D Marinucci

G Peccati

Mean-Square Continuity on Homogeneous Spaces of Compact Groups *

Keywords: Random Processes, Isotropy, Mean-Square Continuity., AMS Classification: 60G05, 60G60

We show that any finite-variance, isotropic random field on a compact group is necessarily meansquare continuous, under standard measurability assumptions. The result extends to isotropic random fields defined on homogeneous spaces where the group acts continuously.

Introduction

The analysis of the spectral representations of stationary and isotropic finite-variance random fields on (subsets of) R d is a classical topic of probability theory, presented in many standard textbooks in the area (see for instance [START_REF] Adler | The Geometry of Random Fields[END_REF][START_REF] Adler | Random Fields and Geometry[END_REF][START_REF] Leonenko | Limit Theorems for Random Fields with Singular Spectrum[END_REF][START_REF] Yadrenko | Spectral Theory of Random Fields[END_REF]). Such representations are sometimes based on Karhunen-Loève constructions (specially under Gaussianity assumptions), realized by first computing the eigenfunctions associated with the covariance kernel, and then by expanding the field into these orthogonal components (see for instance [START_REF] Adler | Random Fields and Geometry[END_REF]). In other cases, the argument proceeds from the construction of an isometry between an L 2 space of deterministic square integrable functions, and some space of finite-variance random variables, with inner product defined in terms of the covariance function of the process to be represented (see for instance [START_REF] Leonenko | Limit Theorems for Random Fields with Singular Spectrum[END_REF][START_REF] Yadrenko | Spectral Theory of Random Fields[END_REF]). In all these approaches, mean-square continuity is assumed as a necessary condition to ensure that the spectral representation holds pointwise.

More recently, considerable attention has been drawn to the case where the process at hand is defined on the homogenous space of a compact group (including the group itself). In this context, one of the most relevant examples for applications is the sphere S 2 , which is well-known to be isomorphic to the quotient space SO(3)/SO [START_REF] Adler | Random Fields and Geometry[END_REF], where SO(d) denotes as usual the special group of rotations in R d . Under these circumstances, spectral representation results take a particularly neat form, as they can be viewed as stochastic versions of the celebrated Peter-Weyl Theorem (see [START_REF] Duistermaat | Lie groups[END_REF]Section 4.6] or [14, Section 2.5]); the latter ensures that the matrix coefficients of the irreducible representations of a compact group G provide an orthonormal basis for the space L 2 (G) of square-integrable functions on the group, endowed with the Haar measure. This is the standpoint adopted for instance by [START_REF] Marinucci | Random Fields on the Sphere: Representations, Limit Theorems and Cosmological Applications[END_REF][START_REF] Peccati | Decompositions of stochastic processes based on irreducible group representations[END_REF] -see also [START_REF] Baldi | On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups[END_REF]. It should be noted that random fields on the unit sphere S 2 play now an extremely important role in many applied fields, for instance in Cosmology -see [START_REF] Dodelson | Modern Cosmology[END_REF][START_REF] Durrer | The Cosmic Microwave Background[END_REF], [START_REF] Marinucci | Random Fields on the Sphere: Representations, Limit Theorems and Cosmological Applications[END_REF] for an overview.

As we shall point out in the sections to follow, the argument based on the group-theoretic point of view does not only provide an alternative proof for classical results, but yields also an unexpected bonus: the assumption of mean-square continuity turns out to be no longer necessary for the spectral representation to hold. More than that, mean-square continuity follows as a necessary consequence of the spectral representation, under standard measurability conditions. The aim of this short note is then to highlight this result, which we can state concisely as follows:

Let T be a measurable finite-variance isotropic random field defined on the homogeneous space of a compact group acting continuously. Then, T is necessarily mean-square continuous.

Our main findings are contained in the statements of Theorem 2 (for fields defined on compact groups) and of Theorem 4 (for fields defined on homogeneous spaces). Section 2 and Section 3 contain preliminary results, respectively on group representations and random fields. Some historical remarks are provided in Section 6.

Preliminaries on group representations

We now provide a brief overview of the results about group representations that are used in this note. For any unexplained definition or result, the reader is referred to [START_REF] Marinucci | Random Fields on the Sphere: Representations, Limit Theorems and Cosmological Applications[END_REF]Chapter 2], as well as to the classic reference [START_REF] Duistermaat | Lie groups[END_REF].

A topological group is a pair (G, G), where G is a group and G is a topology such that the following three conditions are satisfied: (i) G is a Hausdorff topological space, (ii) the multiplication

G × G → G : (g, h) → gh is continuous, (iii) the inversion G → G : g → g -1 is continuous.
In what follows, we use the symbol G to denote a topological group (the topology G being implicitly defined) which is also compact and such that G has a countable basis. We will denote by C (G) the class of continuous, complex-valued functions on G; G is the (Borel) σ-field generated by G; dg denotes the normalized Haar measure of G (in particular, G dg = 1). We shall denote by L 2 (G, G, dg) = L 2 (G) the Hilbert space of complex-valued functions on G that are square-integrable with respect to dg. Plainly, the space L 2 (G) is endowed with the usual inner product

f 1 , f 2 G = G f 1 (g) f 2 (g)dg; also, • G = •, • 1/2 G .
Let X be a topological space. A continuous left action of G on X is a jointly measurable mapping A : G × X : (g, x) → A(g, x) := g • x, satisfying the following properties: (i) g • (h • x) = (gh) • x, (ii) if e is the identity of G, then e • x = x for every x ∈ X, and (iii) the mapping (g, x) → g • x is jointly continuous. Right actions are defined analogously and will not be directly considered in this note, albeit every result concerning left actions proved below extends trivially to right actions. The space X is called a G-homogeneous space if G acts transitively on X, that is: for every y, x ∈ X, there exists g ∈ G such that y = g • x. Group representations (as described in the next paragraph) are distinguished examples of group representations.

Let V be a normed finite-dimensional vector space over [START_REF] Marinucci | Random Fields on the Sphere: Representations, Limit Theorems and Cosmological Applications[END_REF]Proposition 2.25], one sees that it is always possible to endow V with an inner product •, • V such that π is unitary with respect to it, that is: for every g ∈ G and every u

C. A (finite-dimensional) representation of G in V is an homomorphism π, from G into GL (V ) (the set of complex isomorphisms of V into itself), such that the mapping G × V → V : (g, v) → π (g) (v) is continuous. Using e.g.
, v ∈ V , π(g)u, π(g)v V = u, v V . Note that •, • V can be chosen in such a way that the associated norm preserves the topology of V (see [8, Corollary 4.2.2]). The dimension d π of a representation π is defined to be the dimension of V . A representation π of G in V is irreducible, if the only closed π (G)-invariant subspaces of V are {0} and V .
It is well-known that unitary irreducible representations are defined up to equivalence classes (see [14, p. 25]). We will denote by [π] the equivalence class of a given unitary irreducible representation π; the set of equivalence classes of unitary irreducible representations of G is written G, and it is called the dual of G. We recall that, according e.g. to [START_REF] Duistermaat | Lie groups[END_REF]Theorem 4.3.4 (v)], since G is second countable (and therefore metrizable) G is necessarily countable.

To every [π] ∈ G we associate a finite-dimensional subspace M π ⊆ L 2 (G) in the following way. Select an element π : G → GL (V ) in [π], as well as an orthonormal basis e = {e 1 , ..., e dπ } of V . The space M π is defined as the finite-dimensional complex vector space spanned by the functions g → π i,j (g) := π(g)e j , e i V , i, j = 1, ..., d π .

Note that such a definition is well given, since M π does not depend on the choices of the representative element of [π] and of the orthonormal basis of V . The following three facts are relevant for the subsequent discussion:

(i) { √ d π π ij : i, j = 1, ..., d π } is an orthonormal system of L 2 (G) (see [14, p. 34]); (ii) dim M π = d 2 π , and (iii) M π ⊆ C (G), for every [π] ∈ G.
To conclude, we recall that, if two representations π and π ′ are not equivalent, then M π and M π ′ are orthogonal in L 2 (G). One crucial element of our discussion is the celebrated Peter-Weyl Theorem (see [14, Section 2.5]), stating that

L 2 (G) = [π]∈ G M π , (2.1)
that is: the family of finite-dimensional spaces

{M π : [π] ∈ G} constitutes an orthogonal decomposition of L 2 (G). In particular, the class { √ d π π ij : [π] ∈ G, i, j = 1, ..., d π } is an orthonormal basis of L 2 (G).
Plainly, the orthogonal projection of a given function f ∈ L 2 (G) on the space M π is given by the mapping

g → f π (g) = dπ i,j=1 d π G f (h)π i,j (h)dh × π i,j (g) = d π G f (h)χ π (h -1 g)dh, (2.2)
where χ π (g) = Trace π(g) is the character of π. See [14, Section 2.4.5] for a discussion of the basic properties of group characters; in particular, one has that two equivalent representations have the same character, in such a way that the projection formula (2.2) is well-defined, in the sense that it does not depend on the choice of the representative element of the equivalence class [π].

From now on, we shall fix a topological compact group (G, G), and freely use the notation and terminology introduced above.

General setting and spectral decompositions

Let T = {T (g) : g ∈ G} be a finite variance, isotropic random field on G, by which we mean that T is a real-valued random mapping on G verifying the following properties (a)-(c).

(a) (Joint measurability) The field T is defined on a probability space (Ω, F , P ), and the mapping

T : G × Ω → R : (g, ω) → T (g, ω) is G ⊗ F -measurable
, where (as before) G denotes the Borel σ-field associated with (G, G).

(b) (Isotropy) The distribution of T is invariant in law with respect to the action of G on itself. This means that, for every h ∈ G, T (hg) d = T (g), where " d =" indicates equality in distribution in the sense of stochastic processes, that is: for every d ≥ 1 and every g 1 , ..., g d ∈ G, the two vectors (T (g 1 ), ..., T (g d )) and (T (hg 1 ), ..., T (hg d )) have the same distribution. Note that, since the application g → hg is continuous, then the mapping (g, ω) → T (hg, ω) is jointly measurable, in the sense of Point (a) above.

(c) (Finite variance) The field T has finite variance, i.e.: ET 2 (g) = T 2 (g, ω)dP (ω) < ∞, for every g ∈ G. For notational simplicity, and without loss of generality, we will assume in the sequel that ET (g) = 0.

Under the previous assumptions and by virtue of the invariance properties of Haar measures, one has that, for every fixed g 0 ∈ G, E[T 2 (g 0 )] = E G T 2 (g)dg < ∞. This implies that there exists a F -measurable set Ω ′ of P -probability 1 such that, for every ω ∈ Ω ′ , the mapping T (•, ω) : g → T (g, ω) is an element of L 2 (G). For every [π] ∈ G, we now define the quantity T π (g, ω) according to (2.2), whenever ω ∈ Ω ′ , and we set T π (g, ω) = 0 otherwise. It is easily checked that, for every [π] ∈ G, the mapping (g, ω) → T π (g, ω) is G ⊗ F measurable.

According to the results discussed in [14, Section 5.2.1], the following two facts take place.

-The class {T, T π : [π] ∈ G} is an isotropic (possibly infinite-dimensional) square-integrable centered field over G, that is: every T π is centered and square-integrable, and for every m, d ≥ 1, for every

lim n→∞ E    T (g 0 )- n k=1 T π k (g 0 ) 2    = lim n→∞ E    G T (g) - n k=1 T π k (g) 2 dg    = 0, (3.3)
in other words: the sequence n k=1 T π k , n ≥ 1, approximates T in the L 2 (P ) sense, both for every fixed element of G, and in the sense of the space L 2 (G). Note that the equality in formula (3.3) is a consequence of the invariance and finiteness properties of the Haar measure dg and of the isotropy of {T,

T π : [π] ∈ G}, yielding E    T (g 0 )- n k=1 T π k (g 0 ) 2    = E    G T (gg 0 )- n k=1 T π k (gg 0 ) 2 dg    = E    G T (g)- n k=1 T π k (g) 2 dg    .
The proof of the isotropy of {T π : [π] ∈ G} is given in [START_REF] Marinucci | Random Fields on the Sphere: Representations, Limit Theorems and Cosmological Applications[END_REF], see Proposition 5.3 on pages 116-117. It should be noted that the proof of this Proposition implicitly exploits the fact that, under isotropy, for every h ∈ G the scalar products of T (•) and T h (•) := T (h•) with any continuous function have necessarily the same distribution (we thank P. Baldi for raising this point). This result is trivial under mean-square continuity, but in the present general circumstances it is a bit less obvious, and hence we report here a proof for completeness.

Lemma 1 Let T be an a.s. square-integrable invariant random field on G and define, for f ∈ L 2 (X ),

T (f ) := G T (x)f (x) dx (3.4)
Then, for every h ∈ G and every f 1 , . . . , f m ∈ L 2 (G), the two random variables

(T (f 1 ), . . . , T (f m )) and (T h (f 1 ), . . . , T h (f m ))
have the same distribution.

Proof. For the sake of simplicity, we shall only deal with the case m = 1; the general case follows along similar lines. For every n > 0 let

T n = T ∧ n ∨ (-n), which is a bounded random field, itself invariant. Now, if f ∈ L 2 (G), by dominated convergence, T n (f ) a.s. → n→∞ T (f ) and T g n (f ) a.s. → n→∞ T g (f ) ,
and therefore the convergence takes place also in distribution. Hence it is sufficient to prove the statement under the additional assumption that T is bounded. But in this case the two r.v.'s T (f ), T h (f ) are themselves bounded and in order to prove that they are equi-distributed it is sufficient to show that they have the same moments, i.e. that, for every choice of integers p, q ≥ 0,

E[(ReT (f )) p (ImT (f )) q ] = E[(ReT h (f )) p (ImT h (f )) q ] . Now E[(ReT (f )) p (ImT (f )) q ] = E G Re(T (x 1 )f (x 1 ))dx 1 ... G Re(T (x p )f (x p ))
G Im(T (y 1 )f (y 1 ))dy 1 ...

G

Re(T (y q )f (y q ))dy q .

Developing the real and imaginary parts and applying Fubini's theorem, the previous expectation reduces to a sum of terms of the form

G . . . G E [b 1 (x 1 )
. . . b p (x p )c 1 (y 1 ) . . . c q (y q )] d 1 (x 1 ) . . . d p (x p )e 1 (y 1 ) . . . e q (y q ) dx 1 . . . dx m dy 1 . . . dy k where b i (x i ) (resp. c j (y j )) can be equal to ReT (x i ) or ImT (x i ) (resp. ReT (y j ) or ImT (y j )) and d i (x i ) (resp. e j (y j )) can be equal to ±Ref (x i ) or ±Imf (x i ) (resp ±Ref (y j ) or ±Imf (y j )). Now just remark that the quantity E [b 1 (x 1 ) . . . b p (x p )c 1 (y 1 ) . . . c q (y q )] does not change if the random field T is replaced by its rotated version T h , as such an expectation only depends on the joint distribution; this concludes the proof.

4 Mean-square continuity on G

In the next statement we show that isotropic fields such as those of the previous section are necessarily mean-square continuous.

Theorem 2 Let T be a square-integrable centered isotropic field on the topological compact group G, verifying properties (a)-(c) of Section 3. Then, T is mean-square continuous: for every g ∈ G

lim h→g E |T (g) -T (h)| 2 = 0. (4.5)
Remark 3 Since (G, G) is a topological space, equation (4.5) has to be interpreted in the following sense: for every net {h i } ⊂ G converging to g, one has that E |T (g) -T (h i )| 2 → 0. See [7, p. 28ff] for details on these notions.

Proof of Theorem 2. Let {[π k ] : k ≥ 1} be an arbitrary enumeration of G. In view of (3.3), for every ǫ > 0, there exists n ≥ 1 such that

sup g∈G E    T (g)- n k=1 T π k (g) 2    ≤ ǫ 6 ,
in such a way that, for every h, g ∈ G,

E |T (g) -T (h)| 2 ≤ 3    E   T (g)- n k=1 T π k (g) 2   + E   T (h)- n k=1 T π k (h) 2      +3E   n k=1 (T π k (g) -T π k (h)) 2   ≤ ǫ + 3E   n k=1 (T π k (g) -T π k (h)) 2   = ǫ + 3 n k=1 E |T π k (g) -T π k (h)| 2 ,
where in the last relation we have used the fact that, for k = k ′ , the two fields T π k and T π k ′ are uncorrelated (see [START_REF] Marinucci | Random Fields on the Sphere: Representations, Limit Theorems and Cosmological Applications[END_REF]Proposition 5.4]). Now, for every [π] ∈ G, we define T π i,j := G T (h)π i,j (h)dh (it is easily seen that T π i,j is a square-integrable random variable). Using (2.2), we therefore deduce that, for every k ≥ 1,

E |T π k (g) -T π k (h)| 2 ≤ d 2 π k dπ k i,j=1 E[| T π k i,j | 2 ] × |π i,j (g) -π i,j (h)| 2 .
Since π i,j ∈ C(G), this last relation together with the previous estimates implies that

lim sup h→g E |T (g) -T (h)| 2 ≤ ǫ,
and the conclusion follows from the fact that ǫ is arbitrary.

5 Mean-square continuity on homogeneous spaces

We now fix a topological compact group G, and consider a topological space X which is also a Ghomogeneous space, where G acts transitively and continuously from the left. Let T = {T (x) : x ∈ X} be a centered, finite-variance isotropic random field on X. This means that the following three properties are verified: (i) the field T is defined on a probability space (Ω, F , P ), and the mapping T :

X × Ω → R : (x, ω) → T (x, ω) is X ⊗ F -measurable, with X denoting the Borel σ-field associated with X; (ii) for every h ∈ G, T (hx)
where we have exploited the trivial fact that g -1 0 is an element of Iso(x n ), because g 0 is. Hence, (A) is fulfilled. Moreover, since g ′ n → G g 0 , by continuity one infers that h n → G g -1 0 , and consequently h n g ′ n → G e, yielding (B). It follows that the sequence {x ′ n } satisfies the requirements in the statement, and the proof is concluded.

Proof of Theorem 4 Fix x ∈ X and let x n → X x. Using the assumptions on T , one infers that the mapping g → T (g • x) := T x (g) is a centered finite-variance isotropic field on G, in the sense of Section 3. According to Lemma 5, there exist sequences {x ′ n } ⊂ {x n } and {g n } ⊂ G such that x ′ n = g n x and g n → G e. By virtue of Theorem 2, we therefore conclude that

lim n→∞ E |T (x ′ n ) -T (x)| 2 = lim n→∞ E |T x (g n ) -T x (e)| 2 = 0.
This argument shows that every sequence {x n } converging to x in X admits a subsequence {x ′ n } such that T (x ′ n ) converges to T (x) in L 2 (P ), and this fact is exactly equivalent to relation (5.6).

As already recalled, our findings apply to the important case where X equals the n dimensional unit sphere S n , n ≥ 1, on which the compact group SO(n + 1) acts transitively. Assume for notational simplicity that T is zero-mean (ET = 0), and write Γ(x 1 , x 2 ) := ET (x 1 )T (x 2 ) for the covariance function of the random field, Γ : S n × S n → R . By isotropy, there exists a function Γ(.) :

R + → R such that Γ(x 1 , x 2 ) = Γ( x 1 -x 2 )
, where the symbol • stands for Euclidean norm; it is hence straightforward (and well-known, see for instance [START_REF] Adler | Random Fields and Geometry[END_REF][START_REF] Leonenko | Limit Theorems for Random Fields with Singular Spectrum[END_REF][START_REF] Yadrenko | Spectral Theory of Random Fields[END_REF]) that, under isotropy, mean-square continuity is equivalent to continuity of the function Γ at the origin.

Of course, it is not difficult to figure out rotationally invariant, positive-definite functions which violate the continuity of Γ at the origin: a simple example is provided by Γ :

S n × S n → R such that Γ(x, x) = Γ(0) = 1 , Γ(x 1 , x 2 ) = Γ( x 1 -x 2 ) = 0 , for all x 1 = x 2 , x 1 , x 2 ∈ S n .
(5.7)

A consequence of Theorem 4 is that such a Γ(., .) cannot be the covariance function of a measurable isotropic process. More generally, under isotropy it is immediate to see that continuity of Γ at the origin entails continuity everywhere of the covariance function Γ, indeed

Γ(x 1 , y 1 ) -Γ(x 2 , y 2 ) = E {T (x 1 )(T (y 1 ) -T (y 2 ))} + E {T (y 2 )(T (x 1 ) -T (x 2 ))} ≤ ET 2 (x 1 )E(T (y 1 ) -T (y 2 )) 2 + ET 2 (y 2 )E(T (x 1 ) -T (x 2 )) 2 ,
and the last term of the previous chain of inequalities converges to zero, whenever (x 1 , y 1 ) → (x 2 , y 2 ) and the isotropic field T is mean-square continuous (or, equivalently, Γ is continuous at the origin). We can hence state the following:

Corollary 6 The covariance function of a measurable finite-variance isotropic random field on the homogenous space of a compact group is necessarily everywhere continuous.

Some Historical Remarks

Our result can be viewed as a characterization of covariance functions for random fields defined on homogenous spaces of compact groups. In a related setting, the characterization of covariance functions for stationary and isotropic random fields in R d was first considered in a celebrated paper by [START_REF] Schoenberg | Metric spaces and positive definite functions[END_REF], see [START_REF] Schoenberg | Metric spaces and positive definite functions[END_REF]. In this reference, it was conjectured that the only form of discontinuity which could be allowed for such covariance functions would occur at the origin, i.e. given any zero-mean, finite-variance and isotropic random field Z : R d → R, its covariance function should be of the form EZ(0, ..., 0)Z(t 1 , ..., t d ) = Γ( t 1 , ..., t d ) = Γ(t) = Γ 0 (t) + Γ 1 (t) , where as before t := t 1 , ..., t d is Euclidean norm, and Γ 0 (.), Γ 1 (.) : R + → R are such that Γ 0 (t) = γ ≥ 0 , for t = 0 , 0 otherwise , while Γ 1 (t) is nonnegative definite and continuous. In a later paper which went largely unnoticed, Crum (1956) ( [START_REF] Crum | On positive-definite functions[END_REF]) proved the conjecture to be right for d ≥ 2. This result was drawn to the attention of the Geostatistics community by Gneiting and Sasvari in 1997 (see [START_REF] Gneiting | The characterization problem for isotropic covariance functions[END_REF]) who argued then that isotropic random fields could be always expressed as a mean-square continuous component plus a "nugget effect", e.g. a purely discontinuous component. The fact that this latter component should be necessarily nonmeasurable was pointed out (for instance) in an oral presentation by [START_REF] Starkloff | Notes on Stationary Processes and Fields, Oral Presentation[END_REF] -see [18, p. 13], as well as Example 1.2.5 in [START_REF] Kallianpur | Stochastic filtering theory[END_REF]. Our results in this note, though, are obtained in a somewhat different setting (e.g. homogeneous spaces of compact groups), and we leave for future research a complete analysis of the relationship between measurability and mean-square continuity in non-compact circumstances.

[π 1

 1 ], ..., [π m ] ∈ G and every h, g 1 , ..., g d ∈ G, the (m + 1)d-dimensional vector T (g 1 ), ..., T (g d ); T πi (g 1 ), ..., T πi (g d ), i = 1, ..., m has the same distribution as T (hg 1 ), ..., T (hg d ); T πi (hg 1 ), ..., T πi (hg d ), i = 1, ..., m -Let {[π k ] : k = 1, 2, ...} be any enumeration of G. Then, for every fixed g 0 ∈ G one has that

-J.Starkloff for useful discussions. Research by DM is supported by the ERC Grant 277742 Pascal.

d

= T (x), where " d =" indicates as before equality in distribution in the sense of stochastic processes; (iii) ET 2 (x) = T 2 (x, ω)dP (ω) < ∞ and ET (x) = 0, for every x ∈ X. Plainly isotropic fields as those introduced in Section 3 are special cases of the above class, obtained by taking X = G. To simplify the discussion, in what follows we implicitly assume that both X and G are metric spaces. The following result shows that the content of Theorem 2 extends to random fields defined on X.

Theorem 4 Let T be a square-integrable centered isotropic field on the G-homogeneous space X, verifying properties (i)-(iii) above. Then, T is mean-square continuous: for every x ∈ X,

The proof of Theorem 4 is based on the following lemma.

Lemma 5 Let {x n } ⊂ X be a sequence converging to x 0 ∈ X in the topology of the homogeneous space (written x n → X x 0 ). Then, there exists a subsequence {x ′ n } ⊂ {x n } verifying the following property: there exists a sequence {g n } ⊂ G such that g n • x 0 = x ′ n for every n, and g n → G e, where e denotes the identity element of G.

Proof. By transitivity, there exists a sequence {g ′′ n } such that g ′′ n • x 0 ≡ x n for every n. Moreover, because the group is compact, this sequence admits a subsequence {g ′ n } ⊂ {g ′′ n } such that g ′ n → G g 0 . Note that g 0 need not be the identity (otherwise the proof would be finished), but it does need to belong to the isotropy group of x 0 , written g 0 ∈ Iso(x 0 ), meaning that g 0 • x 0 = x 0 . Write x ′ n = g ′ n • x 0 . We claim that there exists a sequence {h n } ⊂ G such that the sequence g n := h n g ′ n , n ≥ 1, satisfies the following two properties: (A) :

(B) :

. Such a sequence is given by h n := g ′ n g -1 0 (g ′ n ) -1 . Indeed, it is immediate to see that


Julien Fatome
email: jfatome@u-bourgogne.fr

Stéphane Pitois

Philippe Morin

Elie Assémat

Dominique Sugny

Antonio Picozzi

Hans-Rudolf R Jauslin

Guy Millot

Victor V Kozlov

Stefan Wabnitz

A universal optical all-fiber omnipolarizer

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The state of polarization is one of the three characteristics of an electromagnetic wave, the others being its energy (or photon number) and frequency. Although the management and measurement of the wave energy and frequency have progressed throughout the laser era up to unprecedented levels of precision, the state of polarization of light still remains largely elusive to control. Indeed, despite the significant progress in optical fiber manufacturing, because of residual birefringence or strain the polarization of light remains unpredictable after propagating a few hundred meters in a fiber. Moreover, in spite of the recent tremendous technological developments in waveguide and fiber-based polarization controllers [START_REF] Martinelli | Polarization stabilization in optical communications systems[END_REF][START_REF] Koch | Record 59-krad/s Polarization Tracking in 112-Gb/s 640-km PDM-RZ-DQPSK Transmission[END_REF], the basic principle of operation of these devices rests upon a combative strategy, consisting in linear polarization transformations followed by partial diagnostic associated with an active loop control driven by complex algorithms [START_REF] Martinelli | Polarization stabilization in optical communications systems[END_REF][START_REF] Koch | Record 59-krad/s Polarization Tracking in 112-Gb/s 640-km PDM-RZ-DQPSK Transmission[END_REF]. Here, we propose a radically different approach which is based on preventing polarization fluctuations in optical fibers, rather that post-compensating them.

Consider for example the action of a linear polarizer and a polarization beam-splitter (PBS) on a sequence of three perfectly polarized pulses, as shown in Fig. 1a,c. The polarizer only passes one component of polarization (SOP) and rejects the other ones. As a result of this rejection principle, in the example of Fig. 1a only two pulses out of three pass through the polarizer (albeit with largely different peak powers), while the last pulse with orthogonal polarization is blocked and removed from the outcoming beam. Hence, due to the rejection principle, all input polarization fluctuations are transformed into output intensity variations, introducing a large Relative-Intensity-Noise (RIN); even if the input beam had a steady intensity in time. Indeed, this property is intrinsic to linear polarizers or any device exhibiting strong polarizationdependent losses. It may seem that a linear PBS is free of RIN because the device is lossless, i.e. the total energy of the transmitted beam is conserved. This is not trueclearly RIN is still present in each individual output channel, as illustrated by Fig. 1c. The principle of operation of a linear PBS (which we call here the principle of continuous splitting), rules that the last pulse must be split in nearly equal portions between the two output channels. Therefore, the peak intensities of pulses in each output stream are modulated in time, even though the pulses had equal intensities in the stream incident onto the PBS.

Here we propose, experimentally validate and theoretically describe the so far unexpected capability of light to self-organize its SOP in optical fibers, with no need for additional control elements. We baptize this new device as the Omnipolarizer. The principle of operation of our Omnipolarizer is schematically illustrated in Fig. 2. Basically, the Omnipolarizer is simply composed by a single span of nonlinear telecom fiber, where a signal beam nonlinearly interacts through a four-wave-mixing process with its own counter-propagating replica produced by means of a back-reflection at the fiber output, see Fig. 2a. Depending on the mirror reflection coefficient (which can be even greater than unity when the back-propagating signal is amplified into a reflective-loop, see Fig. 2b), the Omnipolarizer may operate in (and can be switched among) two distinct RIN-free regimes. Consider first the PBS mode, which is obtained with a below-unity back-reflection coefficient. For any arbitrary polarized input signal, two fixed output SOPs are produced, corresponding to right or left circular states (i.e., the two poles of the Poincaré sphere). These SOPs are universal as they do not depend on the telecom fiber sample or environmental conditions. The sign of the initial signal ellipticity determines which of the two poles is obtained at the device output. Next consider the polarizer mode which is obtained when the back-reflection coefficient is larger than unity (i.e., the back-reflected light is amplified). In this case a single circular output SOP is generated, irrespective of the initial polarization state of the incoming signal. As we shall see, the major advantage of the Omnipolarizer is not just that two polarization control functions may be combined into a single optical device, but rather in the peculiar mode of operation of both the PBS and the nonlinear polarizer.

Let us first characterize the nonlinear polarizer function of the Omnipolarizer and contrast its operation principle with the rejection principle of a linear polarizer, cf. Fig. 1a and Fig. 1b. In the Omnipolarizer, all incident SOPs are nonlinearly trapped and merge into a small spot around a well-defined SOP at its output. Therefore, this device is free of RINa steady in time and varying in polarization input beam is converted into a steady in time and fully polarized output beam. Note however that the Omnipolarizer cannot remove any RIN existing originally in the intensity profile of a signal. Yet, unlike conventional passive linear polarizers, the device does not introduce any additional RIN penalty due to the polarization rejection principle and propagation channel. Next, consider the PBS feature of the Omnipolarizer and contrast its discrete splitting operation principle with the continuous splitting which is inherent to a linear PBS. Depending on its input polarization ellipticity, the signal beam is directed into either one or another channel at the device output. This means that an input pulse is no longer split among the two output channels in amounts which are proportional to the projections of its SOP on the two polarization axes of the PBS. To the contrary, the signal pulse is routed to an output channel as a single digital entity, cf. Fig. 1b and Fig. 1d. Therefore we may conclude that this device, when functioning as a PBS (called here discrete PBS) is free from RIN.

In the quest of more transparency for future optical networks, there is a need for ultrafast alloptical control of light polarization, which has motivated a renewed interest in polarization optics. In this field research moves along three distinct paths: i) the development of linear and nonlinear methods of re-polarization of a partially coherent and initially fully depolarized light with interferometers [START_REF] Martinelli | Polarization stabilization in optical communications systems[END_REF], in a random scattering medium [START_REF] Sorrentini | Enpolarization of light by scattering media[END_REF], via propagation in a Kerr medium [START_REF] Yu | Propagation of partially polarized light[END_REF][START_REF] Yu | A new principle of broken time reversibility in solids using unpolarized light[END_REF][START_REF] Prakash | Change in coherence properties and degree of polarization of light propagating in a lossless isotropic nonlinear Kerr medium[END_REF][START_REF] Picozzi | Spontaneous polarization induced by natural thermalization of incoherent light[END_REF][START_REF] Connaughton | Condensation of Classical Nonlinear Waves[END_REF], or even through propagation of a diffracting beam in free space [START_REF] James | Change of polarization of light beams on propagation in free space[END_REF]; ii) the engineering of dissipative polarizers [START_REF] Thevenaz | All-optical polarization control through Brillouin amplification[END_REF][START_REF] Zadok | Vector analysis of stimulated Brillouin scattering amplification in standard single-mode fibers[END_REF][START_REF] Galtarossa | Polarized Brillouin Amplification in Randomly Birefringent and Unidirectionally Spun Fibers[END_REF][START_REF] Shmilovitch | Dual-pump push-pull polarization control using stimulated Brillouin scattering[END_REF][START_REF] Martinelli | Evidence of Raman-induced polarization pulling[END_REF][START_REF] Kozlov | Theory of fiber optic Raman polarizers[END_REF][START_REF] Ursini | Raman Nonlinear Polarization Pulling in the Pump Depleted Regime in Randomly Birefringent Fibers[END_REF][START_REF] Muga | Broadband polarization pulling using Raman amplification[END_REF]; iii) the design of nonlinear methods of lossless polarization control of fully coherent and depolarized beams [START_REF] Fatome | Observation of light-by-light polarization control and stabilization in optical fibre for telecommunication applications[END_REF][START_REF] Morin | All-optical nonlinear processing of both polarization state and intensity profile for 40 Gbit/s regeneration applications[END_REF][START_REF] Pitois | Polarization and modal attractors in conservative counterpropagating four-wave interaction[END_REF][START_REF] Kozlov | Nonlinear repolarization in optical fibers: polarization attraction with copropagating beams[END_REF][START_REF] Heebner | Conversion of unpolarized light to polarized light with greater than 50% efficiency by photorefractive two-beam coupling[END_REF]. Polarizers belonging to the first class are based on the coherent suppression of phase noise, which is intimately related to the incoherent nature of light beams. These polarizers are not of interest to us here, because they either are inapplicable to fully coherent beams, or when applicable they are not free from RIN. Examples of devices belonging to the second class are the Brillouin amplifiers [START_REF] Thevenaz | All-optical polarization control through Brillouin amplification[END_REF][START_REF] Zadok | Vector analysis of stimulated Brillouin scattering amplification in standard single-mode fibers[END_REF][START_REF] Galtarossa | Polarized Brillouin Amplification in Randomly Birefringent and Unidirectionally Spun Fibers[END_REF][START_REF] Shmilovitch | Dual-pump push-pull polarization control using stimulated Brillouin scattering[END_REF] proposed by L. Thevenaz et al., and Raman amplifiers [START_REF] Martinelli | Evidence of Raman-induced polarization pulling[END_REF][START_REF] Kozlov | Theory of fiber optic Raman polarizers[END_REF][START_REF] Ursini | Raman Nonlinear Polarization Pulling in the Pump Depleted Regime in Randomly Birefringent Fibers[END_REF][START_REF] Muga | Broadband polarization pulling using Raman amplification[END_REF] first demonstrated by M. Martinelli et al. in ref. [START_REF] Martinelli | Evidence of Raman-induced polarization pulling[END_REF]. These devices do not conserve the energy of the beam and, more importantly, because of their intrinsic properties based on polarization dependent gain, they suffer from a large amount of output RIN.

Our Omnipolarizer belongs to the third class. Polarizers belonging to this class are all free of RIN. However most of these devices have the drawback of requiring an additional polarization control intense beam (pump beam), whose state of polarization must be in turn also accurately stabilized [START_REF] Fatome | Observation of light-by-light polarization control and stabilization in optical fibre for telecommunication applications[END_REF][START_REF] Morin | All-optical nonlinear processing of both polarization state and intensity profile for 40 Gbit/s regeneration applications[END_REF][START_REF] Pitois | Polarization and modal attractors in conservative counterpropagating four-wave interaction[END_REF][START_REF] Kozlov | Nonlinear repolarization in optical fibers: polarization attraction with copropagating beams[END_REF]. In contrast, the Omnipolarizer does not need any pump beam or active optoelectronic element, since light is capable of self-organizing its own SOP. Its operation principle is based on the ultrafast Kerr nonlinearity inherent to silica glass, and it is at least six orders of magnitude faster [START_REF] Kozlov | Nonlinear repolarization dynamics in optical fibers: transient polarization attraction[END_REF] than the single-beam polarizers utilizing the slow photorefractive nonlinearity which were pioneered in [START_REF] Heebner | Conversion of unpolarized light to polarized light with greater than 50% efficiency by photorefractive two-beam coupling[END_REF].

Here, the Omnipolarizer is not only a fast single-beam polarizer free of RIN, but most importantly it also demonstrates a RIN-free PBS based on the discrete splitting principle, as exemplified by Fig. 1d. From a fundamental point of view, the experimental demonstration of the Omnipolarizer represents a significant step forward as its operation goes beyond current way of thinking. Indeed, this device provides the first clear experimental demonstration example of the self-organization of light polarization in a nonlinear medium thanks to feedback reflection, which is described by means of a perfectly isotropic model (see Eq. (1) and Ref. [START_REF] Kozlov | Theory of lossless polarization attraction in telecommunication fibers[END_REF]). Similarly to the spontaneous organization which may occur in the spatial domain as it was numerically predicted in ref. [START_REF] Firth | Spontaneous hexagon formation in a nonlinear optical medium with feedback mirror[END_REF], in spite of its symmetry the medium prefers to support a strongly anisotropic polarization pattern -one or two tightly localized spots on the Poincaré sphere for the output signal SOP (see Fig. 2) instead of the expected isotropic distribution, i.e. a uniform coverage of the Poincaré sphere, see Fig. 2. The illustration of the beam SOPs on the Poincaré sphere in Fig. 2 comes handy to visualize the main function of the Omnipolarizerthe spontaneous re-polarization of an initially unpolarized wave.

Experimental set-up and results

We implemented and demonstrated the concept of universal Omnipolarizer as shown in Fig. 3, see the Methods section for additional details. The efficiency of the device was first characterized when operating in the discrete PBS mode. In this case the setup only involves an output reflective element (i.e., the configuration depicted in Fig. 2a), such as a Fiber Bragg Grating (FBG, used here with 95% of reflection, 5% of transmission) or an optical circulator plus a fiber coupler, or end fiber coating. The corresponding experimental results are provided in Fig. 4. The SOP of an initial On/Off Keying (OOK) 40-Gbit/s signal centred around 1564 nm was scrambled so that it spreads over the entire Poincaré sphere. As a result, its eye-diagram signal was completely closed beyond a linear polarizer.

When the input average power was increased up to 27 dBm, we could observe that the output SOP self-stabilized in two orthogonal points on the Poincaré sphere: light self-organized its SOP. Indeed, for a positive (negative) ellipticity of the input signal SOP (i.e., the red (blue) points in Fig. 4), the output signal SOP remained confined around the north (south) pole of the sphere, that is to say close to the right (left)-handed circular polarization. In order to quantify the quality of the attraction process, the residual fluctuations in the output SOP can be contained in a maximum circle-cap characterized by a solid angle of 0.42 sr for the north pole (0.68 sr for the south), corresponding to a maximum fluctuation of the ellipticity angle of 0.37 rad (0.47 rad) peak-topeak or a solid angle of 0.10 sr (0.12 sr) in rms corresponding to a variation of the ellipticity angle of 0.18 rad (0.20 rad), respectively. This residual distribution around a small area is attributed to the fact that the process of self-organization asymptotically converges to the poles in the limit case of very long fibers or input powers. Consequently, nearly orthogonal input SOPs are difficult to attract into a single output point. Moreover, it is important to note that the two output SOPs are universal, in the sense that they do not depend on either the input signal, the telecom fiber sample, the laboratory reference frame or any environmental changes. Indeed, we checked that straining the fiber does not influence the position and width of the output SOP distributions. Moreover, by means of an air-free quarter-wave plate/polarizer setup, we measured the absolute (i.e., in the fixed laboratory frame) value of the SOP which always remains in a circular state. As a result, in the configuration leading to two universal output SOPs the Omnipolarizer behaves as a discrete PBS. Indeed, in spite of the initial polarization scrambling, no intensity fluctuations can be observed in the eye-diagrams of the 40-Gbit/s output intensity profiles. Depending on its initial ellipticity, all of the 40-Gbit/s signal energy is digitally routed to either the right or left-circular SOP without any pulse splitting.

Next we tested the second configuration shown in Fig. 2b and3, which involves the reflective loop set-up and allows the Omnipolarizer to operate in (or switch among) the two different regimesa polarizer or a PBS. Namely, one or two points of SOP stabilization can be observed, depending on the amount of energy that returns back into the fiber upon reflection. An associated short movie, which can be found in the Supplementary Movie, illustrates well the evolution of the output SOP as a function of the average power of the reflected signal for an input power of 27 dBm. The transition between the two regimes may be clearly observed.

The 40-Gbit/s experimental results obtained for the Omnipolarizer in polarizer mode are summarized in Fig. 5. When the back-reflected signal was amplified by means of the reflective loop configuration of Fig. 3 so that its power was just beyond the input power (i.e., when the power of the back-reflected signal was increased up to 28 dBm), a single point of stabilization survived: all of the output SOPs remained localized around a small area of the Poincaré sphere (see Fig. 5) contained in a circle-cap characterized by a solid angle of 0.28 sr, corresponding to a maximum fluctuation of the ellipticity angle of 0.30 rad peak-to-peak or a solid angle of 0.06 sr in rms corresponding to a variation of the ellipticity angle of 0.14 rad. As in the previous PBS case, we would like to notice that since all the input SOPs converge asymptotically to the pole of the sphere, a large number of nonlinear lengths would be theoretically required to reduce this small area to a single point, which is hardly possible in practice. Note that the finite size of the SOP spot can also be interpreted theoretically by the fact that the singularities lie on the boundary of the energy-momentum diagram (see Fig. 2 of the supplementary materials), so that convergence to the singularities cannot take place in an isotropic way, which limits the efficiency of the attraction process [START_REF] Assémat | Hamiltonian tools for the analysis of optical polarization control[END_REF]. We remark that the polarization controller inserted into the reflective loop of Fig. 3 can be used to tune the point of attraction on the Poincaré sphere and thus to select a specific output SOP on demand. Consequently, whatever the initial SOP, the output polarization of the 40-Gbit/s signal remains trapped close to a single SOP and the output eye-diagram is completely open behind a polarizer. In other words, a negligible polarizationdependent loss is obtained from the Omnipolarizer, indicating that the self-polarization process operates in its full strength, free from RIN. More importantly, the corresponding bit-error-rate (BER) measurements (also presented in Fig. 5) show that, in spite of the initial polarization scrambling process, the Omnipolarizer enables clean error-free data recovery behind a polarization dependent component. This completes the demonstration of the powerful SOP stabilization which is achieved by the device.

Theoretical analysis

From a theoretical point of view, the description of the previously discussed self-polarization phenomenon can be established on the analysis of the spatio-temporal evolution of counterpropagating beams (the propagating signal and its own reflective replica) in a standard randomly birefringent telecom fiber, whose dynamics can be modelled by a set of 2 coupled equations [START_REF] Kozlov | Theory of lossless polarization attraction in telecommunication fibers[END_REF]:

    SS v S IJ tz JJ v J IS tz                    (1)
where z is the spatial coordinate along the fiber, v is the group-velocity,  is the nonlinear Kerr coefficient,  denotes the vector product and I is a diagonal matrix with coefficients (-1,-1,1) [START_REF] Kozlov | Theory of lossless polarization attraction in telecommunication fibers[END_REF]. The SOPs of the forward and backward beams are described by the Stokes vectors   ,,

x y z S S S S  and   ,, x y z J J J J 
. In spite of its apparent simplicity, model ( 1) captures all of the essential properties of the Omnipolarizer: excellent quantitative agreement with the experimental results is obtained without using adjustable parameters (see the theoretical supplementary material). In particular, the simulations of Eq. ( 1) confirm the existence of either one or two points of self-organization for the SOP on the poles of the Poincaré sphere (see Figs. 6a-c). The self-organisation mechanism can be qualitatively understood as follows. Because of statistical averaging over its random linear birefringence distribution, the fiber does not favour any particular polarization direction. For instance, this argument indicates that the SOPs of the two beams cannot relax towards, e.g., a linear state, because such a relaxation would violate the symmetry properties of the fiber. In this respect, the left and right circular SOPs are the only ones which satisfy the requirements imposed by the characteristics of the optical fiber. In its passive configuration, the two circular SOPs are completely symmetric, hence the fiber exhibits two distinct points of attraction for the beam SOPs. The switching between the two modes of operation results from the symmetry-breaking induced by the boundary conditions inherent to the active configuration. More precisely, suppose that the energy of the transmitted signal remains larger than its backward replica (i.e., when a FBG only is used in the configuration of Fig. 2a). In this case the numerical simulations confirm well the experimental observation that, depending on the initial ellipticity of the input SOP, the output SOP self-stabilizes in either one or another of the two poles of the Poincaré sphere (see Fig. 6a). In this situation, the Omnipolarizer acts as a digital PBS, just as in the experimental observations of In addition to this qualitative understanding, the physical mechanism underlying the observed self-polarization phenomenon can be described in terms of mathematical techniques associated with Hamiltonian geometric singularities [START_REF] Cushman | Global aspects of classical integrable systems[END_REF] (see the theoretical supplemental). Such singularities are topological structures that are also known as singular tori: these can be viewed as a two-dimensional extension of the concept of separatrix, which is a well-known property of basic one-dimensional physical systems (e.g., a pendulum). It can be shown that singular tori play the role of attractors for the SOP as described by Eqs. (1) [START_REF] Sugny | On the role of singular tori in the spatio-temporal dynamics of nonlinear wave systems[END_REF][START_REF] Assémat | Complete nonlinear polarization control in an optical fiber system[END_REF]. This is schematically illustrated in Fig. 6c, which provides a phase-space representation of the system: The numerical simulations show that the SOP converges toward the singular torus, i.e. the red point in Fig. 6c.

The singular torus for the present model can be represented as a sphere, an object which is topologically singular in the sense that it cannot be transformed into a regular torus by means of continuous deformations.

We briefly illustrate here the results for the passive configuration of the Omnipolarizer and refer the reader to the Supplemental for details. The numerical simulations of Eqs. (1) reveal that, irrespective of the initial conditions, the spatiotemporal dynamics exhibit a relaxation toward a stationary state. This can be understood intuitively by remarking that light reflection on the mirror introduces a supplementary condition for the polarizations states of the forward and reflected waves: S(L,t) -J(L,t) = 0 at any time t. This condition may be viewed as a temporal fixed point on the mirror at z = L. Then the physical picture that one may have in mind is that the system gradually extends this stationary behaviour from the mirror toward the whole fiber. This relaxation process is possible thanks to the reflected wave, which can evacuate the fluctuations of the waves throughout the free boundary condition at z = 0. In this way the counterpropagating waves relax toward an inhomogeneous stationary state in which their SOPs keep a fixed constant value for any z.

This brings us to the study of the stationary system (1), whose main property is that it is

Hamiltonian with H = (S x J x + S y J y -S z J z ). In this expression of H, the three axes play an analogous role, a property which leads to three additional conserved quantities, K x = S x + J x , K y = S y + J y and K z = S z -J z . These functions play a similar role in the system dynamics, so that all stationary states can be described in a reduced two-dimensional phase-space representation, as illustrated in Fig. 6c for H vs K z . We show in the Supplemental that the singular torus of interest is located at the origin of the diagram at K y = K x = K z = 0. These values of the constant of the motion together with the mirror boundary conditions impose that only the circular SOPs can exist on the singular torus. In conclusion, the topological properties of the singular torus impose the universal circular SOPs which are selected by the Omnipolarizer.

Discussion

From the practical side, it is important to answer the questionwhat is the response time of our Omnipolarizer? That is, how fast in time can be the input polarization fluctuations of the beam, so that the Omnipolarizer still maintains its ability to function as polarizer or discrete PBS? Our equipment allowed us to drop the characteristic time of input polarization fluctuations down to 30 microseconds, without any detectable changes in the SOP stabilization performance of the Omnipolarizer. This response time is comparable with the record value of today's polarization controllers based on Lithium-Niobate waveguides [START_REF] Martinelli | Polarization stabilization in optical communications systems[END_REF][START_REF] Koch | Record 59-krad/s Polarization Tracking in 112-Gb/s 640-km PDM-RZ-DQPSK Transmission[END_REF]. In Ref. [START_REF] Kozlov | Nonlinear repolarization dynamics in optical fibers: transient polarization attraction[END_REF] we have analysed the response time of another kind of typical fiber-optic nonlinear lossless polarizer based on the interaction between counter-propagating beams, and found that it is of the order of L NL /c, where L NL =(P) -1 is the nonlinear length,  is the fiber nonlinear coefficient and P is the input signal power. This estimate suggests that for typical operational regimes, the response time is of a few microseconds. The physical reason why the response time of the Omnipolarizer turns out to be much slower than the response time of the Kerr effect in silica (which is of the order of a few femtoseconds) is the counter-propagating nature of this nonlinear polarizer and thus the distributed interaction along the entire fiber length. Hence a finite response time is necessary in order to establish an equilibrium polarization pattern inside the nonlinear medium. In practice, this analysis suggests that the response time of the Omnipolarizer could be reduced by at least an order of magnitude by utilizing fibers with higher nonlinearities, such as Chalcogenide, Tellurites, Bismuth or lead Silicate fibers [START_REF] Granzow | Supercontinuum generation in chalcogenide-silica step-index fibers[END_REF][START_REF] Liao | Fabrication and characterization of a chalcogenide-tellurite composite microstructure fiber with high nonlinearity[END_REF][START_REF] Lamont | Supercontinuum generation in dispersion engineered highly nonlinear (γ = 10 /W/m) As 2 S 3 chalcogenide planar waveguide[END_REF][START_REF] Domachuk | Over 4000 nm bandwidth of mid-IR supercontinuum generation in sub-centimeter segments of highly nonlinear tellurite PCFs[END_REF][START_REF] Conseil | Chalcogenide step index and microstructured single mode fibers[END_REF][START_REF] Feng | Dispersion controlled highly nonlinear fibers for all optical processing at telecoms wavelengths[END_REF][START_REF] El-Amraoui | Microstructured chalcogenide optical fibers from As2S3 glass: towards new IR broadband sources[END_REF].

The counter-propagative configuration and four-wave mixing interaction of the Omnipolarizer also impose high average powers for both forward and backward light beam, typically around 500 mW in Figs. 4 and5, which is high above the standard telecommunication levels of powers and consequently, which could not be applicable to weak input signals. Nevertheless, this drawback could be also overcome by the use of highly nonlinear fiber based on soft glass materials [START_REF] Granzow | Supercontinuum generation in chalcogenide-silica step-index fibers[END_REF][START_REF] Liao | Fabrication and characterization of a chalcogenide-tellurite composite microstructure fiber with high nonlinearity[END_REF][START_REF] Lamont | Supercontinuum generation in dispersion engineered highly nonlinear (γ = 10 /W/m) As 2 S 3 chalcogenide planar waveguide[END_REF][START_REF] Domachuk | Over 4000 nm bandwidth of mid-IR supercontinuum generation in sub-centimeter segments of highly nonlinear tellurite PCFs[END_REF][START_REF] Conseil | Chalcogenide step index and microstructured single mode fibers[END_REF][START_REF] Feng | Dispersion controlled highly nonlinear fibers for all optical processing at telecoms wavelengths[END_REF][START_REF] El-Amraoui | Microstructured chalcogenide optical fibers from As2S3 glass: towards new IR broadband sources[END_REF], which could allow to decrease the required average power by two orders of magnitude thanks to their strong Kerr coefficients. Another important issue for telecommunication applications is the compatibility of the Omnipolarizer with a wavelength division multiplexing (WDM) configuration. Currently, we could emphasize that because of the strong nonlinear regime of propagation occurring in the Omnipolarizer, the power level should be yet higher in a WDM configuration and thus a substantial nonlinearity-mediated cross-talk would be observed between the different channels, providing a non-negligible amount of quality impairments. In order to overcome these impairments and excess of power, we would emphasize that a demultiplexing operation should be required before repolarization process in a WDM environment. This last point is presently under study and requires further investigations.

In summary, this work reports the first experimental observation of self-organization of light state of polarization in optical fiber. The device called Omnipolarizer is based on a nonlinear interaction through a four-wave-mixing process occurring in optical fibers between a signal beam and its own counter-propagating replica produced by back-reflection at the fiber output end. Two modes of operation of the Omnipolarizer were experimentally demonstrated. First as digital circular beam splitter when only a reflective element such as FBG was inserted at the end of the fiber. This configuration allowed us to successfully route an arbitrary polarized OOK 40-Gbit/s signal into a well-defined universal right or left circular SOP without any polarization depending loss induced RIN. Secondly as an ideal polarizer, it corresponds to the implementation of an amplified feedback loop at the end of the fiber. In this configuration, the Omnipolarizer has allowed us to self-trap the state of polarization of an arbitrary polarized OOK 40-Gbit/s without any polarization depending loss, and provided weak penalty when compared to back-to-back measurements. In conclusion, our observations of polarization self-organization in a standard telecommunication optical fiber unveil the possibility for a radically new approach to light polarization control, and open up the path to new exciting applications in photonics including signal processing, imaging, laser, spintronics or sensing measurements.

Methods

The 40-Gbit/s Return-to-Zero signal was generated by means of a 10-GHz mode-locked fiber laser (Calmar Laser) delivering 2.5-ps pulses at 1564 nm. The spectrum of the initial pulse train was frequency sliced by means of a programmable liquid-crystal based optical filter (Finisar Waveshaper) in order to enlarge the pulses to 7.5-ps Gaussian pulses. The resulting 10-GHz pulse train was intensity modulated by a LiNbO 3 modulator through a 2 31 -1 pseudo-random bit sequence PRBS (Modbox Photline technologies and Anritsu pattern generator) before x4 time multiplexing to achieve a 40-Gbit/s optical data stream. A polarization scrambler (Agilent polarimeter) was then used to introduce wide polarization fluctuations at a rate of 0.625 kHz.

Before injection into the optical fiber, the 40-Gbit/s signal was finally amplified by means of an Erbium doped fiber amplifier (EDFA from Manlight) at an average power of 27 dBm. The optical fiber within the Omnipolarizer was a 6.2-km long standard Non-Zero Dispersion-Shifted Fiber (NZDSF) with chromatic dispersion D = -1.5 ps/nm/km at 1550 nm, a dispersion slope S=0.07 ps 2 /nm/km, a nonlinear Kerr coefficient =1.7 W -1 km -1 and a PMD coefficient D p =0.05 ps/km 1/2 . An optical circulator was inserted at the input of the fiber. At the opposite end of the fiber, two configurations were tested. The first one involved a FBG with a 1-THz flat-top bandwidth centred at the signal wavelength with a reflection (transmission) ratio of 95% (5%), respectively. After propagation, the repolarized 40-Gbit/s data signal was optically filtered by means of a 70-GHz Gaussian shape optical bandpass filter (programmable WaveShaper filter from Finisar) shifted by 290 GHz from the initial signal central frequency so as to improve the

Fig. 4 .

 4 On the other hand, whenever the energy of the back-reflected signal becomes equal or slightly larger than the transmitted signal energy, by means of an additional gain as in Fig.2b, the solutions of Eqs. (1) confirm the experimental findings of Fig.5that the Omnipolarizer acts as lossless polarizer (see Fig.6b). In this case, irrespective of the initial SOP at the input of the device, one obtains at the Omnipolarizer output a unique polarization state without any residual polarization depending loss. It can be shown that the selection of either the right or left circular SOP is determined by the angle of polarization rotation in the reflection process. Such angle is at the origin of the symmetry breaking between the two circular SOP, and it can be adjusted experimentally thanks to the polarization controller. More precisely, if one only considers a polarization rotation around the vertical axis of the Poincaré sphere, then a positive (negative) rotation angle favors the left (right) circular SOP. This result can be interpreted intuitively, since a positive angle turns the polarization state in the same sense as the left circular polarization state, which thus favors the attraction process toward this particular SOP (see the theoretical supplementary material).

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Schematics and principle of the Omnipolarizer. a Passive setup: the light beam

Figure 3 .Figure 4 .

 34 Figure 3. Experimental set-up of the Omnipolarizer. The polarization state of an input 40-Gbit/s

Figure 5 .

 5 Figure 5. Experimental results obtained in the polarizer mode when the back-reflected signal

Figure 6 .

 6 Figure 6. a Theoretical Poincaré representation obtained by numerically solving the spatio-

Figure 1 Figure 1 .

 11 Figure 1

Figure 1 c

 1 Figure 1. a, b Illustration of the lossy nature of a linear polarizer based on the rejection principle, as opposed to the Omnipolarizer. Three input pulses, with different initial polarization states are incident on a) the linear (or conventional) passive polarizer; b) the Omnipolarizer. In the first case, we observe strong output intensity variations due to the rejection principle. In the second case all three pulses pass through with no degradation. In both cases the outcoming pulses are vertically polarized.Figure 1 c, d Illustration of the continuous splitting principle of the linear (or conventional) PBS, as opposed to the discrete splitting principle of the Omnipolarizer. Three pulses, with different initial polarization states are incident on c) the linear PBS; d) the Omnipolarizer. In the first case, pulse splitting and intensity variations are observed on each axis of the PBS. In the second case, no splitting is observed: all energy is routed to either one or another channel, simply depending on the initial polarization ellipticity of the pulse. Note that this figure only serves for illustrative purposes. Indeed, the Omnipolarizer splits mutually orthogonal circular (and not linear, as shown in the figure) polarizations.

Figure 2 Figure 2 .

 22 Figure 2

Figure 3 Figure 3 .

 33 Figure 3

Figure 4 .

 4 Figure 4. Experimental results obtained in the digital PBS operation mode, in configuration with a FBG, when the input power was set to 27 dBm. Two orthogonal universal points of attraction are observed depending on the input signal SOP ellipticity. All initial SOPs initially situated in the northern (southern) hemisphere emerge from the fiber in the right (left) circular polarization, as in a discrete circular PBS.

Figure 5 Figure 5 .

 55 Figure 5

Figure 6 Figure 6 .

 66 Figure 6

intensity extinction ratio [20]. At the receiver, a polarizer or an association of quarter-wave plate and linear PBS, were inserted in order to translate the polarization fluctuations into intensity fluctuations. Behind the polarizer, the 40-Gbit/s eye diagram was monitored by an optical sampling oscilloscope (OSO EXFO picosolve) while the data were detected by means of a 70-GHz photodiode (from u2t) and electrically demultiplexed (Centellax 56-Gbit/s demultiplexer) at 10 Gbit/s in order to measure the bit error rate (BER). Note that the BER measurements represented in Fig. 5 are averaged on the 4 resulting demultiplexed 10-Gbit/s channels. The 40-Gbit/s signal SOP was also analyzed by means of a commercially available polarization analyzer (Agilent polarimeter).

Acknowledgements

All the experiments were performed on the PICASSO platform in ICB. The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013 Grant Agreement n°306633, PETAL project coordinated by Julien Fatome). We also acknowledge the financial support from the CNRS, Conseil Régional de Bourgogne: Photcom PARI program, Ministère de l'Enseignement Supérieur et de la Recherche and FEDER. The work of V. K. and S. W. was

Author Contributions

J.F. and S.P. provided equal contributions. J.F., S.P. and P.M. carried out the experiments. A.P., E.A., H.J., V.K. and S.W. contributed to the theoretical and numerical analysis. S.W. provided overall technical leadership. G.M. provided supervision. All authors participated in the analysis of the results and in the paper redaction.

Additional information

The authors declare that they have no competing financial interests. Supplementary information accompanies this paper with details on the theoretical approach describing the Omnipolarizer as well as a movie illustrating the evolution of the Omnipolarizer output state of polarization of the 40-Gbit/s signal injected with an average power of 27 dBm as a function of reflected power (from 20 to 28 dBm). Here we may clearly observe the transition between the initial scrambling state, the PBS mode and finally the polarizer operation mode of the Omnipolarizer.


Roberto Pasqua
email: rpasqua@laas.fr

Damien Foures
email: dfoures@laas.fr

Vincent Albert
email: valbert@laas.fr

Alexandre Nketsa

FROM SEQUENCE DIAGRAMS UML 2.x TO FD-DEVS BY MODEL TRANSFORMATION

Keywords: MDA, meta-modeling, Sequence Diagrams, Finite and Deterministic DEVS, discrete event simulation

This work transforms sequence diagrams to Finite and Deterministic DEVS (FD-DEVS) in Model-Driven Engineering field. The main goal is the formalisation of behaviours, described with UML sequence diagram, to make verification activity by space state exploration and to make validation activity of a set of traces by simulation. In this context, we have chosen to elaborate a model transformation. This paper shows how, after the construction of meta-model for sequence diagrams and for Finite and Deterministic DEVS, it is possible to automate the transformation from one instance of source meta-model to one instance of destination meta-model. The source model is a sequence diagram and the target model is a FD-DEVS component. The destination model is converted into DEVSJava code to simulate its execution.

INTRODUCTION

Model-Driven Engineering offers tools, languages and standard notions to establish and transform models. Today, employment of models lies at the bottom of systems engineering. One of the main purposes in this field is the ability to reuse these models. For that we will have to expand the knowledge about model handling.

Context

System Engineering defines a series of approaches to the development of complex system through the definition of a process from management of requirements to the system release. In every step of the development it is possible to make the simulation predict system behaviour. Verification and Validation (V&V) is a key step towards the development of complex systems and we need formal languages and simulations to do this. The Unified Modeling Language (UML) is a very expressive language which designs the system behaviour but it is not enough formal to provide that this comportment be truly follow after the modelling. The more the language is expressive the less formal it is. The Object Management Group (OMG) introduces important novelties in the 2.0 version of UML. Several of these changes concern the Interaction Diagrams and, in particular, the expressiveness of sequence diagrams. [START_REF] Micskei | The many meanings of UML 2 Sequence Diagrams: a survey[END_REF] give an overview about the proposed formal semantics in the several papers. In this context we will concentrate our regard on the concept of trace which is the main construct of the sequence diagrams semantics. In UML specification the trace is defined like "a sequence of event occurrences, each of which is described by an OccurrenceSpecification in a model". In particular in order to make a V&V process, we need to make the distinction between valid and invalid traces. We want to consider the valid trace and to ignore the invalid trace.

The DEVS formalism has a well defined semantic to the simulation. FD-DEVS is a subset of DEVS and it has a finite state space, so we use this formalism because we can make the verification process through an exploration of its state space. Then, the transformation to sequence diagrams into FD-DEVS (SD2FDDEVS) allows us to verify, with model checking techniques or validate by simulation, the composition of the trace issue from the scenario described by sequence diagrams. Moreover, we consider specification of invalid trace that we transform into linear temporal logic (LTL) constraints. Today it does not exist a tool to check the LTL constraints in DEVS although it is a current research topic [START_REF] Hwang | Reachability Graph of Finite and Deterministic DEVS Networks[END_REF].

Approach

We try to develop a platform to formal verification and validation of a simulation models. A model is a hypothetical description of a complex entity that is used to describe a particular behaviour of a real entity. UML allows to modelling a behaviour by three diagrams: Activity, Interaction and StateMachine. A transforma-tion which comes toward the formalisation of UML behaviour is [START_REF] Foures | Automation of SysML activity diagram simulation with model-driven engineering approach[END_REF], where the authors transform SysML activity diagrams into Petri Nets. With the same approach, we propose the formalisation of sequence diagrams, which are a kind of interaction diagrams. In order to realize it we also develop a transformation based on Atlas Transformation Language (Atlas group and INRIA 2006) to put in practice the specified rules established between two meta-models. This paper is structured as follow: in the next section we discuss about associated work, after this, we present the models and the meta-models in section named preliminaries; in the section named transformation we explain the transformation rules and our method; in the second last section we present the transformation with an example and, finally, in the last section we conclude the paper with some proposal to future works.

RELATED WORKS

In view of their importance in describing the scenarios, several transformations concerning sequence diagrams have been developed. In (Ouardani et al. 2006), the authors transform sequence diagrams into Petri Nets without considering the time explicitly. The sequence diagrams meta-models are reduced to take into account only messages and lifelines (objects in their notation). Another work that transforms sequence diagrams into Petri Nets is [START_REF] Ameedeen | A Model Driven Approach to Represent Sequence Diagrams as Free Choice Petri Nets[END_REF], where the authors focus on the flow of events described with sequence diagrams and they do not consider the constraints which can appear in the scenario. Here the decomposition of sequence diagrams into fragments and mapping them into "Petri Nets blocks" is interesting. We used the same approach. Sequence diagrams have not only been transformed into Petri Nets, but also into State Machines by graph transformation [START_REF] Gronmo | From UML 2 Sequence Diagrams to State Machines by Graph Transformation[END_REF] and into Communicating Sequential Processes (Dan 2010) to enable a formal verification and analysis. We can not forget (Sqali and Trojet 2009) that translates scenario (described by Message Sequence Charts) in state machine (represented by DEVS). In this work, the authors show the advantages of the use of coupled DEVS, which enables the behaviour simulation of every objects of the system.

PRELIMINARIES

Models

Sequence Diagrams

Sequence diagrams are the most used diagrams to display interactions in UML. The sequence diagrams show the interactions between objects and how they are temporally ordered. An interaction describes how the instances interact dynamically among themselves and it describes the messages exchanged. An interaction can represent a scenario or a use case. Figure 1 shows an Interaction composed by two Lifelines, one Message and four OccurrenceSpecification. There are two kinds of OccurrenceSpecification : message and execution. Above all that, they are the CombinedFragment. It can be composed of one or most operators and UML offer a few operation kind (see meta-model class Interaction-OperandKind in figure 6). We focus our attention on alternative (ALT), negative (NEG) and iterative (LOOP) fragments: ALT allows to modelling a conditional construct, NEG is used to show a forbidden behaviour and LOOP represents the recursive part of the diagrams. In this work we consider only the negative fragments.

Finite and Deterministic DEVS

The FD-DEVS formalism is defined in [START_REF] Hwang | Reachability Graph of Finite and Deterministic DEVS Networks[END_REF]. We use two definitions as follows:

Definition 1. An atomic FD-DEVS is a 7-tuple:

A = X, Y, S, s 0 , τ, δ x , δ y (1)
where: X is a finite set of input events; Y is a finite set of output events; S is a non-empty and finite states set;

s 0 ∈ S is the initial state; τ : S → Q +,∞ 0
is the maximum sojourning time of a state where Q +,∞ 0 denotes a set of non-negative rational numbers plus infinity; δ x : S × X → S is the external transition function; δ y : S → S × Y ∅ is the internal transition function where Y ∅ = Y ∪ {∅} and ∅ denotes the silent event.

Definition 2. A coupled FD-DEVS is a 5-tuple:

C = X, Y, {M d }, D, EIC, IOC, IC (2)
where: X is a finite set of input events; Y is a finite set of output events; {M d | d ∈ D} is an index set of FD-DEVS models. {M d } can be either an atomic or an coupled model; D is a finite set of names of subcomponents; EIC is a set of external input connectors; IOC is a set of internal output connectors; IC is a set of internal connectors.

Semantic Figure 2 shows the graphic semantic chosen to represent a FD-DEVS model. phase 1 is a transitive state since t a = 0. Then δ y is immediately triggered after entering in phase 1. An output event is triggered on Y and next state is the passive state phase 2. When an input event occurs on X, δ x is triggered updating the state to phase 3.

phase 3 t a = 5 phase 1 ta = 0 phase 2 t a =∞ δx δy Y ∅ X Atomic FD-DEVS Coupled FD-DEVS FD-DEVS 1 FD-DEVS 2 EIC IOC IC X Y X 11 Y1 X 2 Y21 X 12 Y22 Figure 2: Graphic semantic of FD-DEVS

Meta-Models

The metamodeling process is used in accordance with the OMG standard Meta-Object Facility (MOF). For this, we use Eclipse Modeling Framework (EMF) that uses Ecore as metamodeling language.

Sequence Diagrams OMG defined the sequence diagrams in [START_REF] Omg | OMG Unified Modeling Language (OMG UML)[END_REF].

In accordance with this specification, the sequence diagrams meta-model is shown in Figure 6. We fix our attention on the main classes considered in the transformation : Interaction, Lifeline, OccurrenceSpecification, Message and CombinedFragment. In particular, we focus our interest in GeneralOrdering which allows us to give a temporal order of the specification occurrences present in the interaction. All instances of GeneralOrdering contain the valid trace. To explicitly consider the time in sequence diagram, we use Dura-tionConstraint.

Finite and Deterministic DEVS

The meta-model shown in Figure 7 is a rigorous description of FD-DEVS definition. An FD-DEVS instance can be an Atom or a Coupled and it is composed of Events.

In the case of an Atom, it is also composed of States and Transitions. A State is composed of one Time and a set of Variable. A Transition is defined between two States. In the case of a Coupled, it is composed of FD-DEVSs and Connectors. Every instance of these classes can have a name so we have introduced the Ecore class ENamedElement with inheritance relationships.

TRANSFORMATIONS

They are founded on ATL. The first transformation is a Model-to-Model transformation realized with an ATL module while, the second one, is a Model-to-Text transformation realized with ATL query.

SD2FDDEVS

The transformation rules, defined at the Meta-Model level, allow us to transform an input model (compliant with sequence diagrams meta-model) represented in the XMI (Extensible Mark-up Language Metadata Interchange) format into another model compliant with FDDEVS meta-model represented in the same format.

The input model can be created with a graphic editor that support the UML 2.x specification.

The main rules of correspondence established between sequence diagrams and FDDEVS are illustrated as follows:

• one interaction that contains at least two lifelines generates a coupled;

• every lifeline that is present in the interaction, generates an atom included in the associated coupled;

• every message generates an input and an output event (we consider complete MessageKind and synchCall, asynchCall MessageSort) in the atom. These events will be used to create internal connectors.

• every specification occurrence that is not contained in a combined fragment generates a state in the atom.

To generate the transitions we use the class Gener-alOrdering of the sequence diagram meta-model. All instances of this class give a temporal order to the events associated with the specification occurrences. Two consecutive specification occurrences, that belong in the same lifeline, generate an output internal transition. If two consecutive specification occurrences do not belong in the same lifeline it generates an "artificial" state and one or two transitions, that can be internal or external, depending on the type of specification occurrences. At the same time the states are created, we create the Time and the set of Variables associated. This variables are the atomic propositions that we use to check LTL properties on the model. For the combined fragments we consider the case of negative fragments with one message inside. For each of these fragments we generate two text files that contain a LTL constraint to verify the model construction.

EXAMPLE

In this example we illustrate our transformation by studying a use case of an automatic coffee machine that allows the possibility of understanding the key concepts of this work. The user can choose one product to display the price. The controller cannot accept a second choice. This is specified by the negative fragment in Figure 3. After this, when the interface sends a message to communicate that the money in input is enough, the controller starts the engine to make a coffee. At the end of the process (process of production), the controller sends a last message to the interface.

Target model

After transformation we have a coupled FD-DEVS composed by three atoms (Interface, Controller and Engine) and seven internal connectors (one for each message). Figure 4 shows the graphical representation of the atom Controller. We point out that "artificial" states and transitions depend on instances of GeneralOrdering. With this model, we can make a LTL verification and/or a validation by simulation.

f irst choice R = 1 → (second choice R = 0 ∪ price S = 1)
To simulate the FD-DEVS model we have devellopped the FDDEVS2DEVSJava transformation with ATL query and we use DEVS-Suite (DEVS-SuiteSim 2010) to execute the code. We give in Appendix the code generated by this transformation for the Controller. Figure 5 shows the execution trace of the simulation up to 10 time unit.

CONCLUSION

We have shown how is possible the formalisation of sequence diagrams by means of model transformation. We

ART IF ICIALGO2 M R f irst choice R = 0 f irst choice R = 1 second choice R = 0 second choice R = 0 price S = 0 price S = 0 price ok R = 0 price ok R = 0 start S = 0 start S = 0 stop R = 0 stop R = 0 remove cup S = 0 remove cup S = 0 AES0S M S0 f irst choice R = 1 f irst choice R = 1 second choice R = 0 second choice R = 0 price S = 0 price S = 0 price ok R = 0 price ok R = 0 start S = 0 start S = 0 stop R = 0 stop R = 0 remove cup S = 0 remove cup S = 0 ART IF ICIALGO7 M R1 f irst choice R = 1 f irst choice R = 1 second choice R = 0 second choice R = 0 price S = 1 price = 1 price ok R = 0 price ok R = 1 start S = 0 start S = 0 stop R = 0 stop R = 0 remove cup S =

Figure 1 :

 1 Figure 1: Graphic formalism of sequence diagrams

Figure 4 :

 4 Figure 4: FD-DEVS model of Controller

Figure 7 :

 7 Figure 6: Meta-models Sequence Diagrams

Table 1 :

 1 Table 2 shows binary variables associated to first six states of Controller. To generate these variables we use the events attached on List that represents GeneralOrdering

	OccurrenceSpecification	Lifeline
		ExecOccSpecStart	Interface
		MessOccSpecSend	Interface
		MessOccSpecRecv	Controller
		ExecOccSpec0Start	Controller
		MessOccSpecSend0	Controller
		MessOccSpecRecv0	Interface
		MessOccSpecSend1	Interface
		MessOccSpecRecv1	Controller
		MessOccSpecSend2	Controlleur
		MessOccSpecRecv2	Engine
		ExecOccSpec1Start	Engine
		MessOccSpecSend3	Engine
		ExecOccSpec1Finish	Engine
		MessOccSpecRecv3	Controller
		MessOccSpecSend4	Controller
		ExecOccSpec0Finish	Controller
		MessOccSpecRecv4	Interface
		ExecOccSpecFinish	Interface
	ARTIFICIAL GO 2 t a = infinity first_choice GO 2	MOSR t a = t 1	GO 3	EOS0S t a = 0	GO 4	MOSS0 t a = 0
		start			price	GO 5
	ARTIFICIAL GO 13 t a =infinity	GO 9	MOSS2 t a = 0	GO 8	MOSR1 t a = t 3	GO 7 price_ok	ARTIFICIAL GO 7 t a = infinity
	GO 13 stop		remove_cup
	MOSR3	GO 14	MOSS4	GO 15	EOS0F
	t a = t 5		t a = 0	t a = infinity

Table 2 :

 2 Variables associated to state

holdIn (" MessageRecv1 " ,1);} else if ( phaseIs (" wait4_GO13 ")){ for ( int j =0; j < m . size (); j ++) if ( messageOnPort (m ," stop " , j ))

holdIn (" MessageRecv3 " ,1);} else if ( phaseIs (" wait0_GO2 ")){ for ( int k =0; k < m . size (); k ++) if ( messageOnPort (m ," first_choice " , k )) holdIn (" MessageRecv " ,10);} } }