Philippe Carmona

Directed polymer in random environment and last passage percolation *

Keywords: directed polymer, random environment, partition function, last passage percolation. Mathematic Classification : 60K37

The sequence of random probability measures ν n that gives a path of length n, 1 n times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment.

Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.

Introduction

Last passage percolation

To each site (k, x) of N × Z d is assigned a random weight η(k, x). The (η(k, x)) k≥1,x∈Z d are taken IID under the probability measure Q. The set of oriented paths of length n starting from the origin is

Ω n = ω = (ω 0 , . . . , ω n ) : ω i ∈ Z d , ω 0 = 0, |ω i -ω i-1 | = 1 .
The weight (energy, reward) of a path is the sum of weights of visited sites:

H n = H n (ω, η) = n k=1 η(k, ω k ) (n ≥ 1, ω ∈ Ω n ).
Observe that when η(k, x) are Bernoulli(p) distributed [START_REF] Alexander | Approximation of subadditive functions and convergence rates in limiting-shape results[END_REF] , the quantity Hn n (ω, η) is the proportion of open sites visited by ω, and it is natural to consider for 0 < ρ < 1, N n (ρ) = number of paths of length n such that H n (ω, η) ≥ nρ.

Q(η(k, x) = 1) = 1 -Q(η(k, x) = 0) = p ∈ (0,
The problem of ρ-percolation, as we learnt it from Comets, Popov and Vachkovskaia [START_REF] Comets | The number of open paths in an oriented ρ-percolation model[END_REF] and Kesten and Sidoravicius [START_REF] Kesten | A problem in last-passage percolation[END_REF], is to study the behaviour of N n (ρ) for large n and different values of ρ.

Directed polymer in a random environment

We are going to consider fairly general environment distributions, by requring first that they have exponential moments of any order:

λ(β) = log Q e βη(k,x) < +∞ (β ∈ R) ,
and second that they satisfy a logarithmic Sobolev inequality (see e.g. [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]): in particular we can apply our result to bounded support and Gaussian environments. The polymer measure is the random probability measure defined on the set of oriented paths of length n by:

µ n (ω) = (2d) -n e βHn(ω,η) Z n (β) (ω ∈ Ω n ), ,
with Z n (β) the partition function

Z n (β) = Z n (β, η) = (2d) -n
ω∈Ωn e βHn(ω,η) = P e βHn(ω,η) ,

where P is the law of simple random walk on Z d starting from the origin.

Bolthausen [START_REF] Bolthausen | A note on the diffusion of directed polymers in a random environment[END_REF] proved the existence of a deteministic limiting free energy

p(β) = lim n→+∞ 1 n Q(log Z n (β)) = Q a.s. lim n→+∞ 1 n log Z n (β) .
Thanks to Jensen's inequality, we have the upper bound p(β) ≤ λ(β) and it is conjectured (and partially proved, see [START_REF] Comets | Directed polymers in random environment are diffusive at weak disorder[END_REF][START_REF]Strong disorder implies strong localization for directed polymers in a random environment[END_REF]) that the behaviour of a typical path under the polymer measure is diffusive iff β ∈ C η the critical region

C η = {β ∈ R : p(β) = λ(β)} .
In dimension d = 1, C η = {0} and in dimensions d ≥ 3, C η contains a neighborhood of the origin (see [START_REF] Bolthausen | A note on the diffusion of directed polymers in a random environment[END_REF][START_REF] Comets | Majorizing multiplicative cascades for directed polymers in random media[END_REF]).

The main theorem

The connection between Last passage percolation and Directed polymer in random environment is made by the family (ν n ) n∈N of random probability measures on the real line:

ν n (A) = 1 |Ω n | ω∈Ωn 1 ( Hn n (ω,η)∈A) = P H n n (ω, η) ∈ A . Indeed, N n (ρ) = ω∈Ωn 1 (Hn(ω,η)≥nρ) = (2d) n ν n ([ρ, +∞)) .
The main result of the paper is Theorem 1. Q almost surely, the family (ν n ) n∈N satisfies a large deviations principle with good rate function I = p * the Legendre transform of the free energy of the directed polymer.

Let m = Q(η(k, x)) be the average weight of a path m = Q Hn n (ω, η) . It is natural to consider the quantities:

N n (ρ) = ω∈Ωn 1 (Hn(ω,η)≥nρ) if ρ ≥ m , ω∈Ωn 1 (Hn(ω,η)≤nρ) if ρ < m .

3

A simple exchange of limits β → ±∞, and n → +∞, yields the following

ρ ± = Q a.s. lim n→+∞ max ω∈Ωn ± H n n (ω, η) = lim β→+∞ p(±β) β ∈ [0, +∞] .
Repeating the proof of Theorem 1.1 of [START_REF] Comets | The number of open paths in an oriented ρ-percolation model[END_REF] gives

Corollary 2. For -ρ -< ρ < ρ + , we have Q almost surely, lim n→+∞ (N n (ρ)) 1 n = (2d)e -I(ρ) .
We can then translate our knowledge of the critical region C η , into the following remark. Let

V η = {ρ ∈ R : I(ρ) = λ * (ρ)} . In dimension d = 1, V η = {m} and in dimensions d ≥ 3, V η contains a neighbourhood of m.
This means that in dimensions d ≥ 3, the typical large deviation of Hn n (ω, η) close to its mean is the same as the large deviation of 1 n (η 1 + • • • + η n ) close to its mean, with η i IID. There is no influence of the path ω : this gives another justification to the name weak-disorder region given to the critical set C η .

Proof of the main theorem

Observe that for any β ∈ R we have:

e βnx dν n (x) = P e βHn(ω,η) = Z n (β) Q a.s. . (1)
Consequently, since e u + e -u ≥ e |u| , we obtain for any β > 0, lim sup

n→+∞ 1 n log e βn|x| dν n (x) ≤ p(β) + p(-β) < +∞ ,
and the family (ν n ) n≥0 is exponentially tight (see Dembo and Zeitouni [START_REF] Dembo | Large deviations techniques and applications[END_REF], or Feng and Kurtz [START_REF] Feng | Large deviations for stochastic processes[END_REF]). We only need to show now that for a lower semicontinuous function I, and for x ∈ R

lim δ→0 lim inf n→∞ 1 n log ν n ((x -δ, x + δ)) = I(x) , (2)
lim δ→0 lim sup n→∞ 1 n log ν n ([x -δ, x + δ]) = I(x). (3)
From these, we shall infer that (ν n ) n∈N follows a large deviations principle with good rate function I. Eventually, equation (1) and

lim n→∞ 1 n log Z n (β) = p(β)
will imply, by Varadhan's lemma that I and p are Legendre conjuguate:

I(x) = p * (x) = sup β∈R (xβ -p(β)) .
The strategy of proof finds its origin in Varadahan's seminal paper [START_REF] Varadhan | Large deviations for random walks in a random environment[END_REF], and has already succesfully been applied in [START_REF]Fluctuation exponents and large deviations for directed polymers in a random environment[END_REF]. Let us define for λ > 0, x ∈ Z, a ∈ R

V (λ) n (x, a; η) = log P x e -λ|Hn(ω,η)-a| = V (λ) (0, a; τ o,x • η) ,
with τ k,x the translation operator on the environment defined by :

τ k,x • η(i, y) = η(k + i, x + y) ,
and P x the law of simple random walk starting from x.

Step 1 The functions v

(λ) n (a) = Q V (λ) (0, a; η) satisfy the inequality v (λ) n+m (a + b) ≥ v (λ) n (a) + v (λ) m (b) (n, m ∈ N; a, b ∈ R) . ( 4
)
Proof. Since |H n+m -(a + b)| ≤ |H n -b| + |(H n+m -H n ) -a| we have V (λ)
n+m (x, a; η) ≥ log P x e -λ|Hn-b| e -λ|(H n+m -Hn)-a|

= log P x e -λ|Hn-b| e V (λ) m (0,a;τ n,Sn •η) = log y P x e -λ|Hn-b| 1 (Sn=y) e V (λ) m (0,a;τn,y•η) = V (λ) n (x, b; η) + log y σ n (y)e V (λ) m (0,a;τn,y•η) ≥ V (λ) n (x, b; η) + y σ n (y)V (λ) m (0, a; τ n,y • η) (Jensen's inequality) ,
with σ n the probability measure on Z d :

σ n (y) = 1 V (λ) n (x, b; η) P x e -λ|Hn-b| 1 (Sn=y) (y ∈ Z d ) .
Observe that the random variables σ n (y) are measurable with respect to the sigma field

G n = σ(η(i, x) : i ≤ n, x ∈ Z d ), whereas the random variables V (λ)
m (0, a; τ n,y • η) are independent from G n . Hence, by stationarity,

v (λ) n+m (x, a; η) = Q V (λ) n+m (x, a; η) ≥ v (λ) n (b) + y Q(σ n (y))Q V (λ) m (0, a; τ n,y • η) = v (λ) n (b) + y Q(σ n (y))v (λ) m (a) = v (λ) n (b) + v (λ) m (a)Q y σ n (y) = v (λ) n (b) + v (λ) m (a) .
Step 2 There exists a function I (λ) : R → R + convex, non negative, Lipschitz with constant λ, such that

-lim n→∞ 1 n v (λ) n (a n ) = I (λ) (ξ) (if a n n → ξ ∈ R)) . (5)
Proof. This is a standard subadditivity argument (see e.g. Varadhan [START_REF] Varadhan | Large deviations for random walks in a random environment[END_REF] or Alexander [START_REF] Alexander | Approximation of subadditive functions and convergence rates in limiting-shape results[END_REF]) combined with the Lipschitz property of V (λ) : from |H n -a| ≤ |H n -b| + |a -b| we infer that

V (λ) n (0, a; η) ≥ V (λ)
n (0, a; η) + λ|a -b| .

Step 3 Q almost surely, for any ξ ∈ R,

lim n→∞ - 1 n log P e -λ|Hn-an| = I (λ) (ξ) . (6)
Proof. Since the functions are Lipschitz, it is enough to prove that for any fixed ξ ∈ Q, (6) holds a.s. This is where we use the restrictive assumptions made on the distribution of the environment. If the distribution of η is with bounded support, or Gaussian, or more generally satisfies a logarithmic Sobolev inequality, then it has the gaussian concentration of measure property (see [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]): for any 1-Lipschitz function F of independent random variables distributed as η,

P (|F -P (F )| ≥ r) ≤ 2e -r 2 /2 (r > 0).
It is easy to prove, as in Proposition 1.4 of [START_REF] Carmona | On the partition function of a directed polymer in a Gaussian random environment[END_REF], that the function

(η(k, x), k ≤ n, |x| ≤ n) → log P e -λ|Hn(ω,η)-a|
is Lipschitz, with respect to the euclidean norm, with Lipschitz constant at most λ √ n. Therefore, the Gaussian concentration of measure yields

Q V (λ) n (0, a; η) -v (λ) n (a) ≥ u ≤ 2e λ 2 u 2 2n .
We conclude by a Borel Cantelli argument combined with [START_REF]Fluctuation exponents and large deviations for directed polymers in a random environment[END_REF] Observe that for fixed ξ ∈ R, the function λ → I (λ) (ξ) is increasing ; we shall consider the limit:

I(ξ) = lim λ↑+∞ ↑ I (λ) (ξ)
which is by construction non negative, convex and lower semi continuous.

Step 4 The function I satisfy (2) and (3).

Proof. Given, ξ ∈ R and λ > 0, δ > 0, we have Given ξ ∈ R such that I(ξ) < +∞, and δ > 0, we have for λ > 0, P H n n ξ < δ ≥ P e -λ|Hn-nξ|e -λδn .

Hence, if we choose λ > 0 large enough such that λδ > I(ξ) ≥ I (λ) (ξ), we obtain lim inf

  η)ξ ≤ δ = P e -λn| Hn n (ω,η)-ξ| ≥ e -λnδ ≤ e λnδ P e -λ|Hn-nξ| . logν n ([ξδ, ξ + δ]) ≤ λδ -I (λ) log ν n ([ξδ, ξ + δ]) ≤ -I (λ) (ξ)and we obtain by letting λ → +∞, log ν n ([ξδ, ξ + δ]) ≤ -I(ξ) .

n→+∞ 1 n 1 n

 11 log ν n ((ξξ + δ)) ≥ -I (λ) (ξ) ≥ -I(ξ) log ν n ((ξδ, ξ + δ)) ≥ -I(ξ) .

* the author acknowledges the support of the French Ministry of Education through the ANR BLAN07-2184264 grant


J Schamps

J M Delaval

Faucher

A calculation of radiative lifetimes of electronically excited rovibronic levels of CuF

Radiative lifetimes for rovibronic levels of the excited valence states of CuF are calculated using a model including both spin orbit and Coriolis interactions. According to a discussion concerning their decay mechanisms, the spin-orbit components are classified into three classes with lifetimes differing by several orders of magnitude. Electronic transition intensity patterns are calculated and compared with spectroscopic observations.

Introduction

In a previous paper [ I J electronic transition di pole moments coupling the spin-orbit components of the valence states of CuF of Cu+ (3 d94s)F-(2p6) structure with the X 1 l: + ground state and -typically one order of magnitude smaller -electronic transi tion moments inter-coupling these excited spin-orbit components have been calculated and tabulated (ta ble 6 of ref.

[ 1 J ) . Radiative lifetimes of the valence excited states have been deduced from these transi tion moments. However this work was restricted to the spin-orbit coupling approximation, i.e. only spin orbit interaction was taken into account in perturb ing the Born-Oppenheimer electronic states. Under these conditions, it was shown that radiative decay of all the D= l ( e, f) components essentially took place towards the X 11: + ground state while the other components (D=O(f) , 2(e, f) and 3(e, f)) decayed much more slowly towards lower-lying excited va lence states. But these last components could also ra diate towards the ground state if Coriolis coupling, that mixes rovibronic levels of different �ompo nents through rotation-electronic and spin-rotation interactions, is added to the model. Indeed these ra diative channels are expected to play a significant part in the lifetimes of several of these states and Coriolis coupling need be taken into account. New wavefunc-tions were therefore determined by diagonalizing the total Hamiltonian matrix including the Coriolis part of the rotation Hamiltonian. Their reliability was successfully tested by calculating A-splitting magni tudes in the excited states [ 2] . 'Fhe very good agree ment with spectroscopic data prompted us to use these wavefunctions in further refined calculations of CuF excited state lifetimes.

Lifetime calculations

In comparison with the spin-orbit case treated in ref.

[ l ] , in which only one transition moment is in volved for each state, the mixing of several basis functions with different Q values in the new spin-or bit-Coriolis wavefunctions makes it necessary to modify formula ( 2) of ref. [ I ] . In view of this, the perturbed wavefunctions are developed in terms of the unperturbed Born-Oppenheimer basis functions: ln($All); v,WMe) = L C;in ln(SAQ);v,WME). n Here, the tildes over the S, A, Q symbols [START_REF] Herzberg | Spectra of Diatomic Molecules[END_REF] express the fact that, after matrix diagonalization, these sym bols do not designate rigorously good quantum num bers but only .. predominant" ones; n (and Ii) is an 1 integer number (or even a letter in spectroscopy) to distinguish between different states having same S, A, Q numbers; E is an index to distinguish between e ( E = + I ) and f ( E = -I ) levels. As explained in ref. [2], the C;i,, coefficients are obtained from diagon alization of the energy matrix so as to best fit the lo• cation of the as yet spectroscopically observed states.

Within the electronic dipole approximation (the only one used in this paper except otherwise stated) the transition probability, A (in s-1 ), for a sponta neous radiative transition from an excited state (ft' Q' v' J' E' ) of energy E' towards a lower-lying state (ii" Q" v" J" E") of energy E" is written [ 4-7]:

16 1t3 A;i•tJ•v'J'E'-n"t.i"v"J"<" = ---3 V�•Q•v'J'E'•n•i'J•v•J•c ' 3hEoC '
where v is the frequency and Si s the line strength of the rovibronic transition [START_REF] Herzberg | Spectra of Diatomic Molecules[END_REF], i.e. the sum of the squared modulae of the dipole moment matrix ele ments over all the M' and M" sublevels of the upper and lower states expressed in the Kronig-symme trized basis set (table 2 of ref. [2] ). Some manipu lation leads to:

= (2J" + 1) I L L Cn'n' C;i•n• n'O',n"O" q ( J ' x Q ' ]" ) 1 2 _r.,,, (n'
Q'v' lµ q ln"Q"v") -q ;/11'
The lifetime of a rovibronic (ii' Q' v' J' t') state is:

= ( L A wi'J'v'J'<';il"!J"v"J"E" ) -I w•tJ•v"J"•"
To carry out the calculations, the C;i,, coefficients were taken from table 3 of ref. [ 2]. For the dipole-mo ment transition elements the ab initio values given in tables 4 and 5 of ref.

[ 1 ] were used. Moreover, the Franck-Condon factors are assumed to be either 0

(if v' # v" ) or l (if v' = v"
) . This is justified because the potential curves in CuF are very nearly parallel (similar re and We) and because we are merely inter-ested in overall electronic lifetimes. However one should be aware that the model basically attributes zero intensity to Av# 0 bands. Furthermore it is un able to account for possible though unlikely pertur bations that might accidentally occur between inter acting high vibrational levels of different electronic states. Anyway the locations of these perturbations are indeterminable in the present status of vibra tional data accuracy in CuF. Moreover, on the small range sampled by the vibration, the electronic wave function does not change noticeably, thus it was deemed justified to neglect the dependence of the di pole moment operator on the internuclear separa tion; this would have been an illusionary refinement anyhow as far as even an approximate value of the weak transition dipole moment itself at a given sep aration is so difficult to obtain in the present case [ l ] . Under such conditions, the lifetime of a given n' Q' component does not depend on v' at all.

The ab initio lifetimes thus obtained are given in table l for three different values of Jnamely J= 5, 25 and 50. In principle, these lifetimes, considered as a function of J, should tend towards a rotationless limit equal to the values given in table 7 of ref.

[ l ] in which rotational effects had not been included into the model. However one should notice some deviations due to the use here of new revised wavefunctions [ 2] following upon the recent discovery of the A 3Il 2 component by Brazier, Brown and Steimle [START_REF] Brazier | [END_REF]. This made it possible to add the energy of this state as a new fitting constraint in the determination of the C;i,, coefficients.

In order to understand the origin of the calculated lifetimes quantitatively, spontaneous emission Ein stein coefficients participating in the lifetime values were calculated for each value of the rotational quan tum number J. An example of such coefficients is given for J"" 50 in table 2. The set of similar tables for all J values below I 00 has been used to calculate rovibronic line intensities within the branches of the 0-0 bands of the electronic transitions towards the X •:t+ ground state. This was carried out as follows. Regarding rovibronic frequencies as constant over a given vibronic transition, the intensity of a line is roughly proportional [START_REF] Herzberg | Spectra of Diatomic Molecules[END_REF] to the appropriate A (ii', Q', v ' , J', t'; ii", Q", v", J", t") Einstein coefficient (such as those given as examples in table 2 for J' = 50) and to a Boltzmann exponential factor exp [ -hcF( J) I kT] where F(J) is the rotational term value (J' for emissions J" for absorption) and Tis the absolute temperature taken equal to 300 K as as sumed by Ahmed et al. [ 9] for their hollow-cathode emission spectra. Inspection of tables 1 and 2 shows that all transition probabilities between states both of which belong to the Cu+ ( 3d9 4s) manifold are ex ceedingly small in comparison with most transition probabilities between these states and the ground state. Therefore, as for their lifetimes, it is conve nient to consider three classes of states: the short-lived states (B 1:t+ and C 1Il) with lifetimes shorter than or around l µs, the mean-lived states (Q=Oe and °'2= 1 components) with lifetimes in the 5-500 µs range, and the long-lived states (Q=Of and Q�2) with lifetimes above 1 ms. These three classes will be successively discussed in the next sections.

Short-lived states

The B 1I:+ and C 1Il states are the only ones to be able to radiate towards the X 11:: + gro�nd state within the purely electronic approximation. Although they are somewhat longer than usually found when elec tronic transitions are allowed, the lifetimes of these essentially singlet states are nevertheless at least one order of magnitude shorter than those of the other states of the cu+ (3d94s) structure. The part of Cor iolis interactions in the B 1:t+ and C 1Il wavefunc tions brings no significant relative contribution to the lifetime values which are therefore nearly constant whatever J. Accordingly the discussion about the ra diative decay mechanisms of the B and C states as given in ref.

[ l ] still remains valid: the observed val ues of the lifetimes are due to a subtle conjugation of orbital-polarization, dynamic-rearrangement and configuration-interaction effects. The look of a typi cal spectroscopic band (0-0 band) for of the C-X and B-X systems is given in figs. 1 and 2. The inten sity distributions do not exhibit marked anomalies except that the R branch appears to be a little less intense than the P branch in the B-X transition, due to interference effects involving the 3Il contamina tion in B 11: +. It is interesting to notice that these sys tems are the most intense observed ones in CuF. Nevertheless their electronic oscillator strengths are only 0.01 for C-X and 0.004 for B-X. This means that other intense electronic transitions should exist, for instance 4s-.4p ones, but these transitions have not yet been observed.

Mean-lived states

The D 3Ah A 3Ili. a 3I:t and A 3Il0e states consti tute the second class. Their transition probabilities towards the ground state result from spin-orbit inter actions with C 1Il for the first three states or with B 1 I:+ for the 3Il0e state. It should be noticed that, al though it is small, the 3Iloe character of the X 1I:+ ground state also contributes for about 10% in the A 3Il0e-X 1I:+ transition probability as a result of the large value of the diagonal 3Il dipole moment [ l ] . ) . For the transitions between non-! states (i.e. A� 0 ), the sill Einstein coefficients are arranged on two lines {the upper one for the case of an e upper level, the lower one for that of an f upper level) following the pattern:

AJ+i,e;J •• (R .. ) A1-1.e;J,e(P .. ) AJ,e;J,r(Qa-) AJ+1,r;J,r(R1J) A1-1,r;J,r(P1J) A1.r;J .• (Qre)

(Letters between parentheses refer to conventional spectroscopic branch names [START_REF] Herzberg | Spectra of Diatomic Molecules[END_REF]. Blanks mean that corresponding coefficients are smaller than 5 s • . Note also that because of their accidental quasi-de generacy the A 3Iloe and A 3Il1e states mix through ro tational spin-uncoupling so that their lifetimes, no-ticeably different at low J, get closer to each other as J increases and should tend to a common value at much higher J. On the other hand, the 3Il1r compo nent, which is much farther (at least ten times) in energy from the 3�one than 3II1e is from 3Iloe, keeps a lifetime that does not depend on J. It would be in teresting to carry out rotationally resolved lifetime measurements so as to try and observe these effects .

- 1 or even non-existent) l 3d1) l3d2) l3d 3 ) 11n1> I
Similar comments apply to the case of the a 31: + state:

since there is no e component for a 3I:t state, the a 3I: � component has a lifetime that does not depend on J whereas the progressively enhanced rotational mixing between the A 3I:(f and A 3I:c1f components leads to /-dependent lifetimes for these f components and makes spectroscopic observation of the 3.It component possible [ 10] . Lastly note that the pres ent calculations gratifyingly account for the intensity anomalies (with respect to the standard -"'3PU. .42 perimentally observed feature namely the fact that intensity maximum in the R-branch occurs at lower J value (about ten units less) than in the P and Q branches of the A 3Il 1 -X 1!:+ subband, is correctly reproduced by the calculations. Similar calculated in tensity patterns for the a 3!:+-x 1I:+ and D 3A 1 -X 1!:+ transitions are given in figs. 5 and 6 respectively. As far as they can be compared with the spectrum pho tographic reproduction given by Ahmed et al. [ 9], the relative branch intensities in fig. 5 are in good agreement with experiment.

Long-lived states

The a 3!:6 , A 3Ilorcomponents and all the !2= 2 and !2=3 components, i.e. A 3Il2, D3A2, D3A 3 and 11'.\2 constitute the third class. In the non-rotating approx imation, these states cannot radiate towards the X 1 !: + ground state but only towards other excited states with transition probabilities smaller by several orders of magnitude [ 1 ] than those transition probabilities considered above for the D=Oe and !2= I compo nents. This constitutes a first radiative decay mechanism.

Taking into account rotational motion brings some 1Il 1 character into the wavefunctions of these meta stable states and therefore leads to a second radiative decay mechanism that takes place towards the ground state and accordingly makes the corresponding life time a decreasing function of J.

For so weak electric-dipolar transition probabili ties it was deemed necessary to ascertain that a third mechanism, namely electric quadrupolar radiation, could still be neglected. This is all the more impor tant as the observed molecular transitions in CuF have been shown [ l 0] to be connected with electric dipole-forbidden but electric-quadrupole-allowed 3d-4s transitions in the cu+ atomic limit. The as yet unobserved 1.-i-X 1I+ bands which are due to a spin allowed dipolar-electronically forbidden transition are particularly concerned. Using previously deter mined orbital wavefunctions [ l ) an ab initio calcu lation of the associated quadrupole -electronic-tran sition moment was carried out. The obtained value was small, about 0.5 (in aij e units), from which it can be deduced [START_REF] Shore | Principles of Atomic Spectra[END_REF] a quadrupolar spontaneous emission probability of only 0.033 s-1• This means that quadrupolar radiation can be neglected com pared to the two first mechanisms quoted above, namely dipole-allowed electronic transitions be tween close-lying excited states competing with Cor iolis-induced transitions towards the ground state.

Which of these two mechanisms dominates de pends on the state under consideration. For instance, the radiative decay of the A 3Ilor and D 3A 3 states is essentially due to the first mechanism (electronically allowed transitions): the A 3Ilor state decays toward a 3It and the D 3..i 3 state towards A 3Il2. It is worth while noting that the lifetimes of the A 3Ilorand D 3 .:i 3 states are nevertheless strongly J-dependent. This unexpected effect results from Coriolis contamina tion of the wavefunctions. Although the -J-depen dent -C;l'n coefficients due to this contamination are weak, they introduce terms involving large diagonal dipole-moment elements in the calculations of the line-strengths (see above the expression for S), so large that, in net quantitative magnitude, these terms outmatch the supposedly first-order J-independent contributions of the very weak 3Il-3I+ and 3.:i-3Il electronic transition moments. It is also because of the predominant part of these diagonal dipole-mo ment elements that the as yet unobserved A 3Il01a 3It and D 3.:irA 3Il2 transitions should be expected to exhibit completely anomalous branch in tensity patterns (see table 2) with a very strong asymmetry between P and R branches. In any case, the transition probabilities between both A 3Ilor and D 3A 3 upper states and the X 1I+ ground state are calculated to be exceedingly small and, contrary to Brazier's expectations for A 3Ilor [START_REF] Brazier | [END_REF], it is undoubt edly impossible to observe the corresponding transi-tions through even sophisticated usual spectroscopic methods. On the opposite, as already mentioned in section 4, radiative relaxation of the a 3It state is only due to spin-uncoupling mixing with the close lying a 3It state from which it borrows the 1Il 1 char acter necessary to radiate towards the ground state. Lastly, radiative relaxation of the three !2=2 com ponents takes place through both mechanisms: not only do they decay towards lower-lying states of the same cu+ (3d94s) structure ( 1 .:i2 towards D 3.:i2 and C 1Il 1 ; D 3.:i2towardsA 3Il 1

;A 3Il2towards a 3It) but as soon as J reaches sufficiently high values, these three !J= 2 components can also decay towards the x I I+ ground state, owing to their weak I n I charac ter borrowed through rotational uncoupling. Thus, at J = 50, this last mechanism participates for 40% in the 1 a2 lifetime, for 50% in the D 3 A2 one and for 75% in the A 3Il2 one.

Indeed this 3Il2 state has been observed via laser excitation spectroscopy [START_REF] Brazier | [END_REF], a now usual experimen tal method in which X 1I+ -+A 3Il2 absorption is de tected on looking at the overall light reemitted after more or less complex radiative and collision pro cesses. Fig. 7 represents the calculated intensity pat tern of the 0-0 band of this A 3Ili-X 1 I+ system. The ratio between the intensities of the Q and R branches is satisfactorily found to be the same as in Brazier et al. 's spectroscopic observations [START_REF] Brazier | [END_REF]. But there ap pears a drastic discrepancy as far as the P branch is considered. This raises an interesting point that de serves special discussion. Brazier et al. [START_REF] Brazier | [END_REF] report ob servation of some very faint P lines they estimate to be about five times less intense than corresponding R lines. On the contrary, present calculations predict P branch intensity to be nearly equal to and even slightly more intense than R-branch intensity. Besides this could only but be expected inasmuch as the A 3I l2-X 1:t+ transition mainly gets its intensity from the A 3Il 1 -X 1:t+ one through Coriolis interaction so that it should reflect the P-over-R intensity preponder ance both observed and calculated for this band sys tem (section 4, fig. 4). Admittedly this preponder ance is much attenuated in the A 302-X 1:t+ transition where interference effects are less marked than in the A 3Il 1 -X 1t+ one. But at any rate, from detailed analysis of the contributions to branch in tensities, P-R intensity inversion from one system to the other definitely cannot occur within the present model. On view of this one could indeed wonder whether a limit of this model has not been reached in this particularly delicate case. However extension of the model via admixture of the weak [ l ] configura tion interaction coming from structures other than ( 3d9 4s), for instance ( 3d9 4p), does not bring any significant change: as they stand, the results are re markably stable with respect to manageable im provement of the model. Therefore it is necessary to question about the possibility of an accidental arte fact that would arise from the use of so tricky a method as laser excitation spectroscopy and that would accordingly invalidate comparison between calculated and observed intensities. Indeed it should be noticed that it is odd to say the least that the P branch intensity is so low whereas that of the R branch is normal as compared to the Q-branch one. This ar tefact could not be asymmetric quenching depopula tion of the excited levels since both P and R lines, which necessarily originate from the same e rovi bronic levels, would be affected in exactly the same way. But it could be a selective radiative depopula tion effect: one could imagine for instance a two-step X 1I: +-A 302-+? process whose second step would be resonant when the laser pump is set at P-line wave lengths but off-resonant when it is at R-line wave lengths. For P-line pumping the probe fluorescence would then be mainly ultraviolet radiation (possibly out of the detector range) or even missing if higher states are dissociative or predissociated while it would be (well detected) visible radiation for R-line pump ing. The question of the A 3Ili-X 1:t+ intensity pat tern remains open and new experiments would be welcome.

.•12 i •-
From our calculations, the X 1 I:+ -1 d2 and X 1 I:+ -D 3 a2 transition moments are predicted to be of sim ilar (weak) magnitude as that of the X 1:t+ -A 3Il2one (calculated values at J = 50: respectively 2, 3 and 4X 10-3au). Figs. 8 and9 show the calculated inten sity patterns for these two transitions that again ap pear to have standard appearance. Moreover due to their higher frequency they are expected to be respec tively four times and twice as intense as the A 3Ili X 1 I:+ transition. Therefore the present calculations should give an impetus to spectroscopists in order to try and detect the I a2 and D 3 a2 states by the same method as the one used to discover the A 3Il2 state [START_REF] Brazier | [END_REF], i.e. laser excitation spectroscopy from the ground state even though the predicted spectral regions [ 10] are experimentally less easy to reach.

Conclusion

This calculation of rotationally resolved lifetimes ... states and they may appreciably depopulate these states long before they radiate. Calculation of non radiative state-to-state transfer rates would be quite useful for a complete assessment of the dynamical processes governing the behaviour of CuF excited states. However, for practical reasons this is out of the question for the time being.

Fig. 1 .

 1 Fig. 1. Calculated intensity distribution in brancltes of the 0-0 band ofC 1Il-X 11:+. T::::300 K; only every fifth rotational line is depicted. Intensity units are arbitrary but they are the same for all figures. Wavenumber scales are given for information only since the model does not include second-order corrections such as centrifugal distortion.

Fig. 2 .

 2 Fig. 2. Calculated intensity distribution in branches of the 0-0 band ofB 1I.+ -X 11:+ (see caption of fig. I).

Fig. 3 .

 3 Fig. 3. Calculated intensity distribution in branches of the 0-0 band of A 3fl-O.,-X 1 .t+ (see caption of fig. l ) .

2 • . I I I I 11 Fig. 4 .

 2114 Fig. 4. Calculated intensity distribution in branches of the 0-0 band of A 3Il 1 -X 1:r+ (see caption of fig. 1 ).

Fig. 5 .

 5 Fig. 5. Calculated intensity distribution in branches of the 0-0 band ofa 3I+-x 1I+ (see caption of fi g. I).

Fig. 6 .

 6 Fig.6. Calculated intensity distribution in branches of the 0-0 band ofD 3.i,-X 1I+ (see caption of fi g. l ).

Fig. 7 .

 7 Fig. 7. Calculated intensity distribution in branches of the 0-0 band of A 3Il2-X 1l: + (see caption of fig. 1 ).

Fig. 8 .Fig. 9 .A

 89 Fig. 8. Calculated intensity distribution in branches of the 0-0 of 1.:1rX •:r+ (see caption of fig. 1).

Table 1

 1 Calculated radiative lifetimes (in µs) for selected rovibronic J

	levels of the valence states of CuF
		J=5	J=25	J=50
	1A2e	2.79xl03	2.99Xl03	l.47X 103
	f	2.79x103	2.99xt03	l.47X 103
	3A1 e	6.01	5.97	5.87
	f	6.00	5.97	5.88
	3A2e	9.86X 103	6.50Xl03	3.l3X103
	f	9.86X 103	6.50x103	3.14X IG3
	e 3A 3 f	14.4x 103 14.4X 103	12.l x 103 12.l x 103	8.0x 1G3 8.0xl03
	c1ne	0.400	0.401	0.403
	f	0.399	0.400	0.401
	B 1:t+ e	1.23	l.22	1.21
	A 3 I loe	5.56	5.85	6.39
	f	733XI03	90.8x 103	24.7Xl03
	A 3 Il 1 e f	13.9 14.6	12.4 14.6	10.5 14.6
	A 3 I I2 e	417Xl03	17.5x103	4.39x103
	f	418Xl03	17.7x103	4.53x103
	a 31:6 f	4893	505	330
	a 3l:( e	99	99	98
	f	101	122	140

Table 2

 2

	50 (their sums for each upper-state yield the inverses of the lifetimes
	J' =
	(in s-1) for rovibronic transitions of CuF at
	Spontaneous emission Einstein coefficients	given in last column of table I


Emile Melki
email: emilemelki@hotmail.com

Henri Fares

Aksam Khaddam

Rana Khaddam

Transvaginal Low Intensity Shockwave Therapy in Endometriosis (T-LISTE) : protocol for a pilot trial

Keywords: Endometriosis, Adenomyosis, Pelvic pain, Dysmenorrhea, Dyspareunia, Shockwave Therapy SWT (shockwave therapy), DIE (deep infiltrating endometriosis), HIFU

Background

Endometriosis is a frequent disorder in women of childbearing age. Standard medical and surgical treatments are often unsatisfactory. Shockwave therapy has been suggested as a possible novel treatment. There is no clinical data however to support this approach. This article describes the research protocol for a pilot interventional trial aiming to evaluate transvaginal low intensity shockwave therapy in women with endometriosis.

Methods

Adult women (anticipated N = 60) suffering from endometriosis will be randomly allocated in a 1:1 ratio to an intervention group and a control group. The intervention group will receive 4 sessions of transvaginal low intensity radial shockwave therapy. The control group will receive 4 sham procedures. Assessment will be carried at 1, 3 and 6 months. The primary endpoint is the visual analog score for pelvic pain.

Conclusion

This monocentric single blinded randomized controlled pilot trial will be the first to assess transvaginal low intensity shockwave therapy as a novel treatment for endometriosis.

INTRODUCTION

Endometriosis is defined as the ectopic localisation of endometrium. Prevalence is estimated at 10 % to 15 % of women of reproductive age and 70 % of women with chronic pelvic pain [START_REF] Parasar | Endometriosis: Epidemiology, Diagnosis and Clinical Management[END_REF]. Symptoms include dysmenorrhea, dyspareunia, bladder and rectal pain that can severely alter quality of life. 11% to 19% of women with endometriosis derive no pain relief at all from medical therapy [START_REF] Becker | Reevaluating response and failure of medical treatment of endometriosis: a systematic review[END_REF]. Surgery, especially excision of deep infiltrating endometriosis (DIE) can be challenging, and complications rates can be high. Major complications after colorectal segmental resection of bowel endometriosis occurs in 11 % of patients [START_REF] Cicco | Bowel resection for deep endometriosis: a systematic review[END_REF]. Moreover, long-term urinary and digestive side effects are common following surgery [START_REF] Dubernard | Urinary complications after surgery for posterior deep infiltrating endometriosis are related to the extent of dissection and to uterosacral ligaments resection[END_REF] [START_REF] Roman | Postoperative digestive function after radical versus conservative surgical philosophy for deep endometriosis infiltrating the rectum[END_REF]. There is a need for new, effective and safe therapeutic methods. High intensity focalized ultrasound (HIFU) and radiofrequency ablation are being evaluated [6][7][8] [START_REF] Guo | High intensity focused ultrasound treatment of adenomyosis: a comparative study[END_REF][10] [START_REF] Dubernard | Transrectal high-intensity focused ultrasound as focal therapy for posterior deep infiltrating endometriosis[END_REF]. Shockwave therapy (SWT) has been suggested as a novel approach to treat endometriosis.

RATIONALE

The theoretical basis for the use of SWT in endometriosis has been developed recently [START_REF] Melki | Endometriosis treatment with shock waves: A novel approach[END_REF].

Basically high intensity focalized shockwaves could be directed to the lesions to obtain a disruptive effect, or low intensity radial or focused shockwaves could be applied to achieve pain relief via neuromodulatory, antioxydant and antiinflammatory effects. This trial aims to evaluate the latter approach. The role of myofascial pain in endometriosis has been highlighted [START_REF] Aredo | Relating Chronic Pelvic Pain and Endometriosis to Signs of Sensitization and Myofascial Pain and Dysfunction[END_REF], and could explain standard medical and surgical treatment failure in some patients. By targeting pelvic myofascial trigger points, the nociceptive sensitization feedback loop could be interrupted. SWT should be considered as it proved successful in musculoskeletal diseases with a favourable tolerance profile [START_REF] Gesginaslan | High-Energy Flux Density Extracorporeal Shock Wave Therapy Versus Traditional Physical Therapy Modalities in Myofascial Pain Syndrome: A Randomized-controlled, Single-Blind Trial[END_REF][15] [16][17][18] [START_REF] Walewicz | Effect of Radial Extracorporeal Shock Wave Therapy on Pain Intensity, Functional Efficiency, and Postural Control Parameters in Patients with Chronic Low Back Pain: A Randomized Clinical Trial[END_REF]. Good outcomes have also been achieved in chronic pelvic pain syndrome/chronic prostatitis in males [START_REF] Vahdatpour | Efficacy of extracorporeal shock wave therapy for the treatment of chronic pelvic pain syndrome: a randomized, controlled trial[END_REF] [START_REF] Zhang | Efficacy of Radial Extracorporeal Shock Wave Therapy for Chronic Pelvic Pain Syndrome: A Nonrandomized Controlled Trial[END_REF].

RESEARCH OBJECTIVE

To evaluate the feasibility of a prospective, single-blinded, randomized, controlled trial of transvaginal low intensity SWT as a treatment for endometriosis.

METHODS

Study setting

Monocentric (private hospital in Fort-de-France, Martinique, French West Indies).

Eligibility criteria Inclusion criteria

Women aged 18 years and older, with clinical symptoms and imaging findings on ultrasound scan and MRI confirming the diagnosis of endometriosis. Ovarian endometriomas, deep endometriosis and adenomyosis meet the inclusion criteria. Histological confirmation is not mandatory. Medical treatment (oral progestatives or oestro-progestatives, GnRH analogs, progestin-releasing IUDs) should be deemed unsatisfactory thus warranting the intervention but should not be discontinued during the trial. Pain-relief medication can be prescribed as recommended by standard of care practice.

Exclusion criteria

Women unable to give consent, to understand written and oral French, or to comply with follow-up.

Virgin women.

Women with ongoing uro-genital infection.

Pregnant women (a pregnancy test should be performed 72 hours at least before each session).

Women undergoing fertility treatment.

History of segmental rectal resection with mechanical anastomosis, shaving or discoid resection.

Women with ureterohydrephrosis warranting ureteral stenting and surgical treatment.

History of another pelvic surgery in the last 6 months.

ESSURE tubal contraceptive implants.

Baseline characteristics

Age (< 30, 30-45, > 45 years), Body Mass Index (BMI) (< 18, 18-25, 25-30, > 30), baseline pain level (VAS < 5, 5-7, > 7), menstrual cycle pattern (complete induced amenorrhea, irregular cycle pattern, regular cycle), history of surgery for endometriosis, history of infertility treatment or assisted reproductive technology (ART), parity (para 0, para 1, para 2 or more), ongoing treatment (oral progestatives or oestro-progestatives, GnRH analogs, progestin-releasing IUDs).

Sample size

We aim to recruit 30 patients in each group, according to Browne's recommendations for sample size of pilot trials [START_REF] Browne | On the use of a pilot sample for sample size determination[END_REF]. The results of this study will facilitate the calculation of an appropriate sample size for further randomized clinical trials.

Randomization

Patients will be allocated to the two groups in a 1:1 ratio using block randomization with a computer-generated list of random numbers using block sizes of four.

Consent

All participants will be given clear oral and written information about the trial and will give written consent. Consent can be withdrawn at any moment.

Tolerance will be assessed by the analysis of adverse events (pelvic, vulvo-vaginal, urinary, digestive or other events).

Statistical analysis

Analysis will be carried out on an intention-to-treat (ITT) basis and will be performed using SPSS 23.0 statistical software (IBM SPSS Statistics, New York, USA). The distributed variables will be compared by using the student t test. Statistical analysis of continuous values will be carried through the independent samples t test. The Mann-Whitney U test and the χ 2 test will be used for the comparison of quantitative data. P < .05 will be considered statistically significant.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

DISCUSSION

Efficacy and safety of SWT as a treatment for endometriosis have yet to be demonstrated.

We believe a pilot human trial is ethically and scientifically acceptable, without prior animal experimentation, as this technique is widely used in other indications worldwide in standard clinical settings. The goal is primarily to validate the feasibility of a future larger trial by assessing tolerability, randomization acceptance and implementation, follow-up and evaluation. Although probably underpowered to detect a small difference between the two groups, collected data will help with future study designs (for exemple to determin sample sizes). Subgroup analysis will obviously need larger trials (comparison between DIE, adenomyosis and ovarian endometrioma groups, comparaison between different hormone therapy groups etc.). Technically, the choice of low intensity (equivalent to 0,09 mJ/mm 2) and radial shockwaves aims to maximize tolerability and safety (no systematic anesthesia requirement). The larger biological effect cone of radial shockwaves versus focalized shockwaves can be seen as an asset, as ultraprecise targeting is not required.

Ultrasound targeting of endometrial lesions is possible but remains operator-dependent, time-consuming and increases costs. Radial SWT yelded the same results as focalized SWT in the treatment of muscle spasticity after stroke [START_REF] Wu | Comparison of the effect of focused and radial extracorporeal shock waves on spastic equinus in patients with stroke: a randomized controlled trial[END_REF]. Further evaluation of energy intensities and focalization is needed. In a trial of focused SWT in myofascial pain, both high and low intensity treatment regimens had good outcomes, albeit with superior functional improvement in the high intensity group [START_REF] Park | High-versus low-energy extracorporeal shock-wave therapy for myofascial pain syndrome of upper trapezius[END_REF]. Other questions include the number of optimal repeat procedures, the role of SWT before surgery and ART, and fertility outcomes.

The authors do not declare any conflict of interest. BIBLIOGRAPHY

Authorization

The protocol will be submitted to the regional research ethical committee (comité de protection des personnes CPP Sud-Ouest et Outre-Mer III, Bordeaux).

Intervention

Transvaginal low intensity SWT will be applied on a weekly basis for 4 weeks, using a Masterpuls ® One system with a Sparrow ® handpiece (Storz Medical, Switzerland). Each session will consist of a systematic application of radial shockwaves at the level of the cervix, at the mid vaginal level and at the lower vaginal level. Targeted applications will be directed to the vaginal fornices or vaginal walls depending on the identified lesions (i.e. posterior vaginal fornice in case of a posterior nodule) and trigger point localization by the probe pressure (at each level : 2 bars, 720 pulses, 5 Hz, 2 minutes). A sham procedure will be carried out in the control group (blinded to the patient only). The typical sound signature of SWT should be maintained during the sham procedure. Assessment Clinical evaluation will be carried before treatment, at 1 month, 3 months and 6 months using validated questionnaires.

Primary endpoint : Pain level (Visal Analog Scale) in dysmenorrhea and chronic pelvic pain.

Secondary endpoints

Quality of life (French version of EHP-5 [START_REF] Renouvel | Linguistic adaptation of the endometriosis health profile 5: EHP 5][END_REF]).

Digestive symptoms (French version of Kess questionnaire [START_REF] Snfcp | Recommandations pour la pratique clinique de la prise en charge de la constipation[END_REF]). Urinary symptoms (USP questionnaire [START_REF] Haab | Comprehensive evaluation of bladder and urethral dysfunction symptoms: development and psychometric validation of the Urinary Symptom Profile (USP) questionnaire[END_REF]).

Sexual function (French version of FSFI questionnaire [START_REF] Wylomanski | Psychometric properties of the French Female Sexual Function Index (FSFI)[END_REF]) Use of pain relief medication (type and frequency per week)


Roberto Speicys Cardoso

Mauro Caporuscio

Exploring Multi-path Communication in Hybrid Mobile Ad Hoc Networks

Keywords: Multi-path routing, middleware, ad hoc networks

Ambient computing requires the integration of multiple mobile heterogeneous networks. Multi-path communication, in such scenarios, can provide reliability and privacy benefits. Even though the properties of multi-path routing have been already extensively studied and a number of algorithms have been proposed, implementation of such techniques can be tricky, particularly when resource-constrained nodes are connected to each other through hybrid networks with different characteristics.

In this paper we discuss the challenges involved in implementing multipath communication on a middleware for hybrid mobile ad hoc networks. We present the PLASTIC middleware, some compelling applications of multi-path communication and the main issues concerning their implementation as a middleware-provided communication primitive.

Introduction

Ambient computing requires the seamless integration of heterogenous networks. Resources in an environment may be available through independent networks using different technologies, and users must be able to access them regardless of communication heterogeneity. For instance, a group of collocated resources (e.g. a printer and a projector) may be connected through a Bluetooth network while another remote resource is only accessible through WiFi (e.g. a file server). Fortunately, the convergence of multiple network interfaces into a single mobile user device can greatly simplify this task. Current cellphones featuring heterogeneous network interfaces (e.g. Wi-Fi, Bluetooth and cellular 3G) can provide users with access not only to resources on networks directly connected to the device but also by forming ad hoc networks and accessing resources on remote networks through other mobile devices.

We call these mobile networks that use heterogeneous wireless technologies hybrid mobile ad hoc networks (HMANETs). They are typically formed by independently managed networks connected to each other by multi-homed devices (also referred to as bridges). All nodes in such networks are potentially mobile (as opposed to wireless mesh networks [START_REF] Akyildiz | Wireless Mesh Networks: A Survey[END_REF]) and any two nodes may be directly connected by multiple different links (in opposition to traditional mobile ad hoc networks (MANETs) where any two nodes share at most one direct connection). More importantly, those connections use different technologies and present heterogeneous properties such as delay, throughput, security, energy consumption and cost. Figure 1 shows the differences between MANETs and HMANETs.

WiFi Interface

WiFi Interface Bluetooth Interface B3G Cellular Interface Fig. 1. A MANET and a HMANET One consequence of the network interface diversity is that there are potentially multiple paths, with different properties, between the source and destination nodes of a message. Applications can take advantage of this path redundancy by using multiple paths for a single communication. Nodes can send the same message through different paths to tolerate unpredictable connection failures, which is particularly important in networks that present unstable structures. Nodes can also send parts of a message through different links to enhance communication privacy and resistance against eavesdropping. Messages can be divided into shares in such a way that an attacker has to control a certain number of shares to reconstruct the whole message.

However interesting the multi-path approach might be, the implementation of point-to-point multi-path communication on a hybrid mobile ad hoc network is complex. Our goal is to enable a node possessing multiple interfaces to establish a session with another node equipped with multiple interfaces regardless of the IP addresses they use to communicate; messages that belong to the same session may have different source and destination IP addresses. The Internet Protocol suite, however, does not natively support such communication through multiple paths. It is not possible, for instance, to create a single unicast TCP session that spans multiple source and destination addresses. Even when using connection-less UDP, application-layer code is necessary to determine that packets containing different source and destination IP addresses are part of the same session. Multicast can be used to send the same message to several destination addresses through multiple paths, but it does not support sending from different source IP addresses nor sending different parts of the same message to multiple destination addresses. Additionally, the topology of a HMANET changes frequently and multi-path communication must adapt dynamically to the properties of available connections.

Such characteristics call for a middleware approach for multi-path routing. By managing multi-path routes on a layer above the network, the middleware can profit from the routing protocols already implemented and running locally on each network and create an overlay network responsible for routing packets among hybrid networks. Multi-path routing on this overlay network can take into account higher-level properties such as reliability, trust and security of nodes and networks traversed by packets. The middleware can also easily manage sessions comprising different overlay connections between nodes on hybrid networks.

In this paper we reflect on the practical challenges involved in providing multi-path communication capabilities as part of a middleware for hybrid mobile ad hoc networks. Many research efforts studied the properties of multi-path routing in MANETs [START_REF] Mueller | Performance Tools and Applications to Networked Systems, chapter Multipath Routing in Mobile Ad Hoc Networks: Issues and Challenges[END_REF], and proposed applications such as using multiple path routing for improving the quality of service [START_REF] Liao | A Multi-Path QoS Routing Protocol in a Wireless Mobile Ad Hoc Network[END_REF] and the use of multiple routes for load balancing [START_REF] Pearlman | On the Impact of Alternate Path Routing for Load Balancing in Mobile Ad Hoc Networks[END_REF]. In this paper, however, we investigate scenarios where we are more interested in guaranteeing that a message will reach its destination reliably and privately. For this reason, we will focus on two types of multi-path communication applications: message redundancy to improve communication reliability and message sharing to improve communication privacy.

The paper is organized as follows. In Section 2 we present the PLASTIC middleware, which we will use as a case study for describing implementation issues. In Section 3 we discuss some applications of multi-path communications that are particularly compelling in HMANETs and in Section 4 we detail the issues concerning implementation of such features on the middleware layer. Finally, in Section 5 we present our conclusions and future perspectives.

PLASTIC Overview

The PLASTIC project aims at developing a comprehensive provisioning platform for software services deployed over B3G networks1 . The project builds upon both Web services and standard component-based technologies and integrates methods and tools for service development, from design to validation, and a supporting middleware for service provisioning in B3G networks. In particular, the main objectives for the PLASTIC middleware are (1) allowing the deployment of services over a large diversity of terminals, including (mobile) wireless, resource-constrained ones and (2) supporting advanced functionalities for mobile adaptable services, i.e.: context-aware service management, trust and security management, SLA enforcement, and information dissemination.

Towards the first objective, the PLASTIC middleware (showed in Figure 2) builds upon the Web Service Architecture, so as to benefit from the perva-Fig. 2. PLASTIC Middleware sive nature of Web technologies that makes them available in most digital environments (B3G SOAP layer). The PLASTIC middleware assumes an all-IP environment but without global routing, and enables the effective exploitation of B3G networking capabilities by composing the various networks in reach to improve availability of services and to offer seamless mobility. This specifically calls for routing protocols for the B3G network (Multi-network routing), where routing should meet requirements associated with both the communication protocols used at the application level and the features of the underlying composed networks (B3G unicast and B3G multicast). Also, the middleware manages the various radio interfaces (Multi-radio Device Management) that are embedded on wireless devices, offering the abstraction of an integrated multi-radio interface to the software services of the upper layers (Multi-radio networking). Such abstraction increases the quality of service access by exploiting the underlying redundancy of B3G network connectivity that results from having distinct radio network links directly connecting two nodes in a transparent way for users.

With respect to the work presented in this paper, we are interested in the Multi-radio Networking layer. Its primary role is to enable the following core functionalities: (1) PLASTIC-Address management, (2) providing communication facilities and (3) interface activation and network selection. In order to identify an application in the network, it is associated to a PLASTIC-Address, a unique identifier that resolves into the actual set of IP addresses bound to the device hosting the application. Upper layers can use this address instead of the traditional IP-based addressing scheme. The PLASTIC-Address is automatically generated and managed by the Multi-radio Networking layer. This layer also provides two types of communication facilities: (i) synchronous unicast is used to read/write packets exchanged during the interaction between client and user applications, and (ii) asynchronous multicast allows the user application to send multicast packets to all members of a group. Finally, this layer also activates and selects the best possible networks (among those available) with respect to application-and user-required QoS level.

Multi-path Communication Applications

As discussed in Section 2, we assume an environment with multiple highly heterogeneous networks running the IP protocol, but without global IP routing. Devices featuring multiple network interfaces can independently form ad hoc networks and some of them may volunteer to route packets between different networks, creating an overlay network formed by bridges and heterogeneous wireless networks. This overlay network can provide various independent paths between a source and a destination, that can be explored to balance connection instability.

The goal of the multi-path communication middleware component is to improve the existing PLASTIC single path communication with multiple path routing. Multi-path communication is used autonomously by the middleware based on user-defined criteria, and exposed to applications through a middleware API. This allows applications to use reliable or privacy-preserving communication for critical messages only, reducing the performance cost of multi-path routing.

Different types of route exist on a network. According to [START_REF] Mueller | Performance Tools and Applications to Networked Systems, chapter Multipath Routing in Mobile Ad Hoc Networks: Issues and Challenges[END_REF], routes can be node-disjoint, that have no links or nodes in common, link-disjoint, that have no links in common and non-disjoint, that may have nodes and links in common. In HMANETs, however, we consider two different types of nodes: network nodes and bridge nodes. A route in a HMANET, then, alternates between bridge nodes and network nodes and as such we can have bridge-disjoint routes (do not have bridges in common), network-disjoint routes (do not have networks in common) and totally-disjoint routes. Even though totally-disjoint routes offer stronger reliability, network-disjoint and bridge-disjoint routes are also important. Networks on a HMANET are heterogeneous and present different levels of stability, so network-disjoint paths help to establish a more reliable communication channel. Bridges are also unstable because they are voluntary and may move or cease to work as a bridge so bridge-disjoint paths are necessary to provide alternative routes.

To improve communication reliability, the middleware uses message redundancy. The goal of message redundancy is to send the same message across different paths to increase the probability that it will reach its destination. Based on the characteristics of available routes (such as reliability, latency or throughput) the middleware must decide how many additional routes are required to achieve a certain reliability. The middleware must also dynamically determine which type of route (bridge-disjoint, network-disjoint or totally-disjoint) is necessary to obtain a reliable communication channel.

Message sharing can enhance communication confidentiality and improve privacy. The idea of message sharing is to divide a message into multiple parts that can be sent through different communication channels in such a way that, to recover the contents of a message, an attacker must control a certain number of shares. By spreading those shares through channels with heterogeneous properties, attackers have to use more resources to access the contents of a message than in the traditional single-path communication model. This property can be obtained through a (M, N ) coding algorithm such as [START_REF] Ayanoglu | Diversity Coding for Transparent Self-Healing and Fault-Tolerant Communication Networks[END_REF], where N is the total number of shares and M is the number of shares required to obtain the message. Coding algorithms have the advantage of causing a small overhead as each coded share is smaller than the original message. However, each share reveals something about the whole message, and an adversary controlling X messages, X < M , can guess the M -X remaining shares and recover the whole message, which is undesirable for privacy protection.

Another possibility is to use a (t, n) secret-sharing scheme [START_REF] Shamir | How to Share a Secret[END_REF] where n is the total number of shares and t is the number of shares necessary to recover the whole message. Protocols such as [START_REF] Lou | SPREAD: Enhancing Data Confidentiality in Mobile Ad Hoc Networks[END_REF] that perform multi-path routing using traditional secret sharing schemes, however, incur on an excessive overhead because in those schemes each share must be at least as big as the message itself [START_REF] Csirmaz | The Size of a Share Must Be Large[END_REF]. An attractive solution is to use a scheme with computational secrecy [START_REF] Krawczyk | Secret Sharing Made Short[END_REF] instead of a scheme with perfect secrecy. A secret sharing scheme with computational secrecy has the advantage of resulting in an overhead comparable to coding algorithms but still providing secrecy against attacks from resource-bounded adversaries.

Whenever a message must be shared, the middleware has to decide which paths to use, in how many parts divide a message and how many shares will be necessary to recover the whole message. Path choice involves quality and reliability information, but can also take into account the software platform of each device or the technology of each network in the paths between the message source and its destination. Those parameters allow computation of paths that resist to attacks in different software platforms or network technologies [START_REF] Desmedt | Revisiting Colored Networks and Privacy Preserving Censorship[END_REF]. Since security vulnerabilities are usually platform-dependent, this strategy can assure that the communication remains private even in face of malicious attacks against vulnerable platforms.

Implementation Challenges in the Context of PLASTIC

Implementation of message redundancy and message sharing in hybrid mobile ad hoc networks presents many practical challenges. There are issues related to the physical wireless layer, for instance how to avoid interferences when simultaneously using multiple wireless interfaces or to the architecture such as the integration of multi-path routing with service discovery to rank results according to multi-path availability. In this section, however, we discuss the challenges related to the implementation of multi-path routing as a middleware-provided communication primitive.

Routing Protocol Each network integrated in a HMANET is autonomously organized and runs the routing protocol better adapted to its local requirements. As such, it is unfeasible to define a single routing protocol to be executed by all devices on a HMANET, including bridges. Rather, a more suitable solution to allow for packet routing among networks is to create an overlay network containing bridges that run a bridge-to-bridge routing protocol while local delivery on each network is performed using the network-specific protocol. To explore multi-path routing on the overlay network, thus, each bridge must learn from the routing protocol all available paths to a destination. Traditional MANET routing protocols, however, keep only a single path between nodes, more specifically the path with the smallest number of hops [START_REF] Perkins | Ad-hoc on-demand distance vector routing[END_REF][START_REF] Johnson | Ad hoc networking, chapter DSR: the Dynamic Source Routing Protocol for Multihop Wireless Ad Hoc Networks[END_REF][START_REF] Jacquet | Optimized Link State Routing Protocol for Ad Hoc Networks[END_REF]. Multi-path extensions to those protocols exist, but some of them only use a secondary route if the main route breaks [START_REF] Lee | AODV-BR: Backup Routing in Ad Hoc Networks[END_REF][START_REF] Marina | On-demand Multipath Distance Vector Routing in Ad Hoc Networks[END_REF] or only compute network-disjoint routes [START_REF] Nasipuri | On-demand Multipath Routing for Mobile Ad Hoc Networks[END_REF], while our scenario requires discovery of network-disjoint, bridge-disjoint and totally-disjoint routes.

The routing protocol must also provide additional information about each network and each node such as available bandwidth, technology, security properties, trust and delay. Additionally, bridges must be able to define the whole path used by each packet to reach its destination. This is necessary to ensure that the packets will follow different routes. Protocols based on routing tables, hence, are not appropriate since each node keeps a local routing table and autonomously forwards packets to the next hop towards the packet destination; different packets to the same destination will always be forwarded to the same next hop. A more adequate technique to forward packets is source route [START_REF] Johnson | Ad hoc networking, chapter DSR: the Dynamic Source Routing Protocol for Multihop Wireless Ad Hoc Networks[END_REF], where the source node includes in the packet header the whole path that the packet must follow to reach the destination. As such, the source can define in the packet headers that two packets addressed to the same destination must traverse disjoint paths.

We designed a hybrid ad hoc routing protocol that uses source routes to forward packets and thus ensures that the path defined at the packet source will be respected during packet transmission. The protocol also proactively keeps the shortest route to every node on the network using the efficient flooding mechanism proposed by OLSR [START_REF] Jacquet | Optimized Link State Routing Protocol for Ad Hoc Networks[END_REF] while multiple paths to a destination are discovered on demand by using a technique similar to DSR [START_REF] Johnson | Ad hoc networking, chapter DSR: the Dynamic Source Routing Protocol for Multihop Wireless Ad Hoc Networks[END_REF]. Route announcements contain not only connectivity information but also quality information such as network throughput, delay, software platform and cost. This protocol provides single-path routing without extra cost and reduces the overhead of multiple route discovery only to situations where a node requires multi-path routing.

Multi-path Granularity There are two issues related to granularity in multipath applications. The first issue, which is more often discussed in the literature, is traffic allocation granularity [START_REF] Krishnan | Choice of Allocation Granularity in Multipath Source Routing Schemes[END_REF][START_REF] Lai | Bifurcated Routing in Computer Networks[END_REF], which defines the smallest traffic unit that can be assigned to each path. From the application standpoint, the smallest unit is a message. However, in the network layer, a message can originate a number of packets depending on its size and on the network's maximum transmission unit (MTU). Per message allocation enables different messages to follow different routes while in per packet allocation different packets from the same message can use different paths.

Implementing per message traffic allocation when devices feature multiple network interfaces is not straightforward. Messages from the same connection can arrive through different interfaces from distinct source IP addresses, but the Internet Protocol suite cannot keep a session across different source and destination addresses. In that case, the middleware must implement some type of session management mechanism, on top of the IP stack, to identify related messages and to deliver relevant messages to applications. Per packet allocation is trickier to implement, since different packets from the same message may arrive through different addresses to the destination. For TCP connections, this would require modifications on the IP stack. However, we can use UDP messages to encapsulate each packet and then use the same middleware layer on top of IP to reconstruct application messages.

The second issue is path selection granularity. Paths can be defined per connection, per message or per packet. Considering that nodes in hybrid mobile ad hoc networks are mobile, paths may change frequently. Routes found at the beginning of a connection may not be available towards its end depending on how often nodes move or how long the connection lasts. However, smaller granularities can cause unnecessary overhead. The best strategy, hence, is to select a set of paths when creating a connection, and reselect them after a pre-defined period of time or when a certain number of paths are disconnected.

Network Density Availability of multiple paths between two nodes is highly dependent on bridge deployment. Bridges connect different networks and can increase network density if conveniently deployed. Current PLASTIC strategies for bridge election only take into account resource availability. This strategy can be improved to incorporate the current network topology, for instance, to prioritize bridges that increase network density regardless of the resources they provide. The election algorithm can also consider properties such as stability and trust on networks and bridges to elect new bridges that provide alternative paths to untrusted or unreliable nodes.

Finally, bridges can be temporarily enabled to create a short-lived alternative route to a required destination. Whenever a node not acting as a bridge receives a route request for an alternative route, it may forward the route request to neighbor networks and became a bridge if it discovers an alternative route. An incentive mechanism could be used to stimulate nodes to share resources and act as bridges.

Conclusion

Multi-path communication can provide greater reliability and improve privacy of message transmissions. By simultaneously using multiple paths to send the same message, or parts of one message, applications can increase the probability that a message will reach its destination and make it harder for an attacker to read its contents. Those properties are particularly important in mobile and infrastructure-less networks with unstable topologies where untrusted user devices forward packets.

We propose the introduction of an additional layer to the PLASTIC middleware, responsible for handling multi-path communication issues. Figure 3 shows The multi-path routing component uses a hybrid routing protocol (proactive and reactive) to reduce routing overhead; multi-path routing messages are only transmitted when multi-path communication is necessary. The Message Sharing and the Path Selection components rely on information obtained by the Multipath Routing component, and are also only used when the application requires stronger reliability or privacy.

We are now developing the Multi-path Communication layer. We plan to evaluate the processing and network overhead produced when simultaneously using different paths to transmit a message and we are now evaluating the energy cost imposed by the multi-path routing protocol described in Sect. 4 when compared to traditional ad hoc routing protocols. Finally, we also intend to analyze the availability of multiple disjoint paths in HMANETs and to quantify the reliability and privacy benefits provided by the multi-path approach.

Fig. 3 .

 3 Fig. 3. Multi-path Communication Layer on PLASTIC Middleware

http://www.ist-plastic.org

Acknowledgment This work is part of the IST PLASTIC project and has been funded by the European Commission, FP6 contract number 026955.


Bowei Shao
email: bowei.shao@sorbonne-nouvelle.fr

Rachid Ridouane
email: rachid.ridouane@sorbonne-nouvelle.fr

APICAL VOWEL IN JIXI-HUI CHINESE: AN ARTICULATORY STUDY

Keywords: apical vowels, ultrasound, Jixi-Hui Chinese, syllabic consonant

In this study, we analyze mid-sagittal and coronal ultrasound data from four subjects in order to investigate the articulatory properties of the apical vowel /z/ in Jixi-Hui Chinese (JHC). We seek to determine whether this unique segment, particularly well studied in Standard Chinese (SC), has articulatory characteristics of a vowel or a fricative consonant. Mid-sagittal ultrasound data show that it displays the same tongue shape as consonant [s] and may have a slightly higher tongue tip, which could explain the presence of abundant frication noise. The raising of the tongue dorsum, already observed in SC, varies in JHC across speakers. Coronal ultrasound data show that the tongue is medial-grooved and has an increased grooving corresponding to the decreased frication noise and the appearance of clear formant structure near its offset. These findings are discussed in light of the behavior of this segment within the phonological system of JHC.

INTRODUCTION

Chinese languages are known to have a series of specific segments, often termed as "apical vowels" [START_REF] Karlgren | Etudes sur la phonologie chinoise[END_REF]. Some researchers showed that they could also be defined as "fricative vowels" [START_REF] Ladefoged | The sounds of the world's languages[END_REF], "syllabic fricatives/sibilants" [START_REF] Duanmu | The phonology of standard Chinese[END_REF][START_REF] Yu | Aerodynamic constraints on sound change: The case of syllabic sibilants[END_REF], or "syllabic approximants" [START_REF] Lee-Kim | Revisiting Mandarin 'apical vowels': An articulatory and acoustic study[END_REF][START_REF] Lee | Standard Chinese (Beijing)[END_REF][START_REF] Zee | Vowel typology in Chinese[END_REF].

The apical vowels in Standard Chinese (SC) have been comprehensively studied. Phonologically, they are allophones to /i/, as they only occur following coronal sibilant onsets [s, ts, ts h , ʂ, tʂ, tʂ h ], while [i] occurs following palatal sibilant onsets [tç, tç h , ç] and other consonants [START_REF] Cheng | A synchronic phonology of Mandarin Chinese (Monographs on Linguistic Analysis[END_REF][START_REF] Duanmu | The phonology of standard Chinese[END_REF]. Phonetically, the apical vowels of SC are homorganic to their sibilant onsets; the tongue tip/blade gesture remains nearly unchanged from the preceding sibilants to the voiced part of the syllable, and only a slight lowering of the tongue body is found [START_REF] Lee-Kim | Revisiting Mandarin 'apical vowels': An articulatory and acoustic study[END_REF]. Frication noise during apical vowels is observed [START_REF] Yu | Aerodynamic constraints on sound change: The case of syllabic sibilants[END_REF] and considered the indicator of its sibilant property by some researches; however its presence is not systematic [START_REF] Faytak | Articulatory variability and fricative noise in apical vowels[END_REF][START_REF] Lee-Kim | Revisiting Mandarin 'apical vowels': An articulatory and acoustic study[END_REF].

Based on its homorganic character and its sibilant property, Dell [START_REF] Dell | Consonnes à prolongement syllabique en Chine[END_REF] analyzed the SC apical vowel as a "voiced prolongation of the consonant", and Duanmu [START_REF] Duanmu | The phonology of standard Chinese[END_REF] proposed to analyze it not as an underlying phoneme, but as a segment triggered by an empty nucleus and the spreading of the feature [+fricative] from the onset. In our study, we adopt the [z] transcription following Duanmu and Dell.

While SC apical vowels are studied in much detail, less is known about these segments in other Chinese languages. This current study is based on Jixi-Hui Chinese (JHC), which is a Hui 徽 group Chinese language spoken in Jixi 绩溪 County in Anhui 安徽 province, China [START_REF] Hirata | Huīzhōu Fāngyán Yánjiū [Study on Huizhou dialects[END_REF][START_REF] Zhao | Ānhuī Jìxī fāngyán yīnxì tèdiǎn [Phonological characters of Jixi dialect from Anhui Province[END_REF][START_REF] Zhao | Jìxī Fāngyán Cídiǎn [Dictionary of Jixi Dialect[END_REF]. It has two major variants; our study is based on the city variant, which has 9 phonological monophthong vowels: [i, y, u, ʉ, o, ɤ, ɔ, a], in addition to the apical vowel [z].

Our previous study [START_REF] Shao | La « voyelle apicale » en chinois de Jixi : caractéristiques acoustiques et comportement phonologique[END_REF], based on acoustic data, shows that [z] in JHC has abundant frication noise on the first half of its duration, and displays a clear formant structure on the second half. This is illustrated in Fig. 1 for the items [pz 214 ] 'compare' and [nz 214 ] 'in'. In these two syllables, [p] and [n] being a labial stop and a sonorant respectively, have objectively no frication noise to propagate into the vocalic parts. The HNR pattern of [z] shows that the presence of frication noise is systematic, suggesting that this is an inherent attribute of [z] in JHC [START_REF] Shao | La « voyelle apicale » en chinois de Jixi : caractéristiques acoustiques et comportement phonologique[END_REF]. In this study, we follow up on the study of properties of [z] in JHC, through the analysis of its articulatory properties based on ultrasound data. The aim is to determine whether [z] has articulatory characteristics of a vowel or of a fricative consonant.

p z n z

METHODS

Ultrasound data acquisition

Ultrasound data were acquired at Jixi city, in Anhui province, China. Three females and four males (age: 49±3.8) were recruited based upon the same criteria: they must be born and raised in Jixi city, by two parents born and raised in the same city, and they must speak JHC on a daily basis in both professional and family contexts. None of the speakers was reported to have speech or hearing diseases or difficulties. We had access to the sound-attenuated studio of the local television channel for our recording sessions. A word list was constructed with the following five segments [i, a, u, ʉ, z] occupying the rhyme of real monosyllabic words starting with /p_, p h _, m_, n_, ts_, ts h _, s_/ with different tonal structures. Each word was produced within the carrier phrase [ki 44 ɕᴐ 213 _ ɕᴐ 213 sᴐ 44 fa 44 ] 'He writes _ three times'. The word list was repeated three times for mid-sagittal recordings and three times for coronal recordings, yielding 1218 total target syllables. Both mid-sagittal and coronal ultrasound data were recorded with an Ultrasound Stabilisation Headset [START_REF]Ultrasound Stabilisation Headset User's Manual: Revision 1.5[END_REF] and the Articulate Assistant Advanced software (AAA, V217.03) [START_REF]Articulate Assistant Advanced User Guide: Version 2.14[END_REF]. The ultrasound image was recorded with a field of view of 92°; due to speakers' morphological specificities, the depth was adjusted to have a maximum view of the tongue, causing different frame rates. The synchronization of ultrasound recordings and corresponding audio was done automatically by AAA software.

Ultrasound data analysis

Data from two female (FS1, FS2) and two male (MS1, MS2) speakers are presented in this study. Ultrasound data were segmented in AAA manually using the corresponding audio and traced manually with the help of the built-in tracing algorithm. The tongue contour of the nearest-central image of every [i, a, u, ʉ, z] token was extracted in XY coordinates; we also extracted the tongue contour of fricative [s]. The data obtained were then analyzed using smoothing-spline analysis of variance (SS ANOVA) in R environment with Package gss [START_REF] Davidson | Comparing tongue shapes from ultrasound imaging using smoothing spline analysis of variance[END_REF][START_REF] Gu | Smoothing Spline ANOVA Models: R Package gss[END_REF]. The palatal traces were generated by averaging the palate contours extracted from six separate water-swallow tasks.

While studies based on mid-sagittal ultrasound data are rather frequent, coronal ultrasound data are rarely studied. One difficulty in coronal ultrasound data is that, in a field study, the coronal plane of the tongue can almost never be optimally positioned on the tongue [START_REF] Stone | A guide to analysing tongue motion from ultrasound images[END_REF]. Nevertheless, a comparison between coronal tongue shapes can still give a straightforward view of the possible differences between segments, namely between [s] and [z] which are the focus of this study.

RESULTS

Mid-sagittal ultrasound

We can see that the four speakers have two different mid-sagittal tongue contours. FS1 and MS2 have a raised tongue tip, but the tongue dorsum is also raised and higher than the tongue tip; FS2 and MS1 have the highest point of the tongue in the front, the tongue dorsum is flat, thus the tongue apex is higher than the tongue dorsum.

Mid-sagittal ultrasound images are generalized into SS ANOVA curves, presented in Fig. 2. The general observation is that [z] and [s] have the same tongue shape, but the tongue dorsum in [z] could be lower than in [s].

Based on SC apical vowel data reported by Zhou and Wu [START_REF] Zhou | Pǔtōnghuà Fāyīn Túpǔ [Articulatory diagrams of Standard Chinese[END_REF], Cheng [START_REF] Cheng | A synchronic phonology of Mandarin Chinese (Monographs on Linguistic Analysis[END_REF] noted that in the production of the apical vowels, the highest point of the tongue is slightly more front, and the back of the tongue is slightly raised. He then suggested that "the SC apical vowel has two simultaneous points of articulation, one at the tongue tip and the other at the body of the tongue." (p.13). This observation on SC apical vowel is confirmed in pattern 1, but not observed in pattern 2, in which the tongue tip is indeed the highest point, but there is no raised tongue body. Interestingly, when [s] has a raised tongue body, [z] has the same gesture, as in FS1 and MS2. Similarly, if [s] does not have a raised tongue body, [z] does not either, as in FS2 and MS1. That is to say, apical vowel [z] and fricative consonant [s] display the same tongue shape in the sagittal plane for each speaker.

FS2 and MS1 seem to produce apical vowel [z] with a lifted tongue tip compared to [s]. This specific pattern could be related to one case reported by Faytak [START_REF] Faytak | Articulatory variability and fricative noise in apical vowels[END_REF], where he observed a raised tongue tip in a coronal plane for one speaker; while all other speakers' tongue tips do not differ from [s] to [z]. Since the apical vowel [z] in JHC is reported to have more frication, we may consider this tongue tip constriction to be one of the factors related to the observed turbulent noise on acoustic signals.

Mid-sagittal tongue shape of [z] in different consonantal contexts

It is shown that the SC apical vowels are homorganic to their sibilant onsets [START_REF] Lee-Kim | Revisiting Mandarin 'apical vowels': An articulatory and acoustic study[END_REF]. Considering that in our word list of JHC, [z] is not only preceded by coronal sibilants, but also by [p, p h , m, n], we compared the tongue contours of [z] in these different contexts to show that the resemblance between the tongue contours of [z] and [s] was not due to the influence of the preceding coronal consonants.

In Fig. 4, we can see a comparison between [z] segments preceded by labial consonants and by coronal consonants for the two speakers. Clear enough, there is no observable difference between [z] in the two contexts. This provides additional evidence that the resemblance of tongue contours between [z] and [s] is not due to the context where [z] occurs. Rather, the specific tongue shapes displayed by [z] seem to be an inherent property of this segment.

Coronal ultrasound

Coronal ultrasound tongue contours are generalized into SS ANOVA curves and shown in Fig. 3. The general pattern is also that [z] has the same tongue shape as [s]. Although we cannot determine where exactly the coronal contours of the tongue were obtained, by comparing the relationship between tongue contour lines in mid-sagittal view and coronal view, we can still see that our coronal view is not captured at the tongue tip, but rather at the frontal part of the tongue body.

The four speakers all have a medial-grooved tongue shape. For MS1 and MS2 [z] is more grooved than [s], which should correspond to the lower tongue body in our mid-sagittal contours. For FS1 and FS2, though the tongue contours are medialgrooved in both [s] and [z], we do not observe a more grooved [z] compared to [s]. This is also consistent with their mid-sagittal contours where the difference between [s] and [z] is rather small.

Compared to other vowels, this medial-grooved shape of the tongue in [z] segments can be considered as the indicator of its sibilant nature, as no vowel in JHC presented in Fig. 3 has this narrowed air passage. It is worth noting that this medial-grooved tongue shape for [z] is observed in all speakers, who have different articulatory strategies for [z] reported in mid-sagittal view.

DISCUSSION AND CONCLUSION

The objective of this study was to determine whether [z] in JHC has articulatory characteristics of a vowel or of a fricative consonant. The resemblance of tongue contours between [s] and [z], together with previous results showing the presence of frication noise during [z] [START_REF] Shao | La « voyelle apicale » en chinois de Jixi : caractéristiques acoustiques et comportement phonologique[END_REF], provide evidence that this segment is best analyzed as a fricative consonant.

There are important articulatory differences between JHC [z] compared to SC [z] [START_REF] Faytak | Articulatory variability and fricative noise in apical vowels[END_REF][START_REF] Lee-Kim | Revisiting Mandarin 'apical vowels': An articulatory and acoustic study[END_REF]. The JHC [z] could be produced with or without a raised tongue body, and the tongue tip could be more raised as for [s]. There are also similarities between SC and JHC [z]: on the acoustic level, they display the same formant structure [START_REF] Shao | La « voyelle apicale » en chinois de Jixi : caractéristiques acoustiques et comportement phonologique[END_REF][START_REF] Zee | An acoustical analysis of the vowels in Beijing Mandarin[END_REF]; on the articulatory level, the tongue body could be raised and medial-grooved while systematically resembling the articulatory configuration observed for [s]. The presence of frication noise in JHC [z] has also been observed in other Chinese languages [START_REF] Hou | Héféi fāngyán dàicāyuányīn de shíyàn yánjiū [An experimental study on [ɿ] in Hefei Dialect[END_REF][START_REF] Ling | A phonetic study of the vowel system in Suzhou Chinese[END_REF]. The absence of frication noise in SC [z] is likely an exception rather than a norm [START_REF] Faytak | Articulatory uniformity through articulatory reuse: insights from an ultrasound study of Sūzhōu Chinese[END_REF].

The fundamental difference between JHC [z] and SC [z] concerns their phonological status and phonotactic distribution. As already stated, SC [z] is an allophonic variant of the phoneme /i/ and is always homorganic to its sibilant onsets. In JHC it is a distinct phoneme from /i/, as both /z/ and /i/ occur after [ts, ts h , s] onsets (e.g. [tsz 31 ] 'chicken' vs. [tsi 31 ] 'a family name'; [ts h z 31 ] 'deception' vs. [ts h i 31 ] 'autumn'; [sz 31 ] 'silk' vs. [si 31 ] 'repair'). Moreover, this apical vowel has a larger distribution than in SC, as it occurs not only after coronal sibilant onsets, but also after [p, p h , m, n] onsets that have no sibilant properties to propagate. [z] in JHC is also a tone-bearing unit (TBU) and can undergo tonal sandhi processes (e.g. /sz 31 / 'west' + /ko 31 / 'melon' → [sz 33 ko 31 ] 'watermelon'). Syllables containing apical vowel [z] are lexically developed, counting for 7,2% of the monosyllabic entries of JHC dictionary [START_REF] Zhao | Jìxī Fāngyán Cídiǎn [Dictionary of Jixi Dialect[END_REF].

Studies on SC [z] tend to treat the homorganic property of [z] as being triggered by the sibilant onsets. We have shown that this analysis cannot apply to JHC [z]. Indeed, the fact that it can occur following both coronals and labials rules out the possibility of considering it as a mere prolongation of the onset consonant.

To sum up, both the articulatory and acoustic characteristics of /z/ support a consonant analysis. The fact that it can be a TBU nucleus does not rule out this analysis, since other consonants in JHC can also function as TBUs (e.g., /m, n, v/). Its limited distribution -appearing only at syllable nucleus position -is most probably a consequence of its diachronic evolution from /i/ [START_REF] Zhu | Hànyǔ yuányīn de gāodǐngchūwèi [Sound changes of high vowels in Chinese dialects[END_REF].

Figure 1 :

 1 Figure 1: Examples of [z] containing frication noise on more than half of the duration, with [p] (left) and [n] onsets in syllables [pz 214 ] 'compare' and [nz 214 ] 'in' [17].

Figure 2 :

 2 Figure 2: SS ANOVA splines of mid-sagittal ultrasound tongue contours, extracted in XY coordinates (mm) at the nearest center image of each segment, the tongue tip is on the right and tongue root on the left. Dotted lines represent [i, u, a, ʉ], dashed red lines represent [s] and solid green lines [z]; the thick grey lines represent the palatal traces; the grey areas represent 95% Bayesian confidence intervals.

Figure 4 :

 4 Figure 4: SS ANOVA splines (mm) of all [z] segments. Solid lines represent [z] preceded by labial consonants and dotted lines represent [z] preceded by coronal consonants. Tongue tip on the right and tongue root on the left.

Figure 3 :

 3 Figure 3: SS ANOVA splines of coronal ultrasound tongue contours, extracted at the nearest center image of each segment. Line type as in Figure 2.

ACKNOWLEDGEMENTS

We thank all speakers who have kindly participated in all recording sessions. This study is part of the French program "Investissements d'Avenir" run by the Agence Nationale de la Recherche ANR-10-LABX-0083 (Labex EFL).