Florent Guichard
email: florent.guichard@institutoptique.fr

Marc Hanna

Laurent Lombard

Yoann Zaouter

Clemens Hönninger

Franck Morin

Frédéric Druon

Eric Mottay

Patrick Georges

Two-channel pulse synthesis to overcome gain narrowing in femtosecond fiber amplifiers

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Ytterbium-doped femtosecond fiber amplifiers have numerous advantages, including high optical-to-optical efficiency, good thermal handling, and high spatial beam quality, making them suitable for the generation of high average power. However, the peak power available from these sources is limited by the ability to efficiently transfer the stored energy to a pulse as short as possible. The tight confinement of the beam inside the fiber and the large interaction length can lead to strong nonlinear effects, leading to a degraded temporal profile or optical damage. High-energy femtosecond fiber amplifiers therefore use the chirped-pulse-amplification (CPA) technique [1], along with various nonlinearity management strategies to limit the impact of a moderate amount of selfphase modulation in the system [2][3][4].

In these amplifiers, the achievable pulse width is limited to around 300 fs, mostly due to the gain profile of Ybdoped fibers, which presents a strong peak at 1030 nm with a ∼40 nm bandwidth [5]. This spectroscopic property induces an important limitation on the achievable bandwidth during high-gain amplification, a phenomenon denoted as gain narrowing. The typical available output bandwidth is around 12 nm in 10-20 dB low-gain amplifiers [5,6]. To attenuate such problems, amplification at larger wavelengths, where the gain profile is broader, is possible [7] usually at the expense of other amplifier characteristics.

For other purposes, ultrafast physics now benefit from the development of ultrashort single-cycle sources with spectra spanning more than two octaves [8]. This was achieved by coherently adding two pulses with distinct spectra produced by two optical parametric amplifiers, where stable and long-term coherent combining was ensured through an active stabilization feedback loop. This idea was recently implemented in Yb fiber amplifiers to increase the available gain bandwidth while allowing power scaling through the coherent combination of several amplifiers. Recent results report on the development of femtosecond spectral combining and pulse synthesis with an actively stabilized feedback loop in a multichannel configuration [9] or in a multicore fiber experiment [10]. Although these results demonstrate the feasibility of spectral combining in the context of fiber amplifiers, they do not show an improvement over a single amplifier setup in terms of optical bandwidth. Quantitative results on short pulse amplification in a high gain system, comparative study of gain narrowing, and operation at high average power are still lacking.

In this Letter, we report on the development of a twochannel pulse synthesizer, which is able to amplify two distinct parts of an incoming spectrum in order to mitigate gain narrowing. For gain values larger than 30 dB, 130 fs pulses with a spectral width of 19 nm are generated in microstructured large mode area (LMA) Yb-doped fibers, at an average power level of 10 W and a repetition rate of 35 MHz. These results are compared to a single amplifier setup to quantify the benefits of spectral coherent combing to mitigate gain narrowing. The combining efficiency and active stabilization loop are also characterized.

The experimental setup is presented in Fig. 1 and starts with an air-cooled Yb:KYW oscillator generating Fouriertransformed-limited 270 fs pulses centered at 1030 nm with an average power of 0.5 W at 35 MHz repetition rate. Output pulses are sent into a 1 m length standard polarization-maintaining passive fiber with 6 μm mode-fielddiameter (MFD) with 50% coupling efficiency, in order to generate a 50 nm spectrum through self-phase modulation. It is followed by a 400 ps grating-based (1600 lines • mm -1 ) stretcher designed to support 50 nm bandwidth pulses. This ∼1500 stretching ratio is chosen to avoid nonlinear effects during amplification since spectral splitting and gain narrowing will strongly decrease the pulse duration inside the amplifiers. The coherent spectral combining setup is a Mach-Zehnder interferometer, where input/output couplers are replaced by two identical sharp-edge dichroic mirrors to ensure proper spectral separation and recombination. These long-wave-pass filters are angularly adjusted to split the incident spectrum with a cutoff wavelength around 1035 nm. A fiber-pigtailed LiNbO 3 integrated phase modulator is inserted in the reflected interferometer arm (short-wavelength arm, SWA) and followed by a 2m-long Yb-doped double-clad polarization-maintaining 40∕200 μm LMA fiber amplifier. The phase modulator is driven by a feedback loop to ensure active stabilization of the relative optical phase between the two interferometer arms. In the second arm (transmitted arm or longwavelength arm, LWA), 2.4 m length of passive singlemode fiber, a free-space delay line, and another 2 m Yb-doped double-clad 40∕200 μm LMA fiber amplifier are implemented. The aim of the single-mode fiber is to compensate the second-order phase accumulated in the short-wavelength arm through propagation in the phase modulator and thus match the group-velocity dispersion of both arms while the delay line acts as a fine delay-matching element. The average power seeded to each amplifier is approximately 5 mW. Finally, the output of each amplifier is collimated, and the optical paths from the LMA fibers end to the combining dichroic mirror are set as close as possible in order to match the spatial profiles and wavefronts of the two beams. The beam is then sent to a 60% efficiency 1600 lines • mm -1 grating-based compressor.

The active stabilization loop is based on frequency tagging [11] and is similar to that described in [12]. The beam rejected by the combining dichroic mirror is detected by a photodiode, allowing us to measure the interference signal corresponding to the overlapping spectral part that is common to both arms. The tagging frequency is 250 kHz and is added to the signal driving the phase modulator. The small induced-phase modulation is translated into an intensity modulation at the output of the interferometer and is detected by a lock-in amplifier followed by a PID controller to generate the error signal applied to the phase modulator. The overall bandwidth of this control loop is 1 kHz, which is sufficient to compensate for the environmental noise that dominates the phase fluctuations. The photodiode signal versus time is shown in Fig. 2, both in a free-running and lockedphase regime. In the free-running regime, maximum interference contrast of 43% in the overlapping spectral part is obtained when proper group velocity and group delay matching are achieved between both interfering spectra. The system was disturbed on purpose to reveal this contrast clearly. In phase-locked operation, the signal shows residual intensity fluctuations. Taking into account the measured visibility, this corresponds to an RMS phase noise of λ∕30 over the period of 8 s, assuming that the intensity noise is negligible. Slight readjustments of the delay line are necessary when the pump power is significantly changed due to thermo-optical and Kramers-Krönig induced refractive index change inside the amplifiers.

The setup is operated at different average-power levels in order to study the spectral coherent combining process. Figure 3(a) shows the amplifier characteristics with the combined, uncombined, and compressed average power together with the combining efficiency, defined as the ratio between the combined average power and the total output power, as a function of pump power. The combining efficiency remains constant at around 86% for all pump powers. It is slightly lower than in the case of previously demonstrated femtosecond coherent combining systems that do not perform spectral separation [12,13]. This is related to differences between the splitting and recombining dichroic mirrors that lead to efficiency drop and to spectral modifications during amplification that change the relative amplitude of spectral components compared to the incident spectrum, thereby affecting the combining efficiency [9]. These effects could be reduced by using steeper spectral-edge filters.

The spectral characteristics of the combined output is also studied as a function of gain value and compared to a single amplifier setup to quantify the benefit of spectral synthesis with regard to gain narrowing. The spectral FWHM of both configurations as a function of gain are plotted in Fig. 3(b). The spectral FWHM is clearly larger for the two-channel systems, highlighting the main advantage of spectral coherent combining. Gain narrowing is mitigated because narrower spectra are seeded to each amplifier, and pump levels can be adjusted to generate the largest bandwidth possible.

After careful adjustment of the pump power levels to match the average power of both arms, the obtained combined average power is 10 W at a 35 MHz repetition rate. At this power level, corresponding to a gain of 30 dB, the combined spectrum exhibits a FWHM of 19 nm, which outperforms any reported high-gain Ybdoped linear CPA fiber amplifier around 1030 nm. Corresponding spectra are shown in Fig. 4(a), as well as the stretched spectrum and the amplified spectra of each channel prior to recombination. The recombined spectrum fits with the superposition of the reflected and transmitted amplified spectra; each exhibits a FWHM of approximately 7 nm, corresponding to a stretched pulse duration of approximately 60 ps inside each fiber amplifier. These results can be compared to the spectral width [Fig. 3(b), green-colored curve] when the full incident spectrum is seeded in a single amplifier at the same gain level. In such conditions, despite having a much broader input spectrum to seed the amplifier with, the output spectrum FWHM is only 12 nm.

Temporal characteristics are shown in Fig. 4(b). The autocorrelation FWHM of each arm is minimum at the same compression point, indicating that the group velocity dispersion and spatial angular superposition mismatch between both beams are negligible. Indeed, minimum compressed pulse duration of both arms occurs for different compressor lengths when both beams are not perfectly superposed since different incident angles in the compressor unit change the induced spectral phase. An imperfect group-velocity dispersion matching between both arms leads to the same observation. The autocorrelation FWHM are equal to 410 fs for the SWA and 390 fs for the LWA, confirming that both arms contribute equally to the combined spectral content. The combined autocorrelation trace exhibits a FWHM of 180 fs compared to 280 fs when the entire incident spectrum is seeded in a single amplifier. Independent secondharmonic generation frequency-resolved optical gating (FROG) measurement reveals a combined pulse duration of 130 fs [Fig. 5(a)] and corresponding peak power of 1.9 MW. Figure 5(b) shows the retrieved autocorrelation trace, which is in good agreement with the measured autocorrelation. The FROG-retrieved spectrum is also in very good agreement with the independent spectrum measurement. The spectral phase [Fig. 5(c)] shows a low-amplitude residual structure, but overall compression quality is satisfactory as indicated by the Fourier transform-limited pulse duration of the recombined spectrum, equal to 115 fs with a corresponding peak-power of 2.4 MW. The time-bandwidth product of the synthesized pulse is 0.70, and the temporal Strehl ratio is 0.79.

Finally, we analyze the beam quality at the output of the compressor for both arms separately and for the combined beam (Fig. 6). M 2 measurement revealed values of, respectively, M 2

x 1.4 and M 2 y 1.3 in all cases. These values are slightly higher than those expected with such LMA amplifiers, for which M 2 ∼ 1.1 is a typical value, but are caused by nonoptimal end-facets surface quality.

In conclusion, we demonstrate a two-channel spectral coherent combining system, which is able to mitigate gain narrowing in a fiber amplifier. This is possible by spectrally separating a broadened incident spectrum of an ultrafast oscillator into two beams with distinct central wavelengths, which are amplified independently. Coherent spectral recombination is then ensured by a dichroic mirror and active phase stabilization. This scheme allows the generation of 19 nm 130 fs pulses at 10 W average power and 30 dB gain. These results represent, to our knowledge, the first demonstration of spectral coherent combining of femtosecond fiber amplifiers at high average power and with a combined bandwidth larger than for a single channel. This experiment clearly demonstrates the advantage over a single amplifier setup in terms of output pulse duration. Shorter pulse widths can be obtained by increasing the number of spectral channels together with the implementation of a high-order dispersion management device, such as a spatial modulator-based pulse shaper. This would potentially lead to full coverage of the gain bandwidth in Yb-doped fibers. A large number of channels also would give finer control of each spectral part, enabling spectral intensity and phase-shaping capabilities in addition to the amplification function. This technique could be implemented in large-scale, coherent combining systems to deliver ultrashort pulses at high energy and high average power.

Fig. 1 .

 1 Fig. 1. Experimental setup of the pulse synthesis experiment. DM1-1: dichroïc mirror (HR: 1000-1035 nm). ϕth: nth phase order term correction.

Fig. 2 .

 2 Fig. 2. Photodiode signal as a function of time in free-running and phase-locked operation.

Fig. 3 .

 3 Fig. 3. (a) Combined, uncombined, and compressed average power and coherent combining efficiency as a function of pump power at 35 MHz repetition rate. (b) Spectrum FWHM evolution for different gain levels in the two-channel (red curve) and one-channel (green curve) configurations.

Fig. 4 .

 4 Fig. 4. (a) Experimental stretched, combined, and separately amplified arm spectra at 10 W combined average power together with the output spectrum in single amplifier configuration. (b) Autocorrelation trace of both arms and combined pulse.

Fig. 5 .

 5 Fig. 5. (a) Retrieved temporal profile at maximum combined power. (b) Corresponding experimental and retrieved autocorrelation trace. (c) Retrieved spectrum and spectral phase together with the experimental spectrum at maximum combined power. (d) Corresponding measured (top) and retrieved (bottom) FROG traces.

The authors acknowledge the financial support of Agence Nationale de la Recherche through the Multi-Femto project and RTRA/Labex PALM through the PulseSynth project.


Emmanuelle Petelet-Giraud

Philippe Négrel

Nicole Baran

Maritxu Saplairoles

Lise Cary

Wolfram Kloppmann

Agnès Brenot

Surface water -groundwater interactions in alluvial contexts: constraints using single and multi-isotope tracers

come

Surface water -groundwater interactions in alluvial contexts: constraints using single and multi-isotope tracers Petelet-Giraud, E. 1 , Négrel, Ph. 1 , Baran, N. 1 , Saplairoles, M. 2 , Cary, L. 3 , Kloppmann, W. 1 Brenot, A.

___________________________________________________________________________

Surface water and groundwater have been considered as separate waters bodies for a long time by hydrologists and also decision makers, so that the relevance of interconnections between groundwater and surface water has frequently been underestimated. Since the last decades, rivers and aquifers are no more considered as physically independent entities, and are now approached as integral components of an aquifer-river-continuum with strong mutual influences between river, aquifer and the interconnecting hyporheic zone. Numerous research approaches were developed to identify and quantify the river-aquifer exchanges that can be in both directions depending on the river profile and the seasonal water levels and hydraulic gradients. The understanding of the links between hydrological, biogeochemical, and ecological process dynamics in hyporheic zones and their implications for river water quality and ecology are also of major concern for water management and decision makers. Furthermore, isotopes of dissolved elements (Sr, B, S, …) also constitute valuable tracers to identify and, in optimal conditions, to quantify the exchanges between surface-and ground-water bodies.

Through various examples we present the contribution of isotopic tracers to our understanding of groundwater recharge in alluvial valley-fillings, riverbank aquifers counting worldwide among the most important water resources for drinking water supply, irrigation and industrial activities. Sr isotopes are excellent tracers of water-rock interactions as 87 Sr/ 86 Sr reflect the signature of the minerals that constitute the drained aquifer lithologies. Boron isotopes also differ according to the drained lithologies, but, contrarily to strontium, signatures can be affected by secondary processes like adsorption/desorption onto clays; in an urban context, they are good tracers of wastewaters whose signature are characteristic of the boron origin used in detergents. Sulphur and oxygen isotopes of sulphates are also potential tracers of drained lithologies (solid sulphates in evaporites, sulphides in igneous and sedimentary rocks) in cases where fertilizers do not provide the majority of dissolved SO 4 . The understanding of surface-and ground-water relations, gained through a multitracer approach, can thus explain groundwater quality but also the spatial variability of surface water quality along a river section, information of primary importance for water managers.

Figure 1 :

 1 Figure 1: Strontium isotopes to evidence and quantify mixings from various origins (Allier, France)

  4* 1 BRGM (French Geological Survey), Laboratory Division, BP 36009, 45060 Orleans, France; e.petelet@brgm.fr

	2 BRGM, Toulouse Agency, 31520 Ramonville-Saint-Agne, France 3 BRGM, Lille Agency, 59260 Lezennes, France 4 BRGM, Lyon Agency, 69100 Villeurbanne, France; *Present address: EDF, 38040 Grenoble, France


Hye-Jin Cho
email: hyejin.cho@malix.univ-paris1.fr

Speculative Bubble Burst 1

Keywords: externalities, speculative bubbles, heterogeneous beliefs, overconfidence, speculative bubble burst, equilibrium with inertia. JEL: D01, D52, D62, D84

Central to market fundamentals are three ideas: (1) Nominal money (2) Dividend (3) Existing stock. In connection with the cumulative dividend stream criterion of fundamental and noise movement, the conception of sequentially stable Markov process is grounded on the theory of bubbles. This paper firstly embodies the origin of speculative bubble burst with overconfidence. Then, unique equilibrium with inertia is re-illuminated by the overconfidence.

Introduction

By applying the equilibrium-conceptual approach to stability of markets, economic and financial crisis from 2008 illuminates familiar problems from a new angle, to pose provocative questions that the price level changes rapidly even though economic fundamentals are seemed to have no problem. Neoclassicals, mainstream scholars in the Microeconomic foundation, ferrot out this key problem by the axiomatic approach. In Microeconomics, classically, economic equilibrium is a state where economic forces such as supply and demand are balanced without externalities and equilibrium values of variables are stabilized. In what follows, we explore the case with externalities with (1) shocks (2) speculative bubbles and (3) an inertia mechanism.

We offer a new interpretation of overconfidence as motivation not to participate in economic activities immediately. Risk-averse agents provide a much more rational way to solve the puzzle of externalities. If beliefs of traders are concordant [START_REF] Milgrom | Information, trade and common knowledge[END_REF]: traders agree about how information should be interpreted, Concordant beliefs arise naturally in statistical problems where there is an unknown parameter about which traders may hold different views. In connection with this belief, the basic framework of trading can take the financing source as a focal point:

(1) Fiat money [START_REF] Santos | Rational Asset Pricing Bubbles[END_REF].

(2) Self-enforcing debt [START_REF] Hellwig | Bubbles and Self-Enforcing Debt[END_REF].

Let us initiate a discussion of the separation between asset pricing bubbles and speculative bubbles. For example, during the German hyperinflation, [START_REF] Flood | Market Fundamentals versus Price-Level Bubbles: The First Tests[END_REF] had immense appeal for hypothesis of no speculative bubbles. However, theoretically, financial market setting emphasizes on finite wealth, the finite horizon and short-selling behavior of risk-averse finitely lived agents. Hence, the bubble condition cannot be paralleled with transversality or boundary conditions. The methodological part of speculative bubble has come to debate on a Markov process and an equilibrium. The article "Sunspots and Cycles" [START_REF] Azariadis | Sunspots and Cycles[END_REF]) is devoted to the detailed technique with sequentially stable Markov process in accordance with perfect foresight. Unstable beliefs in the form of the transition probability matrix are intended to yield a regular stationary sunspot equilibrium (SSE). In this article, market fundamentals becomes apparent with three concepts of O. [START_REF] Blanchard | Speculative bubbles, crashes and rational expectations[END_REF] in the section 2. The section 3 introduces the infinitesimal generator to cumulative dividend process. In the section 4, psychological factors of agents like aggregated beliefs, overconfidence carry implications for a speculative bubble crush with an infinitesimal generator. In the section 5, origin of speculative bubble burst and unique trading point are addressed. In the section 6, existence of unique equilibrium allocation with inertia at the speculative bubble crush is described. I shall proceed in the following with regard to the literature.

Axiomatic Foundations of Speculative Bubbles

Market fundamentals rank the price level x t in a discrete-time framework according to

x t = f t + aE(x t+1 | Ω t ), a ∈ (0, 1), t ∈ N
where x t is the price level, f t is fundamentals and Ω t is the information set.

It becomes apparent with three examples suggested by O. [START_REF] Blanchard | Speculative bubbles, crashes and rational expectations[END_REF]:

1. Reduced form of a money market equilibrium: x t is price level and f t is nominal money [START_REF] Flood | Market Fundamentals versus Price-Level Bubbles: The First Tests[END_REF] 2. Arbitrage equation: x t is price of share and f t is dividend [START_REF] Scheinkman | Overconfidence and Speculative Bubbles[END_REF] 3. Equilibrium of a material market (for example the gold market): x t is price and f t is existing stock [START_REF] Harrison | Speculative Investor Behavior in a Stock Market with Heterogeneous Expectations[END_REF] No single answer about the definition of bubbles will suffice but it is highly probable that bubbles are price deviations from their fundamentals similar to [START_REF] Allen | Churning Bubbles[END_REF]. Still, it bristles with ambiguity how price simply goes down when bubble bursts even though bubble is itself a rich assembly of determinants. [START_REF] Scheinkman | Overconfidence and Speculative Bubbles[END_REF] formulated a criterion linked to the continuoustime cumulative dividend stream. The buyer's willingness to pay is a function of the value of the option that he acquires, the payoff from stopping is [START_REF] Stokey | Recursive Methods In Economic Dynamics[END_REF], in turn, related to the value of option like American options with no maturity. The continuous-time cumulative dividend process D t is given by

dD t = f t dt + σ D dZ D t , (1)
where f is the fundamental random variable, σ D is a constant volatility parameter and Z D t denotes the standard Brownian motion of the process D t . The value of process f t is defined by

d f t = -λ( f t -f )dt + σ f dZ f t , (2)
where mean reversion is non negative such that λ ≥ 0, and f denotes long-run mean of f . The idea centered on the speculative behavior if the right to resell stock makes them willing to pay more. Relevant to "churn bubbles" of [START_REF] Allen | Churning Bubbles[END_REF], fund managers are ready to negotiate trading above their fundamental even though bubble can crash by a bad luck.

Dividend Process and the Infinitesimal Generator "A" of the parameter T t

Let us induce the fundamental and noise movement from the dividend stream in equation (3) with an infinitesimal generator "A" defined by

A f = lim t→0 (T t f -f ) t ,
where T t satisfies the property of a feller semigroup. Thus, I define the cumulative dividend process by

dD t = A f t dt + σ D dZ D t , (3)
where the value of f is given in equation ( 2).

It is worth noting that the collection of parameters denoted by {T t } t≥0 is a feller semigroup (≡ T t+s = T t • T s ). It is positive linear and contracting mapping from the space of all real-valued continuous functions on a locally compact topological space (bounded borel function, [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] with a countable base that vanish at infinity. The contraction mapping is equipped with the sup-norm • for all t ≥ 0 such that

T t f ≤ f ,
and satisfies properties like resolvent and spectrum [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF].

The resolvent of a feller process (or semigroup) is a collection of maps (A λ ) λ>0 for any λ ∈ R + defined by

A λ f = ∞ 0 e -λt T t f dt
It can be shown that it satisfies the identity.

A λ A µ = A µ A λ = (A µ -A λ )/(λ -µ).
The spectrum is the complement of the resolvent set:

σ(A) = R -A λ .
If an infinitesimal generator A fails to maintain the uniform limit of fundamental random variable A f t dt, it means that A is not resolvent and no spectrum of A existed in

A f = lim t→0 (T t f -f ) t .
Fleshing out the infinitesimal generator takes a highly microscopic approach to the question of bubble phenomenon that fundamentals have the tendency to become noise factors during crisis. In examining the speculative bubble crush, I take up memory-less property of Brownian motion.

• A Brownian motion is an example of feller processes.

• Every feller process satisfies the strong Markov property (memory-less).

• A stochastic process has the Markov property if the conditional probability distribution of future state of the process depends only upon the present states defined in terms of a random variable known as a stopping time.

Aggregation of Beliefs

Consider two groups of agents A and B who observes a vector of their own signals s A and s B . Assume that heterogeneous beliefs increased by overconfidence offer joint dynamics of the D, f, s A , s B . All agents observe a vector of each signal s A and s B that satisfy:

ds A t = A A f t dt + σ s dZ A t , similarly for ds B t .
Here, agents of group A(B) believe that innovations dZ A (dZ B ) in the signal s A (s B ) are correlated with the common volatility of signals σ s and the innovation dZ f in the fundamental process. This process is by Bayesian analysis of the normal distribution.

a(dD t -ds A t ) 2 +b(dD t -ds B t ) 2 ≈ (a+b)       x - (ads A t + bds B t ) (a + b)       2 + ab (a + b)(ds A t -ds B t ) 2
The factor (ads

A t + bds B t ) (a + b)
has the form of a weighted average of ds A t and

ds B t . ab (a + b) = (a -1 + b -1 ) -1 .
Remark 4.1. The factor (ads A t + bds B t )/(a + b) has the form of a weighted average of ds A t and ds B t along ab/(a + b) = (a -1 + b -1 ) -1 . For combination of a and b, it's necessary to reciprocate, add, and reciprocate the result again to get back into the origin units. ab/(a + b) is one-half of the harmonic mean of a and b.

If agents in group A trust more their own signal s A than the true information, agents are highly confident to undertake some speculative action. Intentionally, the overconfidence about fundamentals is divided into the part of fundamentals, signal σ s valued partial fundamentals dZ f t correlated with overconfidence φ (0 < φ < 1) and signal σ s valued noises dZ A t which is fluctuated by the residual oscillation of determined correlation with overconfidence. Hence,

ds A t = A A f t dt + σ s φdZ f t + σ s 1 -φ 2 dZ A t . and ds B t = A B f t dt + σ s φdZ f t + σ s 1 -φ 2 dZ B t .
Regardless of overconfidence = 0, differences of beliefs are according to:

g A = A B f B -A A f A , g B = A A f A -A B f B = -g A .

5

Documents de travail du Centre d'Economie de la Sorbonne -2016.21 Whence:

dg A = A A d f B -A B d f A = -ρg A dt + σ g dW A g .
Overconfidence is newly coming up after heterogeneous beliefs' movement when it comes to need to distinguish fundamentals and noises depending to ρ and σ f . Coefficients of Sheinkman and Xiong depend upon parameters of the model (λ, σ D , σ f , σ s , φ) with an infinitesimal generator A according to:

ρ = A(λ + φ σ f σ s 2 + (1 -φ 2 )σ 2 f 2 σ 2 s + 1 σ 2 D , σ g = √ 2φσ f
The expressions for the second agent are analogously derived as:

dg B = dA A f A -dA B f B = -ρg B dt + σ g dW B g .
Without overconfidence, noises of heterogeneous beliefs can degenerate to 0. Then, differences of groups concerning the volatility σ f , σ s , σ D and the combination of an infinitesimal generator A and mean reversion λ are still considered. However, with φσ f overconfidence on fundamental volatility, we can find the fluctuation point of φσ f from ρ to σ g . Differences in beliefs across agents that will lead to trading. Let us suppose that the optimistic investors A want to bid up prices and happen to hold the whole supply of A and B, for:

p A t = sup τ≥0 E A t t+τ t exp[-r(s-t)] f + exp[-r(s -t)]( f A t -f ) dD s +exp(-rτ) p B t+τ -c ,
for the trading cost c : a seller B pays c ≥ 0 per unit of the asset sold. [START_REF] Scheinkman | Overconfidence and Speculative Bubbles[END_REF] assumes the following form for the equilibrium price function of group A :

p A t = p A A A f A t , g A t = ff + A A f A - f r + λ + q(g A t ), q(g A t ) = sup τ≥0 E A t g A t+τ r + λ + q(g B t ) -c exp(-rτ) .
In that, the bubble concept relates to the difference between the demand price of the current owner (group A), i.e., group B in their exposition, and his fundamental valuation along

b = q(-k * ) = q(g A ),
where k * denotes the minimum difference in opinion-between group A and group B-that generates a trade.

4.1.

Can it be the Speculative Bubble or Overreaction to Surprises Definition 4.2. Overconfidence parameter φ increases as a larger φ increases, agents attribute to their own forecast of the current level of fundamentals where 0 < φ < 1.

It is important to note that dD t ≈ ds A t ≈ ds B t ≈ A f A dt where dD t denotes the cumulative dividend process , ds A t is inferred value with signals of group A and A f A dt is mean reverted fundamentals with mean reverted conditional belief and an infinitesimal generator A. Then, conditional mean of beliefs of agents in group A satisfies

dA f A = -λA( f A -f )dt + φσ s σ A f + γ σ 2 s (ds A -A f A dt) + γ σ 2 s (ds B -A f A dt) + γ σ 2 D (dD -A f A dt)
where agents in group A overreact to surprises in s A because of overconfidence (φ > 0) related to "surprises" with the common volatility of signals σ s and related Wiener processes are denoted as

dW A A = 1 σ s (ds A -A f A dt) (4) dW A B = 1 σ s (ds B -A f A dt) (5)
and

dW A D = 1 σ D (dD -A f A dt). (6)
Similar representation of equations ( 4), ( 5), ( 6) can be obtained for group B by simply replacing A f A by A f B .

4.2. Overconfidence φ and an infinitesimal generator A as a speculative bubble crush index Definition 4.3. Speculative Bubble Crush Index A : The infinitesimal generator A as the Speculative Bubble Crush Index parameter is newly modeled from [START_REF] Scheinkman | Overconfidence and Speculative Bubbles[END_REF]. Regardless of A, agents attribute to their own forecast of the current level of fundamentals as a larger overconfidence φ increases (stationary solution agreed by [START_REF] Rogers | Diffusions, Markov Processes, and Martingales[END_REF]sec. 6.9, Lipster and Shiryaev 1977;theorem 12.7).

γ ≡ (Aλ + φ σ f σ s ) 2 + (1 -φ 2 )(2 σ 2 f σ 2 s + σ 2 f σ 2 D ) -(Aλ + φ σ f σ s ) 1 σ 2 D + 2 σ 2 s Lemma 4.4.
Stationary variance γ decreases with φ.

Proof. Letting

θ(φ) = (Aλ + φσ f /σ s ), Ψ(φ) = (1 -φ 2 )[(2σ 2 f /σ 2 s ) + σ 2 f /σ 2 D ]
, one derives:

dγ dφ = [(2θ(φ)θ (φ) + θ (φ)]/[2 θ(φ) 2 + Ψ(φ)] -θ(φ) 1/σ 2 D + 2/σ 2 s = [θ(φ)/ θ(φ) 2 + Ψ(φ) -1]θ (φ) + Ψ (φ)/2 θ(φ) 2 1/σ 2 D + 2/σ 2 s ≤ 0.
This is the same result as Lemma 1 in [START_REF] Scheinkman | Overconfidence and Speculative Bubbles[END_REF]. Further, we can extend to the origin of speculative bubble crush.

Origin of Speculative Bubble Burst and Unique Trading Point

Upper Boundary of Speculative Bubble Burst with Overconfidence

Lemma 5.1. If A (= θ (λ)) is infinitesimal and B (= φσ f /σ s ) is significantly larger than A, the stability Variance γ does not increase as mean reversion λ increases and:

dγ dλ = A (Aλ + B) -A < 0.
Hence, B (= φσ f /σ s ) > Aλ.

Proof. Letting

θ(λ) = Aλ + φσ f /σ s , γ ≡ θ(λ) 2 + ∼ -θ(λ)/( 1 σ 2 D + 2 σ 2 s ),
it is derived that:

dγ dλ = 1/2 [θ(λ) 2 ] 2θ (λ) -θ (λ) 1 σ 2 D + 2 σ 2 s = θ (λ) √ θ(λ) 2 -θ (λ) 1 σ 2 D + 2 σ 2 s = θ (λ) 1 θ(λ) -1 1 σ 2 D + 2 σ 2 s
Omitting the constant terms 1/σ 2 D + 2/σ 2 s , the sign of dγ/dλ is the one of

A 1 θ(λ) -1 .
Remark that

θ (λ) = A = A 1 Aλ + f /σ s ) ] -1 .
If A is a nonnegative real number, then

A Aλ + B -A < 0.
If A is infinitesimal, after the mean reversion,

Aλ > A. If B > Aλ, then A 2Aλ -A < 0, hence 1/Aλ < 2 and A/(Aλ + B) -A < 0.
Even though the overconfidence parameter φ makes constant volatility of fundamentals allocated to the signal of each agent σ f smaller by σ f /σ s , if φσ f /σ s is bigger than the mean reverted infinitesimal generator Aλ, stability is not kept by mean reversion process.

Origin of Speculative Bubble Burst

Theorem 5.2. (Origin of Speculative Bubble Crush) Mean reversion λ doesn't work to maintain the stability variance γ of random process, the infinitesimal generator A is positive oscillation of fundamental value A to maintain the Brownian motion affects mean reversion λ, then threshold fundamental oscillation Aλ is captured by the noise movement between B(= φσ f /σ s ) and -(σ f /σ s ), hence

-σ f /σ s < Aλ < B(= φσ f /σ s ). (7)
Proof.

If A is nonnegative and less than 1, then

dγ dλ = A (Aλ + B) -A < 0 It's same with 1 < Aλ + B and 1 -φσ f /σ s < Aλ Forσ f > 0, σ s > 0, 0 < (1 -φ)σ f + σ s , 0 < σ f -φσ f + σ s , -σ f < σ s -φσ f , - σ f σ s < σ s σ s -φ σ f σ s -σ f /σ s < 1 -φσ f /σ s < Aλ
The Breaking point of the forward process can be defined when the speculative bubble crushes , the forward process cannot be completed and the backward induction process is required. If the infinitesimal generator "A" is operated with mean reversion λ, forward process perturbing from the final position which has "bubbles" cannot be processed within the threshold -σ f /σ s < Aλ < B(= φσ f /σ s ) in A f = lim t↓0 (1/t)(P t ff ) where P t is semi-group. Forward process can be obtained by perturbing the final position, P t A f is the limit of

P t (1/ε)(P ε f -f )) as ε → 0.
Otherwise, backward induction procedure can be obtained by perturbing the initial position:

∂P t f = AP t f = lim ε→0 (1/ε)(P ε -I)P t f.
Origin of Speculative Bubble Crush is spotted when dividend noise σ D dZ D t is affected by the decrease of A f t dt in the equation (3). 5.3. Individual Trade using properties of American Option, Smooth Pasting condition. Unique "Trading Point" k * Existence For each trading cost c ≥ 0, there exists a unique trading point k * where c(r + λ) is the expected present value of cost and h(•) is the solution of Ito's lemma condition that solves [START_REF] Wilmott | Paul Wilmott on quantitative finance[END_REF] Depending upon above properties of American option, optimal stopping should be an equilibrium of optimal value function q executed if g 0 > k * , wait for the first if the resale option g 0 is bigger than the trading point k * . Before pursuing the equilibrium setting, by symmetry of heterogeneous beliefs, it's clear to look the [START_REF] Scheinkman | Overconfidence and Speculative Bubbles[END_REF] as followings.

k * -c(r + λ) h (k * ) + h (-k * ) -h(k * ) + h(-k * ) = 0. Continuity of h' If c = 0, then k * = 0. Optimal Exercise Point If c > 0, k * > c(r + λ).
[Bubbles] (Scheinkman and Xiong ( 2003)

q(-k * ) = h(-k * )/(r + λ) h (k * ) + h (-k * )
[Resale option] ( [START_REF] Scheinkman | Overconfidence and Speculative Bubbles[END_REF] for

x < k * q(x) = (q(-k * )/h(-k * )) × h(x)
for x ≥ k * as below,

x/(r + λ) + q(-k * ) × h(-x)/h(-k * )c

Dana and Le van (2014) pervades an individually rational efficient allocation and an equilibrium setting under two conditions of (1) no arbitrage prices and (2) no unbounded utility arbitrage. It is intended to broaden our viewpoints of understanding the non-negative expectations with respect to risk adjusted probabilities. In connection with this issue, I wish to stretch newly to the argument of individual and collective absence of arbitrage with overconfidence.

The unique equilibrium allocation with inertia at the speculative bubble crush

The price system is the l-tuple p = (p 1 , ..., p h , ..., p l ). The value of an action a (arrow, 1959) relative to the price system p is l h=1 p h a h where a h is accumulation factor.

A feasible allocation (x 1 , ..., x I ) induced from i x i ≤ i w i where i are finitely many agents indexed by i = 1, ..., I and a non-zero price vector p ∈ R S + are an equilibrium with inertia [START_REF] Rigotti | Uncertainty and Risk in Financial Markets[END_REF] where S are possible states of nature indexed by s = 1, ..., S if for all i,

x i x i =⇒ p • x > p • w i (8)
and endowment w i ∈ R S + w i ∈ R S + is started at date 1 assumed that consumption doesn't occur at date 0 and consumption balance left after trading Arrow securities and for all i, p • x i = p • w i , and for each i, either

x i = w i , or E π i u i (x i ) ≥ E π i u i (w i ) .
where E π • denotes the expected value with respect to the probability distribution π, each π i ∈ Π i , ∃ a closed, convex set Π i and u(x) denotes the vector u(x 1 ), ..., u(x S ).

Inertia drives an equilibrium in trading only at which is the status quo for all traders so T is the set of agents who are involved in trading in this equilibrium, T = {i : x i w i }.

The number l of commodities is a given positive integer. An action a of an agent is a point of R l , the commodity space. A price system p is a point of R l . The value of an action a relative to a price system p is the inner product p • a. The equilibrium allocation (w 1 , ..., w I ) is supported by p as below,

p • a • x i = p • a • w i ,
where a is accumulation factor. In case of x < k * where k * is a trading point,

p • ψ(-k, x) • x i = p • ψ(-k, x) • w i . In case of x ≥ k * , p • ψ(r, λ, -k, x) • x i = p • ψ(r, λ, -k, x) • w i . Hence, speculative bubble crushes at B(= φσ f /σ s )/A > λ > -(σ f /σ s )/A.
Here by, the standard simplex denoted as is the smallest convex set containing the given vertices in R S + . Moreover, given a set , rint is the relative interior of .

Theorem 6.1. We assumed that u i is strictly concave and a closed, convex set of the probability distribution Π i ⊂ rint . If (w 1 , ..., w I ) is an equilibrium allocation, then there is the unique equilibrium allocation with inertia at the speculative bubble crush when x < k * .

Proof. Two cases are to be considered in their turn.

1. For x < k * , given the resale option function ψ(-k, x), there exists a unique equilibrium allocation:

p • ψ(-k, x) • x i > p • ψ(-k, x) • w i , x i + (1/n)w i i w i , p • ψ(-k, x) x i + (1/n)w i i p • ψ(-k, x) • w i , that implies p • ψ(-k, x) • x i > p • ψ(-k, x) • w i
Agents are restricted to the traders involved in trading. Hence, {x i w i φ}.

For example, given a set X, rint(X) denotes the relative interior of X. But Π i ⊂ rint( ) = {π ∈ : π s > 0, ∀s} .

For a subset C of and C ⊂ , one obtains rint(C) = {q ∈ C : a neighborhood V of q in R s such that V ∩ rint( ) ⊂ C}.

Now suppose that

p • ψ(-k, x) • x i = p • ψ(-k, x) • w i .
Fix α ∈ (0, 1). Then: Whence:

p • ψ(-k, x) • w i = αp • ψ(-k, x) • w i -αp • ψ(-k, x) • w i + p • ψ(-k, x) • w i = αp • ψ(-k, x) • w i + (1 -α)p • ψ(-k, x) • w i = αp • ψ(-k, x) • x i + (1 -α)p • ψ(-k, x) • w i
E π i u i (x i ) ≥ E π i u i (w i ) ,
that in turn implies, from the definition of an equilibrium with inertia:

E π i u i (αx i + (1 -α)w i > E π i w i , ∀π i ∈ Π i ,
that implies, from the strict concavity of u i , E π i u i (αψ(-k, x)x i + (1 -α)ψ(-k, x)w i = E π i ψ(-k, x)w i , a contradiction. 2. For x ≥ k * , given the resale function p • ψ(r, λ, -k, x) • x i = p • ψ(r, λ, -k, x) • w i , ψ(r, λ, -k, x) cannot get value at the origin of speculative bubble crush B(= φσ f σ s ) > Aλ > -σ f /σ s , that completes the statement.

=

  p • (αψ(-k, x)x i + (1 -α)ψ(-k, x)w i .13 Documents de travail du Centre d'Economie de laSorbonne -2016.21

This work was achieved through the Laboratory of Excellence on Financial Regulation (Labex ReFi) supported by PRES heSam under the reference ANR-10-LABX-0095. It benefitted from a French government support managed by the National Research Agency (ANR) within the project Investissements d'Avenir Paris Nouveaux Mondes (invesments for the future Paris-New Worlds) under the reference ANR-11-IDEX-0006-02.

Documents de travail du Centre d'Economie de la Sorbonne -2016.21

Varian, Hal R. (1992) Microeconomic Analysis. New York: Norton.


Edy Veneziano
email: edy.veneziano@univ-paris5.fr

Effects of conversational functioning on early language acquisition: When both caregivers and children matter

Introduction

An English version of Grace Wales Shugar's original article (1975) was published in Waterson & Snow (1978), a collection of papers selected from the Third International Child Language Symposium held in London in 1975. At this meeting the statuses of the International Association for the Study of Child Language, officially created in 1972 in Florence, were approved. It was a time of effervescence in the study of child language when researchers were turning their attention to the communicative, interactional and pragmatic aspects of language acquisition and use. The Florence conference lied down important new guiding principles (although not necessarily prophetic):

"That conference signaled the end of the study of language acquisition as if it were a linear development of innate grammatical rules ... A strong trend emerged to see language as a social phenomenon with explicit attention drawn to input factors and to non verbal communication".

Grace Wales Shugar (hitherto GWS) was an active protagonist of this renewal (see also, e.g., 1976, 1981). Her work, advocating a very dynamic view of language acquisition, was even more innovative. Several important and still very actual points were made in her article. Particularly noteworthy is her position on the nature of early adult-child interaction. She considered conversational activity as a highly interactional affair to whose outcome both adult and child contribute and are mutually influenced by the contribution of their partner. Not only she clearly states such a principle but also gives several hints for analyzing concretely and showing the existence of such a mutual influence, for example when she reports that "confirmation of the child's utterance ... appeared to be an important condition for the child to engage further in active text construction " (1978: 237).

The deep meaning of a genuine interactional approach where the contributions of one partner are seen to depend on and concurrently shape the contributions of the other, is still difficult to grasp entirely and to measure satisfactorily. More common approaches within learning-oriented theories of language acquisition remain single-partner oriented, placing most of the emphasis either on the child's contribution (for example, in the cognitively-based "intentional" approach of [START_REF] Bloom | The transition from infancy to language: Acquiring the power of expression[END_REF] or on the adult's skillful framing of interaction, as in the "scaffolding" approach within which most work on social interaction situates itself, going back to [START_REF] Garvey | Some properties of social play[END_REF] and Bruner's early work on formats (1975) and on Bruner's more general socio-cognitive model of development (e.g., 1983).

In much the way that GWS was advancing, the present chapter shows the intricacies of interaction and reports results bearing on immediate and long-term effects of dyadic conversational moves. It will be shown, for example, that early in the second year relating to the topic of one partner crucially depends on the previous response of the other, resulting in exchanges that GWS called "Variant I" (Adult-Child-Adult or longer) and "Variant II" (Child-Adult-Child or longer) where the contributions of the individual partners can hardly be considered independently of each other.

Studying the effect of conversational functioning: some methodological considerations

Finding evidence for the effect of conversational functioning on early language acquisition and use requires thorough developmental and discursive approaches advocated in GWS's paper and in other groundbreaking work of that same period (e.g., Bloom,1973;[START_REF] Greenfield | The structure of communication in early language development[END_REF][START_REF] Slobin | Imitation and grammatical development in children[END_REF][START_REF] Snow | The development of conversations between mothers and babies[END_REF].

All conversational activity where minimal requirements for conversation (turn taking and topic relatedness) are met, need to be retained for analysis, including in this way children's behaviors that present some but not necessarily all of the characteristics of the competence under study. For example, if word learning is aimed at, conversational exchanges involving children's productions not yet easily identifiable as words by an observer, are of utmost interest. If word combination is the target, single word utterances produced in succession and referring to the same event (usually known as SSWUs, e.g., [START_REF] Bloom | One word at a time[END_REF], have to be included, particularly those that unfold during the exchange itself. Excluding these early manifestations and the dynamic processes within which they occur greatly limits the understanding of conversation as a process of acquisition.

The "sequentiality" of extended discourse is another essential aspect to be taken into account when studying the effects of conversational exchanges on language acquisition. Indeed, conversational sequences, though "created from individual utterances, are more than the sum of their parts" (Clark, 1996:318). The sequence is likely to create new events that the partners may have not foreseen beforehand. Such an "unforeseen" novelty of the encompassing totality of the sequence may be the crucial original contribution of conversations to language progress.

Finally, to test the impact of conversational exchanges on later mother-child functioning, the study should dispose of longitudinal data gathered at short intervals. Time intervals greater than a couple of weeks can provide snapshots of development but are unlikely to find specific traces of earlier conversational activity from one time data point to another.

The work presented here attempts to implement these methodological guidelines.

The data

The data presented here come from two longitudinal studies of mother-child dyads observed in their homes for at least one hour approximately every two weeks during naturally-occurring interaction (including free play, book-reading and occasional snack-times). In the first study (referred hereafter as HS, for "Hebrew study"), 6 dyads were observed when the children were between 9/10 and 17/19 months old, and were in the process of acquiring Hebrew. The families, considered "middle-class", lived in Rehovoth, Israel, and spoke Hebrew at home.

The other study (referred hereafter as FS, for "French study") concerns 8 dyads in the process of acquiring French. Here we relate the results of one of these dyads, where the child, a girl, was followed between the age of 15 and 26 months.

Some of the analyses presented are based on methods used in earlier work (e.g., [START_REF] Veneziano | Vocal-verbal interaction and the construction of early lexical knowledge[END_REF][START_REF] Veneziano | Early lexical, morphological and syntactic development in French: Some complex relations[END_REF]. Other analyses are based on methodology originally developed for the present chapter and will be discussed in the corresponding sections.

Topic-related sequences in early mother-child interaction

Mother's and child's adjacent productions are considered to share a topic when they share some content: one partner uptakes and repeats all or part of the other's previous utterance sharing at least sonorities (imitative response 1 ), or relates only semantically to it (semantic-only response 2 ) (see, for more details, [START_REF] Veneziano | Vocal-verbal interaction and the construction of early lexical knowledge[END_REF]; for a similar definition, see more recently, Bloom, Margulis, Tinker & Fujita,1996). Content-related conversational exchanges occur both in Child-Mother and in Mother-Child initiated sequences. In both cases, thematic continuity can be discontinued after the second turn (sequences with "simple" thematic continuity) or it may be kept on, giving rise to a sequence containing at least three-turns bearing on the same topic/content (sequences with "extended" thematic continuity). Examples of Child-Mother, and of Mother-Child, initiated sequences are provided in the next section. They present imitative or semanticonly relations and the whole sequence realizes either simple or extended thematic continuity. It should be noted that in a genuine dyadic approach, exchanges are no more child than mother initiated. In Child-Mother sequences, the child's first turn may indeed solicit the mother's response -which would make the exchange childinitiated -but it is the mother's response that transforms the child's initial turn into a thematically-linked exchange -and in this respect the exchange is motherinitiated. The same can be said of Mother-Child sequences, where again their realization is due as much to the mother's initiation as to child's responding. If interaction is taken seriously, it is not clear that results reporting for example on mothers being more likely to respond than to initiate exchanges (e.g., [START_REF] Harris | Language experience and early language development: From input to uptake[END_REF] concern exclusively mothers' rather than mothers and children's functioning. In accordance, in the rest of the chapter, sequences will thus be referred to as "CM-initial" and "MC-initial".

CM-initial simple sequences: topic continuation with and without imitative uptake

The mother may simply repeat the child's production without altering it. This is mainly the case when the child's production is already on target:

Example Whether or not the mother really believes that the child meant to produce the particular word, the linking established by this sort of conversational functioning creates or strengthens such a language value of the child's production.

Linking and meaning inferences are allowed also when the mother disapproves the child's form (ex. 4), or considers the word inappropriate to the situation at hand (ex. 5 and6 In examples 1 to 6 above, the relation between the child's production and the target is established explicitly providing the child with the opportunity to find similarities and differences between them and eventually correct the form and reshape the boundaries of its meaning.

In other topic-related exchanges the relation between the mother's and the child's utterances rest upon the semantic link only. Pronominal reference is one of the means to attain it :

Example 7 -FS Child : /'pu'pe/ /'pe/ (looking at mother) Mother : va la chercher, elle est restée à la cuisine 'go find it, it remained in the kitchen' Semantic-only links rest upon immediate clear recognition of the child's uttered words that can then be taken as an established starting point for further talk.

MC-initial simple sequences: topic continuation with and without uptake

Topic continuation by children via imitative uptake, usually selective, indicates they attend actively to the mother's language production, while creating an impression of sharing the context as well as its verbal representation:

Example 8 -HS (the child holds a piece of paper) Mother : efo haniyar? efo haniyar shel Tali? (looking around in the room) 'where is the paper? where is Tali's paper? ' Child : /'ni'a:/ (looking ahead)

Example 9 -FS Mother : voilà le bouchon (offering a toy bottlecap to the child) 'here is the bottlecap' Child : /∫õ/ (taking the toy bottlecap from mother's hands)

Semantic-only linking by children rests upon some understanding and shows their ability to use, as GWS might say, "stuff from their store" while taking the lead from offered hints:

Example

CM-initial and MC-initial extended topic-related sequences

In examples 1 to 11 the exchanges are "simple". Often however thematic continuity extends beyond the second turn and CM-initial and MC-initial sequences present at least a third topic-related turn. Example 12 provides an example of MC-initial sequence where both the mother and the child keep focalizing on the same lexical item (saute, 'jump(s)) -in M2 and C2 -resulting in a four-turns topic-related sequence:

Example 12 -FS (mother and child look at a mechanical frog running and jumping on the floor) Mother 1 : t'as vu la grenouille comment elle saute? 'have you seen the frog how it jumps?' Child 1 : /'sot/ 'jump(s)' Mother 2 : elle saute 'she jumps' Child 2 : /sot/ 'jump(s)' Example 13 provides an example of CM-initial sequence where mother and child remain on the same topic in four further turns. In this sequence the child modifies her initial production, in C2 and also with the further unfolding of the exchange, in C3:

Example 13 -HS (the child plays with the wheels of a toy truck) Child 1 : /'∫e a: 'u'lεl/ Mother 1 : maze? galgal? 'what's that? '(a) wheel?' Child 2 : /'a'lo/ (holding the wheel) Mother 2 : galgal #galgal galgal '(a) wheel # '(a) wheel '(a) wheel' Child 3 : /'a'lal/ (holding the wheel) Mother 3 : efo galgal? hine #hine# hine galgal 'where is (the) wheel? here# here # here is (the) wheel'

Exchanges having a minimum of three turns and where, like in examples 12 and 13 above, one lexical item is kept in force through the exchange and is produced across partners, are called reciprocal exchanges.

When more than one lexical item circulates across the turns of both partners, the extended thematic sequences are called discursive. They can be realized:

-by the child's imitative uptake (of M1, in C2) of a word different from the one produced in the child's earlier turn (C1):

Example 14 -HS Child 1 : /'aba/ 'daddy' Mother 1 : aba alach 'daddy left' Child 2 : /'a'la/ 'alach ≈ left' Mother 2 : alach 'left'

-by a child-produced semantic-only relation co-existing with another link within the same thematic sequence. In example 15, an imitative link (established by the mother between M1 and C1) is followed by a semantic-only relation established by the child between C2 and M1. In example 16, the semantic-only relation between C1 and M1 is followed by an imitative link (established by the mother between M2 and C1), followed in turn by an immediate semantic-only relation established by the child between C2 and M2. In M3 the mother pulls everything together:

Example 15 -FS (the child points towards a mechanical frog jumping on the floor) Child 1 : /gu/ Mother 1 : oui c'est la grenouille 'yes it is the frog' Child 2 : /sot/ 'jump(s)'

Example 16 -FS (mother and child look at a picture book) Mother 1 : où sont les enfants? 'where are the children?' Child 1 : /zεn/ 'carrousel' ≈ 'merry-go-round' Mother 2 : en carrousel? ils sont dans le carrousel?

'on the merry-go-round? they are on the merry-go-round?' Child 2 : /afã/ 'enfant(s)' ≈ 'child(ren)' Mother 3 : les enfants sont en carrousel?

'the children are on the merry-go-round?' Child 3 : hein

Developmental results

In the Hebrew study topic-related exchanges present two significant increases, one at 14 months (χ

2 =5     .0 , df=4 3        s  χ  17.21     , df=2) (see figure 1), Figure 1
Overall, mothers relate to the children's production as much as children relate to the mother's (matched t-test, n.s.). Individually, in only one dyad the mother related significantly more to the child than the other way around (matched t-test performed on 8 longitudinal data points: t=3.92, p<.01, df=7). Throughout the period, thematic continuity is implemented mostly by imitative uptakes, as much by children (mean of 90.3%) as by mothers (mean of 88.7%). Semantic-only relations show a mean increase at 16 months (χ 2 13.01, p<.05, df=6) 4 .

The profile of the three main types of topic-related exchanges -simple, reciprocal and discursive -also changes with children's development (see Figure 2).

Although simple exchanges remain proportionally most frequent throughout the whole period, up to about 13 months their proportion greatly outweighs that of extended sequences; simple exchanges decrease starting at 14-15 months and, at 16-17 months, extended sequences are proportionally the most frequent. Reciprocal exchanges increase at 14-15 months and discursive exchanges at 16-17 months.

Similar results have been found in the French study. Also there, thematic continuity is implemented mostly by imitative uptakes (between 75 and 95% of all the exchanges), as much by the child (mean of 82%) as by the mother (mean of 91%). At 19 months the child relates more than the mother does via semantic-only links (see Figure 3).

Reciprocal exchanges are already well represented at 15 months, the time the study began for this dyad, when they are proportionally more frequent than simple exchanges (see Figure 4).

As in the Hebrew study, simple exchanges remain proportionally important afterwards but at 18-19 months extended sequences are the most frequent again. After 15 months, reciprocal exchanges continue to occur with proportions similar to those of the Hebrew study (around 40%); discursive exchanges appear at 16 months and increase at 18 months. The profile of exchange types remains the same at 19 months but, at this time, semantic-only relations are most common in the realization of discursive sequences (see Figure 5).

We turn now to consider the effect of dyadic conversational functioning on children. At first, we will look at its immediate effects during the exchange and then at longer-term effects on the use and early acquisition of language.

Effects of conversational exchanges 1. Immediate effects of mothers' imitative uptakes

4.1.1. Opportunities for improved matching [START_REF] Chouinard | Adult reformulations of child errors as negative evidence[END_REF] found that after a corrective reformulation by mothers, children modify immediately after their productions. To measure this effect, in the HS we looked for sequences where the second intervention of the child was better on target than the previous child's intervention (i.e., C 2 is a better approximation than C 1 in (M)C 1 MC 2 sequences). Excluding sequences where the child's initial production is already on target, we found that adaptive accommodation occurs very rarely before 14 months. In the 14 to 17 months period adaptive accommodation is observed on the average in 34% of the exchanges where accommodation was possible (with a range between 21% and 50%, according to dyads). Significant increases in adaptive modifications were observed at 14 months for three of the children, at 16 months for one child, and at 17 months for the remaining two children.

Thus the first immediate effect of topic-related conversations is to provide children opportunities to actively conventionalize their emerging lexical forms (see also [START_REF] Otomo | Maternal responses to word approximations in Japanese children's transition to language[END_REF].

Opportunities to remain on topic

A second immediate effect of topic-related conversations is to produce further interaction on the same topic [START_REF] Tomasello | Joint attention and lexical acquisition style[END_REF]Veneziano,1988). In the HS we found that, starting at 14 months, children are three times more likely to continue the topic if the mother's contribution is itself related to the child's previous utterance than if the mother's utterance initiates a new topic (44.9% vs. 11.5%, at 14-15 months; and 46.8% vs. 17.4% at 16-17 months, the difference between the proportions being significant for the corresponding z-scores with p<.001). In other words, children are more likely to attend to an adult's utterance that relates to their own focus of attention than to one that doesn't. Moreover, in this conversational context, children are more likely to continue the topic by restating the initial utterance (giving rise to a reciprocal exchange, see examples 13 and 14 above), than to utter a topic-related verbalization different from the one produced earlier (resulting in a discursive exchange like in examples 15 and 16 above). Again, the difference between proportions, measured by z-scores, is significant (with p<.001) at all three periods [START_REF] Veneziano | Vocal-verbal interaction and the construction of early lexical knowledge[END_REF].

Interestingly, also mothers' topic-relatedness shows the same trend. Mothers are much more likely to relate to the child's preceding utterance when this relates to their own previous utterance then when it does not so relate (55% vs. 14.6%, at 14-15 months; and 52.4% vs. 19.1% at 16-17 months, the difference between the proportions being significant for the corresponding z-scores with p<.001). Between 14 and 17 months also mothers are more likely to stay on topic by reinstating explicitly the child-produced word, giving rise to a reciprocal rather than to a discursive exchange (42.8% vs. 12.2%, at 14-15 months; and 40.8% vs. 11.6% at 16-17 months, the difference between the proportions being significant for the corresponding z-scores with p<.001).

Children's reinstatement of their own initial production is not independent of the adult's behavior: the child is more likely to repeat his own previous production if the mother has related imitatively to it than otherwise. Indeed, at the 14-15 months, children's self-repetition is significantly more likely to occur when there is an intervening mother's topic-related turn between the two productions (C2 in C1MC2, for C1=C2), then when no such intervention is present (C2 in C1//C2, for C1=C2)(χ 2 0 1        . Moreover, the child's second production is more liely to present adaptive modifications in the former than in the latter case (χ 2 6 83       ). When either the mother or the child take the first step in topic-related responding, a conversational process gets started such that the partner is likely to stay on topic and the child -in M1C1M2C2 or in C1M1C2 sequences -has the opportunity to compare productions and eventually to attain better matching to targets (see also [START_REF] Chouinard | Adult reformulations of child errors as negative evidence[END_REF][START_REF] Otomo | Maternal responses to word approximations in Japanese children's transition to language[END_REF].

The influence, on both partners, of the conversational activity of the other is completely in line with results already reported in GWS' original paper as well as with those of other authors (e.g., [START_REF] Folger | A pragmatic analysis of spontaneous imitations[END_REF][START_REF] Tomasello | Joint attention and lexical acquisition style[END_REF].

Focalization contributes to extended joint attention

Topic-relatedness by the mother may not be the only relevant variable for children's increased tendency to continue staying on topic. Mothers' topic-related imitative uptakes present, to a greater extent than other turns (i.e., topic-initiating and non imitative topic related turns), the characteristic of focalization [START_REF] Veneziano | Vocal-verbal interaction and the construction of early lexical knowledge[END_REF]. Mothers' imitative uptakes of children's productions tend to be focalized within the mother's turn by either being the only elements of the turn or by being separated from the rest of the turn by manually measurable pauses (greater than half a second). Moreover, focalized elements are also often repeated:

Example 17 -HS -focalization with repetition Child : /'a'ma/ (while trying to pull apart a toy drum) Mother : ima ima 'mommy, mommy'

Example 18 -FS -focalization with repetition (The child touches the prickling spot on the image of a face in a picture book) Child : /pik/ 'sting(s)' Mother : pique! ça pique oui, comme la barbe de papa 'sting(s)! it stings yes like father's beard At 14 months, when topic-related exchanges increase significantly, focalization occurs in 39% of mothers' turns. However, while focalization occurs in 30% of unrelated turns, as much as 76% of imitatively related turns focalize exactly the uptaken lexical item (the difference between the proportions is significant: p(z=12.95)<.001). Focalization is thus a specific characteristic of mothers' imitations in topic-related turns. Although focalized elements can also be repeated, repetition is not specific to imitative uptakes, as it occurs as much in imitatively-related (32%) as in unrelated mothers' turns (34%).

Children are more likely to relate to focalized than to non focalized lexical items. Indeed, while 26.5% of words focalized in mothers' turns are followed by a child's imitative relation, only 14.7% of non focalized items are uptaken, a difference between proportions that is significant (p(z=5.14)<.001).

Focalization in mothers' turns then positively interacts with topic-relatedness, contributing to children's continuation of the topic, to the production of reciprocal exchanges and, through the modifications than can occur in these conversational contexts, to improved correspondence between children's productions and target words.

Long-term effects of conversational functioning

4.2.1.Changes in the interactional language value of children's productions

This analysis looks at whether children's productions involved in imitative CM-initial or MC-initial, simple or extended sequences, function at later sessions as the words to which they have been related to in earlier conversational exchanges. In this case, considering that semantic-only relations conversationally treat the corresponding productions as having relatively conventional forms and meanings, at later sessions, mothers should relate to them more often with semantic-only relations than is the case of children's productions not involved earlier in imitative topic-related exchanges. Moreover, children's productions involved earlier in topic-related exchanges, should also be easily identified by an observer as the corresponding words when occurring outside of conversational sequences.

To test this hypothesis we first distinguished, for each child, and at each session of the longitudinal studies, lexical items that were observed already at earlier sessions from those appearing for the first time at the session under scrutiny (starting from the session where three "previous" and three "new" items could be identified). Previous and new lexical items were then classified according to whether they were exclusively involved in imitative relations, were involved also, or exclusively, in semantic-only links, or, if not involved in any conversational exchange, where considered by the observer to be used appropriately.

Results show that "new" lexical items (appearing for the first time at the session under scrutiny) are involved significantly more in imitative relations than "previous" lexical items; instead, the latter are involved significantly more in semantic-only relations, or are used appropriately when not involved in any topicrelated exchange (see Table 1). This is the case for each dyad, at each session where the measures could be performed. For dyad Yuval the proportion of semantic-only relations is lower than for the other dyads (this dyad using highly the imitative link). However the difference between "new" and "previous" lexical items remains significant and goes in the same direction as for the other dyads. Between 61% and 97% of "previous" lexical items were earlier involved, at least once, in imitative exchanges, and between 30.4% and 72.2%, at least once in extended reciprocal exchanges. Moreover, when they occur again, these same "previous" lexical items are produced less in imitative exchanges than at previous sessions (for the difference between proportions comparing overall imitative exchanges: p(z=6.35) <.001; comparing reciprocal exchanges: p(z=4.72)<.001).

These results suggest that imitative relations contribute to the smooth recognition of children's productions as the particular words they have been linked to, allowing straighter discursive semantic relations to occur, and their identification as those same words by an observer. This finding is consistent with earlier studies showing that child-imitated words are used later spontaneously [START_REF] Bloom | Imitation in language development: if, when and why[END_REF][START_REF] Ramer | The function of imitation in child language[END_REF].

Effects of extended reciprocal exchanges on later lexical acquisition

In the late sixties [START_REF] Slobin | Imitation and grammatical development in children[END_REF] already stated explicitly that sequences in which children imitate mothers' expansions (what we have called here "reciprocal" exchanges) are of great relevance since children are there confronted with a correct model of their own production at a time their attention is already engaged and the situation remains unchanged (1968:440). However, although several other authors, among which GWS (1975,1976,1978,1981), pointed out the particular interest of extended discourse for early language acquisition, most studies have taken into account mothers' expansions and children's imitations independently of each other and very few have tried to assess the effect of extended discourse relative to that of simple pairs of topic-related turns. This analysis has been performed on our data and is presented in this section.

In order to assess the relative effect of "simple" and of reciprocal sequences we looked for a time period in which children had a similar number of words in their repertoire (this was the case at 14 months) -and for a later time in which this measure differentiated the children (in our sample this happened at 17 months) (see Table 2).

Results show a specific effect of reciprocal exchanges on the rhythm of lexical acquisition by the children (see Table 3).

Both MC-initial and CM-initial reciprocal exchanges correlate significantly with gains in vocabulary, while simple imitative exchanges -expansions by the mother not followed up by a reciprocal imitation by the child (simple CM-initial) and imitative uptakes by the child not followed up by a reciprocal imitation by the mother (simple MC-initial) -, do not correlate.

Overall measures of mothers' expansions do not correlate significantly with gains in vocabulary, while overall measures of children's imitative uptakes do. However, when simple MC-initial exchanges are distinguished from extended Cumulatif vocabulary at 14 and 17 months, by child, and gains in vocabulary in the 14-17 months period Table 2 ones, only the latter remain significantly correlated with later vocabulary acquisition.

Mothers' imitative topic-related turns

These results underscore the utmost importance of working within a dyadic approach. Indeed, they show that mothers' imitations and expansions are only potentially beneficial to lexical acquisition: their value is realized only through children's back-channel relation to them. Also children's imitative uptakes have greater value for lexical acquisition if the mother provides a back-channel relation to them. These conversational events offer children opportunities to actively compare their own production to that of the mother and to modify their phonological form and possibly adjust their meaning in the immediately subsequent topic-related turn.

The overall sequence confers different roles to similar thematic relations

As we have seen above, extended sequences can be reciprocal or discursive (see examples 12 to 15 above). Reciprocal sequences, which explicitly maintain one lexical item in the focus of joint attention, are developmentally prior to discursive sequences, where the child remains on topic through the production of at least one lexical items different from the one produced earlier in the sequence. Both the reciprocal (in example 19) and the imitative discursive (in example 20) exchanges can be schematized to have a C 1 M 1 C 2 structure constituted of two pairs of topic-related turns, C 1 M 1 and M 1 C 2 . These pairs, considered in themselves, can be described similarly in the two sequences: the first pair contains an imitative uptake by the mother of the child's preceding production; the second pair, the child's imitative uptake of one of the words contained in the mother's preceding turn. However, when the pairs are considered as components of extended sequences, they are rather different systemically and what differs specifically is the relationship that C 2 holds to C 1 . In the reciprocal exchange, C 1 and C 2 are the same (C 1 =C 2 ) since the relation between C 1 and M 1 , and that between M 1 and C 2, concern one and the same lexical item "tourne" 'turn(s)'. In the imitative discursive exchange of example 20, C 1 differs from C 2 (C 1 ≠C 2 ) since the relation between C 1 and M 1 concerns one lexical item ("tourne" 'turn(s)'), and that between M 1 and C 2 concerns another lexical item (toupie, 'spinning top'). While in reciprocal exchanges there is mutual focus on the same lexical item establishing it as a shared word, discursive exchanges give rise to a Successive Single-Word Utterance (SSWU), a transitional phenomenon between single-word and two-word utterances [START_REF] Bloom | One word at a time[END_REF][START_REF] Greenfield | The structure of communication in early language development[END_REF]Ochs, Schieffelin & Platt,1979;[START_REF] Scollon | A real early stage: an unzippered condensation of a dissertation on child language[END_REF][START_REF] Greenfield | The structural and functional status of single-word utterances and their relationship to early multiword speech[END_REF][START_REF] Barrett | Early language development[END_REF][START_REF] Veneziano | From one word to two words: repetition patterns on the way to structured speech[END_REF]. Discursive exchanges, containing imitative, or imitative and semantic relations, are in fact particular kinds of SSWUs that are generated by conversational functioning. They have been considered pivotal in the transition and found to be among the first kinds of SSWUs to occur (e.g., Schwartz, Chapman, Prelock, Terrell, & Rowan, 1985;[START_REF] Veneziano | Early lexical, morphological and syntactic development in French: Some complex relations[END_REF]. It appears then clearly that the same type of relation between adjacent turns (for example, an imitative uptake by the child) may have different roles in the overall sequence and may thus have different potentialities for language acquisition. When it is a component of a reciprocal exchange it creates favourable conditions for lexical acquisition; when it is a component of a discursive exchange it contributes to progress in combinatorial speech.

Concluding remarks

Starting in the early part of the second year, what children learn of their environmental language they use it to communicate with familiar partners and their interventions occur progressively more frequently in topic-related conversational exchanges where mothers interpret, reformulate or expand their children's productions, and children imitatively uptake words from their caregiver's utterances or relate only semantically to them. These communicatively-driven conversational exchanges are one of the major source of information about the "languageness" of children's early productions, both for the actors (caregivers and children) and for the observer. At the same time they offer children opportunities to learn more about words, the way they are used to express intentions effectively, as well as the way they combine with other words. Although two years old can learn new words even if not directly addressed to [START_REF] Akhtar | Learning words through overhearing[END_REF], conversational functioning may be essential at younger ages, and remain a viable source of information throughout development.

Our results show that conversational functioning has the effect of producing further on topic conversation by mothers and by children. Moreover, when children produce a second turn in reciprocal sequences (C2, in C1M1C2 or M1C1M2C2), they are much more likely to modify their production than if they were simply repeating it without the mother's corrective uptake in between them (M1 or M2 above).

Our findings show that these immediate effects of conversation have also longer-term effects. On the one hand, productions involved in exchanges at earlier sessions are recognized, at later sessions, as the words to which they have been linked to. On the other, a specific effect of reciprocal sequences, but not of simple ones, was found on the rate of lexical acquisition.

In reciprocal exchanges several important characteristics converge: joint attention on forms and often on referents (e.g., [START_REF] Tomasello | Constructing a language: A usage-based theory of language acquisition[END_REF], opportunities for articulatory practice and auditory feedback (e.g., [START_REF] Locke | Vocal learning and the emergence of phonological capacity[END_REF] with likely modification of forms and adjustments in meaning, and the exchange of signs of mutual recognition. This builds up, on the one hand, mother's belief that her child's early vocalizations represent word attempts, a variable found to correlate with lexical acquisition [START_REF] Murphy | Predicting rate of lexical development[END_REF]; on the other, an increasing body of children's productions having a selective entry into mothers' responsiveness. Both aspects can be considered to be the underlying and subtle mental basis for mutual understanding.

The specific effect of reciprocal compared to simple exchanges shows the importance of considering extended discourse instead of limiting the analyses to two-turns relations, as is the case of mothers' expansions, imitations and corrections, and of children's different kinds of imitations. It also underscores the fact that what matters is not so much what each partner does individually but what mothers and children do together, the activity of one boosting that of the other, and vice versa. This empirical finding completely supports GWS' idea that mutual influence is central to conversational activity and takes it one step further in showing that it is the essential variable in assessing the effects of conversation on early language acquisition 5 .

The types of conversational structures occurring with children's development are another essential variable to be considered. They point out that conversational sequences have "a life of their own" conferring different roles to the individual contributions of which they are composed. In this early period at least, jointly constructed sequences give rise to original events that are properly conversational: mutual recognition of forms and meanings, on the one hand, and semantic links between successively produced one-word utterances, on the other. We argue that the crucial and specific contribution of conversational activity to progress in language acquisition and use resides in such properly interactional effects of the whole discourse sequences. While children use their usual conversational skills (for example, establish an imitative relation with the preceding intervention of the caregiver), the result provided in the overall sequence may be new and unplanned in advance.

Particularly interesting are discursive exchanges in which children are led, through the unfolding of conversation, to successively utter words that can be meaningfully related one to the other. The new events emerging from such a conversational functioning are SSWUs that children could not have produced at that time, had they relied exclusively on their competences.

Compared to reciprocal sequences, children's functioning for the obtainment of imitative discursive sequences is different: it requires a shift in attention from a previously uttered and related to item (probably focalized in the caregiver's utterance), towards another word that, albeit contained in the mother's utterance, has not been previously attended to. This change doesn't seem to require new conversational or properly linguistic skills. It appears to demand progress in cognitive abilities, like greater representational skills, increased inhibitory capacities 6 and an opening towards variation in the expression of events and intentions (e.g., [START_REF] Veneziano | The emergence of expressive options in early child language: a constructivist account[END_REF]. It is such a shift in attention by children that is ultimately responsible for generating different conversational outcomes having

  Figure 5 Distribution of discursive exchanges Imitative vs. semantic by child, by child's age

Hebrew study -mean number of topic related exchange per child's age

	120
	100
	80
	60
	40
	20
	0
	10	11	12	13	14	15	16	17
				age of the child
			initial MC		initial CM	total exchanges

Figure 2 Proportion of simple, reciprocal and discursive thematic cotninuity exchanges in the Hebrew Sample: Data averaged over children at three time periods

	100%
	80%
	60%
	40%
	20%
	0%
	11 -13	14 -15	16 -17
		age in months
	Simple	Reciprocal	Discursive

Figure 4 Proportion of simple, reciprocal and discursive exchanges over total topic-related exchanges, by child's age

			Figure 3
	Proportion of imitative links in topic-related exchanges by mother and by child,
			by child's age
	100%
	80%
	60%
	40%
	20%
	0%
	15	16	17	18	19
			age in months
	100%
	80%
	60%
	40%
	20%
	0%
	15	16	17	18	19
			age in months
	% simple		% reciprocal	% discursive

% M's imitative links over all topic-relations by M % C's imitative links over all topic-relations by C

  Proportion of "previous" and "new" words involved in semantic-only relations, exclusively or not, at the session under scrutiny, per dyad

		Table 1
		"previous"	"new"	Chi-square
	Individual dyads	words	words	(2x2)
	Oded
	number of analyzed word	23	24
	% exclusively	or also in semantic-
	relations only		82.6%	50.0%	4.20*
	Shimon
	number of analyzed word	15	21
	% exclusively	or also in semantic-
	relations only		93.3%	28.6%	12.35***
	Nili
	number of analyzed word	54	46
	% exclusively	or also in semantic-
	relations only		79.6%	39.1%	15.46***
	Yuval
	number of analyzed word	36	70
	% exclusively	or also in semantic-
	relations only		27.8%	8.6%	5.43**
	Tali
	number of analyzed word	31	66
	% exclusively	or also in semantic-
	relations only		64.5%	33.3%	7.13**
	* p<.02; **p<.01; *** p<.001

different potentialities for progress. It is only gradually that these initial steps can lead to durable changes, as the relations revealed by the overall discourse sequence need to be apprehended and integrated into other habitual functioning.

Wales Shugar, G. 1981. Early child discourse analyzed in the dyadic interaction unit. International Journal of Psycholinguistics 8-2 ( 22), 55-78.

1 These types of relations have been called in previous work Sonorically-Related Turns -SRTs [START_REF] Veneziano | Vocal-verbal interaction and the construction of early lexical knowledge[END_REF] 2 These types of relations have been called in previous work Nonsonoric Semantically-related turns -NSRTs (op.cit))

3 Tested by the one-sample chi-square test at each point of the distribution after the first. The value at one session has been compared to the expected value taken to be the mean of the distribution up and including the tested value. As it takes into account the tested value, the test is conservative. 4 Three dyads contribute most to this mean increase; two others show a progressive increase and yet another a late start (at 17 months).

5 Joint conversation has wider and far-reaching influence even at later ages. For example, [START_REF] Haden | Motherchild conversational interactions as events unfold: Linkages to subsequent remembering[END_REF] found it to be a relevant variable in 30 to 42 months old children's recall of experienced activities. 6 Research should establish whether such a shift in attention is first linked to particular properties of caregivers' utterances. For example, M1, in C1M1C2 sequences, might be linked with a semantic-only relation to C1, or the imitative uptake of C1 could be non focalized. Moreover, while for the dyad studied here, imitative discursive sequences precede semantic ones, other children might better attain a discursive sequence by purely semantic relations that do not require to inhibit the repetition of the child's previous production.