T Divoux

V Vidal

F Melo

J.-C Géminard

Acoustic emission associated with the bursting of a gas bubble at the free surface of a non-newtonian fluid

We report experimental measurements of the acoustic emission associated with the bursting of a gas bubble at the free surface of a non-newtonian fluid. On account of the viscoelastic properties of the fluid, the bubble is generally elongated. The associated frequency and duration of the acoustic signal are discussed with regard to the shape of the bubble and successfully accounted for by a simple linear model. The acoustic energy exhibits a high sensitivity to the dynamics of the thin film bursting, which demonstrates that, in practice, it is barely possible to deduce from the acoustic measurements the total amount of energy released by the event. Our experimental findings provide clues for the understanding of the signals from either volcanoes or foams, where one observes respectively, the bursting of giant bubbles at the free surface of lava and bubble bursting avalanches.

INTRODUCTION

A host of broad-interest phenomena involve bursting bubbles at fluid surfaces. In daily life, jam or purée cooking produces sonic bubbles that can project fragments at bursting. In the geophysical context, giant bubbles bursting at the top of a volcano vent, or at the surface of a lava lake, are examples whose understanding might be crucial for predicting volcanic activity [1,2]. Although less considered, bubble bursting also occurs at the surface of aqueous foams [START_REF] Weaire | The Physics of Foams[END_REF] typically produced by wash-or beauty-products or even by poured beer. The analysis of acoustics emission is then a natural way of investigating bursting systems, revealing the collapse or bursting mechanism and properties of the fluid gas mixture [START_REF] Prosperetti | [END_REF][START_REF] Leighton | The Acoustic Bubble[END_REF][START_REF] Müller | [END_REF][7][8][9].

On the one hand, various statistical analysis of bursting noise have been carried out [START_REF] Müller | [END_REF]7,10]. For instance, the sound pattern of collapsing foams was recently analyzed, revealing a log-normal distribution for the energy of events. For the events of highest acoustic energy, the distribution is however a power law suggesting that a "wide variety of bubble membranes areas is exploding" [7]. Consistently, the film rupture event seems to be independent of the bubble size and exhibits instead a correlation in space due to cascade bursting [9]. In turn, the typical frequency of the acoustic signal has been statistically correlated to the bubble size [7,10]. The acoustic emission of a single standing spherical bubble has been recorded as well [10] and sophisticated high speed techniques have been used to elucidate the bursting dynamics of spherical smectic films [11]. However, detailed correlations of high speed images of foam bursting-films with the features of the acoustic emission have not yet been performed.

On the other hand, volcanologists have recorded the sound produced by astounding burstings, and have tried to infer, from the signal characteristics, the dynamical processes involved in these natural phenomena [12]. Previous laboratory experiments have intended to reproduce bubbles formation, rising and bursting in geometries presenting similarities with bubbles in magma conduit or lava lakes [13,14]. Since the forces applied on lava have a time scale much larger than its relaxation time (about 4 × 10 -8 s for Strombolian magma [15]), most of model experiments have been performed in newtonian fluid [13,14,16]. However, departure from this newtonian behavior can occur if the magma contains crystals [16], which can be observed for example on Strombolian ejecta [17]. Some experiments, performed in more 'exotic' fluids like japanese curry or tomato sauce, have demonstrated the complex behavior of such fluids, and brought to the fore the non-trivial physical processes leading to sound generation -from Helmholz resonator type to bubble oscillation inside the fluid [START_REF] Ichihara | Japan Earth and Planetary Science Joint Meeting[END_REF].

To understand the geometrical and dynamical aspects of sound produced by bursting bubbles having an elongated shape, we described recently the acoustic emission from an overpressurized cylindrical cavity, closed at one end by a fluid film [START_REF] Vidal | [END_REF]. We showed that the cavity geometry governs the frequency, the viscous dissipation and radiation are responsible for the wave damping and the acoustic energy depends not only on the energy initially loaded inside the cavity but also on the characteristic time associated with the film bursting.

Here, we present the analysis of the high-frequency acoustic-wave emitted by a bubble bursting at the free surface of a non-newtonian fluid. In such fluids, the complex rheology [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF] is the source of puzzling phenomena including surface instabilities due to elastic effects [START_REF] Podgorski | [END_REF], cusp at the tail of elongated rising bubbles [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF][22][23][START_REF] Chhabra | Bubbles, drops, and Particles in Non Newtonian fluids[END_REF] and oscillations of falling spheres or rising air-bubbles [25][26][27]. We choose an experimental situation in which the bubble, generally elongated, exhibits a nearly-conical steadyshape during the rise toward the free surface. When the (c) As a result of the outgoing air flow and of the capillary forces, the remaining part of the bubble head is blown upwards and shrinks. The width of each image is 3 cm whereas the time-difference between them is 0.8 ms. Here, the acoustic emission associated with the event does not last more than 2 ms, during which the bubble body clearly does not significantly deform.

bubble reaches the free surface, a liquid film separates the bubble body from the surrounding air. This film thins, eventually breaks, and can display a complex behavior (see for instance [28][29][30][31] for viscous films, and [START_REF] Warszyński | Colloids Surfaces A: Physicochem[END_REF][START_REF] Angarska | Colloids Surfaces A: Physicochem[END_REF] for soap films).

In the chosen experimental conditions, due to the nonnewtonian fluid properties, the bubble body does not significantly deform during the bursting event. We first link the characteristics of the radiated sound with bubble volume and shape, which is governed by the rheological properties of the fluid. Despite the conical shape of the bubbles, acoustic signals show a narrow frequency spectrum whose characteristic wavelength is linear in the bubble length, exhibiting a well-defined offset that is due to acoustic radiation [34]. Both the results are theoretically explained using linear acoustic [START_REF] Landau | Course in Theoretical Physics: Fluid Mechanics[END_REF][START_REF] Pierce | Acoustics -An introduction to its physical principles and applications[END_REF][START_REF] Rayleigh | Theory of Sound[END_REF]. Differences and similarities with the newtonian-fluid case are underlined. We then show that, if the rupture time of the viscoelastic film does not control the wave generation (i.e., it is fast enough), an optimal size for a 'sonic' bubble exists: The largest amplitude of the acoustic signal is recorded for a bubble having this optimal size. We then show that, without a detailed knowledge of film-bursting dynamics, acoustic measurements are not a reliable method to access the total amount of energy released. Indeed, the amount of the energy transferred to the acoustic wave drastically depends on the characteristic time associated with the opening of the bubble [START_REF] Vidal | [END_REF], which is not controlled experimentally. This result might find interesting applications to aqueous foams as well, indicating that the statistic of energy released by bubble bursting-avalanches, recently characterized by acoustic emission [7], might be not only influenced by the distribution of the bubble sizes, but also by that of the rupture times. Consistently, our results suggest that bursting cascades might be triggered more likely by silent bubbles than by noisy ones. In this case, most of the potential energy loaded inside the cavity would contribute to larger distortions of bubble network.

EXPERIMENTAL SETUP AND PROCEDURE

The experimental setup consists of a vertical plexiglas container (square section 30 × 30 mm, height 88 mm) filled with a transparent non-newtonian fluid up to the upper plane. Thanks to the transparent and planar walls of the container, the fluid can be imaged from the side without any optical distortion by means of a fast-camera (HiSIS 2002, KSV Instruments Ltd., up to 1220 images/sec, Fig. 1)

In order to produce bubbles having a well-defined volume V (from 0.1 to 1.5 mL), a chosen amount of air is rapidly injected by means of a syringe pump connected to the container by a hole drilled at the center of the lower plane. After injection, the bubble rises in the fluid, reaches the upper free surface (Fig. 1 & 2) and finally bursts, producing a characteristic audible sound.

To characterize this phenomenon, we record the acoustic emission by means of a microphone (ATM33a, Audio Technica associated with a preamplifier, Eurorack UB802) which is located 3 cm away from the gel freesurface, with a 45-degrees inclination from the vertical. The position of the microphone shall remain identical for all the experimental results reported herein.

The chosen non-newtonian fluid is obtained by diluting a commercial hair-dressing gel (Gel fijador de cabello, for men, Camel White R ) in pure water. This latter choice is mainly justified by the fact that, in such a fluid, the air bubbles usually exhibit a nice vertically-elongated shape, terminated by a cusp at the bottom, which significantly differs from the rounded shape usually observed in a newtonian fluid [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF]. In addition, one can easily be supplied with large quantities of fluid, reproducible mixtures are rather easy to prepare and they are stable in the time. The non-newtonian character of the fluid is more or less pronounced depending on the concentration, c, of gel in the mixture (from 25 to 40% in volume). All the solutions are obtained after the mixing of the two components during half a day by means of a magnetic stirrer. Afterwards, the small bubbles that still remain trapped in the fluid are eliminated by placing the solution in an ultrasonic bath for several hours. In order to avoid any memory effect, the gel is stirred and let at rest for a few minutes between two bubble rises. The mixtures are likely to be subjected to drying: In order to avoid any significant change in the overall concentration c, each sample is used only for 4 days.

PRELIMINARY OBSERVATIONS : BUBBLE SHAPE AND BURSTING DYNAMICS

Before analyzing in details the acoustic emission associated with the bursting of a single bubble at the free surface, one must first pay attention to the bubble shape and dynamics.

First, depending on the concentration c and on the volume V , one observes two qualitative different steady shapes of the bubbles which rise up in the bulk of the fluid. Indeed, for low gel concentration (typically c < 30%), the bubbles exhibit an almost spherical shape, similar to that observed in newtonian fluids. On the contrary, for larger concentration (typically c > 30%), the bubbles exhibit an elongated shape with a cusped tail, as already observed in non-newtonian fluids [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF][22][23][START_REF] Chhabra | Bubbles, drops, and Particles in Non Newtonian fluids[END_REF].

During the rise, one also notices oscillations in the bubble shape, as previously pointed out in the literature [27].

The shape of the bubble which bursts at the free surface qualitatively exhibits almost the same transition. For small concentration c, one observes a rounded bubble on the last image previous to the bursting (Fig. 2a) whereas, for a larger concentration c, a cusp is clearly observed (Fig. 2d). However, we point out that at intermediate concentration (typically c ∼ 30%) the shape of the bubble, when bursting, also depends on its dynamical behavior when reaching the free surface and, thus, on its volume V : For small volume (V < 0.25 mL), only rounded bubbles are observed whereas, for large volume (V > 0.40 mL), the bubbles always exhibit a cusp; In the intermediate range (0.25 ≤ V < 0.40 mL), both types of bubble can be observed for the same volume V . Qualitatively, the capillary forces are large enough to maintain the small bubbles in equilibrium at the free surface so that a bubble, which initially exhibits a cusp while rising, deforms and equilibrates before the thin film that encloses the inside air breaks due to the drainage. No cusp is then observed. On the contrary, the capillary forces are not large enough to maintain the largest bubbles in equilibrium at the free surface. As a consequence, the tail of the bubble does not significantly deform at the free surface before the thin film at the top breaks, due to the increase of its surface area and not to the drainage, in this case. A cusp is thus systematically observed. In the intermediate case, the capillary forces are likely to maintain the bubble at the free surface but, because of some premature ruptures of the thin film at the top, one can observe a bubble exhibiting either a rounded bottom or a cusp.

The equilibrium shape or the dynamical behavior of the bubble at the free surface, which would deserve an extensive study to be accounted for, are not the aim of the present work. From the qualitative description of the bubble behavior at the free surface presented above, we shall only remember that the bubble are elongated and exhibit a cusp in most of the experimental conditions and that, depending on the volume, the thin film at the top might break either because of the drainage or of the increase in its surface area. In what follows, we shall only analyze the acoustic signal associated with the bursting event with regard to the shape and dynamics of the bubble at the free surface.

THE ACOUSTIC SIGNAL

The bursting event is systematically associated with the emission of a sound wave, characterized by a welldefined frequency (Fig. 2).

Qualitatively, the sudden bursting of the thin film at the top excites a resonant pressure-wave in the bubble body which is initially overpressurized. The phenomenon is similar to that thoroughly described in Ref. [START_REF] Vidal | [END_REF], where the acoustic emission associated with the bursting of a thin soap film that initially closes a cylindrical overpressurized cavity is analyzed in details. In our experimental case, due to the viscoelastic properties of the fluid, the bubble body is generally elongated and the bubble wall does not significantly deform during the characteristic duration of the sound emission (Fig. 1). The opened bubble-body thus selects resonant modes among which the fundamental is the most intense. Due to the radiation at the open end, one records outside the cavity a sound wave exhibiting well-defined frequency and duration which are the subject of the analysis presented below.

Acoustic wavelength

In figure 3, we report the wavelength, λ, associated with the acoustic wave in air as a function of the bubble length, L, as defined in figure 4. Taking the whole set of the experimental data obtained for different gel concentration c and bubble volume V into account, one observes experimentally that λ increases almost linearly with L according to λ ≃ λ 0 + (2.8 ± 0.1)L with λ 0 ≃ (1.6 ± 0.1) cm. The experimental slope dλ/dL ≃ 2.8 deserves to be contrasted with the slope dλ/dL = 4 obtained in the case of a cylindrical cavity [START_REF] Vidal | [END_REF][START_REF] Pierce | Acoustics -An introduction to its physical principles and applications[END_REF]. In the same way, a straightforward analysis of the acoustic problem leads to dλ/dL = 2 in the case of a conical cavity. In order to account for the experimental slope, let us now consider the acoustic In the images (scale bar: 1 cm), one notices that the bubbles are more elongated when the gel concentration c is larger. In addition, we report the signal from the microphone. We observe that the typical frequency decreases and the characteristic duration of the acoustic emission increases when the bubble length is increased. wave inside the bubble, taking into account the cusp at the tail and the overall shape of the cavity.

Because of the cusp at the tail, the bubble body resembles a cone. As a consequence, we shall work in a system of spherical coordinates centered in O, at the cusp. Let M be a point of the bubble wall. The shape of the bubble, assumed to be axisymmetric, can be accounted for by the angle α(r) between the vector --→ OM and the symmetry axis (r ≡ OM , Fig. 4a). For instance, a conical bubble would be described by α(r) = α 0 , a constant. In a first approximation, assuming that the variation of the bubble cross-section does not depend too rapidly (in comparison to the wavelength) on the radius r, we can write the equation for the pressure field P (r, t), assumed to depend only on the distance r,

1 v 2 ∂ 2 P (r, t) ∂t 2 = 1 r 2 ∂ ∂r r 2 ∂P (r, t) ∂r (1)
+ ∂P (r, t) ∂r

d dr log [1 -cos α(r)]
where v stands for the velocity of the sound in air. The equation (1), written in spherical coordinates, governs the propagation of a pressure wave in an acoustic horn [START_REF] Pierce | Acoustics -An introduction to its physical principles and applications[END_REF][START_REF] Rayleigh | Theory of Sound[END_REF] whose profile is described by the function α(r). First, a velocity node locates at O. Second, neglecting the radiation at the open end, we can, in a first approximation, assume that a pressure node locates in the plane of the aperture [P (L) = 0, note here that the condition of zero pressure in the output plane is not com-patible with the geometry of the pressure field. However, in a first approximation, to within a term of the order of Φ/L, where Φ stands for the aperture diameter, this assumption provides a good estimate of the resonant wavelength.] For instance, in the case of a conical cavity (α = α 0 ), the solution at the frequency ω is P (r, t) = sin kr kr e jωt with kL = π, which leads to λ = 2L for the fundamental k ≡ 2π λ . Experimentally, the bubble is not conical and its profile is successfully interpolated by the phenomenological function α(r) = α 0 cos (β r L ), where α 0 denotes half the angle at the bubble tail (Fig. 4a). In this case, the equation ( 1) must be solved numerically. The coefficient β accounts for the diameter Φ of the aperture at the top surface, according to α 0 cos (β 1 + ( Φ 2L ) 2 ) = arctan Φ 2L . In Fig. 4b, we report the slope dλ dL as a function of α 0 for different values of the ratio Φ/L. For instance, one obtains dλ dL = 2.8 for α 0 = π 4 and Φ L = 1 2 , and dλ dL = 2.9 for α 0 = π 8 and Φ L = 1 4 , which corresponds to the typical values of these two parameters in our experimental conditions. Thus, in spite of the slight dependance of the slope dλ dL on the α 0 and Φ/L, the experimental wavelength λ is observed to depend almost linearly on the bubble length L with dλ dL ≃ 2.8 for the whole set of experimental data reported in Fig. 3. However, the simple acoustic model presented above fails in accounting for the finite offset λ 0 ≃ 1.6 cm clearly observed in figure 3. As already pointed out, we arbitrarily assumed that a pressure node locates in the plane of the open end of the bubble, this latter condition being incompatible with the geometry of the pressure field inside the bubble. In order to recover the value λ 0 of the offset, one must consider the diffraction of the sound wave by the aperture. Indeed, the boundary condition in the output plane is imposed by the continuity of the pressure and velocity fields in this very plane, and one must also consider the acoustic wave outside the cavity. The structure of the acoustic wave resulting from the diffraction of a planar wave by a circular aperture (diameter Φ) has been determined by several authors. As a result, to the first order in Φ/λ, one obtains the acoustic impedance of the outer acoustic wave Z out ≃ ζjkΦ, where ζ = 4 3π for of a flanged aperture [START_REF] Rayleigh | Theory of Sound[END_REF] and ζ = 0.3 for an unflanged aperture [34]. In the case of the diffraction of a spherical wave, we expect these results to hold true [the correction due to the curvature of the pressure field is expected to be of the order of Φ L 2 .] In addition, we can estimate the acoustic impedance of the inner acoustic wave, Z in , by calculating the average pressure and velocity in the aperture plane from the solution of equation ( 1), the boundary condition at O being taken into account. Then, insuring the continuity of the pressure and velocity fields in the aperture plane, one can determine the resonant wavelength λ. In the case of a conical cavity, writing Z out /(ρv) ≃ ζjkΦ, one gets λ = 2L + 2πζΦ. Thus, in the case of a conical cavity and of a flanged aperture, we estimate λ 0 ≃ 8 3 Φ (We remind here that one would expect λ 0 ≃ 4 3π Φ for a cylindrical cavity [START_REF] Vidal | [END_REF].) In the case of the bubble, the impedance Z in must be evaluated numerically as the equation ( 1) does not exhibit any simple analytical solution. Here, we only aim at elucidating the physical origin of the offset λ 0 . As a consequence, we only point out that, if due to the radiation at the open end, we expect λ 0 = (1.5 ± 0.5) cm for the experimental range of the aperture diameter (Φ ∈ [0.4, 0.7] cm). From the agreement of this last estimate with the experimental value of λ 0 , we conclude that the offset originates from the radiation of the resonant acoustic wave at the open end.

Damping of the acoustic signal

In order to account for the damping of the acoustic wave, we report in Fig. 5 the typical duration, in number of periods, n, of the acoustic signals. This choice is explained by the difficulty in defining precisely the signal duration because of the rather complex envelop of the acoustic signal. For an exponential decay over the characteristic time τ , we would expect n = n 0 + ωτ 2π , so that the experimental value of n provides an estimate of the characteristic time τ to within an offset. The damping of the acoustic signal might be governed by several physical processes, among them the partial reflection at the cusp, the viscous dissipation at the side walls and the diffraction at the open end. Each of the processes would lead to a different dependence of the duration on the geometry of the bubble. The partial reflection at the cusp would lead to constant ωτ whereas the viscous dissipation would lead to ωτ ∝ Φ √ ω [START_REF] Vidal | [END_REF] and, thus, to a decrease in n for increasing L. On the contrary, one observes experimentally an increase in n with the bubble length L. In order to account for the radiation, we can further expand the acoustic impedance Z out to the second order in kΦ and write Z out /(ρv) ≃ ζjkΦ + ξ (kΦ)

2 where ξ is a constant. For the radiation of a planar wave at a flanged (resp. unflanged) aperture, ξ = 1 8 (resp. 1 16 ) [34,[START_REF] Rayleigh | Theory of Sound[END_REF]. As already mentioned, we expect the curvature of the wavefront to slightly alter the value of the impedance to within a term of the order of (φ/L) 2 but we do expect ξ to remain of the same order. The solution of the equation 1, taking into account the boundary condition at the open end, then leads to ωτ = 1 ξ L 2 πΦ 2 and thus to n

= n 0 + 1 ξ L 2 2π 2 Φ 2 .
Our measurements are not accurate enough to make it possible to determine the experimental slope 1 ξ but we point out that they are compatible with the value ξ = 1 8 (Fig. 5). We can thus conclude that the damping of the acoustic signal is mainly governed by the radiation at the open end of the bubble.

Acoustic energy

At this point, it is particularly interesting to focus on the energy associated with the acoustic emission. From the pressure signal, P (t), provided by the microphone, one can estimate the total amount of energy released in the acoustic signal at the fundamental frequency, E a . Assuming that the acoustic wave outside the bubble is almost spherical, centered at the bubble aperture, in the half-space above the free surface, we can write:

E a ≃ 2πd 2 ρv ∞ t=0 P (t) 2 dt (2)
where, we remind, d stands for the distance from the microphone to the bubble aperture. As E a is expected to depend on the volume V and of the initial overpressure δP of the air inside the bubble before the bursting, let us now consider an estimate of the total amount of released energy, E T , assuming a rapid expansion of air:

E T = 1 2 V δP 2 ρv 2 . (3)
Experimentally, the volume V is obtained from the injected volume of air but the overpressure δP is rather difficult to estimate. However, we measured the surface tension γ = (25 ± 5) mN/m of the gel-water interface for all concentration c and we can estimate the tension of the thin film at the top, previous to the bursting event, to be about 2γ. Measuring the radius of curvature of the bubble head, R, from the image, we can estimate δP ≃ 4γ/R. We point out that the corresponding value of E T must be considered with caution (Laplace's law might not be valid for bubbles bursting without reaching equilibrium at the free surface). However, it makes possible to estimate -and therefore, to further discuss -the energy of the acoustic emission. FIG. 6: Ratio Ea/ET vs. volume V . At a given volume V , the acoustic energy Ea does not account for the amount of energy released by the bursting bubble. Inset: Ratio Ea/ET vs. frequency ω: The experimental results clearly demonstrate that the bubbles can be separated in two categories: small bubbles that sit at the free surface before bursting (open symbols) and larger ones that cross the interface without stopping (full symbols).

We report first E a /E T as a function of the bubble volume V (Fig. 6). One clearly observes that the data points are widely distributed, which indicates that the acoustic energy E a does not image the total amount of energy released by the bursting event, E T . However, the envelop of the data points indicates that, at intermediate volume, in some cases, a large part of E T is transferred to the acoustic mode, which explains why some of the bubbles are so loud whereas others are barely audible. As already discussed in Ref. [START_REF] Vidal | [END_REF], the transfer of the energy intially loaded in the bubble to the acoustic modes is mainly governed by the characteristic time associated with the rupture of the bubble head: The opening of the cavity is efficient in exciting the inner resonant modes only if rapid enough. For the smallest volumes, the bubble rises and stops at the free surface. The film that closes the cavity bursts after drainage. Even small, the characteristic time associated with the cavity opening is long compared to the acoustic period because the bubble body is very short. As a consequence, E a /E T is relatively small. In the opposite limit, the bubbles are large and cross the interface without stopping. The bursting results from the breaking of the bubble head which is torn apart due the bubble dynamics. In this case, the film is not as thin as one would obtain as the result of the drainage and the characteristic opening-time is long compared to the period associated with the resonant modes, even if the bubble body is long. Again, the ratio E a /E T is relatively small. The optimal conditions are reached when the volume of the bubble is such that the bubble crosses the interface slowly enough for the film to have time to thin but rapidly enough for the bubble tail not to disappear (this leads to a significant increase of the resonant frequency). This latter conclusion is supported by the data reported in Fig. 6, inset. One observes, reporting E a /E T as a function of the resonant frequency ω, that the data can be separated in two groups: bubbles that sit at the interface before bursting and bubbles that cross the interface without stopping. We clearly note that E a /E T is maximum for bubbles crossing dynamically the free surface.

From these remarks on the acoustic energy, we would like to point out that the acoustic energy in the fundamental mode drastically depends on the dynamics of the thin film rupture and that, as a consequence, measuring the acoustic energy is not enough for obtaining a good estimate of the total energy release.

CONCLUSION

Motivated both by the will to understand the physical processes involved when a bubble bursts at the surface of a non-newtonian fluid, and the hope to use the acoustics as a tool to investigate natural systems such as volcanoes the relationship between the acoustic wave and the rheological properties of lava, we have investigated the bursting of bubbles at the free surface of a gel solution, diluted at different concentration.

We have shown that, at large-enough gelconcentration, the bubble, which is elongated, acts as a motionless resonator and, thus, exhibits a welldefined acoustic-frequency at bursting. The amplitude of the acoustic wave emitted at bursting depends on various parameters: gel concentration, volume of the bubble, and film rupture time. In spite of the observed clear transition between a static regime, where the small-volume bubble remains trapped at the surface, and a dynamical regime, where the high bubble-rising velocity makes the bubble go through the surface and burst directly, the problem remains rich and complex, in particular, due to the unpredictable film rupture time, which directly influences the amplitude and energy of the acoustic signal.

From a practical point of view, we can raise the following question: What pieces of information can we infer from acoustic measurements, if they are the only available data? From our study, we can conclude that the frequency of the signal gives a direct access to the bubble length. However, any attempt to interpret the amplitude and energy of the acoustic signal would surely lead to strong misinterpretation. Indeed, we have shown that the same experimental conditions (gel concentration, bubble volume) can lead to completely different acoustic-signals at bursting, due to the high sensitivity to the film bursting dynamics.

FIG. 1 :

 1 FIG.1: Bursting event in the fast camera. (a) Initially, the bubble sits at the free surface. The film at the top is thinning due to the drainage. (b) The thin film suddenly breaks. One observes, in this specific example, that the film tears at its base. (c) As a result of the outgoing air flow and of the capillary forces, the remaining part of the bubble head is blown upwards and shrinks. The width of each image is 3 cm whereas the time-difference between them is 0.8 ms. Here, the acoustic emission associated with the event does not last more than 2 ms, during which the bubble body clearly does not significantly deform.

FIG. 2 :

 2 FIG. 2:Images of the bubble right before bursting and associated acoustic signals. In the images (scale bar: 1 cm), one notices that the bubbles are more elongated when the gel concentration c is larger. In addition, we report the signal from the microphone. We observe that the typical frequency decreases and the characteristic duration of the acoustic emission increases when the bubble length is increased.

FIG. 3 :

 3 FIG. 3: Wavelength λ vs. bubble length L. The resonant wavelength increases linearly with the bubble length, estimated from the open aperture to the tail. The offset λ0 is accounted for by the radiation at the open-end.

FIG. 4 :

 4 FIG. 4: (a) Image of a bubble and definitions. The dotted curve corresponds to the proposed interpolation of the bubble profile by α(r) = α0 cos (β r L ) with α0 = 0.378 and β = 1.11. (b) Slope dλ dL vs. α0 for different ratio Φ L . The slope dλ dL is obtained from the solution of Eq. 1 taking into account the profile of the bubble wall. The full diamond corresponds to the bubble in (a).

FIG. 5 :

 5 FIG.5: Characteristic duration of the sound emission n, in number of periods, vs. L 2 /Φ 2 . The dependence of n on the ratio L/Φ demonstrates that the damping of the acoustic signal is mainly governed by the radiation at the open end (The slope of the grey line is 8, thus corresponding to ξ =1 8 , see text.)

ACKNOWLEDGEMENTS

The collaborative research was supported by CNRS/CONICYT project #18640 and Conicyt-Chile FONDAP project #11980002.


domain: math info

text:
David Allouche
email: allouche@toulouse.inra.fr

Thomas Schiex
email: tschiex@toulouse.inra.fr

Simon De Givry
email: degivry@toulouse.inra.fr

Martin Sanchez
email: msanchez@toulouse.inra.fr

TagSNP selection using weighted CSP and russian doll search with tree decomposition

  L'archive ouverte pluridisciplinaire

Introduction

In bioinformatics, uncertainty and variability are usually ubiquitous and combinatorial problems modelling biological objects or phenomenon usually involve a combination of a large number of local uncertainties or correlations which must be incorporated in a global criteria. It is therefore not surprising to see that graphical models (HMM, Bayesian nets, (conditional) Markov random fields. . . ) are massively used in bioinformatics.

The constraint satisfaction problem and its implementation as constraint programming languages represent a deterministic variant of graphical models which allows to represent combinatorial problems on discrete variables. Optimization (if any) is often considered as a second order target that can be reached by explicitly modeling a cost variable which is then iteratively bounded until an optimum is reached. This however requires the use of global constraints that cover all variables, thereby hindering all the problem structure.

In this paper, we consider the TagSNP problem, a specific form of lossy compression problem arising in genetics. Given a very large set of SNP (genomic positions where polymorphism is observed in a given population), the aim is to select a smallest subset of SNPs which represents the complete set of tagSNP reliably. This problem can be naturally defined as a variant of the set covering problem. By modeling the problem as a weighted CSP (or cost function network), we get a model which still offers opportunities to exploit the problem structure described through a tree-decomposition of its graph.

Specifically, this enables the fruitful application of a recent algorithm [START_REF] Sanchez | Russian doll search with tree decomposition[END_REF] called "Russian Doll Search with Tree Decomposition" that exploits problem structure for solving cost function networks. As the BTD algorithm [START_REF] Terrioux | Bounded backtracking for the valued constraint satisfaction problems[END_REF], this algorithm is based on the identification of conditionally independent subproblems, which are solved independently, using a lower bound based on local consistency, and whose optimum is cached during Branch and Bound search. The time complexity of the algorithm obtained is only exponential in the largest subproblem size instead of the global problem size. These conditionally independent subproblems are called clusters ad form a tree. The intersection between two clusters, if not empty, is called the separator between these two clusters (see Fig. 1 for an illustration). Such a decomposition can be obtained by computing a so-called tree-decomposition of the problem. The main novelty of our algorithm lies in the incorporation of a "Russian Doll Search" like approach enabling the computation of stronger initial local bounds by inductively solving a relaxation of each subproblem identified in the tree-decomposition.

The algorithm obtained, BTD-RDS, generalizes both RDS and other treedecomposition based algorithms such as BTD or AND-OR Branch and Bound. As BTD, it uses a restricted dynamic variable ordering which must be compatible with the tree decomposition exploited : a variable from a cluster cannot be instantiated before variables from its parent cluster.

This algorithm has been applied to radio link frequency assignment problem instances defined in the CELAR benchmark [START_REF] Cabon | Radio Link Frequency Assignment[END_REF], closing a very hard frequency assignment instance which has been open for more than 10 years. It has also been used to tackle a problem from bioinformatics, defining a new benchmark for constraint-based approaches : the tagSNP selection problem.

In this paper, we will present this problem with associated instances defining benchmarks and the main results obtained in our experiments.

Modeling the TagSNP problem

TagSNP selection occurs in genetics and polymorphism analysis. Single nucleotide polymorphisms, or SNPs, are DNA sequence variations that occur when a single nucleotide (A,T,C,or G) in the genome sequence of an individual is altered. For example a SNP might change the DNA sequence AAGGCTAA to ATGGCTAA. For a variation to be considered as a SNP, it must occur in at least 1% of the population. There are several millions SNPs in the 3 billions nucleotides long human genome, explaining up to 90% of all human genetic variation. SNPs may explain a portion of the heritable risk of common diseases and can affect response to pathogens, chemicals, drugs, vaccines, and other agents. The TagSNP problem is a sort of lossy compression problem which consists in selecting a small subset of SNPs such that the selected SNPs, called tag SNPs, will capture most of the genetic information. The goal is to capture a maximally informative subset of SNPs to make screening of large populations feasible [START_REF] Hirschhorn | Genome-wide association studies for common diseases and complex traits[END_REF]. From the combinatorial point of view the TagSNP problem is equivalent to a set covering problem (NP-hard) with additional quadratic criteria.

By sampling a first relatively small population, it is possible to compute a link (correlation) measure r 2 between each pair of SNPs. A tag SNP is considered as representative of another SNP if the two SNPs are sufficiently linked. The simplest TagSNP problem is to select a minimum number of SNPs (primary criteria) such that all SNPs are represented. This is captured by the fact that the r 2 measure between the two SNPs is larger than a threshold θ (often set to θ = 0.8 [START_REF] Carlson | Selecting a maximally informative set of single-nucleotide polymorphisms for association analyses using linkage disequilibrium[END_REF]). We therefore consider a graph where each vertex is a SNP and where edges are labelled by the r 2 measure between pairs of nodes. Edges are filtered if their label is lower than the threshold θ. The graph obtained may have different connected components. The TagSNP problem then reduces to a set covering problem on these components. This simplest variant has been studied in [START_REF] Choi | Efficient Genome Wide Tagging by Reduction to SAT[END_REF] where it is solved using the d-DNNF compiler c2d with good results.

In practice the number of optimal solutions may still be extremely large and secondary criteria are considered by state-of-the-art tools such as FESTA [START_REF] Qin | An efficient comprehensive search algorithm for tagsnp selection using linkage disequilibrium criteria[END_REF]. Between tag SNPs, a low r 2 is preferred, to maximize tag SNP dispersion. Between a non tag SNP and its representative tag SNP, a high r 2 is preferred to maximize the representativity. To optimize these criteria, FESTA uses two incomplete algorithms, the simplest is called FESTA-greedy, and it uses a simple reedy approach. The second, called FESTA-hybrid combined the greedy approach with a limited exhaustive approach.

For a given connected graph G = (V, E), we build a binary weighted CSP with integer costs capturing the TagSNP problem with the above secondary criteria. For each SNP i, two variables i s and i r are used. i s is a boolean variable that indicates if the SNP is selected as a tag SNP or not. The domain of i r is the set of neighbors of i together with i itself. It indicates the representative tag SNP which covers i. For a SNP i, hard binary cost functions (with 0 or infinite costs) enforces the fact that i s ⇒ (i r = i). Similar hard cost functions enforce (i r = j) ⇒ j s with neighbor SNPs j in G. A unary cost function on every variable i s generates an elementary cost U if the variable is true. The resulting weighted CSP captures the set covering problem defined by TagSNP.

To account for the representativity, a unary cost function is associated with every variable i r that generates cost when i r = i. In this case, the cost generated is 100.

1-r 2 i,ir 1-θ
. For dispersion between SNPs i and j, a binary cost function between the boolean i s and j s is created which generates a cost of 100.

r 2 ij -θ 1-θ
when i s = j s = true The resulting WCSP captures both dispersion and representativity. In order to keep these criteria as secondary, we just use a large enough value for U (the elementary cost used for tag selection). This problem is similar to a set covering problem with additional binary costs. Such secondary criteria are ignored by [START_REF] Choi | Efficient Genome Wide Tagging by Reduction to SAT[END_REF]. Here, c2d yields a compact compiled representation of the set of solutions of the pure set covering problem, but the number of solutions is so huge (typically more than billions) that applying the second criteria on solutions generated by c2d would be too expensive. A direct compilation of the criteria in the d-DNNF does not seem straightforward and would probably necessitate a Max-SAT formulation as the authors acknowledge in their conclusion.

Experiments

The algorithms tested (BTD and RDS-BTD) have been implemented in toulbar2 C++ solver 1 . Note than when the tree decomposition used reduced to a single cluster, BTD is equivalent to a Depth First Branch and Bound algorithm.

The min domain / max degree dynamic variable ordering, breaking ties with maximum unary cost, is used inside clusters (BTD and RDS-BTD) and by DFBB. The dynamic variable ordering heuristic is modified by a conflict backjumping heuristic as suggested in [START_REF] Lecoutre | Last conflict based reasoning[END_REF]. EDAC local consistency is enforced [START_REF] De Givry | Existential arc consistency: Getting closer to full arc consistency in weighted CSPs[END_REF] during search. Tree decompositions are built using the Maximum Cardinality Search (MCS [START_REF] Rose | Tringulated graphs and the elimination process[END_REF]) heuristic, with the largest cluster used as root. A variable ordering compatible with the rooted tree decomposition used is used for DAC enforcing [START_REF] Cooper | Arc consistency for soft constraints[END_REF].

All the solving methods exploit a binary branching scheme. If d > 10, the ordered domain is split in two parts (around the middle value), else the variable is assigned to its EDAC fully supported value or this value is removed from the domain. In both cases, it selects the branch which contains the fully supported value first, except if a previous solution is available (the corresponding value is used in this case). Reported CPU times correspond to finding the optimum and proving its optimality.

The instances we considered have been derived from human chromosome 1 data provided by courtesy of Steve Qin [START_REF] Qin | An efficient comprehensive search algorithm for tagsnp selection using linkage disequilibrium criteria[END_REF]. Two values, θ = 0.8 and 0.5 have been tried. For θ = 0.8, a usual value in tag SNP selection, 43, 251 connected components are identified among which we selected the 82 largest ones. These problems, with 33 to 464 SNPs, define WCSP with domain sizes ranging from 15 to 224 and are relatively easy. Solving them to optimality selects 359 tag SNPs in 2h37' instead of 487 in 3' for FESTA-greedy (21% improvement) or 370 in 39h17' for FESTA-hybrid (3% improvement, 15-fold speedup).

To get more challenging problems, we lowered θ to 0.5. This defined 19, 750 connected components, among which 516 are not solved to optimality by FESTA. We selected the 25 largest one. These problems, with 171 to 777 SNPs have graph densities between 6% and 37%. They define WCSP with max domain size ranging from 30 to 266 and include between 8000 to 250, 000 cost functions. The decomposability of these problems, estimated by the ratio between the treewidth of the original MCS tree-decomposition (without any cluster merging, i.e. with s max = +∞) and the number of variables varies from 14% to 23%.

In theory, algorithms exploiting tree-decompositions are only exponential in the maximum cluster size and the ideal tree-decomposition that should be used is therefore a tree decomposition minimizing the size of the largest cluster (also called the treewidth or induced width of the problem). In practice however, when a Branch and Bound like approach such as BTD is used, small separators are also attractive because the worst-case space complexity of the algorithm is exponential in the separator size. Even if this space complexity is usually far from being reached because of the pruning induced by lower bounds, small separators are also attractive because the removal of large separators creates larger clusters that gives more freedom to dynamic variable ordering. There is therefore a compromise to reach : small clusters are desirable but large separators should be avoided. Starting from an original tree-decomposition, it is always possible to reach a decomposition with a maximum separator size below a given threshold s max by just merging any pair of clusters which has a separator of size above s max [START_REF] Jégou | Dynamic management of heuristics for solving structured CSPs[END_REF].

All the problems have been tackled with an initial upper bound found by FESTA-greedy on 2.8 Hz CPU with 32 GB RAM. To better show the importance of bounded separator size (using s max ), we considered values ranging from 0 (DFBB), 4, 8, 12, 24, 32 to +∞ for both BTD and RDS-BTD. We report both the number of problems solved within a 2-hour limit per instance and the total amount of CPU time used (an unsolved instance contributes for 2 hours).

Using s max = 4, our implementation improves the compression ratio of FESTA-greedy by 15% (selecting 2952 tag SNPs instead of 3477 for the 516 -13 solved instances). Note that the differences in CPU time between BTD and RDS-BTD would increase if a larger time limit had been used. From a practical viewpoint, the criterion of the TagSNP problem could be further refined to include : sequence annotation information (e.g. preferring tag SNPs occurring in genes), and measures between triplets of markers as proposed in [START_REF] Choi | Efficient Genome Wide Tagging by Reduction to SAT[END_REF] (SNPs covered by a pair of tag SNPs). The good performances of RDS-BTD may allow to tackle this more complex problem with realistic θ = 0.8. Problems which are successfully solved (with an optimality proof) are materialized as continuous line,while dashed lines represent failures. The instance 17034, which could be solved only for s max = 4 is represented by a gray square.

The Figure 3 allows to materialize the 12 instances which could be solved for s max = {4, 8}. This underlines the fact that the nature of each solved instance may be different depending of the s max value. For example, instance 17034 is only solved for s max = 4, which is balanced by the resolution of instance 14359, solved only on the interval 8 -12 of s max .. Overall, on the whole range of variation of s max , 13 instances are solved but only 12 can be solved for an optimal choice of s max = 4, 8. This may plead for an adaptive choice of the parameter s max = 4, 8. Moreover, a significant decrease of the normalized treewidth is often correlated with the occurrence of a successful resolution.

Indeed, for most of the tested instances, reflecting the structure of the problems, the curves show an important decrease for s max ∈ {0, 8}, which is very likely one of the significant events explaining the high number of successful resolution in this range. The decrease of the treewidth is then lower. It gradually reaches (s max > 12), a plateau with a gentle slope, asymptotically reaching the ratio corresponding to a full decomposition of the problem (without cluster merging). In the experimentation, this area is often associated to failure.

This behavior is also reflected in the resolution speedup. Considering for example instance 15757, an optimal solution is found for s max = {4, 8, 12, 24} with respective cpu-time of of 514, 7, 299 and 3810 seconds. Simultaneously, the tree width ratio varies from 0.63 for s max = 4 to 0.30 for s max = {8, 12} and then to about 0.2 for s max = 24. The optimal resolution time occurs for s max = 8. Several instances follow a similar behavior. Most of solved instances are optimally resolved for a specific s max value . This behavior follows very likely from the compromise between the gains provided by the decomposition and the losses related to a decrease in the freedom of choice during search in dynamic variables ordering in smaller clusters.

Conclusion

In this paper , we specifically addressed the tagSNP selection problem, a variant of the covering problem coming from bioinformatics. Our WCSP model allows to take into account additional criteria such as dispersion and representativity. The data used in our experimentation is likely the largest data set available in the public domain. With a usual threshold (θ = 0.8), our model provides good results. But the situation quickly become more challenging if less stringent (more artificial) filtering conditions such as θ = 0.5 are used. In the near future, we intend to extend our study to include an integer linear programming model solved using CPLEX.

Beside this, from the methodological point of view, this work underlines the fact that the practical exploitation of tree decompositions to solve structured combinatorial optimization problems is not straightforward. Our experiments show that, even on problems that have a visible structure, it is often very profitable and sometimes crucial to restrict the maximum size of the separators of the decomposition used. Theory says that separator size influences the space complexity of structure-based algorithms such as BTD and RDS-BTD but in practice, the improvement in efficiency is mostly explainable by the added freedom in variable ordering allowed by cluster merging, an observation consistent with jegou et al. conclusions [START_REF] Jégou | Dynamic management of heuristics for solving structured CSPs[END_REF].

Our current algorithm still leaves areas for improvements. A attractive direction would be to try to design a dedicated heuristic allowing for dynamic adaptation of the maximum separator size of the decomposition. This could be based on the specific structure of the input instance and could be achieved, for a given decomposition, by analyzing treewidth variations as a function of s max . Another more challenging direction would be to provide decomposition algorithms aimed at producing small separators in the graph decomposition.

Fig. 1 .

 1 Fig. 1. The graph of a WCSP and its tree decomposition.

Fig. 2 .

 2 Fig. 2. CPU-time and number of solved instances for different values of the maximum separator threshold (smax).

Fig. 3 .

 3 Fig. 3. Evolution of treewidth normalized by variable number for different maximum separator size threshold smax.

Figure 3

 3 Figure 3 presents the evolution of the ratio between the tree width and the problem size for different values of s max . All instances (θ = 0.5) solved at least once during the experimentation are shown in this graph.

http://mulcyber.toulouse.inra.fr/projects/toulbar2, version 0.8.

Acknowledgments This research has been partly funded by the French Agence Nationale de la Recherche (STALDECOPT project).


Who's Afraid of the Sophists?

Against Ethical Correctness SOPHISTIQUE, subst. A. Ensemble des doctrines, ou plus exactement attitude intellectuelle commune des principaux sophistes grecs (Protagoras, Gorgias, Prodicus, Hippias, etc.). B. (Nom commun.) Se dit d'une philosophie de raisonnement verbal, sans solidité et sans sérieux. SOPHISTICS, substantive noun. A. The set of doctrines, or more precisely, the shared intellectual attitude of the main Greek sophists (Protagoras, Gorgias, Prodicus, Hippias, etc.). B. (Common noun.) Used to refer to a philosophy of verbal reasoning, lacking solidity or seriousness. (A. Lalande, ed., Vocabulaire technique et critique de la philosophie).

I. The Occasion

The set of doctrines or teachings associated with the individuals known as the sophists is termed sophistike, in French, sophistique. The expression is lacking in English, which puts one in the position of either using the adjectives 'sophistic', 'sophistical', or of using the dismissive expression 'sophistry'. As I shall be arguing for a systematic role of these doctrines, I will ask your indulgence and introduce the neologism 'sophistics' for now. The question is, why should one be interested in sophistics today?

As occasional causes are by far the most significant and the most efficient, I would like to explain first of all where my own interest in sophistics stems from. It arose from the encounter of two trajectories which were rivals in all senses of the word. The first phase of study, both triumphant and confused, took place under the sign of Heidegger. Because everything possessed a renewed intelligibility, everything also fit neatly into the palm of one's hand. The history of philosophy was philosophy itself, which the question of Being enabled one to scan and rework into epochs and turnings, with a hitherto unequaled skill which knew when to be self-effacing. It is very difficult to rid oneself of the idea that philosophers today do anything else besides rework Heidegger's gesture, even the anti-Heideggerians who sought their training in Kant, with a different point of entry than the Kantbuch. In order to move out from this circumscribed territory, no less is required, doubtless, than (a) a redefinition of philosophy throughout its history, in such a way that this widening of the scope does not produce a mere analytic restriction or moral rigidity which can immediately be traced back destinally to the technical and technological nature of our epoch, and (b) probably some new conceptual characters, to use Deleuze's expression. But the most frequent approach, which Deleuze himself initiated or at least made use of contemporaneously (using the Stoics, Spinoza and Bergson), is to draw attention to the readings Heidegger failed to perform, or did not perform, inasmuch as they are held to be strategically impossible.

My own growing rigid, in this context, has to do with the determination of the origin and the dawn. The Greek morning which Heidegger arranged for us is a monomaniac and a kleptomaniac. It robs an entire array of texts and possibilities so that they may fit under the aegis of Parmenides' poem, such that the Parmenides of 1942-1943 reads polis merely as pelein, the old Greek verb for einai: if the polis in itself is only the "pole of pelein," then "it is only because the Greeks are an absolutely non-political people" that they could found the polis, and did. The first reading that I found impossible to perform using Heidegger alone, in the truly grandiose perspective of Parmenides' unveiling, was Gorgias' Treatise of Non-Being, a treatise which, approximately a half-century after the dawn, provides a full-fledged demonstration of the mechanisms or strategies thanks to which the Poem conforms to Heidegger's dream; it is a text which critically exceeds ontology in its nascent state. Thus there was a different way of being Pre-Socratic.

The teachings of the sophists serve as a good tool, maybe even the best of the available tools, to produce something like a new narrative of the history of philosophy --the tale of a new morning which makes one want to count the fingers of the dawn --. as well as a new delimitation of the entity called 'philosophy' in relation to the other entities it constructs (sophistics as rhetoric, and then as literature).

The following remarks, then, will take the form of a philosophical and literary stroll, which responds to an aspect of things that does not belong to our ordinary habits (but which, once one responds to it, is as good as they are at monopolizing our attention), a stroll which gives itself the right to stop, along the course of time, and collect texts of a different lineage than that which runs from Parmenides, Plato, and Aristotle, to Hegel, Heidegger, or Habermas. Once these texts have been gathered up, they begin to resonate with one another.

This should surely enable us to glimpse how artificial the border between the rational and the irrational is, and perhaps to reorganize the cosmos of philosophy, starting from a richer, more complete, more contemporary, or otherwise effective point of view.

II. Sophistics as Historical Fact and Structural Effect

As the state of the corpus of translations and stubborn retranslations of the same ancient texts reveals, we have until quite recently, in France in any case, been prisoners of the image that antiquity sought to project of itself, namely, a succession of moments of excellence. Yet my choice of sophistical doctrine as an object of research does not, at least initially or not solely, stem from an antiquarian zeal for ill-known texts which require a great deal of philological and historical work. Nor, thereby, does it originate in an interest in the margin where these roughly crossed-out texts lie, an interest which would make the margin into an area of research, legitimizing a militant pathos in favour of accursed thinkers, against debarments and exclusion. Hence I will not be proposing a "rehabilitation," and especially not one of these rehabilitations which are grounded, by means of a circular argument, in the modifications and improvements they enable one to introduce into the persistent frame of the most traditional of all histories.

Indeed, the singularity of sophistical doctrine is to already be, inasmuch as it is a historical fact, a structural effect, that is, an effect of the structure: the "doctrine of the sophists" is a philosophical concept, whose model is truly provided by the real practice of those who called themselves, and were called, sophists (individuals such as Protagoras, Gorgias, and Antiphon) --this is definition A in Lalande's Vocabulaire --but it is also used in philosophy to refer to one of the possible modalities of non-philosophy --this is definition B, which is as magisterial as it is mysterious: "a philosophy of verbal reasoning, lacking any solidity or seriousness." I will frequently return to this definition. This is Plato's starting-point, and the point where he welds together the two halves of the problem: sophistic doctrine, which is a historical reality, is at the same time artificially produced by philosophy. The essence of this artefact is simply to construct the sophist as the negative alter ego of the philosopher: his bad Other. They have resembled each other ever since the Stranger's comment in the Sophist that the sophist resembles the philosopher "as the wolf does the dog, as the most savage resembles the most tame" (231b). They are so much alike that even when one reaches out with both hands to catch one, one catches the other.

Socrates's cathartic midwifery (maieutics), his practice of refutation end up belonging to the genei gennaia sophistikê (which one translates as "the genuine and truly noble art of sophistics," ignoring the emphasis which cements the bond). Conversely, at the end of the dialogue, when all dichotomies must be recapitulated, the final arborescence provides us with the same pair, when the issue is how to describe the demagogue or speech-maker: "wise man or sophist?" The decision is only reached when Theatetus states a thesis: "We posited him as not knowing" (268c). In any case, the sophist, who is an "imitator of the wise man," is his paronym to exactly the same extent as the philosopher himself is.

If we consider the Platonic dialogues as a whole, we can indeed discern the figure of sophistics, which will henceforth belong to the tradition. It is devalued on all grounds; ontologically, because the sophist is not concerned with being, but seeks refuge in non-being and what is accidental; logically, because he is not in pursuit of truth or dialectical rigor, but merely opinion, seeming coherence, persuasion and victory in the oratorical joust; ethically, pedagogically, and politically: his goal is not wisdom and virtue, for the individual or for the city, but rather personal power and gain; the sophist is even devalued on literary grounds, since the figures of speech he makes use of, his style, are merely the bulges of an encyclopedic vacuity. If one makes use of the standard of being and truth in order to judge the teaching of the sophist, it must be condemned as pseudo-philosophy: a philosophy of appearances and a mere appearance of philosophy.

On this basis, however, another dimension of the Sophist comes to light which shakes up its strict organization: the artefact itself becomes a producer of philosophy. The sophist is the other of the philosopher, whom philosophy never ceases to expel from its domain, and even from humanity itself, as we will see with Aristotle, because the philosopher in turn defines himself as (merely) the other of the sophist, an other whom he pushes further and further into a corner. Philosophy is the child of wonder, and, according to the first sentence of the Metaphysics, "all humans naturally wish to know." Yet "those who ask if one should honour the gods and love one's parents, or not, simply need a good lesson, and those who ask if snow is white or not only need to look" (Aristotle, Topics). The sophist exaggerates, as

Protagoras does regarding the gods, Antiphon regarding the family, and Gorgias regarding that which is and that which one perceives. He always asks one question too many, he always derives one consequence too many. Due to this insolence, philosophy is literally beside itself, the love of wisdom is forced to transgress the limits that it assigned itself, and to make gestures which surely do not belong to the rest of its procedure, such as pulling out the stick.

In what follows, I should like to analyze some of these gestures, as a goal of my research. The doctrine of the sophists is indeed an operator which serves to circumscribe and define the scope of philosophy.

Thus the point of view I have chosen makes use of a philosophico-sophistical perception (Novalis, in his Logological Fragments, suggested the term "philosophistize"), the nature of which may be conveyed by the term 'effect'. It is an effect, first of all, because sophistics is an artificial creation, a by-product of philosophy [as the difficulty with terminology itself shows]. But it is an effect, second, because sophistics, a fiction of philosophy, reverses the direction of things and shocks philosophy, never ceasing to have an effect on it. This sums up the structural version of an infinitely complex history of ideas which, in my view, historians of philosophy are wrong to ignore.

III. The Sophistical Effect

My work over the last ten years has focused on showing how sophistical doctrine obliges philosophy to reflect on itself, and reflect itself, starting from three moments, three privileged bodies of text, which enable one to express the sophistical effect, and reconstitute a history of the limit.

A Treatise Against a Poem

The confrontation between Parmenides' On Nature, or On the Entity and Gorgias' On What is Not (or, On Nature) shall provide us with our primal scene. As Heidegger has shown, but not only as he has shown, everything hangs on the way in which being and saying are connected (recall Lalande's definition: "a philosophy of verbal reasoning").

"Nothing is; if it is, it is unknowable; if it is and it is knowable, it cannot be said // communicated." The three theses of the Treatise present themselves as a reversal of Parmenides' Poem, which, from Plato to the present day, has been taken to say, firstly, that there is being, since "Being is" and "non-Being is not," secondly, that this Being is essentially knowable, since "being and thinking are one and the same." Thereby philosophy was able to embark on the right path, to know being qua being, and diversify itself commercially itself into doctrines, disciples and schools. This series of reversals should not, however, be viewed as an idle little game, for it is a radical critique of ontology.

Gorgias's strategy, as displayed in his first thesis, is to have us realize that Being, which is the Parmenidean hero just as Ulysses is the Homeric hero, is never anything besides an effect of the Poem. By following the way in which, at the starting-point of the Poem, the "word of the path" (Is) suffices to secrete the full subject, which is identified thereafter as the article (the entity), by means of a series of infinitives and participles, the sophist dissects the way in which syntax creates semantics. If there is a "sophism," it resides in the "is" and its ontological treatment, since the mere statement of the identity of being, which leads one to confuse the copula with existence, derives its entire worth from an amphiboly and a homonymy.

The second postulate of the poem --which Heidegger translates paratactically as "being, thinking, the same" (fragment 3) --marks the mooring-point of truth as unconcealment, and then as adequation. Here the catastrophe is perfect. It is enough for me to think something, and moreover, to speak it, for that thing to be, by that token: if I say "tanks battle in the open sea," then tanks are battling in the open sea. There is no room for non-being, or for errors and lies. It is Parmenides's ontology alone, taken at face value and pushed to the limit, which guarantees the infallibility and efficaciousness of the discourse, which thereby is sophistical.

That being is a speech effect now takes on a twofold meaning: we are not simply faced with a critique of ontology --your purported being is nothing but an effect of the way you speak --, but with a claim which is characteristic of "logology," to use a term of Novalis (and also of Dubuffet). What matters from now on is not a being which would supposedly be already there, but the being produced by the discourse; one must assess the magnitude of the shift in landscape, from the primal scene onwards. The safest identity principle is no longer formulated as 'Being is', or 'The entity is', but, to quote another sentence from the treatise, "he who speaks, speaks." The presence of Being, the immediacy of Nature and the evidence of a speech which aims to express them adequately all vanish at the same time; the physics discovered by speech makes way for the politics created by discourse. Indeed, it is here, thanks to the sophists, that we reach the dimension of the political as agora for an agôn: the city as the continuous creation of language. The discourse of the sophists is to the soul what the pharmakon (poison and remedy) is to the body: it induces a change of state, for better or for worse. But the sophist, like the doctor, knows how to use the pharmakon, and can transmit this knowledge; he knows and teaches how to move, not, according to the bivalency of the principle of non-contradiction, from error to truth or from ignorance to wisdom, but, according to the inherent plurality of comparison, from a lesser state to a better state. In the Theatetus, Protagoras, who professes virtue, states this through the mouthpiece of Socrates, who then defends him: "one has to effect a change from another condition to the better. But the physician effects a change by drugs, the sophist by speeches" (167a), and wise and good orators like him "make cities be of the opinion that the good things in place of the poor things are just" (167c). The entire rhetoric of the sophists is thus a vast performance which, time after time, by means of praise and counsel, produces the consensus required for the social bond. This consensus is minimal, even minimalist, because far from requiring a uniform unity, the sophistical consensus does not even require that everyone think the same thing (homonoia), but only that everyone speaks (homologia) and lends their ear (homophônia). In this way, it is hinted that the final motor [ressort] of political discourse is homonymy ("that the sentence have a meaning for each of the meanings of each of its terms"), which alone can have an effect on "the metaphysician" as much as on "the cook," to speak with Ponge. 1

At this point, the distance that has been achieved from the Heideggerian dawn begins to sink in. In the philosophizing Greece of alêtheia, the invention of the city is non-political, because the political qua political is in no way political; rather, it is always subordinate to Being, the True, and the Good. But in a philosophisticizing Greek, where the ontological immediately reverts to, backtracks towards the logological, the logos enables us to grasp the very immanence of the political, in its condition of possibility, in a perception which is more

Aristotelian and Arendtian than it is Platonic and Heideggerian. "Humans live together according to the mode of speech" 2 : the specificity of the political lies in the competition of the logoi, governed by the norm of taste (in the sense this term has had since Kant), which "seeks to obtain the consent of the other," in the midst of a plural condition. Arendt emphasizes that this is why doing "political philosophy," that is, "to look upon politics from the perspective of truth [...] means to take one's stand outside the political realm." Under the aegis of the first statement, namely, Antiphon's assertion in On Truth that "one citizens" (politeuetai tis) no longer refers to nature except as something that has escaped from a private crevice into the public realm, the autonomy of the logos which creates the legal sphere and the autonomy of the political are henceforth intertwined.

The Logos of a Plant

All of Aristotle's regulation of language, in which modernity is invented, may justifiably be seen as a rejoinder to sophistics: how can language be tamed, how can it be rendered ethical, both prior to and during its rhetorical-political life?

The response or come-back to the primal scene can easily be located: it is the demonstration of the principle of non-contradiction which is performed at the beginning of The real dynamic of the refutation lies in a series of equivalent relations which, once they are stated, are as self-evident as ontology itself: to speak is to say something that has a meaning, and this meaning is the same for oneself and for another. Thereafter, indeed, I need only speak ("Hello") for the principle of non-contradiction to be proven and instantiated in the process: it is impossible for the same (word) to have and not have the same (meaning), at the same time. All I need to do is speak, or, alternately, "so long as the opponent says something":

Aristotle finalizes the procedure by ensuring that this necessary and sufficient condition is present in the very definition of man, and excluding all those who do not fall under his demonstration from humanity, from the outset, "for such a man, as such, is like a plant, from the outset." The requirement of meaning, once it is equated with the aim of univocity in this fashion, is first of all a formidable war machine against homonymy. All those who refuse to submit to this requirement have been reduced to so much silence or noise, something prior to language, ever since Aristotle. They are free to inquire into "what is in the sounds of the voice and in words," to drivel or nonsense (blabla), or in other words, into the signifier inasmuch as it does not signify. This freedom will not even extend to the conventional sphere of human language.

From Parmenides to Gorgias, and from Gorgias to Aristotle, let us grant that the sophist has been check-mated. But much is to be learned from the game, up until the present day.

It is up to us, in particular, to decipher the ambiguous status of psychoanalysis in light of this teaching --this speech which one pays for, this pharmakon which is bought and sold, just as the sophist were reproached for. For it is obvious that Freud, then Lacan, occupy the site that Aristotle assigned to the recalcitrant sophists, the site of the signifier. Yet there is a crucial difference, which changes everything: they occupy this site as Aristotelians. Thanks to psychoanalysis, even drivel, even homonymy fall into the embrace of meaning. Freud's definition of the "pun" (one of the categories of which, let's not forget, is the "sophism") as "sense within non-sense" stands as sufficient witness that speaking for one's pleasure or, as Lacan puts it, "speaking to no end" (parler en pure perte) is today a sophistical activity which has been embraced or taken over by Aristotelianism. ground; just like with the principle of all principles, one must both ground and cease to ground. The resolution of the aporia is also the same: one takes a step back, a regress towards the transcendental condition of human language. The proof-strategy is the same: an opponent is required, be it Popper or Protagoras, who is refuted by showing him that "the rules of the transcendental language-game" are such that he has "always already implicitly acknowledged their validity." 3 In other words --those of Aristotle, precisely --: "If one is to destroy the logos, one must surely have a logos of one's own." Lastly, and above all, one finds the same exclusion of the radical, evil other, who persists in denying the "meta-institution of all possible human institutions," whether it is termed language-game or decision of meaning, and must pay for this refusal by "losing his own identity as a meaningful agent," namely, suicide or dementia. In short, Aristotle's plants today would be headed for the morgue or the asylum.

But, as you will have understood, the point of contention of this exclusion remains the same:

by making an entire dimension of speech philosophically and ethically inaudible, one has confused otherness with nothingness.

From the Deer-Goat to the Novel

The consequence of the decision of meaning is twofold. First, and most radically, as we have seen, there is the exclusion from the community of all those who "speak for the pleasure of speaking," in the terms of Gamma 5. The sophist, whom Aristotle and Apel find so disagreeable, is not human. But there is a second consequence which is more subtle, that has to do with the topology of meaning sketched out by Aristotle's statement of equivalence.

To speak is to say being; such was Parmenides's thesis, and such, again, mutatis mutandis, is the ordinary regime or daily lot of Aristotelian language. Ordinarily, indeed, when one speaks, the meaning of the word expresses the essence of the thing. This is the case when the thing exists; the essence of the entities is the meaning of the word that refers to them --if "man" means "animal with two feet," then, providing man exists, man is an two-footed animal --, and it is precisely in this that the ontological necessity of excluding "idle talk" is rooted, all the way to Heidegger. Now, faced with the sophist, the Aristotelian response in kind opens up a radically new possibility. One is no longer forced to speak of something which exists in order to mean something; one can very well speak of the "deer-goat" (which is the great example in De Interpretatione) without putting ontology at risk. One can speak non-being because one can speak non-being, because with the language of possibility comes a meaning which is no longer bound to reference. This is in fact why truth values are nothing more than a question of sun, syntax and synthesis, between a subject who is or is not, and a predicate which is or is not. Thus we may utter the truth when we speak about things which do not exist (the statement 'A deer-goat is not a cow' is true), things which, when one speaks about them, continue not to exist. Aristotelian semantics, on its part, produces parallel worlds in which true sentences assign non-existent predicates to non-beings --stemming not from the false which is, but from the true which is not. From the ontology which has thus been reassured by Aristotle, the possibility flows of an assumption of logology: by speaking of things which have no existence, and therefore, in Aristotle's terminology, no essence or definition, by discarding the physical or phenomenal reference, one has opened up the possibility of promoting meaning alone, meaning itself. Just as Parmenidean ontology had always already become sophistical logology, here, on Aristotelian ground, sophistical logology is assigned to a new dwelling-point: literature, as a case which is legitimized by the interpretation of the legein ti as sêmainein ti.

In the wake of rhetoric, which had already chosen the expression "to speak to" over the expression "to speak of," it is an entirely different corpus which becomes more intelligible, that of the second wave of the sophists, still Greek yet already Latin, and this marks the inception of the novel. If the novel is considered in this light, it shows itself to be an extremely original response to philosophical prohibition. For the novel is self-consciously a pseudos, and presents itself as such; it is a discourse which forgoes any ontological adequation and follows its own demiurgical path, namely, speaking for the pleasure of speaking rather than speaking in order to mean something, and thus producing a world-effect, a novelistic "fiction." And the popularity of novels, which resumes the foundational tradition of the Homeric poems, ends up constituting the cultural avatar of a political consensus, extended to the limits of the inhabited world by means of the pax Romana. As Dion said, not everyone sees the same sky, but even the Indians know Homer.

Thus the paradigm of truth has been transformed. From now on, sophistics no longer has to do battle with philosophical authenticity, but rather with the exactness of historical fact.

It is now the gaze of the historian which brings the accusation of the pseudos to bear against sophistics, and its literary kin. Lucian's How to Write History, in particular, bears witness to the existence of this new conflict: the historian as mirror is completely opposed to the poet, who is entitled to "overthrow the fortress of the Epipoles with a stroke of the pen," and defines himself by not being a novelist. However, Lucian practices the art of the sophists on sophistics itself; in The Genuine History, his irony ends up jeopardizing his own practice: "I will tell the truth on one point, which is that I tell lies." Here, "genuine" history takes up the liar's paradox again, and against the history of the chroniclers and the faithful account of events, it responds with the unmatchable power of invention.

When confronted with philosophy, the early sophists preferred a consensus-building discourse to that which sought to conform to the entity, or to the being of the entity. This very shift from adequation to political and cultural consensus also impacts on the shift in the relevant opposition: the later sophists had to confront history rather than philosophy. We have thus moved from ontology to the human sciences, and from the art of the sophists to literature.

IV. Ethical Correctness

The epicentre of the history of philosophy, once it is rewritten under the influence of sophistics as the history of the delimitation of philosophy, becomes the problem of the regulation of language. Ordinary philosophy (of which I speak with the same respect as of ordinary language) musters up all its forces, at all levels and by all the means at its disposal --Platonic, Aristotelian, Kantian, Heideggerian, Levinasian, analytic, Habermasian, Foucauldian and even Perelmanian --to reinforce this regulation ethically. As ideology is lacking, we are given moral lectures instead. As a result, from Allan Bloom to Vaclav Havel, the consensus that sophistics should be the prime target has reached remarkable dimensions, extending as far as politics.

Allow me to return once more to the mutual demarcation of the space between the philosopher and his bad other. Its most tangible effect is to relinquish the distinction between good and evil, the foundation of ethics, and hand it over to the most slippery genre, that of resemblance; as Nietzsche noted, "the border between good and evil disappears --that is sophistics." As a result, the two major themes which philosophy develops in order to distinguish the good side from the bad side of resemblance are cancelled out by sophistics, in such a way that ethics is displaced.

The first theme is that of the mask of "hypo-crisy" which one slips on. We are familiar with the accusation of flattery and deceit which runs throughout the Gorgias: rhetoric and sophistry are eidôla which "slip beneath" justice and legislation. But generally one fails to notice that Protagoras, who proudly describes himself as showing his face, literally undoes this accusation, when he shows, at the end of his myth, why the only mask one need wear is indeed that of justice:

As for the other excellences [...] if one someone wishes, for instance, to be good at playing the flute or some other art, while he is not, he is mocked and becomes angry, and those who are close to him come and reproach him for his folly. If, on the contrary, the issue is justice and the political virtue in general, when a man who is known to be unjust, when this man ends up saying the truth about himself in public, this frankness which previously was viewed as wisdom, is in the present case considered to be madness, and one states that all must declare themselves to be just, whether they are or not, and that he who does not put on the mask of justice is mad.

The social bond is created by pretending it exists, and when philosophy reproaches the sophists for being hypocritical, this hypocrisy is none other than the essence of political virtue. Hence the sophistical alter ego is the one who drags ethics onto the political terrain.

The second elaboration in lieu of a distinction is, in Aristotelian terms, that pertaining to intention, proairesis. The sophists once again "slip" into "the same form as the philosopher," but this time, the difference between the two has to do with "their choice of a way of life." The intimate character of intention makes it even harder to grasp the difference. This is seized upon by Latin rhetoric and the second wave of sophistics, in order to lead the moral problematic to its other point of reversal. "Philosophia enim simulari potest, eloquentia non potest" (Quintilian, Oratorical Institution). In the second part of the sentence, Quintilian forces one to definitively reevaluate the first part; he leaves the question of intention to philosophy, both as regards its contestable criterial status, and its concretely or effectively ungraspable [inassignable] character (this is why philosophy "always presents simulations // counterfeit material"), whereas eloquence, once again, as always, produces the effect which is unfailingly index sui. Philosophy never relinquishes its claim to unmask sophistics, by banking on the concept of intention; sophistics never ceases to distinguish itself from philosophy by emphasizing the accounting of effects. The consideration of effects can match that of intention because the effect is no longer at the mercy of a dichotomy: faced with the polarized duplicity of intention, there is or there is not an effect, de facto, precisely. When one responds to the question of intention with the effect, one has shifted the grounds of the ethical problematic, the definitional shield of philosophy, onto another terrain, that of aesthetics.

This twofold destabilization or upsetting of ethics is obviously what leads one to cry wolf. The most common name for the wolf today is "relativism." This underscores in eminently philosophical, that is, Platonic fashion that the ethical upset is dependent on the refusal to subject discourse to the criterial verdict [instance] of truth. Unfortunately for whoever ends up repeating him, Joseph Moreau is one of those who has maintained this most consistently, for example when, regarding the media in particular, he denounces "the great failing of a sophistical civilization," one defined (like ours) by "technological power and the power of the word": "the absence of a function which can regulate the usage of technology and of logos, without which the one is in the service of boundless appetites, and the other consents to justify undisclosable interests."4 This should be taken quite literally, for "sophistic reason" is responsible in particular for the demise of the gold standard, in economics: "To be sure, gold is just an idol, but one which was indispensable for lack of universal good faith. We are merely beginning to see the damage that this sophistic challenge to common sense and honesty can cause."

What is interesting in Moreau's argument is that he is intelligent enough to plead guilty at the right moment, thereby justifying the impression one might have from the outset: one feels annoyed, but also stifled by his position, which I would venture to term 'ethical correctness'. As he writes, "one cannot avoid being accused of intolerance, when one invokes the right of truth, faced with those who proclaim the sovereignty of opinion. However, the accusation of intolerance is unjustified, for if I assert that truth must live up to absolute demands, of which one must take account, I do not thereby claim that I am in possession of it, or that I must impose my view at any cost." This reflection enables one to distinguish two ways of being an anti-relativist, in the current debate. One is more "crude" or unbearable than the other, because it is dogmatic and non-critical. A position is dogmatic when it operates as if truth were not merely a regulative idea, a site, but also a content that must be respected, that one must make people respect. François Furet's reaction against the "denunciation of European culture" he finds characteristic of political correctness most likely falls under this category. Furet regrets that this new political vulgate leads one, on the intellectual level, to "relativize all works of art and of the spirit, to wreck the idea of the universality of truth, without which the notion of education itself loses its meaning."5

The second kind of anti-relativist takes care to leave the site of truth empty.

Dogmatism and relativism then become symmetrical dangers for him. Allan Bloom, waging war against political correctness, thus specifies that "there are two threats to reason, the opinion that one knows the truth about the most important things, and the opinion that there is no truth in these matters; the first one asserts that the quest for truth is unnecessary, and the second one asserts that it is impossible." 6 Hence the right attitude, with due classicism, is the Socratic knowledge of ignorance. Alain Badiou's modality of anti-relativism makes for a radically different perspective, insofar as he strives both to demarcate himself from the consensual discourse on ethics and to make room for a structural perception of sophistics.

Notice in passing that it is hardly inconsequential for such different standpoints to come together on this issue. In Conditions, 7 Alain Badiou reenacts a Platonic gesture, faced with "modern sophistry" (Nietzsche and Wittgenstein) and its "contemporary" form (Lyotard, Rorty, Vattimo and Derrida). There is no philosophy without sophistics, and no philosopher who does not define himself as an opponent of the sophist ("every definition of philosophy must distinguish it from sophistics"). However, it is not only the definition of philosophy in general which is dependent on sophistics, but also the distinction between good and bad philosophy:

Philosophy must never give way to anti-sophistical extremism. When it nourishes the dark desire to do away with the sophist once and for all, philosophy goes astray. This is precisely what defines dogmatism, in my view: claiming that the sophist should not exist, simply because he is a perverse twin of the philosopher. No, the sophist must simply be assigned to his place.

Faced with "disaster," with "dogmatic terror," which lies in declaring that "the sophist must not be," and hence decreeing the "eradication of its other," the "ethics of philosophy," on the contrary, maintains "the sophist as its opponent," and "preserves the polemos": "At all times, the sophist is required for philosophy to keep hold of its ethics. For the sophist is the one who reminds us that the category of Truth is empty. To be sure, he only does so in order to deny truths, and in this regard he must be fought. But this fight takes place on the ground of ethical norms." For a dogmatic Platonism, the first type of anti-relativism, the sophist as such is a danger to ethics; for a "Platonism of the manifold," which refuses to give up the category of truth as understood through sophistics (which then becomes ethically necessary, as "operation" and as "void"), to keep a firm hold on philosophy can only mean to withdraw the manifold "from the authority of language," to not be obliged to "recognize the constitutive character of linguistic variation."

In my opinion, it is exactly at this point that a more intelligent form of terror rears its head, if we are to believe the philosophistical rewriting of the history of philosophy. Indeed, it teaches us that the margin, the degree of freedom separating the act of eradication from that of assigning a place, is perilously slim. How might one separate the initial gesture, which is good, from its chain of effects, which are bad, when from Plato to Habermas, the issue is always Book X of the Laws, where the place that is assigned to those who speak as atheists is prison, followed by death without burial, and book Gamma of the Metaphysics, in which he who denies the principle of non-contradiction is a plant, whose place is outside of humanity?

The difference between morals and moralism, or ethics and morals, regardless of the chiasmas with added value, has yet to be fulfilled, from this standpoint.

Anti-Platonism seems like such an outdated notion! Anti-sophistics, too, seems so outdated! Haven't we moved beyond these exchanges of threats long ago? Well, neither a Heideggerian problematic, nor the return to Kant appear to be sufficient to me. They fail, in any case, to prevent the French scene, within philosophy and without, from reducing ethical discourse to something dangerously inane.

  It is also up to us to grasp in what way Aristotle's gesture in the Metaphysics is repeated in front of our own eyes. Meaning, consensus and exclusion is precisely the structure of what Karl-Otto Apel calls "the a priori of the communicative community." The basic question for Apel, who views himself as following in the footsteps of Wittgenstein and Peirce, but is really the follower of Aristotle, is "the condition of possibility of meaningful discourse or meaningful argumentation." It is the same problematization of the notion of an ultimate

  , yet can be carried out dialectically, in the shape of a refutation of the opponents who claim to deny it. Now, refutation is the paramount weapon of the sophists; Aristotle, following Socrates, borrows it here for the occasion. If Gorgias's treatise is read in the way we just did, it stands in toto as a refutation of Parmenides's poem. It starts with the speech of the other, as enunciative procedure and thematic statement, and brings to light its catastrophic consequence --ontology taken literally means logology, or in other words, if Parmenides, then Gorgias. Quite symmetrically, the demonstration by refutation starts with the opponent's statement of the principle, if only so as to express his denial, and brings to light the unsettling consequence that the opponent obeys the principle at the very moment he denies it. Sophistics taken literally is Aristotelian; if Protagoras speaks (which sophists normally do), he can only speak the way Aristotle does.

1 I am paraphrasing Francis Ponge's formidably dry statement, in Pratiques d'écriture ou l'inachèvement perpétuel (Paris: Hermann, 1984), p. 40. 2 These quotations are drawn from "Philosophie et politique," Les Cahiers du Griph 33 (Printemps 1986), and "Vérité et politique," in La crise de la culture (Paris: Gallimard, 1972); the original of the second quotation is from "Truth and Politics," in Between Past and Future (New York: Viking Press, 1968), p. 259.

principle

"La question d'une fondation ultime de la raison," trans. S. Foisy & J. Poulain, Critique (octobre 1981).

Joseph Moreau, Platon devant les sophistes (Paris: Vrin-Reprise, 1987), especially the 1979 article "Qu'est-ce qu'un sophiste ?".

"L'utopie démocratique à l'américaine," Le Débat

(mars-avril 1992).6 Giants and Dwarfs(Touchstone, 1990); the quotation is from the preface, "Western Civ." 7 (Paris: Seuil, ?). See also Manifeste pour la philosophie(Paris: Seuil, 1989).


domain: sdv info

text:
Jaap Van Milgen
email: jaap.vanmilgen@rennes.inra.fr

Roberto Barea

Ludovic Brossard

Nathalie Le Floc'h

Mathieu Gloaguen

Yolande Jaguelin

Anne Pasquier

Nadine Mézière

Yvon Colléaux

George Guillemois

Jean-François Rouaud

Patrick Touanel

Chrystèle Cornée

Bernard Carrissant

Hervé Demay

Maurice Alix

Jérome Liger

Mallory Prigent

Amélie Cotard

Valine and isoleucine requirements in piglets

•

  To control nitrogen excretion: • control the CP content of the diet • use crystalline amino acids • improve the amino acid balance • More amino acids become co-limiting (

Determination of a SID Lys level limiting performance (exp. 1)

	Isoleucine response/requirement studies
	Crude protein, % SID Val:Lys, % SID Lys, % 1.00 57 SID Lys, % SID Ile:Lys, % 50 Experiment Objective 17.3 57 1.00 1.00 60 SID Ile:Lys, % 48 60 Feed intake, g/d 820 Feed intake, g/d 689 695 3 Interaction between Ile and Lys 18.3 70 70 1.15 1.15 1.20 50 60 48 60 818 814 808 Feed intake, g/d 800 819 815 827 Daily gain, g/d 486 a 530 b Daily gain, g/d 401 393 501 484 4 Interaction between Ile and other BCAA 21.1 1.20 RSD RSD 803 55 123 513 b 41 Daily gain, g/d 443 478 501 499 76	Probability Probability RSD Val Ile Val×Ile Ile Lys Ile×Lys P 27 <0.01 0.99 0.63 0.61 0.70 0.91 0.18 36 <0.01 <0.01 0.25 0.65 0.39 0.04 0.33
	Gain:Feed Gain:Feed Gain:Feed	0.58 0.56	0.59 a 0.57 0.58	0.65 b 0.62 0.62	0.60 0.60	0.64 b 0.03 <0.01 0.03 0.05 0.85 <0.01 0.09 <0.01 0.11 1.00

E N V I R O N N E M E N T SID Lys: 1.0% Other essential amino acids: ≥ ideal protein Response to L-Val and L-Ile supplementation (exp. 2)

Interaction between Ile and Lys (exp. 3) content (% in DM)

	Lys	Ile	Val	Leu

Interactions between Ile and excess BCAA (exp. 4)

	Protein source	Corn gluten	Spray-dried			Probability
		meal	blood cells
	SID Ile:Lys, % Experiment Objective 50	65	50	65	RSD source Ile	source×Ile
	SID Val:Lys, % 5 Response to L-Val supplementation ("control" BCAA) 70 70 105 105
	SID Leu:Lys, % 6 Response to L-Val supplementation (excess Leu; corn gluten meal) 180 180 168 168
	Feed intake, g/d 7 Interaction between Val and excess Leu 744 744 736 803 75	0.20	0.09	0.08
	Daily gain, g/d	442	444	449	482	62	0.16	0.28	0.35
	Gain:Feed	0.59	0.60	0.61	0.60 0.04 0.27	0.65	0.43

Response to Val supplementation (exp. 5 and 6)

		Linear-plateau	Curvilinear-plateau
		ADFI	ADG	F:G	ADFI	ADG	F:G
	"Requirement", % SID Val:Lys
	Exp. 3 (control BCAA)	73.7	70.2	67.7	80.5	74.6	71.7
	Exp. 4 (excess Leu)	68.8	68.8	-	73.8	73.5	-
	Response at 60% SID Val:Lys (% of the plateau value)
	Exp. 3 (control BCAA)	87.5	82.3	94.3	87.4	82.9	94.6
	Exp. 4 (excess Leu)	53.2	48.1	-	48.4	42.3	-