomial ring R := k[x, y].

2. Suppose that for every facet F l we have either:

(a) x (l) 0 y (l) n,1 ∈ Gm + B ∆ for any 2 ≤ n ≤ k l . (b) x (l) 0 ∈ F • .
Then we have :

m 2 △ = Gm △ + B △ .
In particular, the reduction number of R/B △ is 1.

Proof.-We consider a monomial xy ∈ m 2 . We want to prove that xy ∈ Gm + B ∆ .

• We have two cases:

A. The variables x, y are distinct. We remark that if x, y ∈ m are not in the same facet then xy ∈ B ∆ , hence xy ≡ 0.

• We can assume that the variables x, y belong to the same facet F l . We fix this facet and we call it F , we also delete all scripts l from the variables defining vertices in F and the associated matrix M . By applying the Proposition 2.14, we have to study the cases :x = x 0 , and y = x m , for some k < m ≤ d or x = x l , and y = x m , for some 1 ≤ l < m ≤ d.

B. The variables x, y coincide.

We consider now in detail both cases.

A. If x m doesn't appears in the matrix M or m = 1, then the edge x 0 , x m belongs to G ′ := E e G (△,B) ∩ E △ . So it is enough to consider a monomial xy = x l x m , such that x l , x m belongs to G ′ . In this case C(x) ∩ C(y) = ∅ and the edge x, y belongs to G ′ . We have

xy = g x y - z∈C(x),z =x zy, that is xy ≡ - z∈C(x),zy ∈B ∆ zy.
Let remark that the condition zy ∈ B ∆ implies that y, z belongs to the same facet and are distinct since they have distinct colors. Taking care of all solved cases we can assume that z = x l ′ s , y = x l ′ t belongs to some facet F l ′ and the edge z, y belongs to G ′ . By applying again the same argument we have:

xy ≡ z∈C(x),zy ∈B ∆ w∈C(y),zw ∈B ∆ zw.
If it rests some monomials in the sum, we redo the algorithm. We will prove that the algorithm stops after a finite number of steps. Assume the opposite that the algorithm will never stop. Then, there exists an infinite chain of edges x, y , y, z , z, y 1 , y 1 , z 1 , . . . . Since the number of the variables is finite, we must have a cycle in this chain, i.e. we have a cycle in G ′ . Moreover, each point of this cycle is colored by either the color of x or the one of y (here, we have C(x) = C(y) because x and y form an edge of G ′ . This is a contradiction to the fact that G ′ admits a good coloration.

Hence, the algorithm will stop, i.e.:

xy ∈ (g 1 , g 2 , . . . , g d+1 )m △ + B △ . B. To finish the proof, we consider the case x = y.

If x is a vertex colored, then by replacing x by g x , one has

x 2 = xg x - z∈C(x),z =x xz.
We have that z and x have the same color, but they are distinct, so xz ≡ 0 by the case A). Hence, x 2 ≡ 0. If x is not colored, then x appears in some scroll matrix and we have a binomial x 2 -yz ∈ B ∆ , with y = z, so x 2 ≡ yz ≡ 0, by the above cases. The proposition is proved. 2

Example 2.17. Our Proposition applies to the example 2.11 colored as follows.

Example 2.18. Our Proposition applies to the following complex 2.18 colored as showed, and extended by the scroll matrices:

M 1 := a b y c , M 2 := f c z b , M 3 := d f x e , M 4 := g a w e
Example 2.19. Our Proposition applies to the complex represented by the picture in the left, but our Proposition cannot be applied to the complex represented by the picture in the right, in this case we have that the degree of the projective variety defined by this extended complex is 8, the codimension is 7. On the other hand by a computation we can check that the ideal (a + c + d, b + e, f + v + y) is a reduction with reduction number two. The complex is extended by the following matrices:

g i = x∈Ci x.
Then, we have:

(g 1 , g 2 , . . . , g d+1 )m △ + B △ = m 2 △ , where m △ = (x, y) is the irrelevant maximal ideal of the polynomial ring R := K[x, y].

Proof: The proof will be by induction on the number λ of facets of △.

The case λ = 1 is a particular case of the Proposition 2.16. See also [START_REF] Barile | On certain algebras of reduction number one[END_REF].

Assume that the proposition is true for λ ≥ 1, we will prove it for λ + 1. Since G △ is a generalized d-tree, one can find a facet F such that its associated vertex in H(G △ ) is a leaf. Consider (△ ′ , B ′ ) the extension complex constructed by the λ facets different from F in △. We put U = F ∩ V △ ′ . The graph G △ ′ is also a generalized d-tree. By induction, the graph G (△ ′ ,B ′ ) admits a good (d + 1)-coloration as in the proposition, and for all xy ∈ m 2 △ ′ , we have:

xy ∈ (g ′ 1 , g ′ 2 , . . . , g ′ d+1 )m △ ′ + B △ ′ ,
where C ′ i , g ′ i are the i th class of color and the correspondent sum. We have two cases:

I) F ∩ V △ = F . II) F ∩ V △ = F .
For each case, we will color the points in

F • ∩ V e G (△,B)
, and we will define the sums g i .

I): F ∩V △ = F . We can suppose that F = {x 0 , x 1 , x 2 , . . . , x n }, and U = {x 1 , x 2 , . . . , x m } (1 ≤ m ≤ n ≤ d), and x i ∈ C ′ i for all i = 1, m. To obtain (d+ 1)-coloration which verifies the proposition, it is sufficient to color the points x i / ∈ U by arbitrary colors C ′ j with j > m. For example, we can color x i by the i th color (∀i = m + 1, n), and x 0 by the (d + 1) th color. Hence, we have:

C i = C ′ i and g i = g ′ i for all i = 1, m or i = n + 1, d, C j = C ′ j ∪ {x j } and g j = g ′ j + x j , j = m + 1, n, C d+1 = C ′ d+1 ∪ {x 0 } and g d+1 = g ′ d+1 + x 0 . II): F ∩ V △ = F .
Let M be the matrix associated to F : We can assume that

M := x 0 y 1,1 . . .
F ∩ V △ = {x 0 , x 1 , x 2 , . . . , x l } (1 ≤ k ≤ l ≤ d).
Let remark that since by construction the propers edges of F are not in △ ′ , we have either x 0 ∈ F • , or x i ∈ F • ∀i = 1, k. So we have to consider two sub-cases:

II-1) x 0 ∈ F • , i.e.
x 0 / ∈ U : In order to color G (△,B) , we color each point x ∈ V △ \ {U, x 0 } by a color not used in U , and we define :

• the color of x 0 is the same of x 2 ;

• the color of y (j-1)1 is the color of x j for all j = 3, k.

• the color of y k,1 is the (l + 1) i-th color not used in F .

We can renumbering the classes of colors in such a way that x i ∈ C ′ i for i = 1, l. Then, we have

g 2 = g ′ 2 + x 0 , if x 2 ∈ U, g ′ 2 + x 2 + x 0 , if x 2 / ∈ U ; g j =     
g ′ j + y (j-1)1 for j = 3, l such that x j ∈ U, g ′ j + x j + y (j-1)1 for j = 3, l such that x j / ∈ U, g ′ j + x j for j ∈ {1, l + 1, ..., k}, such that x n / ∈ U ; g l+1 = g ′ l+1 + y k1 ; g j = g ′ j for all other indices j ;

II-2) If x 0 / ∈ F • : In this case x i ∈ F • for all i = 1, k.
We can suppose that U = {x 0 , x s , . . . , x l } with k < s ≤ l, and that x j ∈ C j for all j = s, l, and x 0 ∈ C 2 . We put:

C

1 = C ′ 1 ∪ {x 1 }; C 2 = C ′ 2 ∪ {x 2 }; C i = C ′ i ∪ {x i , y (i-1)1 } for all i = 3, k; C t = C ′ t ∪ {x t } for all t = k + 1, s -1; C j = C ′ j for all j ≥ s; C l+1 = C ′ l+1 ∪ {y k,1 }. Then we have: g 1 = g ′ 1 + x 1 ; g 2 = g ′ 2 + x 2 ; g i = g ′ i + x i + y (i-1)1 for all i = 3, k; g t = g ′ t +
x t for all t = l + 1, s -1; g j = g ′ j for all j ≥ s and j = k + 1; g l+1 = g ′ l+1 + y k1 .

Let us remark that in all the cases, for all j the support of g jg ′ j is contained in F • .

A) First we will prove that xy ≡ 0 for any x ∈ F • , x = y and y ∈ F .

• Case I) We have that C(y) ∩ F contains only y, so by applying the Remark 2.15, we have xy ≡ 0.

• Case II-1) since x 0 ∈ F • , then by the Proposition 2.14 xy ≡ 0 for any variables x = y ∈ F , excepts for the products xy, for x, y ∈ F ∩ △ \ {x 0 }. Since we are interested in the monomials x ∈ F • , and y ∈ F ∩ △, we have to consider the following cases:

II-1-a) x ∈ F • , and x = x u , y = x v appear in M , u, v = 0. By applying the Remark 2.15, since x u ∈ F • , we have either • Case II-2) First we prove that x 0 y n,1 ≡ 0, for any n ≥ 2. By the Remark 2.15, we have x 0 y n,1 ≡ x 2 y n,1 . If n = k since x 2 ∈ F • , and C(y k,1 ) ∩ F = {y k,1 }, Remark 2.15 we have x 2 y k,1 ≡ 0. If n < k then x 2 y n,1 ≡ x 2 x n+1 , but x n+1 ∈ F • so x 2 x n+1 ≡ x 0 x n+1 . By using the binomial relations in the matrix M , we will have that

x u x v ≡      0 if v = 1, x u x 0 ≡ 0 if v = 2, x u y (v-1)1 ≡ 0 if v > 2. II-1-b) x ∈ F • . If
x 0 x n+1 ≡ x 1 q q ∈ Y n+1 py n+1,1 p ∈ Y 1 . Now by the Remark 2.15 x 1 q ≡ 0, py n+1,1 ≡ 0 if n + 1 = 0, and py n+1,1 ≡ px n+1,1 if n + 1 < k. by applying the binomial relations in the matrix and the Remark 2.15 after a finite number of steps we will have x 0 y n,1 ≡ 0. By using the Proposition 2.14, we have xy ≡ 0 for any variables x = y ∈ F l , excepts for the products x 0 x 1 , y 1,u y 1,v , x 0 y, for y ∈ F but not appearing in M , and xy for x, y ∈ F l ∩ △ \ {x 0 }. So we need to consider the following cases:

1. x 0 x 1 ≡ -x 2 x 1 ≡ -x 2 gx 1 ≡ 0. This case also will imply that 2. x 0 x m , x m ∈ F • doesn't appears in M . x 0 x m ≡ -x 2 x m ≡ -x 2 gx m ≡ 0.

3. x m x n , m, n > 0, x m ∈ F • , both x m , x n appear in M , this implies x n ∈ F • .

We can assume that n ≥ 2, so x m x n ≡ x m y n-1,1 if n > 2, or x m x n ≡ x m x 0 , both monomial are equivalent to 0.

4. x m x n , m, n > 0, x m ∈ F • , x m appears in M but x n doesn't appears in M . Then x m x n ≡ -x m gx n ≡ 0.

5. x m x n , m, n > 0, x m ∈ F • , x m doesn't appears in M but x n appears in M . Then x m x n ≡ -gx m x n ≡ 0.

6. x m x n , m, n > 0, x m ∈ F • , both x m , x n don't appear in M . Then x m x n ≡ -x m gx n ≡ 0. m i (g ig ′ i ).

Since the support of g ig ′ i is in F • and Supp(g ig ′ i )∩Supp(m i ) = ∅, due to the precedent cases m i (g ig ′ i ) ∈ (g 1 , g 2 , . . . , g d+1 )m △ + B △ .

It implies that xy verifies ( * ).

C) • x = y ∈ F • , In this case, if in addition x is not colored, modulo B △ (see Lemma 2.12), we re-obtain one of cases above. If x is colored, we replace x by the sum g x of all variables in the class of color of x, we will be in the case x = y. Hence the proposition is proved. 2

  in F l ∩ △ and by y (l) ijm the vertices in F l \ △.

  l) d l be an extension facet of △ by the points {y (l) ijm }in the proper edges x

  facet [ a, b, c] extended by x, y ; facet [ b, c, d] extended by u, v. If G △ is a generalized d-tree, then in [BM1] it was constructed an explicit reduction for the quotient by the Stanley-Reisner ideal associated to △. The aim of the following proposition is to prove that the reduction number of R/B △ is 1, and to give an explicit expression of the reduction. Proposition 2.20. Let G △ be a generalized d-tree. Then G (△,B) admits a binomialcoloration. Let C 1 , C 2 , . . . , C d+1 denote the classes of colors. Put

  x appears in M , and y doesn't appears in M , since C(y) ∩ F = {y}, we get by the Remark 2.15,that xy ≡ 0.II-1-c) x ∈ F • . If x doesn't appears in M and y appears in M , since C(x) ∩ F = {x}, then due to the Remark 2.15, it is sufficient to check the case where y = x v with 2 ≤ v ≤ k. But in this case, one has also that eitherxx v ≡ xy (v-1)1 ≡ 0 or xx v ≡ xx 0 ≡ 0.II-1-d) x ∈ F • , both x, y don't appear in M : One has C(y) ∩ F = {y}, so xy ≡ 0 by to the Remark 2.15.

B

  ) • x, y ∈ V △ ′ , By induction, one has xy = d+1 i=1 m i g ′ i mod B △ ′ with m i ∈ m △ ′ .


Gaëlle Ferré
email: gaelle.ferre@univ-nantes.fr

Testimonials on emotions -a multimodal speech analysis

 and the conclusion of the paper is that speech is felt to be more emotional when a set of prosodic and gestural parameters is modified from more neutral speech, e.g. smiles or laughs do not in themselves convey happiness, but must be combined with other gestural and prosodic features to do so.

Introduction

This paper proposes a multimodal analysis of emotional speech compared to more neutral parts of discourse in podcast films recorded by the association "GoodPlanet" for a public exhibition directed by photographer Yann Arthus-Bertrand. The corpus description (see section 2) addresses the issue of the use of podcast recordings for research purposes. Among the films recorded by the association, I chose to analyze a collection of testimonies which where semantically linked with emotions, the underlying principle being that while speaking of their emotions, the speakers show a bit of these emotions in their expression. According to Ekman (1999:50) expressions "are part of an emotion; they are a sign that an emotion is occurring". This principle illustrates what Caffi & Janney, quoted in Plantin (2003:9), call emotional communication, i.e. "a type of spontaneous, unintentional leakage or bursting out of emotion in speech". And this is exactly what occurs in the videos, where speakers narrate some emotional experience to the public in a rather neutral tone, but at times, their emotions burst out in their intonation and facial expression. In section 4, I present a qualitative analysis of some salient prosodic and gestural features related to cold anger, sadness and happiness.

Corpus description

The corpus I worked on is a series of podcasts made available online by the association "GoodPlanet" headed by photographer Yann Arthus-Bertrand. The aim of the association's project "6 billion others" is to collect a series of testimonials from people around the world on a vast quantity of topics. Ordinary people from different parts of the world were asked to give their opinion on questions such as "what do you fear most?", "what has been the biggest joy in your life?, "on what occasion did you cry?", "what makes you particularly angry?" or "have you ever felt discrimination?". They were video-taped by a team of filmmakers of the association and the videos were edited into clips, the utterances said being inserted as a French subtitle into the clip. Each clip shows up to five or six people speaking different languages and expressing an opinion on a common topic, and lasts about 5 minutes. The project has been running for five years now and the final aim is to show the videos (more than 5000 interviews filmed in about 75 countries) in a free exhibition to be held in the Grand Palais in Paris in January and February 2009. The association has given consent for the video-clips to be used in a research project on emotions, provided the persons are treated with dignity and respect. This doesn't completely answer the issue on ethics, since such research work was not initially covered by the association's project, but the participants were recorded of their free will and accepted that the recordings be used in a public exhibition and displayed on the internet. No judgement will be made in this paper on the participants' physical appearance or on the relevance of their speech.

The major difficulty while working on podcasts is the issue of the naturalness of the data. The clips I worked on have been edited by professional filmmakers who obviously cut the sequences of hesitations and false starts always present in spontaneous data, but the resulting films are very close to natural data. It just means, in terms of prosody, that the clips can't be used for a study of speech rhythm, silent pauses and hesitation discourse markers, which is not the purpose of this paper. Another drawback of the clips is that the persons have been filmed during their testimonials but no unemotional speech of the same speakers has been recorded for comparison and although each clip contains testimonials on the same topic, speakers do not say exactly the same thing. Consequently, the object of the paper will be to study the shifts between neutral and emotional speech within each testimonial rather than comparing emotional speech between speakers.

When using podcasts, the major drawback lies in the often poor quality of the recordings. Most of the time, the recordings use a compression rate which is too low for a good image quality in terms of pixel number, a quality which is nevertheless necessary to allow even a manual detection of fine facial details. This is not the case of the GoodPlanet podcasts which are of a high quality despite compression. The clips were edited in m4v, but I changed the codec into MPEG1 to be able to read the clips in ELAN (see section 8). Another drawback of podcasts is that speakers are usually filmed in action, with numerous zooms and close-ups which make it difficult for the analyst to annotate the movements of a particular part of the body, not always visible on the video. The testimonies of GoodPlanet are all filmed in the same condition, with an extreme close-up on the speaker's face. This is particularly convenient for the annotation of fine details of the face, although it means, of course, that other parts of the speakers' bodies are not visible on the screen. At last, a prosodic analysis of speech requires an extremely good quality of the sound of the recordings, and this is far from being the case in most podcasts. In the vast majority of podcasts, the sound-track is full of background noise, either music added to the initial recording, or surrounding noise during the recording which prevents any serious detection of prosodic parameters such as F0 or intensity. Another problem lies in the fact that in many recordings, several people speak in overlap and are recorded with a single microphone which also hinders the detection of prosodic parameters. GoodPlanet podcasts present none of these faults since the speakers are recorded professionally with no overlapping speech and background music or noise. The sound tracks of the clips are of a good quality and allow prosodic treatment of the data.

Annotation of the corpus

Phonology and prosody

The first step in the annotation process consisted in a prosodic annotation of about 15 mn of the corpus collection with PRAAT (see section 8). The sound files, extracted from the videos, were first converted into wave files to be readable in PRAAT and transcribed into current orthography in what could be termed intuitive prosodic groups. The translations made by the filmmakers could not be used since they didn't correspond to the exact words uttered by the speakers and were not aligned with the sound signal. From this initial transcription, words, syllables and phonemes were then annotated using the SAMPA convention of annotation, entirely manually for English and with the help of EasyAlign for French (see section 8). I then noted focal accents on a separate tier and on yet another tier, prosodic units (intermediate vs. intonational phrases, see Cruttenden, 1997:59-60) were noted as well as the general pitch contour on each phrase, using the contours described in [START_REF] Cruttenden | Intonation. Cambridge, CUP[END_REF]: high and low fall, high and low rise, rise-fall, fall-rise, and flat. In the cases where PRAAT displayed F0 detection errors, I used the Prosogram (see section 8) to obtain a better representation of the contour. The phonemic and prosodic annotation is shown in Figure 1 below:

Gestures

These annotations were then imported in ELAN, a tool for the annotation of video files (see section 8). For the annotation of gestures, I used an adapted version of the MUMIN coding scheme, fully described in [START_REF] Bertrand | Le CID: Corpus of Interactional Data -protocoles, conventions, annotations[END_REF]. The scheme was developed to describe gestures made in interactional data, which is quite poor in emotional load [START_REF] Bouchard | M'enfin !!! Des "petits mots" pour les "petites" émotions[END_REF]. Yet, the description of gestures using this scheme is extremely precise and a tier has been devoted to the annotation of emotions/attitude. I compared this scheme with the one recommended by the Network of Excellence HUMAINE and the annotation types of the two schemes are very close. Since the films were close-ups of the speakers' faces, I only annotated facial movements in a first step: gaze direction (front, left, right, up, down), head direction and head movements (shake, nod, tilt, beat...), eyebrow movements (frown, raise), mouth movements (smile, laugh...). In the file concerning sadness, I also added a track to annotate eye moisture (wet eyes) which was not initially thought of in the scheme, since this parameter seemed to play a role when the speaker displayed this emotion. At last, in each file, I annotated the parts where speech seemed to be more loaded with emotion, simply stating the emotion (cold anger, sadness, happiness). This was made rather intuitively and by only one annotator for the moment. It is clear that in the future, several annotators will be needed for this annotation which is more subjective than the mere annotation of gestures, and their agreement checked by Kappa measurements.

Correlations of the different parameters were made with the help of the search function of ELAN.

Results and discussion

Although the files were too short to present any statistical analysis here, the results obtained are nevertheless interesting in a qualitative approach which may constitute the basis for further work.

Prosody

As stated earlier in this paper, the edited nature of the files does not allow work on rhythm (filled and silent pauses as well as speech rate) since parts of the file have probably been deleted and we do not have access to the original recordings. Yet, much can be done on pitch and stress. Previous studies have mostly described the links between emotional speech and speech span as well as F0 average height. Working on Japanese and French, [START_REF] Yamasaki | Perception des émotions "positives" et "négatives" chez les auditeurs français et japonais à travers le contour de F0[END_REF], for instance, found that positive emotions were rather perceived with rising F0, high average F0 being associated with gaiety. On the contrary, negative emotions were rather perceived when the F0 was falling, low average F0 being associated with sadness. She didn't however annotate contours from a phonological perspective. Devillers and Vasilescu (2004) studied the span and height of the intonation contour and found that in two negative emotions, fear and anger, the amplitude of the parameters depended upon subjects, but tended to be larger in emotional than in neutral speech. Both studies agree on the important weight of lexical/semantic content in the perception of emotional speech. In this study, I concentrated on phonological F0 contours, comparing the distribution of contours for each speaker on the total testimony on one hand, and in correlation with emotions on the other hand. Results showed that there is a larger proportion of rising-falling contours in sadness as in the total testimony for this speaker. Happiness, on the contrary, was more correlated with low rising contours. This is in agreement with the findings of Yamasaki (op.cit.) associating lower intonation to negative emotions and higher intonation to positive ones. Looking at cold anger, however, an emotion which could be classified as negative, one finds that there is a higher proportion of high rises and rising-falling contours which do not correspond to Yamasaki's results. Yet, they confirm Devillers and Vasilescu's results (op. cit.) stating that the pitch span is larger in emotional speech, since I didn't find any high rising contours in declarative utterances in other contexts.

What makes sense however is that emotional speech implies a higher involvement of the speaker in his discourse and this is shown prosodically speaking with the use of a higher proportion of modulated tones such as the rising-falling contour. According to [START_REF] Morel | Grammaire de l'intonation : L'exemple du français. Paris, Ophrys[END_REF], rising contours show a greater appeal to the co-participant in conversational data on the part of the speaker. In this context, which is not a conversation, the expectancy of empathy is nevertheless still present since the speakers may be said to convey a message to the public, represented by the eye of the camera. Emotional speech then can be described as conveying more involvement on the part of the speaker who also expects a greater involvement on the part of the addressee. [START_REF] Bagou | Validation perceptive et réalisations acoustiques de l'implication emphatique dans la narration orale spontanée[END_REF] and [START_REF] Plantin | Structures verbales de l'émotion parlée et de la parole émue[END_REF] find that emotional involvement is regularly associated with emphasis in spontaneous speech. I also found that there was a high quantity of focal accents in the files I treated, especially in the one showing cold anger. The focal accent is mainly realized through lengthening of word initial onset consonants. Yet, speakers in this file switched from rather neutral speech to speech showing cold anger and the proportion of focal accents was not higher in the speech parts perceived as more emotional. In other words, I could not find a direct link between prosodic emphasis and emotional speech, which shows that other parameters are needed for speech to be perceived as emotional. This is even reinforced by the gestural analysis (Cf. Analysis of "beats") in the next section.

Gestures

To follow my thread of thinking, the gesture annotation of the corpus showed that in the cold anger file, most focal accents were accompanied by a head beat of speakers (e.g. a rhythmic downward rapid movement of the head produced on accented syllables). These beats were not either associated with the perceived emotional parts of the file. This shows that a strong focal accent, marked both prosodically and gesturally, is not sufficient for speech to be perceived as loaded in emotional weight, although the utterance is not perceived as neutral, but as emphatic. In my opinion, emphasis and emotion are not necessarily linked.

Other gestures and physical properties were however typically associated with the three emotions under study. What appears immediately in Table 1 is that each gestural parameter allows a distinction between two emotions. For instance, head movements allow a distinction between sadness and happiness (shakes being associated with sadness and nods with happiness), whereas no particular head movement is found in correlation with cold anger.

Gesture Cold anger

Sadness Happiness

Raising X Eyebrows Frowning X Shakes X Head Nods X Away X Gaze Front X Narrowed eyes X Eyes Wet eyes X Laughs X X Mouth Smiles X
Table 1: Relevant gestures and physical properties related to the three emotions.

Table 1 corroborates the results of [START_REF] Smith | A Componential Approach to the meaning of facial expressions[END_REF], partially based on [START_REF] Darwin | Détection des émotions à partir d'indices lexicaux, dialogiques et prosodiques dans le dialogue oral[END_REF], especially concerning eyebrow movements (raised eyebrows conveying sadness as opposed to frowns which convey cold anger). I also found like Smith & Scott that laughs and smiles are associated with happiness. What is different though is that laughs are also met in cold anger in this corpus, which is not however surprising. As stated in [START_REF] Bertrand | L'observation et l'analyse des affects dans l'interaction[END_REF], laughter reveals a distance of the speaker from his feelings. Laughter can then be met in other feelings than happiness, such as in cold anger and may be induced by the fact that the speaker suddenly realizes his greater involvement in a negative emotion from which he needs to take a distance. This is directly linked as well to the question of politeness evoked in Kerbrat-Orecchioni (2000:51) for whom « la civilité n'admet pas les manifestations émotionnelles intempestives et incontrôlées » and « la politesse et les rites sociaux ont précisément pour fonction principale de canaliser le flux affectif, de juguler les débordements émotionnels (...) ». This is probably also why speakers conveying sadness tend to look away from the camera when the emotion is too strong, whereas in happiness, they look straight at the camera, this latter emotion being a positive one. No regularity in gaze direction could be observed when the emotion conveyed is cold anger. At last, as far as eyes are concerned, this work corroborates Smith & Scott's findings (op. cit.) in which anger was associated with narrowed eyes. So far, the results for cold anger and happiness also corroborate the studies made by [START_REF] Cosnier | Les mimiques du créateur, ou l'auto-référence des représentations affectives[END_REF] on the cognitive representation of emotions, a study in which they find that the mental representation of anger triggers movements in the subjects' eyes and eyebrows, whereas the representation of happiness triggers mouth movements.

Smith & Scott did not test eye moisture, but in the file I worked on, it was obvious that sadness was perceived in a stronger way with greater eye moisture of the speaker.

Figure 4: Eye moisture in the perception of sadness. This parameter is however not as reliable as the other ones though, since eye moisture on a video file may depend much on the eye colour of the speakers, as well as on the lighting used during the recording.

Conclusion

As a conclusion, I may say that this paper addressed several issues currently under discussion in the research community on emotions: firstly, concerning the data which lacks cruelly, it was shown that some podcast recordings may be used although work on such recordings cannot answer all the questions raised (but is this not the case of any corpus?). The corpus of the project "6 billion others" is of a quality which allows a prosodic and gestural treatment and I showed how speakers, through emotional semantic content, switch from rather neutral narration of their feelings to more emotionally involved speech. The emotions conveyed were cold anger, sadness and happiness, which I understand as attenuated forms of anger, deep sorrow and joy. They were expressed through certain intonation contours: high rises and rising-falling contours for cold anger, rise falls for sadness and low rises for happiness. These preferred contours show more involvement on the part of the speaker who also seeks some empathy from the addressee. The speaker's involvement is however not linked to the presence of focal accents in this corpus. In terms of gestures, emotions are conveyed with a set of different parameters: cold anger is associated with eyebrow frown, narrowed eyes and even laughter at times; sadness is associated with raised eyebrows, averted gaze, eye moisture and head shakes; happiness is associated with head nods, direct gaze, smiles and laughter. What both the prosodic and gestural analyses show is that prosody and gestures are not redundant in emotional speech, as already stated by [START_REF] Aubergé | Prosodie et émotion[END_REF], and that both participate in the perception of emotion. It also shows that analyses of video films are useful to complement studies based on pictures, since a laughing face in itself does not necessarily convey happiness, for instance. Yet, a still photograph of a laughing face would nevertheless be interpreted as a happy face without the context of the film and other parameters as already pointed out by Beavin [START_REF] Beavin Bavelas | Faces in dialogue[END_REF]. The limits of this study are also important. The quantity of data treated was quite small and an annotation of more recordings would be needed first to confirm the results of this pilot study, then to be really able to compare the results obtained on English and French which were treated here as equivalent.

Figure 1 :

 1 Figure 1: Textgrid of the orthographic, phonemic and prosodic annotation of the corpus in PRAAT.

Figure 2 :

 2 Figure 2: Snapshot of the annotation board in ELAN showing orthographic, phonemic, prosodic and gesture tracks.

Figure 3 :

 3 Figure 3: Still pictures of speakers showing cold anger (left) and happiness (right).

Acknowledgements

I am particularly indebted to A.-L. Charriot and F. Gilard, producers of the project "6 billion others", for allowing me to use the recordings of the association "GoodPlanet", headed by photographer Yann Arthus-Bertrand. I hope the 2008 exhibition in Paris is a success and that useful work can be done in linguistics using their material. Any error in the description of the corpus or the aim of the association is mine.


Aurélie Bissay

Philippe Pernelle

Arnaud Lefebvre

Abdelaziz Bouras

Knowledge integration through a plm approach

come

INTRODUCTION

Companies have to conjugate customers satisfaction and the increase of their productivity. To respond to these constraints, companies need to be reactive and to reduce the development cycle of their products. Furthermore, the optimal re-use of the know-how is still not enough exploited within PLM systems. Any objective of cycle's reduction by the optimal re-use of the knowledge implies that the information system must be able to identify elements of knowledge but also be able to estimate them.

Figure 1. PLM system and knowledge

The objective of this paper is to study product lifecycle information systems within the contex of a knowledge capitalization. We wish to propose more particularly a methodology which allows the knowledge integration around a system PLM.

The industrial context of this work lies in the industrial sector in the plastics industry. In this industry, the needs involving the optimization and the development of products are quintessential. Indeed, the plastic manufacturing context is made of many small and medium-sized firms operating in a highly competitive sector.

The Marmillon SAS company involved in this work is our industrial partner. This company develops and manufactures plastic parts for the automotive industry. To respond to changes in its external environment and as a result of internal findings, the company has chosen to remain competitive by focusing on two areas of progress: optimize the development of new products, propose request for quotation adjusted to score on a reactive basis.

First we present the context of knowledge capitalization and business process modelling (BPM). Then we present our methodological approach for knowledge integration in PLM systems.

Lastly, we present the setting works with an industrial case in the plastic manufacturing.

KWNOLEDGE CAPITALISATION CONTEXT 1 Needs of corporate memories

Knowledge contributes to the preservation of the know how of the company. The corporate memory explicitly represents the company's information and knowledge [START_REF] Van Heijst | Organizing corporate memories[END_REF]. It constitutes also the company's resources implemented by knowledge and collective data [START_REF] Prasad | Corporate memories as distributed case librairies[END_REF]. This knowledge is interlinked with various elements handled within the company's activity such as products, customers, marketing strategies, the expertise stemmed from projects.

The exploited knowledge within a medium-sized company can concern several elements treated within an activity such as products, design and production processes, return on experience on the projects, expertise resulted from these projects. Supports of these collections can take various forms as databases, electronic documents, reports, breviaries of knowledge, descriptive index cards.

Companies have difficulties to choose methods to well capitalize and protect their knowledge.

The need to establish corporate memories is justified by the re-use of knowledge, reasoning and behavior. This aim meet those of the PLM systems. Indeed products life cycle management promotes the know-how re-using. In companies where innovation is the incremental process, mastering its know-how enables the increase of innovation capacity as well as the decrease of the products development time.

Capitalisation methods

The re-using of knowledge requires that enterprises are able to formalize it. From a general viewpoint the existing approaches coincide with knowledge life cycle defined by Grundstein.

Figure 2. Extract of knowledge lifecycle

Methodologies from the knowledge engineering [START_REF] Dieng | Methods and tools for corporate knowledge management[END_REF] aim at proposing a framework with methods to preserve and serve as a base for the knowledge formalization. Several methodologies exist: methods based on interviews (KOD Knowledge Oriented Design), methods of knowledge capitalization (REX and MEREX [START_REF] Corbel | Méthodologie et retour d'expérience: démarche MEREX de Renault[END_REF]), and methods of structured acquisition (MSKM [START_REF] Ermine | MKSM méthodes pour la gestion des connaissances, Rapport technique[END_REF] [6] [START_REF] Ermine | Capitaliser et partager la connaissance avec la méthode Mask[END_REF]).

The knowledge formalization allows to assure a coherence and an exploitation of knowledge bases. There are various formalisms of knowledge representation. The use of graphs to represent knowledge is usual [START_REF] Quillian | Semantic memory[END_REF] [START_REF] Hendrix | Encoding knowledge in partitioned networks[END_REF]. Graphs are used to abstract the relevant information and concentrate only on the topology of a problem.

With the Conceptual Graph (CG), Sowa [START_REF] Sowa | Conceptual structures : information processing[END_REF] proposes a generic formalism for any type of representation of the knowledge. The meaning of a concept is reduced to its relative position according to the other concepts. It thus takes a sense only if a semantic network modelling the general knowledge of the system exists. Furthermore, ontologies allow to specify a conceptualization of a specific domain of knowledge [START_REF] Gruber | Formal ontology in conceptual analysis and knowledge representation, Chapter: Towards principles for the design of ontologies used for knowledge sharing[END_REF]. They present the advantage to propose a formal description which constitutes a structured set by concepts allowing to give sense to the information. The concepts are organized in a graph whose relations may be represented by semantic relations or relations of subsomption.

Even if they are not used in link with PLM systems, the ontologies or the abstract graphs allow to describe numerous concepts to be implemented in the PLM [START_REF] Barcikowski | Vers une évaluation de la robustesse des connaissances au sein d'une base de connaissances[END_REF]: individuals; classes sets( for types of objects), attributes (characteristics of the object), relations (the way objects are bound), events (relations change or attributes). It is thus possible to transcribe (in a PLM compatible format) a part of the knowledge formalized with these representations [START_REF] Pernelle | Modélisation intégrée et pérennisation des connaissances dans une approche PLM, Ingénierie de la conception et cycle de vie des produits, Traité IC2 -série Productique[END_REF].

For companies, the question is how to define an information system that is the most adapted to capitalize knowledge. In a lot of companies, the information system is composed of heterogeneous components covering all sectors of the company. Each element covers a particular function (ERP, PLM, SCM, CRM). In this context, the manufacturing management (ERP), the management of the customer relation software (CRM) and the supply chain management software (SCM) are aiming to optimize the information flow linked to the products and traks. These applications are designed to optimize repetitive operations and are especially effective to handle processes which appeal to the same task. The PLM is defined as: "an organizational device allowing to regulate the creation, the traffic, the use and the evolution of the informative heritage of definition of the product, that is all the information which define how the product is designed, made and used " [START_REF] Pernelle | Système d'Information Produit pour les PME/PMI : modélisation multi niveaux d'entreprises engagées dans un travail coopératif[END_REF]. These systems are thus in theory the most capable to take care of the totality of the product development process [START_REF] Cimdata | Cimdata glossary[END_REF] [START_REF] Sudarsan | A Product Information Modeling Framework for Product Lifecycle Management[END_REF]. Actually, PLM systems have an implicit vocation to capitalize the know-how.

CONTEXT OF BUSINESS PROCESS MANAGEMENT

The BPM (Business Process Management) has been managing the companies' businesses processes for several years. Despite the increasing number of tools, implementation remains delicate for several reasons : impact on organizations, reluctance to change, scope and granularity of the process. Whatever the difficulties are, a BPM approach is always in line with a company's rationalization process exactly as the implementation of a PLM system. This is the reason why we choose to link theses approaches described in the following paragraphs.

Processes representation models are commonly used in companies. Traditionally, we find the approaches around IDEFX, SADT, etc.... Even if these models are often used to represent processes, they have shown their limits with less structured or progressive processes, but above all, they do not allow execution. Approaches around the WfMc's Workflows are suggesting to model and run these processes by means of a runtime software. These different models are generic and can go from the business representation to the instantiation in the information system. In fact they are poorly adapted for implementation in the scope of the companies industrial tools like PLM or ERP.

INTEGRATION APPROACH WITHIN PLM SYSTEMS 1 Global approach

A PLM system can be represented by the figure 3.

Figure 3. The three components of a PLM system

Our goal is to integrate and capitalize the knowledge generated by a process into a PLM system. Our approach (figure 4) is devided in two phases: process integration and knowledge integration. These two steps are described in next sections.

Figure 4. Global approach to BP and Knowledge integration

The business processes integration in the PLM system is the first step in our approach. PLM systems can build validation circuits and tasks to achieve between the various actors in a process.

Currently, PLM processes are focused on the life cycle and not on activities. This point can not allowes processes instantiation with tools such as BPMN, XPDL or BPML.

Our approach (figure 5) is composed of four stages: identification, modeling, instanciation, and evaluation.

Figure 5. BP integration

Identification: from the tasks / activities identified in business processes, this step is to identify classes of objects as well as their state periphery. Peripheries statements define the life cycles of objects. The purpose of this step is to get the product model in the PLM.

Modelling: within the PLM systems, processes modelling is made by means of workflows centred on the life cycle. From the previous identification, the processes are restructured into executable elements (usine BPML model for instance).

Instantiation: once processes modelling achieved within the PLM system, processes are instantiated and used by users. This step of use takes place continuously and activities tracks generated by the system are used for assessment.

Evaluation : In order to integrate business processes and evaluate them within a PLM system, we propose a typology of processes with associated indicators. The objective is to evaluate them at the time of their execution by the PLM system.

There are different approach to caracterize processes: approach based on activities, on the types of inputs / outputs or on the degree of structuring. The typology of indicators suggested in this paper is based on a decomposition following four criteria: time base (limited process, cyclical,...), stability (stable process, evolutionary, unstable), genericity (single instance, multiple instances), measurability (measurable process, un-measurable).

3 Knowledge integration within a PLM system

From the process modelling in the system, the second major step is to integrate elements of knowledge in the PLM system. The aim is to allow a knowledge capitalization from the process activities to expand PLM system capacities to manage knowledge. In the context of the knowledge lifecycle, the main steps (Grundstein) are identified by different stages as shown in figure 6: Identify : identification of knowledge to build up a PLM system. This step is to identify more precisely elements of knowledge to be manage within the PLM system. The modelling design should lead a data model which could be instantiated in the PLM system. UML class diagrams can be used to model objects and links between them.

Identification of business rules: this step is to identify behavioral rules that will directly depend on identified objects in the previous stage.

Formalise : it is necessary to create convertion rules in XML to use knowledge in the system. Formalization aims at transforming conceptual graphs models (if CG models choosen) in order to integrate them into the PLM system.

Use : the PLM system framework includes a presentation server and an application server that are communicating with a data base as shown in figure 7. As seen previously, the information system designer will have to identify knowledge and set up the mapping in order to use it within the system. The user gets access to knowledge objects, rated by indicators, through the system interface.

INDUSTRIAL CASE

As specified previously, the industrial application is in the field of plastics manufacturing within Marmillon SAS company. The aim of a PLM system deployment in this company is to optimize the product development process. The company's business is based on responses to request for quotation. However, we note that this request for quotation process is failing. Today, the response time to the customer as well as the acceptance rate of products cost estimation is not in accordance with strategic requirements. For these reasons, we chose to focus our analysis on this business process and knowledge it manages.

Based on business process integration steps (see figure 5 and paragarphe 4.2, identification and modelling steps), we were able to identify the product model and its the status as shown in figure 8.

Figure 8. Example of processes using BPMN

The resulting model (Figure 9) is instantiable in the meta-model solution Audros. parameters such as margins, hourly rates which fluctuate according to the situation rates providers such as the material price, shipping prices which are regularly updated estimations such as purchase price of the mold, number of working hours of the research department. These estimations depend on the part characteristics.

Knowledge to capitalize falls into this third category called "estimations". Indeed, these cost estimations made by the person in charge of this task require significant expertise and knowledge trade. In this expertise, the mold made by a subcontractor of the company is a very important element. The mold also depends on the characteristics of the part to be manufactured as shown in figure 10.

Figure 10. Part and mold properties (Class Diagramm)

To complete the identification step of knowledge integration, identified elements of knowledge must be formalized within the PLM system. These elements are: the cost estimation, the mould, the part, the estimation interface. This last object is going to contain the calculation rules based on mold parameters and and part characteristics. Furthermore, the use case (figure 11) of this estimation allows to identify the skill of different actors (realization of cost estimations, validation estimation).

Figure 11. use case for request for quotation

From the knowledge identification, we must define business rules. We propose a set of predicates for one of identified objects. Here is an example of a rule that can be identified: "If type of part = , if weight = 10g, if cavities number = 3, if type of steel = tempered steel, then purchase price is between X € and Y €".

Indicators will estimate knowledge to know its reliability. In the previous example, we want to know if the mold estimated price for this part is reliable. It will therefore establish a quantitative indicator between the estimated purchase price and the actual purchase price. The information quoted by a quantitative indicator is to be weighed. The work on knowlege indicators is still in progress.

The following diagramm shows the link between the different elements identified within each step of our approach.

CONCLUSION

Through an industrial example, we show the possibilities of a knowledge capitalization approach within a PLM system. The aim is to use capabilities of structuring and traceability of these systems to build and supply elements of knowledge on products development. The proposed approach aims at showing that crutial knowledge can be extract from processes modelling easily useful by people who are not knowledge engineers.

PLM systems presents two advantages for knowledge management:

The first point concerns the extension of meta-models to model entities related to technical knowledge.

The second point concerns the management track that allows the establishment of indicators to evaluate this knowledge. However, it seems difficult to provide funding methods and tools for the generation of knowledge.

The areas for improvement and validation remain steps upstream and downstream of construction business wokflows ie the identification of knowledge and indicators to evaluate them.

Figure 6 .

 6 Figure 6. Knowledge integration

Figure 7 .

 7 Figure 7. Use of the PLM system

Figure 9 .

 9 Figure 9. Extract of product model (Class Diagram)

Figure 12 .

 12 Figure 12. Knowledge integration


Benoit Merit

Valérie Lemarquand
email: guy.lemarquand@univ-lemans.fr

Andrzej Dobrucki

Motor Nonlinearities in Electrodynamic Loudspeakers: Modelling and Measurement

Keywords:

This paper studies the motor nonlinearities of a classical electrodynamic loudspeaker. Measurements show the dependence of the voice-coil inductance with its position and with the driving current as well as the force factor dependence on the coil position. They allow the tuning of the parameters of a model proposed for these dependences. Time and frequency analysis of the model help in the explanation of both the harmonic and intermodulation distortions observed in classical loudspeakers.

Introduction

The nonlinearities of electrodynamic loudspeakers lead to harmonic and intermodulation distortions in the sound production. Their prominent sources of nonlinearity are suspensions and the motor. In this paper, the motor nonlinearities have been studied.

Classical loudspeakers are traditionally represented by the Thiele&Small parameters [START_REF] Thiele | Loudspeakers in Vented Boxes, Parts I and II[END_REF][START_REF] Small | Direct-Radiator Loudspeaker System Analysis[END_REF]. These parameters are used to simply model the functioning of a loudspeaker: the electrical domain is represented by a resistor Re and an inductance Le; the mechanical domain is represented by the mechanical losses Rms, the moving mass Mms and the stiffness of the suspensions Kms; and the acoustical domain is represented by an acoustical impedance that depends on the loudspeaker conditions in use and on its dimensions. When only both mechanical and electrical domains are of interest, a loudspeaker can be represented by the diagram of Fig. 1. This diagram includes the modelling of eddy currents emergence via the resistor Rµ. This modelling is justified by the fact that eddy currents increase with the frequency of the input signal.

The coupling between the two domains is made thanks to the force factor Bl. In the mechanical domain, this parameter defines the motion force F = Bl.i that makes the voice-coil move when a current i runs through its wire. In the electrical domain, the force factor defines the back electromotive force V emf = Bl.v that is created by the movement of the voice-coil with a velocity v = dx/ dt in the magnetic field, B, created by the magnetic circuit of the loudspeaker. When all the parameters are assumed to be linear, this coupling between the two domains leads to a system of two linear differential equations in which the current i L (t), is the current running through the voice-coil:

u(t) = Le(Re + Rµ) Rµ di L (t) dt + Re • i L (t) + Bl • v(t), M ms • d 2 x(t) dt 2 + Rms - Bl 2 Re + Rµ • dx(t) dt + Kms • x(t) = Bl Re + Rµ • (Rµ • i L (t) + u(t)) .
(

In reality, it is known that a loudspeaker is a highly nonlinear system. The acoustic pressure it generates is not an accurate image of the driving voltage. Among the principal sources of nonlinear distortions, one finds the suspensions and the motor of the loudspeaker [START_REF] Klippel | Speaker Auralization -Subjective Evaluation of Nonlinear Distortion, 110th AES Convention[END_REF][START_REF] Dobrucki | Nontypical effects in an Electrodynamic Loudspeaker with a Nonhomogeneous Magnetic Field in the Air Gap and Nonlinear Suspensions[END_REF]. The nonlinearities due to the suspensions are well-known, and loudspeaker manufacturers have developed materials and designs to minimize the drawbacks of the suspensions.

The nonlinearities causes due to a loudspeaker motor have also been identified [START_REF] Klippel | Tutorial: Loudspeaker Nonlinearities -Causes, Parameters, Symptoms[END_REF][START_REF] Lemarquand | Ironless Loudspeaker[END_REF][START_REF] Vanderkooy | A Model of Loudspeaker Driver Impedance Incorporating Eddy Currents in the Pole Structure[END_REF][START_REF] Merit | In Pursuit of Increasingly Linear Loudspeaker Motors[END_REF][START_REF] Remy | Ironless and leakage free voice-coil motor made of bonded magnets[END_REF][START_REF] Ravaud | Magnetic pressure and shape of ferrofluid seals in cylindrical structures[END_REF][START_REF] Ravaud | Radial Stiffness of a ferrofluid seal[END_REF]. In classical loudspeaker motors, the iron is a source of eddy currents, and makes the inductance dependent on the displacement of the voice-coil as well as on the driving current. And since the magnetic induction seen by the voice-coil varies with the axial position in the airgap, the force factor Bl is also dependent on the displacement of the voice-coil.

The effects of all these nonlinearities are well-known: they lead to harmonic and intermodulation distortions. Therefore, the aim of this paper is to highlight the role of the motor alone in the generation of these distortions, thanks to a temporal resolution of the differential equations that govern the loudspeaker functioning.

Measurement of the motor nonlinearities

Evidence for nonlinearity

To help the study, each parameter of a loudspeaker motor is measured independently. Measurements are done on a typical loudspeaker whose moving part displacement can be ±4 mm. First, to point out the nonlinear nature of the loudspeaker, the velocity of the diaphragm when a simple sinusoidal signal drives the loudspeaker is measured thanks to a laser vibrometer (Fig. 2). At 43 Hz, the increase of the input voltage from 0.5 V to 2 V generates high amplitude harmonics.

At 134 Hz, the amplitude of the harmonics does not vary significantly from the 0.5 V to the 2 V input voltage. At this frequency, the displacements of the moving part are small compared with those at 43 Hz. Thus, we can conclude that the nonlinearities of the studied loudspeaker are related to the moving part displacement only.

In fact, the dependences of the stiffness, the inductance and the force factor on the displacement are at issue, but this paper plans to measure and model the contribution of the motor nonlinearities only.

Force factor nonlinearity

The most consequent source of nonlinearities in a loudspeaker motor is the dependence of the force factor Bl on the displacement x of the moving part [START_REF] Klippel | Tutorial: Loudspeaker Nonlinearities -Causes, Parameters, Symptoms[END_REF]. To observe this dependence for the studied loudspeaker, the magnetic induction B in the airgap is first measured thanks to a Hall probe, and the force factor is deducted.

Figure 3 shows the force factor's non-symmetric dependence on the coil displacement. To take into account this dependence in the model, one has to approximate it thanks to a well-controllable mathematical function. The polynomial form is chosen to model the force factor variation. Its coefficient optimization must lead to a good conformity to the original curve and to its derivative. Figure 3 shows that a 4th order polynomial is necessary to reach an ideal conformity:

Bl(x) = Bl • (1 + Bl 1 • x + Bl 2 • x 2 + Bl 3 • x 3 + Bl 4 • x 4 ), (2)
with Bl 1 = -54.01 m -1 , Bl 2 = -31438.2 m -2 , Bl 3 = 1.312 • 10 6 m -3 and Bl 4 = 4.24 • 10 8 m -4 .

Inductance nonlinearity

Another source of nonlinearities is the dependence of the coil inductance on the current intensity, i L , and on the coil position in the air gap, x [START_REF] Ravaud | Time-varying Non Linear Modelling of Electrodynamic Loudspeakers[END_REF][START_REF] Mowry | Simplified Simulation Method for Nonlinear Loudspeaker Parameters[END_REF]. To measure these dependences, a tailor-made equipment allows us to completely block the loudspeaker moving part -so, the voice-coil-in a required axial position x. For each position, for various current intensities from 20 mA to 180 mA, and for various frequencies from 30 Hz to 1000 Hz, the impedance is measured thanks to a Wayne-Kerr bridge. We model it as a resistor Re in series with an inductance Le in parallel with a resistor Rµ, which represents the eddy currents (Fig. 1).

Figure 4 presents the dependence of the inductance on the current intensity, for various positions of the voice-coil. The general trend shows that this inductance is not clearly dependent on the current intensity. Even if this observation agrees with previous observations on the velocity measurements, it would have been interesting to be able to supply the loudspeaker with higher current intensities, the bridge being limited to 200 mA. In addition, we assume that Rµ depends only on the axial position x, as Le, and show its values in Fig. 6. This is consistent with Fig. 5: a variation in the

Le(x, i L ) = Le.L x (x) = Le • (1 + L x1 • x + L x2 • x 2 + L x3 • x 3 + L x4 • x 4 ), (

Harmonic distortions

For the calculations, the loudspeaker input u(t) is chosen to generate the maximal 4 mm displacement to emphasize the related nonlinearities. The model used to calculate these contributions shows that dependence of Le and Rµ on the displacement does not induce clear harmonic distortions. On the other hand, Fig. 7 shows that the non-uniform Bl (x ) leads to consequent harmonic distortions. The temporal response is highly distorted and high-amplitude harmonics are produced in the frequency response. Especially the third harmonic at 90 Hz is only 12 dB lower than the fundamental one.

Intermodulation distortions

In this section, the loudspeaker is virtually excited by a two-tone sinusoidal signal with frequencies 30 Hz and 513 Hz. As in the previous section, influences of displacement-varying Le and Rµ are indistinguishable. Figure 8a shows that their dependence creates intermodulation distortion (thick line). But the level of the first intermodulation components is at least 36 dB lower than the fundamental at 513 Hz. These observations are true for various couples of frequencies. Assuming that the model well represents the loudspeaker, we conclude that the dependence of Le and Rµ on the displacement is not critical in generation of the intermodulation distortions.

Looking at the dotted-line of Fig. 8a, one can see that the dependence of the force factor Bl on the displacement leads to quite consequent intermodulation distortions. The level of the first intermodulation component is 22 dB lower than the fundamental at 513 Hz. This phenomenon is amplified at low frequencies and when the frequencies of the two-tone signal are close to each other (Fig. 8b). Adding the production of harmonic distortion, one can understand that the nonuniform Bl (x ) is a great source of intermodulation distortions.

Discussion and conclusion

This paper highlights the audible distortions created by the motor of an electrodynamic loudspeaker of classical construction. For the studied loudspeaker, the most consequent source of nonlinearities seems to come from the displacementvarying force factor. The weaker, but still significant, effect is caused by the nonlinearity of voice-coil inductance. The nonlinearity of the voice-coil is usually neglected in the consideration of the nonlinear distortion in loudspeakers. The paper shows that this nonlinearity should be taken into account. Particularly, it produces bothersome intermodulations. However, these observations and conclusions are valid for the actual studied loudspeaker. More general conclusions can be obtained from further measurements and modelling of other loudspeakers. These conclusions could help loudspeaker designers in conceiving their loudspeaker motors, with the aim of reducing their nonlinearities, enough to be sure that no sound distortion would be audible. Another way to get rid of the distortions generated by a loudspeaker motor is to use a completely different construction of the motor, which consists in the total removal of the soft-iron magnetic circuit [START_REF] Ravaud | Design of ironless loudspeakers with ferrofluid seals: Analytical study based on the coulombian model[END_REF][START_REF] Ravaud | Modeling an ironless loudspeaker by using three-dimensional analytical approaches[END_REF][START_REF] Merit | Ironless Low Frequency Loudspeaker Working Under its Resonance Frequency[END_REF]. In this construction, eddy currents disappear and the voice-coil inductance is low and constant. A good design of such a motor can also lead to a constant force factor.

Fig. 1 .

 1 Fig. 1. Classical block diagram showing electrical and mechanical-mobility circuits of a loudspeaker and including the modelling of the eddy currents thanks to the resistor Rµ.

Fig. 2 .

 2 Fig. 2. Diaphragm velocity: input voltage 0.5 V (continuous line) and 2 V (dashed line) at a) 43 Hz and at b) 134 Hz. The reference is the measurement for the diaphragm at the rest position.

Fig. 3 .

 3 Fig. 3. Force factor Bl of the studied loudspeaker in the air gap vs. position x.

Fig. 4 .

 4 Fig. 4. Inductance Le of the loudspeaker vs. driving current intensity i, for various positions of the voice-coil from x = -0.004 mm to x = 0.004 mm.

Figure 5

 5 Figure5shows the variation of the inductance with the axial position of the coil, for various current intensities. The grouped curves confirm that the inductance is not dependent on the current intensity. Nevertheless, the inductance

  3) with L x1 = -112.65 m -1 , L x2 = -11095.3 m -2 , L x3 = 1.501 • 10 6 m -3 and L x4 = 2.607 • 10 8 m -4 .

Fig. 5 .

 5 Fig. 5. Measured inductance Le of the loudspeaker vs. position in the airgap x.

Fig. 6 .

 6 Fig. 6. Measured resistor Rµ of the loudspeaker vs. position in the airgap x.

Fig. 7 .

 7 Fig. 7. Temporal pressure responses (a) and sound pressure spectra (b) of the loudspeaker with Bl (x )-nonlinearity (black line), with Le(x )-nonlinearity (red line) and with Rµ(x )-nonlinearity (blue line).

Fig. 8 .

 8 Fig. 8. Loudspeaker sound pressure spectrum in response a) -to a two-tone excitation signal (30 Hz and 513 Hz), with Bl (x )-nonlinearity (dotted line) and with Le(x )-nonlinearity (thick line) and b) -to a two-tone excitation signal (10 Hz and 50 Hz) with Bl (x )-nonlinearity.

Author manuscript, published in "archives of acoustics 34, 4 (2009) 579-590"

Acknowledgment

This article is an extended version of the paper presented at the 56th Open Seminar on Acoustics -OSA2009, September 15-18 in Goniądz.

magnetic flux means an increase in the source of eddy currents, so Rµ is related to the slope of Le. Thus, the variation of Rµ with displacement x, is modelled by (4):

with Rµ 1 = -45.15 m -1 , Rµ 2 = -17081.4 m -2 , Rµ 3 = 536430 m -3 and Rµ 4 = 3.855 • 10 8 m -4 .

3. Modelling of loudspeaker with the motor nonlinearities

Differential equations

When the inductance Le, the force factor Bl and the resistor Rµ are not assumed to be constant anymore but dependent on the displacement, the model of Fig. 1 does not lead to (1) anymore.

Especially, contrary to Bl (x ) and Rµ(x ), the nonlinear model of inductance cannot simply substitute the linear parameter Le in the equation of equilibrium of the voltages in the electrical side of the equivalent circuit. The voltage on the inductance u L (t) is not expressed by the inductance times the derivative of current over time anymore, but it is the derivative of the magnetic flux Φ(t):

Considering that Le(x, i L ) varies independently with x and i L , which are the two time-dependent unknowns of the problem, yields:

Note that the second differential equation of ( 6) takes into account the reluctance force created by the dependence of the inductance on the displacement. It is proportional to the square of the current i L (t) in the voice-coil [START_REF] Cunningham | Non-linear Distortion in Dynamic Loudspeakers due to Magnetic Effects[END_REF]. By switching the nonlinear parameters Le, Bl and Rµ, the resolution of (6) leads to the variables i L (t) and x(t), used to calculate the acoustic pressure generated by the modelled loudspeaker.


Benoit Merit

Guy Lemarquand Laum

IRONLESS LOW FREQUENCY LOUDSPEAKER WORKING UNDER ITS RESONANCE FREQUENCY

Keywords: flat diaphragm, ironless motor, under resonance frequency functioning

A soon achieved low frequency loudspeaker (10 Hz to 100 Hz) is described. To suppress the non-linearities and the drawbacks due to the motor, its structure is totally ironless. The large flat diaphragm and the high force factor of the loudspeaker lead to its high efficiency. Efforts have been made for reducing the non-linearities of the loudspeaker for a more accurate sound reproduction. Especially we have developed a motor totally made of permanent magnets, which create a uniform induction across the entire intended displacement of the coil. The motor linearity and the high force factor of the loudspeaker make it possible to work under its resonance frequency. This functioning needs the use of a DSP to raise the 12 dB/octave slope of the pressure response of the loudspeaker under its resonance frequency and to protect the moving pieces from large displacements. Results presented here are to be completed as soon as the loudspeaker will be achieved.

Introduction: functioning of the loudspeaker

Nowadays most woofers manufacturers are confronted with the problem of decreasing their mechanical resonance frequency, to extend the bandwidth of the woofer. Since a suspension whose stiffness tends to be null does not exist, the widening of this bandwidth implies an increase in the mobile mass, at the expense of the efficiency of the woofer. It is also met loudspeakers whose bandwidth comprises this resonance, despising phase problems among other things.

For these reasons it has been decided to make up a loudspeaker (Fig. 1) working under its resonance frequency. In this bandwidth, current, velocity of the diaphragm and radiated pressure are in phase. Moreover, this way of functioning means a coherent work on making the mobile mass be light, to increase the bandwidth of the woofer at the same time as to increase its efficiency.

Yet, let's remind that, in a simple way, the pressure radiated by a loudspeaker can be assimilated to a high-pass filter for the radiated pressure (Fig. 2). This means that an embedded electronics (DSP) is necessary to raise the 12 dB/octave slope of the pressure response of the loudspeaker under its resonance frequency f 0 . In order to protect each piece of the loudspeaker, the DSP is also used to restrain the displacements of the mobile mass at very low frequencies. To conserve all advantages of working under the resonance frequency of the woofer, it is important to make it efficient as well as the most possible linear. Efforts have been made to ideally conceive the motor and the diaphragm of the woofer.

Ironless and linear motor

The drawbacks due to the iron in classical loudspeakers have already been described [START_REF] Lemarquand | Ironless loudspeakers[END_REF][START_REF] Vanderkooy | A model of loudspeaker driver impedance incorporating eddy currents in the pole structure[END_REF]. Iron is the source of reluctant effect and Eddy currents, and is the origin of a magnetic field that is a function of the axial position in the airgap. Accordingly, the iron is a major source of non-linearities and as a result of harmonic distortion. That is why, in addition to considering or reducing these effects, it is important to try to remove them.

The first step of this research is the use of a motor entirely made of magnets, i.e. the removal of iron. The benefits of these whole magnet structures have been demonstrated [START_REF] Lemarquand | Ironless loudspeakers[END_REF] and some manufacturers have already used this technology [START_REF] House | Transducer motor assembly[END_REF][START_REF] Ohashi | Magnetic circuit and speaker[END_REF]. But all presented structures are aimed to create an intense magnetic induction; this induction still varies with the axial position in the airgap.

Searching for a uniform induction, analytical calculations of the magnetic field created by permanent magnets have been made using the Coulombian model [1-3, 8, 9]. Associated to an algorithm of rotation, this model can be applied to every permanent magnet, whatever is their shape.

The structure we find uses two sets of three superimposed rings whose radial magnetization differs in order to give the intended shape to the observed induction. The space between the two sets defines the airgap where the coil moves. Figure 3 shows an example of the contribution of each ring and the resulting induction. The dimensions of the magnets are arbitrary chosen to show clearly the contributions of the three rings and their effect on the result. When ideal set of dimensions and magnetizations are found, this assembly of three rings creates a uniform induction across the entire height of the central ring. The structure we use for our first speaker is not optimized in terms of uniformity because we have to use a low volume of magnet. Nevertheless we reach a quite uniform induction whose intensity is 0.58 T across a 40 mm height, while the total height of the magnetic structure is 52 mm (Fig. 4). Today the force factor Bl reaches 35 N•m, but it could be optimized to reach 45 N•m and more.

The structure allows large and linear displacements, while being quite flattened. The reduced height is an advantage for the use of a cylindrical support of coil whose height is also reduced. Eigen modes are then moved to high frequencies.

Circular flat diaphragm

The march in material research gives rise to increasingly rigid and light materials. This progress makes possible the use of a flat diaphragm instead of using a classical cone-shaped diaphragm for the loudspeaker. A circular flat diaphragm is easier to manufacture than a conical one. It is also in agreement with the radiation model of a flat piston, and analytical calculations of its radiation and its eigenmodes can be lead. And finally, it allows a more faithful reproduction of sound through the elimination of interference phenomena which are subject conical shapes [START_REF] Quaegebeur | Influence of material and shape on sound reproduction by an electrodynamic loudspeaker[END_REF].

Considering the support of coil

The diaphragm of the first prototype speaker has a diameter a = 20 ; it is 3 mm thick and is made up of a composite honeycomb sandwich whose flexural modulus is higher than 2 GPa and whose density is less than 400 g/m 2 . The first modes of this diaphragm alone appear at frequencies lower than 100 Hz, i.e. in the intended bandwidth of the woofer.

Because the diaphragm is flat, we can stiffen it using a well sized cylindrical support of coil. To study its influence, the support is assumed to be rigid. Figure 5 shows that the frequency of the first appearing eigenmode is maximal when the ratio b/a equals 0.7. That is why the diameter of our support of coil equals 14 . With this support, the first eigenfrequency of the diaphragm with its previous characteristics reaches about 80 Hz.

Figure 6 gives the shape of the first three eigenmodes of the diaphragm when fixed to the support of the coil. Here the modal deformation of the first mode corresponds to the modal deformation of the second mode of the diaphragm without support. To stiffen the diaphragm, a diametrical reinforcement could be used and would make the first eigenfrequency increase.

Considering the suspension

The surround suspension brings also an additional stiffness to the diaphragm. Given the material used for this suspension, eigenfrequencies of the diaphragm should then become higher, especially if a suspension that does not allow rotation is used. We have not estimated yet a vibration model including a suspension. But when the contact between the diaphragm and the suspension is considered as a clumping, the first eigenfrequency reaches 220 Hz. So in reality, considering the previous results, the first eigenfrequency of the diaphragm can be expected to be higher than 100 Hz.

Conclusion

First calculations and measurements are quite promising for the conception of an efficient and accurate woofer. Its coming assembly will lead to new measurements that will give us the definitive values of the loudspeaker parameters.

Fig. 1 .

 1 Fig. 1. Composition of the ironless flat loudspeaker.

Fig. 2 .

 2 Fig. 2. Behaviour of the pressure response of a loudspeaker vs. frequency.

Fig. 3 .

 3 Fig. 3. Radial magnetic induction, Br (T), created by the three rings structure (bold line), by its central ring (full line) and by its two external rings (dashed line). The vertical dashed lines reflect the heights of the three rings.

Fig. 4 .

 4 Fig. 4. Magnetic structure used in our first woofer and its created radial magnetic induction, Br (T), along the observation axis.

Fig. 5 .

 5 Fig. 5. Evolution of the eigenfrequencies of an arbitrary circular plate (diameter a), when fixed to a circular support whose diameter b varies.

Fig. 6 .

 6 Fig. 6. First three modal deformations of the diaphragm in free-supported conditions.