Marion Cornet

Jérôme Degert
email: jerome.degert@u-bordeaux.fr

Emmanuel Abraham

Eric Freysz

Frequency resolved cross-correlation between optical and terahertz pulses: application to ultrashort laser pulse characterization

Keywords: Ultrafast nonlinear optics, (320.7100) Ultrafast measurements, (300.6495) Spectroscopy, terahertz

Introduction

Nonlinear interaction of terahertz (THz) and ultrashort optical pulses in Zinc Blende crystals is widely used to perform electro-optic (EO) sampling of THz pulses [START_REF] Duvillaret | Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques[END_REF][START_REF] Duvillaret | Electro-optic sensors for electric field measurements. II. Choice of the crystals and complete optimization of their orientation[END_REF]: the THz pulse, due to the Pockels effect, induces a phase modulation of the optical pulse which is measured for different time delays τ between the two pulses. In most experiments, the Pockels-induced phase is detected with photodiodes and very few attention is paid to the evolution of the spectrum of the probe pulse. Actually, according to Gallot and Grischkowsky [START_REF] Gallot | Electro-optic detection of terahertz radiation[END_REF], the spectrum of the optical pulse resulting from the interaction between the THz and optical fields in conventional EO sampling can be understood as the generation of phase-coherent sidebands, by sumand difference-frequency mixing (SFG-DFG), on the spectrum of the incident optical pulse. These sidebands can also be regarded as the result of a Pockels-induced cross-phase modulation (XPM). For intense THz pulses, it has been shown that higher order XPM, namely Kerrinduced, adds to this Pockels XPM to produce large modification of the optical spectrum [START_REF] Shen | Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses[END_REF][START_REF] Shen | Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses[END_REF]. Unfortunately, in these experiments performed in a 110 ZnTe crystal, the interaction of the THz wave, polarized along the 001 -axis, with the probe pulse, cross-polarized with respect to the latter, leads to sidebands which are superimposed to the spectrum of the incident optical pulse. As a consequence, and unless very intense THz fields are applied, one can hardly infer from such an experiment what is actually the spectral content of the sidebands resulting from XPM, whatever its origin. From this vantage point, spectrally resolved EO sampling with such a geometry leads to a loss of information concerning the sampled THz wave since the original spectral content of the incident probe pulse overwhelms the information related to the interaction of the optical and THz pulses. So, one needs a specific detection scheme which makes it possible to get rid of the spectrum of the probe pulse and preserve the sidebands due to Pockels XPM.

In the present paper, we report on the spectral analysis of the optical sidebands produced by the interaction of optical and THz pulses in a ZnTe crystal. To this end, we make use of a special geometry of the polarization states of both beams which results in an amplitude modulation of the optical pulse, unlike in [START_REF] Shen | Nonlinear cross-Phase modulation with intense single-cycle terahertz pulses[END_REF][START_REF] Shen | Spatiotemporal control of ultrashort laser pulses using intense single-cycle terahertz pulses[END_REF], making it possible to separate the optical pulse generated by the Pockels XPM from the incident pulse. Measured as a function of the delay τ it yields a spectrogram which can be viewed as a frequency resolved cross-correlation between the optical and THz pulses and enables to characterize the optical pulse. Hereafter we demonstrate that this technique can be easily applied to the measurement of the group-delay dispersion of an ultrashort optical pulse. The experiment is the following (Fig. 1(a)): an intense linearly polarized THz pulse spanning the 0.3-7 THz frequency range is emitted from air ionized by a two-color, namely 400 nm and 800 nm, femtosecond laser field [START_REF] Thomson | Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications[END_REF]. This THz pump field is then collimated and focused onto a 110 ZnTe crystal, with a thickness L = 300 µm, by two off-axis paraboloidal mirrors with a 150 mm-focal length. Inside the crystal, the ŷ-polarized THz beam, with an amplitude E THz = 8 kV/cm, propagates collinearly with a weak 800 nm probe beam, ẑ-polarized thanks to a halfwave plate and a polarizer, the ẑ-axis coinciding with the 001 -axis of the crystal (Fig. 1(b)). After the crystal, a polarizer transmits the component of the optical pulse generated along the ŷ-axis by the THz-induced Pockels effect. A delay stage makes it possible to delay the probe pulse with respect to the THz pump pulse. The temporal evolution of the optical pulse intensity transmitted by the polarizer is measured by a photodiode connected to a lock-in amplifier, whereas the temporal evolution of its spectrum is detected with a spectrometer. Finally, the THz intensity can be adjusted by a pair of high density polyethylene polarizers.

Experimental setup

Experimental results

We first measured the intensity, S 1 , of the optical pulse transmitted by the polarizer after the ZnTe crystal as a function of the delay τ. The result is displayed on Fig. 2(a), together with the intensity of the THz wave measured by conventional EO sampling, i.e. measurement of the Pockels-induced phase modulation by ellipsometry [START_REF] Gallot | Electro-optic detection of terahertz radiation[END_REF]. Both intensity profiles are in good agreement except near the zeros of the THz wave, due to both the lack of phase sensitivity of our detection scheme and the convolution of the THz wave with a probe pulse intensity profile of finite duration. Figure 2(b) confirms the quadratic evolution of the peak value of S 1 with respect to the THz electric field strength, E THz , as expected from the use of a quadratic detector, indicating thus that the THz wave is weak enough to modulate the optical pulse spectrum only through the Pockels effect.

Figure 3(a) displays the evolution of the spectrum of the ŷ-polarized optical pulse versus τ, recorded for a pulse compression optimizing the peak of the THz pulse. To reduce the subsequent probe pulse temporal broadening we took care of having the same dispersive media (i.e. polarizer and half-wave plate) on the 1.5 mJ beam used to generate the THz and on the probe beam (not shown in Fig. 1). As can be seen, the spectrogram of Fig. 3( trace given by the well-known Cross-correlation Frequency-Resolved Optical Gating (XFROG) characterization technique [START_REF] Linden | XFROG -A new method for amplitude and phase characterization of weak ultrashort pulses[END_REF]. However, here, one pulse is in the optical range, whereas the other one is in the THz range. Hereafter, we demonstrate more precisely the connection between the spectrogram of Fig. 3(a) and the XFROG technique.

Spectrally resolved EO sampling in a 110 ZnTe crystal

Let us write

E(t, x) = 1 2 Ê(t, x) exp i(k pr x -ω pr t) + c.c.
the electric field of the probe pulse, and

P (2) (t, x) = 1 2 P(2) (t, x) exp i(k pr x -ω pr t) + c.c.
the second-order polarization of the crystal, assuming the THz pulse is weak enough to avoid higher order nonlinear phenomena like Kerr effect [START_REF] Cornet | Terahertz Kerr effect in gallium phosphide crystal[END_REF]. Here, ω pr and k pr stand for the angular frequency and the wave vector of the probe beam. Then, in the frame of the laboratory, within the slowly varying amplitude approximation, the probe wave equation writes [START_REF] Butcher | The Elements of Nonlinear Optics[END_REF]:

∂ ʵ (t ′ , x) ∂ x = i π λ pr n pr ε 0 P(2) µ (t ′ , x), ( 1
)
where t ′ = tx/v g , n pr is the refractive index of ZnTe at the probe wavelength λ pr , ε 0 is the permittivity of vacuum, v g is the group velocity of the probe pulse, and µ = y, z.

Assuming that we are far from resonance, in the frame of the crystal the second-order polarization corresponding to the interaction of the optical probe and THz pulses through the Pockels effect is given by [START_REF] Butcher | The Elements of Nonlinear Optics[END_REF]:

P(2) (t ′ , x) = 2ε 0 χ (2) : Ê(t ′ , x) E THz t ′ + x v g , x ,
where χ (2) is the second-order nonlinear susceptibility tensor of the crystal. The latter is related to the common electro-optic tensor r by χ

(2)

i jk = -1 2 n 2 i n 2 j r i jk
, where i, j, k = X,Y, Z, and n is the refractive index in the optical range [START_REF] Boyd | Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities[END_REF]. For an isotropic crystal like ZnTe, n i = n pr ∀ i, so that this relation reduces to χ (2) = - 1 2 n 4 pr r, with r 41 = r i jk (i = j = k) the only nonzero element of the electro-optic tensor.

In the frame of the laboratory, the THz and optical electric fields write E THz (t, x) = E THz (t, x)ŷ and Ê(t, 0) = Êz (t, 0)ẑ, respectively, so that P(2) becomes:

P(2) µ (t ′ , x) = ε 0 n 4 pr r 41 Êz (t ′ , x)E THz t ′ + x v g , x δ µ,y (µ = x, y, z). ( 2
)
Thus, with the peculiar geometry considered here, the only nonzero component of P( 2) is along the y-axis. As a consequence, to the first order, the interaction of the optical probe pulse and the THz wave via the Pockels effect gives rise to an optical pulse at angular frequency ω pr , cross-polarized with respect to the incident optical pulse, so that, at a point x within the crystal, the optical pulse envelope now writes Ê(t, z) = Êy (t, x)ŷ + Êz (t, x)ẑ = Êy (t, x)+ Êz (t, x). However, this new component can also couple to the THz wave and, in turn, impacts on the propagation of Êz (t, x) through to the second-order nonlinear polarization P ′(2) (t ′ , x) = 2ε 0 χ (2) :

Êy (t ′ , x) E THz t ′ + x v g , x ,
which, in the frame of the laboratory, writes

P ′ (2) µ (t ′ , x) = ε 0 n 4 pr r 41 Êy (t ′ , x)E THz t ′ + x v g , x δ µ,z (µ = x, y, z). (3)
Finally, according to Eqs. ( 1), ( 2) and (3), the probe pulse envelope, Ê(t, z) = Êy (t, x)ŷ + Êz (t, x)ẑ, obeys to the following coupled wave equations:

∂ Êy (t ′ , x) ∂ x = iκ Êz (t ′ , x)E THz t ′ + x v g , x , (4a)
∂ Êz (t ′ , x) ∂ x = iκ Êy (t ′ , x)E THz t ′ + x v g , x , (4b)
where κ = πn 3 pr r 41 /λ pr . Solving Eq. ( 4) requires to solve also the wave equation for the THz field. However, in the weak field regime, one can consider that the THz wave is left unaltered by the optical pulse during its propagation. Finally, the THz wave is just affected by dispersion/absorption in ZnTe, an effect which can be readily calculated as shown later. In order to integrate Eq. ( 4), let us introduce the new fields ʱ (t ′ , x) = Êy (t ′ , x) ± Êz (t ′ , x), with ʱ (t ′ , 0) = ± Êz (t ′ , 0). Then, Eq. ( 4) lead to the following uncoupled wave equations

∂ ʱ (t ′ , x) ∂ x = ±iκ ʱ (t ′ , x)E THz t ′ + x v g , x ,
whose integration is straightforward and finally leads to

Êy (t ′ , L) = i Êz (t ′ , 0) sin κ L 0 E THz t ′ + x v g + τ, x dx , Êz (t ′ , L) = Êz (t ′ , 0) cos κ L 0 E THz t ′ + x v g + τ, x dx ,
where we have taken into account the fact that the probe pulse is delayed by an amount of time τ with respect to the THz wave. Since in our experiment κ L 0 E THz (t ′ + x/v g + τ, x) dx ≪ 1, one gets Êz (t ′ , L) ≃ Êz (t ′ , 0) and

Êy (t ′ , L) ≃ iκ Êz (t ′ , 0) L 0 E THz t ′ + x v g + τ, x dx. ( 5
)
There is a one to one correspondence between the field given by Eq. ( 5) and the optical pulse generated through sum-and difference-frequency by interaction of the probe pulse with the THz wave. Consequently, the Fourier transform of Eq. ( 5), denoted Êy (ω, L), fully describes the optical sidebands mentioned in [START_REF] Gallot | Electro-optic detection of terahertz radiation[END_REF]. However, with the geometry of our experiment, these sidebands can be detected independently from the spectrum of the incident optical pulse, Êz (ω, 0), since these two spectra are cross-polarized.

Let us now calculate Êy (ω, L) whose squared modulus corresponds to the spectrogram of Fig. 3. In order to account for dispersion/absorption in the THz range, one needs, first, to express Eq. ( 5) as a function of the spectral amplitudes E THz (Ω, 0) of the input THz pulses, defined as the Fourier transform of E THz (t ′ , 0). We have:

E THz t ′ + x v g + τ, x = +∞ -∞ dΩ 2π e -iΩt ′ E THz (Ω, 0)e iΩ[ ñ(Ω)-n g] x/c e -iΩτ , (6)
where ñ(Ω) = n(Ω) + iκ(Ω) is the complex refractive index of the crystal in the THz range, and n g = n g (ω pr ) is the group index for the optical pulse. In Eq. ( 6), the phase factor Ω ñ(Ω)x/c simply corresponds to the phase accumulated by the THz pulse in the course of its propagation within the crystal. Inserting Eq. ( 6) into Eq. ( 5), and expressing Êz (t ′ , 0) as the Fourier transform of Êz (ω, 0) then leads to

Êy (ω, L) = +∞ -∞ dt ′ e iωt ′ Êy (t ′ , L), = iκ L 0 dx +∞ -∞ dΩ 2π Êz (ω -Ω, 0)E THz (Ω, 0)e i Ωx c [ñ(Ω)-ng]-iΩτ . ( 7
)
Assuming that the input optical pulse may have some residual group-delay dispersion (GDD), φ

in , Êz (ω, 0) then writes:

Êz (ω, 0) = Êz (ω) exp[iφ (2)
in ω 2 /2)], where Êz (ω) can be determined from the input pulse spectrum. However, Eqs. ( 1) and ( 4) neglect the dispersion within the crystal. To the first order, the latter can be accounted for by replacing, in Eq. ( 7), Êz (ω, 0) by Êz (ω, x) = Êz (ω, 0) exp[iφ (ω, x)], with φ (ω, x) = k ′′ pr ω 2 x/2, where k ′′ pr = d 2 k/dω 2 ω pr , k (ω) = ωn(ω)/c, and n(ω) is the refractive index of ZnTe in the optical range, taken from [START_REF] Marple | Refractive index of ZnSe, ZnTe, and CdTe[END_REF]. All calculations done, these substitutions lead to:

Êy (ω, L) = iκL +∞ -∞ dΩ 2π sinc ∆k ′ L 2 Êz (ω -Ω, 0)E THz (Ω, 0)e i 1 2 φ (2) in (ω-Ω) 2 -Ωτ+ 1 2 ∆k ′ L , (8)
with sinc(x) = sin(x)/x and ∆k

′ = ∆ k + 1 2 k ′′ pr (ω -Ω) 2 , where ∆ k = Ω [ ñ(Ω) -n g ] /c
. The latter equation can be written as follows:

Êy (ω, L) = iκL +∞ -∞ dΩ 2π sinc ∆k ′ L 2 Êz (ω -Ω, 0)E THz (Ω, 0)e -1 2c κ(Ω)ΩL × e iϕ(ω,Ω) , (9)
with

ϕ(ω, Ω) = 1 2 φ (2) in (ω -Ω) 2 -Ωτ + 1 4 φ (2)
ZnTe (ω -Ω) 2 + 1 2 ∆k(Ω)L. (10)
Here, φ

ZnTe = k ′′ pr L is the GDD resulting from the propagation of the optical pulse through ZnTe, and ∆k(Ω) = Ω [n(Ω)n g ] /c corresponds to the wave vector mismatch between the optical and THz pulses. Thus, Êy (ω, L) corresponds to the integral of a rapidly oscillating term, exp[iϕ(ω, multiplied by a slowly varying amplitude term. Note that the latter spans the 0 -4 THz range due to both the cardinal sine function and the absorption in ZnTe [START_REF] Wu | Design and characterization of traveling-wave electrooptic terahertz sensors[END_REF]. Moreover, within this spectral range, the phase matching condition is fulfilled in ZnTe [START_REF] Nahata | A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling[END_REF]. As a consequence, in Eq. ( 10), one has ∆k(Ω)L ≃ 0.

Let us now discuss about the differences between our spectrally resolved EO sampling and the XFROG technique. In the latter, one needs two pulses, one of which is well characterized to serve as a reference, with the same order of duration. Here, we have a femtosecond pulse interacting with a picosecond THz wave. The THz wave could be seen as the reference since it can be fully characterized through conventional EO sampling. But this is precluded by its too long duration which blurs any temporal information (amplitude and phase) concerning the ultrashort optical pulse. From this point of view, the spectrally resolved EO sampling is not equivalent to the XFROG technique, and cannot be used to reconstruct the optical pulse (amplitude and phase) knowing the electric field of the THz wave. However, it would be wrong to conclude that no information concerning the spectral phase of the optical pulse can be inferred from the spectrogram of Fig. 3. Indeed, Eq. ( 8) indicates that a photon generated by Pockels XPM at a frequency ω pr + ω results from a spectral interference, over the whole spectrum of the THz pulse, between a photon in the optical range, at frequency ω pr + ω -Ω, and a photon in the THz range at frequency Ω. So, the spectral components shown in the spectrogram of Fig. 3 are sensitive to the relative spectral phase between the incident optical pulse and the THz wave. Upon the 0 -4 THz range transmitted by the ZnTe crystal, this relative phase is given by ϕ(ω, Ω), with ∆k(Ω)L ≃ 0. Since the phase matching condition in ZnTe cancels out the dependence of ϕ(ω, Ω) with respect to the dispersion in the THz spectral range (term Ω n(Ω)/c in ∆k(Ω)), it means that the spectral content of the spectrogram is only sensitive to the dispersion in the optical range (φ

(2) in , φ (2)
ZnTe ), that is to the spectral phase of the optical probe pulse, as we will see in the next section.

Discussion

We are now ready to see what kind of information about the probe pulse one can infer from the spectrogram of Fig. 3(a). Actually, since the latter is slightly tilted with respect to the angular frequency axis, we can already say that the probe pulse is not Fourier-transform-limited. It is linearly chirped due to some small positive GDD after compression of the laser output to optimize the THz generation and propagation through the ZnTe crystal. From Eqs. ( 9) and [START_REF] Boyd | Microwave nonlinearities in anisotropic dielectrics and their relation to optical and electro-optical nonlinearities[END_REF], one can deduce the GDD of the probe pulse from the inverse of the "slope" of the spectrogram. Indeed, as already mentioned, Eq. ( 9) indicates that Êy (ω, L) corresponds to the integral of a rapidly oscillating term, exp[iϕ(ω, Ω)], multiplied by a slowly varying amplitude term. According to the stationary phase approximation [START_REF] Born | Principles of optics[END_REF], the only significant nonzero contributions to Êy (ω, L) then occur for angular frequencies ω for which the phase ϕ(ω, Ω) is stationary. Consequently, the spectrogram will have significant values around angular frequencies such that ∂ ϕ/∂ Ω = 0, that is for ω

= ω pr + Ω -τ/ φ (2) in + φ (2) ZnTe /2 ≃ ω pr -τ/ φ (2) in + φ (2)
ZnTe /2 , since ω pr ≫ Ω. From the straight line superimposed to the spectrogram of Fig. 3(a), we obtained φ

(2) in + φ (2)
ZnTe /2 = -∆τ/∆ω = 1210 fs 2 . From [START_REF] Marple | Refractive index of ZnSe, ZnTe, and CdTe[END_REF] and [START_REF] Diels | Ultrashort Laser Pulse Phenomena[END_REF], one gets φ

(2) ZnTe = 640 fs 2 , leading to φ

(2) in = 890 fs 2 . This value is in good agreement with the one deduced from an autocorrelation of the probe pulse performed in front of the ZnTe, which gives φ

(2) in = 860 fs 2 . At first glance, our determination of φ [START_REF] Duvillaret | Electro-optic sensors for electric field measurements. II. Choice of the crystals and complete optimization of their orientation[END_REF] in is rather rough due to the uncertainty on the position of the straight line used to quantify the tilt of the spectrogram. Moreover, the latter is also slightly curved owing to the presence of some cubic dispersion in the optical pulse which is not accounted for by the straight line "fit" used here. However, despite its simplicity, our method gives rather good values of GDD as confirmed by the following experiment: we inserted on the path of the probe beam, in a region where there is no overlap between optical and THz beams, a 3 mm thick window of ZnSe. The latter adds to φ

(2) in a GDD φ

(2) ZnSe = 3096 fs 2 , as calculated from the refractive index given in [START_REF] Marple | Refractive index of ZnSe, ZnTe, and CdTe[END_REF]. The spectrogram resulting from the interaction of this probe pulse with the THz wave in ZnTe is shown in Fig. 3(b). Due to the probe pulse broadening, the amplitude of the recorded signal decreases, leading to a slight degradation of the spectral sensitivity of the spectrogram. We then calculated from the inverse of the slope of the straight line superimposed to the spectrogram that φ ZnTe /2 = 4250 fs 2 , leading to φ

(2) ZnSe = 3040 fs 2 , in good agreement with the value given above. From repeated experiments, we estimated the uncertainty on the value of φ (2) to be about 7%.

Conclusion

We have measured and modeled the evolution of the spectrum of the optical pulse generated during EO sampling in ZnTe of a THz wave by an optical pulse versus the time delay between them. We have also demonstrated that this spectrogram can be used to measure the GDD of the optical pulse. This characterization technique strongly depends on the EO crystal used, since it requires that the phase matching condition is fulfilled upon the spectral range transmitted by the crystal. Note that this method could be potentially extended to the determination of higher orders of dispersion in the optical range, provided that the spectrogram is fitted with more elaborate functions than a straight line. From a practical point of view, our experimental setup is also suited for the characterization of THz pulses by detecting the optical pulse generated via the Pockels effect by ellipsometry. Therefore, just by adding a spectrometer to a conventional THz time-domain spectroscopy (THz-TDS) experiment, one can measure the phase and amplitude of both optical and THz pulses used to perform THz-TDS spectroscopy. As a consequence, one does not need another setup, such as FROG or SPIDER, to characterize the optical pulse.

Fig. 1 .

 1 Photodiode or spectrometer

Fig. 2 .Fig. 3 .

 23 Fig. 2. (a) S 1 : intensity profile of the Pockels-induced optical pulse; S EO : intensity profile of the THz wave measured by conventional EO sampling. (b) Peak value of the Pockelsinduced optical pulse with respect to the THz electric field strength E THz .

Acknowledgments

The authors acknowledge financial support of the Conseil Régional d'Aquitaine through the project Hi ter (2011 1603 004). M. C. is grateful for thesis funding from the Direction Générale de l'Armement (DGA) and the Région Aquitaine.


Ying Florence

Wang

Arnaud Sallaberry

Karsten Klein

Masahiro Takatsuka

Visualizing Time-varying Twitter Data with SentimentClock

Keywords:

Microblogs such as tweets contain useful information for sentiment analysis. In this work, we introduce SentimentClock for visualizing the sentiment of time-varying Twitter data on 2D affective space. To illustrate our visualization design, two case studies are conducted. They demonstrate the effectiveness of SentimentClock in visualizing the temporal sentiment variations of tweets and comparing the sentiment of tweets on different topics. A demo video of our system is available at: http://youtu.be/JvQFAFW

INTRODUCTION

The growth of online microblogging services, particularly Twitter, have enabled people to express and spread opinions by posting short text messages such as "tweets". Therefore, the collection of tweets contains an enormous amount of information that are related to people's opinions and sentiments on various topics. Given such an advantage, many research studies use tweets as a basis for sentiment analysis. Usually, the analysis aims to gauge people's opinions (e.g. positive or negative) or complex emotional states (e.g. happy, sad, ⇤ e-mail: florence8627@vislab.net † e-mail: arnaud.sallaberry@lirmm.fr ‡ e-mail: karsten.klein@sydney.edu.au § e-mail: masa.takatsuka@sydney.edu.au stressed) conveyed by their tweets using text analysis methods.

To visualize the sentiments of tweets, many previous works, e.g. [START_REF] Hao | Visual sentiment analysis of customer feedback streams using geo-temporal term associations[END_REF], abstracted "sentiment" as a unidimensional variable, e.g. from negative to positive. However, recent psychological studies underline the more complex nature of sentiments. For instance, "sentiment" in the circumplex model [START_REF] Barrett | Independence and bipolarity in the structure of current affect[END_REF] are considered to form a multidimensional polar space, i.e. the affective space, having two principal axes: valence (i.e. from unpleasant to pleasant) and arousal (i.e. from deactivation to activation). Therefore, the unidimensional abstraction of sentiment cannot truly represent the distances between different emotional states. To overcome this, Ramaswamy [START_REF] Ramaswamy | Visualization of the sentiment of the tweets[END_REF] combined scatter plot, heat map and tag cloud with the affective model proposed by [START_REF] Barrett | Independence and bipolarity in the structure of current affect[END_REF] for visualizing the two dimensional sentiment of tweets using multiple views. With his visualization, the representation of "sentiment" has been improved, however, the temporal information of tweets are not associated with the visualization of 2D sentiment. Therefore, temporal variation of 2D sentiments of tweets (modeling complex emotions expressed by tweets) cannot be visualized. In this work, we aim to address this problem by proposing a visualization design called SentimentClock.

TASKS

Many interesting tasks related to sentiment analysis of microblogs require the consideration of the temporal dimension. We have compiled a list of these tasks and use them as the guidelines for designing and evaluating our visualization: [T1] Visualize and compare temporal variations of sentiments for collected tweets.

[T2] Compare sentiments variations of tweets on different topics.

[T3] Visualize the distribution of tweets on 2D affective space. [T4] Visualize both dimensions of sentiments (i.e. valence, arousal) of tweets and their semantic meanings (e.g. elated, stressed).

DATA ABSTRACTION

We adopt the ANEW dictionary [START_REF] Bradley | Affective norms for english words: Instruction manual and affective ratings[END_REF] to estimate sentiment of tweets. It contains affective ratings for 1034 English words obtained from empirical studies. For each word, the ratings consist mean and standard deviation for both valence and arousal. To calculate the 2D sentiment (i.e. valence and arousal) of tweets, two types of signaling words are searched: (1) affective words recorded in ANEW; (2) negation words, e.g. "should not, don't". We use the method suggested by [START_REF] Ramaswamy | Visualization of the sentiment of the tweets[END_REF] to statistically weight the ratings of multiple affective signaling words in a tweet. Meanwhile, if a negation word exists, we estimate the sentiment to be opposite to what are indicated by affective words. Eventually, for each tweet, polar coordinates consisting of the angle specifying a sentiment (Russell's model [START_REF] Barrett | Independence and bipolarity in the structure of current affect[END_REF]), and the radius indicating the strength of the sentiment are obtained.

VISUAL MAPPING

The features of our visualization design are shown in Figure 1.

(1) For [T1, T2], Time rings are used for adding the temporal dimension. Linked rectangle view and Tool tips showing statistics are used to allow better comparison.

(2) For [T3], Tweets distribution histogram is calculated based on the angle between valence and arousal. Color of the histogram shows positivity (red) or negativity (green) of the sentiment, which is mainly associated with valence [START_REF] Barrett | Independence and bipolarity in the structure of current affect[END_REF][START_REF] Ramaswamy | Visualization of the sentiment of the tweets[END_REF]. Sentiment strength belt shows the strength of sentiment (i.e. radius under polar coordinate system). Position of the belt shows the average strength. Thickness of the belt indicates standard deviation of such strength.

(3) For [T4], Radial layout is used to follow Russell's affective model [START_REF] Barrett | Independence and bipolarity in the structure of current affect[END_REF]; Sentiment wheel is labeled with affective words and maps two pairs of complimentary colors to valence and arousal, i.e. green to red shows pleasant level (valence), blue to yellow shows activation level (arousal).

CASE STUDIES

To illustrate our visualization, we collected tweets via Twitter streaming API for two case studies. Case study 1 Our first case study is based on tweets collected during 2013 Australian election period. 36016 related tweets posted on the election day (0:00 to 24:00, 7th Sep, 2013, GMT+10) are selected to visualize the temporal variation of sentiments. As shown by Figure 1, sentiments of tweets in the morning and afternoon (0:00 to 18:00) follow similar distribution ([T1, T3]). In the evening (18:00 to 22:00), which is the vote counting and result releasing period (highlighted in blue), tweets are found to have both high arousal and high valence, primarily falling into the elated and excited range (highlighted in purple) with high strength low standard deviation shown by the strength belt and the tool tip ([T1, T4]). At night (22:00 to 24:00), which is just after result announcement, sentiments split into two parts (with more being happy and elated) but both having high arousal ([T3, T4]). Case study 2 Our second case study is based on tweets collected from 15th Jun to 17th Jun, 2014 tracking two topics: "Australian Politics" and "World Cup 2014". Around 71200 tweets are found to contain affective signaling words for sentiment visualization. As shown by Figure 2, tweets on the topic "Australian Politics" are more spread out along the sentiment wheel and express more negative sentiments, e.g. upset and stressed, than the topic "World cup 2014" ([T2, T4]). Meanwhile, the sentiments of "World cup 2014" are mainly concentrated within the range of content and elated, whereas sentiments for "Australian Politics" split into two parts with similar strength ([T3, T4]).

Figure 2 :

 2 Figure 2: Comparing the sentiment variations of tweets on two topics: "Australian Politics" (left) and "World Cup 2014" (right).

CONCLUSION

In this work, we introduced SentimentClock for visualizing the sentiments of time-varying microblog data. We illustrated our visualization design with two case studies to show its effectiveness in achieving various tasks related to sentiment analysis.


InfoVis for the Masses : Easy Navigation and Interactive Browsing of a Digital Catalog

Interactive graphics can turn raw data into intuitive graphical pieces that make information understandable for millions of readers/users. Graphs and charts can have a decisive impact and convince users about complex issues. Designing Information Visualization for the masses relentlessly calls for a trade-off between communicating complex data and packaging it in ways that a large number of people will be able to read and manipulate. We propose a visualization targeted at large audiences and public open spaces. Intuitive layout equipped with basic

Browsing an e-catalog through an interactive visual interface

interaction makes the whole system suitable for a touch screen display. The system is currently being used in a public setting at the InnovaNews exhibit taking place at "La cité des sciences" in Paris (www.cite-sciences.fr and look for InnovaNews).

visual interface

The visualization is designed to browse through categorized data. Lower level information is organized into categories and sub-categories, a map showing links between categories helps users dig into more details. A network showing links between categories is initially displayed based on a force-directed layout. After a category has been selected, it moves towards the center of the screen. Neighbor categories are kept visible while all others fade away. Links are induced from items belonging to more than one category, in turn inducing links between (sub)-categories. Reduction of edge crossings follows a multi-circular sifting heuristic, somehow extending the greedy-switch algorithm to radial layout and improving over the two layer barycentric approach.

InnovaNews also offers geographic map, a TreeMap and a timeline.

GRAVITÉ is a joint INRIA, CNRS, and Université de Bordeaux project. Contact: Guy.Melancon@inria.fr, Arnaud.Sallaberry@pikko-software.com


Viscosity of multimodal suspensions predicted from solid fraction model for mixtures G.

Theoretical relative viscosity versus normalized total solid volume fraction:

References:

Viscosity of a suspension:

η = η 0 1 - ψ i ψ i MAX $ % & & ' ( ) ) -C E β i=1 n ∏ ψ i = y i 1 φ -y j j=1 i-1 ∑ C E = 2,5 ψ i MAX = β 1 φ - 1 γ i & ' ( ( ) * + + + y i 1 -β ( ) 1 φ - β γ i -1 -β ( ) y j j=1 i-1

∑

Wall effect: Loosening effect:

d hyp = d 1 1 + 2 k P2 x β' 2 x ( ) = π 1 + x ( ) 4 x 1 + 2 k P2 x ( ) 3 2 -1 $ % & & ' ( ) ) arcsin x 1 + x $ % & & ' ( ) ) . . 2 1 + 2 k P2 x ( ) 3 2 -3 1 + 2 k P2 x ( ) 1 + k P2 x 1 + x $ % & & ' ( ) ) + 1 + k P2 x 1 + x $ % & & ' ( ) ) 3 -x 3 + 3 k P2 x 3 1 + x + x 3 1 - k P2 1 + x $ % & & ' (
) )

The viscosity of a multimodal suspension is predicted by taking into account geometrical interactions and hydrodynamic interactions between particles. The model initially proposed by Farris is modified to include interactions used in the 4-parameter compressible packing model (CPM) to predict the compactness of a dry granular mixture. The wall effect and the loosening effect are the subject of a new theory. A spherical reference cell, concentric with a foreign sphere lost in a monodimensional mixture, can be used for both types of geometrical interactions. Its diameter is adjustable, depending on fine/coarse diameter ratio and on the specific packing density of the elementary dominant class. x 0 ≈ 0,2

& & &

∑ β: compactness value, supposed to be constant, of each elementary class ; y j : volume proportion of each elementary class, by reference of the total solid volume; b ij : wall effect coefficient; a ij : loosening effect coefficient; γ i : compactness value of the mixture when the class i is dominant.

The 4-parameter CPM is apt to account for the loosening effect on big particles from interstitial small ones and for the wall effect within assemblies of small particles near a big one. The new theory is based on a specific treatment of configurations of one secondary class particle surrounded by dominant class neighbours. Four parameters are: wall effect and loosening effect coefficients, critical cavity size ratio and compaction index. Compactness of small spheres in the disturbed area by wall effect:

Wall effect coefficient for a binary mixture:

( ) Delimitation of the spherical reference cell:

Compactness of large spheres in the reference cell:

Compactness of large spheres in a more global interference zone by taking into account the critical cavity size ratio x 0 :

Volume proportion of small particles at the « eutectic » point:

Loosening effect coefficient for a binary mixture:


Nils Caillerie

Large deviations of a velocity jump process with a Hamilton-Jacobi approach

We study a random process on R n moving in straight lines and changing randomly its velocity at random exponential times. We focus more precisely on the Kolmogorov equation in the hyperbolic scale (t, x, v) → t ε , x ε , v , with ε > 0, before proceeding to a Hopf-Cole transform, which gives a kinetic equation on a potential. We show convergence as ε → 0 of the potential towards the viscosity solution of a Hamilton-Jacobi equation ∂tϕ + H (∇xϕ) = 0 where the hamiltonian may lack C 1 regularity, which is quite unseen in this type of studies.

Résumé

Grandes déviations pour un processus à sauts de vitesse avec une approche Hamilton-Jacobi. Nous nous intéressons à un processus aléatoire sur R n qui alterne des phases de mouvements rectilignes uniformes et change de vitesse à des temps exponentiels. Nous étudions plus précisément l'équation de Kolmogorov après rééchelonnement hyperbolique (t, x, v) → t ε , x ε , v , ε > 0, puis nous effectuons une transformée de Hopf-Cole qui nous donne une équation cinétique suivie par un potentiel. Nous montrons la convergence pour ε → 0 de ce potentiel vers la solution de viscosités d'une équation de Hamilton-Jacobi ∂tϕ + H (∇xϕ) = 0 où le hamiltonien peut présenter une singularité C 1 , ce qui est assez inédit dans ce type d'études.

Version française abrégée

Nous nous donnons une densité de probabilité M ∈ L 1 (R n ) et nous notons V son support. Nous supposons que V est compact et que 0 appartient à l'intérieur de l'enveloppe convexe de V , que l'on note Conv (V ). Pour p ∈ R n , nous notons µ (p) = max {v • p | v ∈ Conv (V )}. Nous étudions le mouvement de particules dans R n suivant le processus de Markov déterministe par morceaux défini comme suit : une particule donnée se déplace de manière rectiligne uniforme avec une vitesse v ∈ V tirée aléatoirement en suivant la loi de probabilité M (v ′ ) dv ′ . À des temps exponentiels de paramètre 1, la particule change de direction en tirant une nouvelle vitesse tirée selon la loi M (v ′ ) dv ′ . Afin d'étudier des résultats de larges déviations du processus similairement aux techniques développées dans [START_REF] Bressloff | On the Hamiltonian structure of large deviations in stochastic hybrid systems[END_REF]- [START_REF] Faggionato | Averaging and large deviations principles for fully piecewise deterministic Markov Process and applications to moleular motors[END_REF], nous nous intéressons à l'équation de Chapman-Kolmogorov forward suivie par la densité de particules après un rééchelonement hyperbolique (t, x, v) → t ε , x ε , v , ε > 0 :

∂ t f ε + v • ∇ x f ε = 1 ε (M (v) ρ ε -f ε ) , (t, x, v) ∈ R + × R n × V.
Nous étudions plus particulièrement l'équation vérifiée par un potentiel ϕ ε obtenu après passage par une transformée de Hopf-Cole :

f ε (t, x, v) = M (v) e -ϕ ε (t,x,v) ε
. Nous cherchons alors une éventuelle limite pour ϕ ε . Nous procédons à un développement WKB : ϕ ε = ϕ + εη, ce qui amène, en posant p = ∇ x ϕ et H = -∂ t ϕ, à la résolution d'un problème spectral dans l'espace des mesures positives : chercher (H, Q) un couple valeur/vecteur propres associé à

l'opérateur Q → (v • p -1) Q + V M ′ Q ′ dv ′ . On obtient une équation de Hamilton-Jacobi ∂ t ϕ + H (∇ x ϕ) = 0. Pour n = 1 et M ≥ δ > 0 sur son support, le vecteur propre Q a une densité et conduit à un hamiltonien H défini par l'équation implicite V M (v) 1 + H (p) -v • p dv = 1.
La positivité de Q garantit que H (p) ≥ µ (p) -1. En dimension supérieure toutefois, et même si M ≥ δ > 0, cette équation peut ne pas avoir de solution H (p) lorsque p devient grand. Cela se manifeste pour le vecteur propre par une concentration de la mesure Q autour des valeurs v qui annulent

1 + H (p) -v • p, ce qui force H (p) = µ (p) -1.
Cette transition entraîne une singularité C 1 du hamiltonien. Nous démontrons la convergence de ϕ ε vers ϕ, où ϕ est solution de viscosité [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF] de l'équation de Hamilton-Jacobi en utilisant la méthode de la fonction test perturbée [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF].

Introduction

We continue the work initiated in [START_REF] Bouin | A Hamilton-Jacobi approach for front propagation in kinetic equations[END_REF]- [START_REF] Bouin | A kinetic eikonal equation[END_REF]. Let M ∈ L 1 (R n ) be a probability density function. We suppose that the support of M , which we denote V , is compact and that 0 belongs to the interior of Conv (V ), the convex hull of V . We denote by |•| the euclidian norm in R n and by • the canonical scalar product. For p ∈ R n , we define

µ (p) := max {v • p | v ∈ Conv (V )} , (1)
Argµ (p) := {v ∈ Conv (V ) | v • p = µ (p)} and Sing (M ) := p ∈ R n , V M(v)
µ(p)-v•p dv ≤ 1 . We focus on the motion dynamics in R n of particles given by the following piecewise deterministic Markov process: a particle moves successively in straight lines with velocity v, chosen randomly with probability distribution M (v ′ ) dv ′ . At random exponential times (with parameter 1), the particle changes its velocity, choosing randomly a new velocity with distribution M (v ′ ) dv ′ . The Chapman-Kolmogorov forward equation associated to the probability density function f (t, x, v) of this process is given by:

∂ t f + v • ∇ x f = M ρ -f, (t, x, v) ∈ R + × R n × V, (2)
where

ρ (t, x) = V f (t, x, v) dv.
In order to investigate large deviation principles for the process, one can study the large scale hyperbolic limit (t, x) → t ε , x ε with ε > 0. In this scale, the kinetic equation ( 2) reads:

∂ t f ε + v • ∇ x f ε = 1 ε (M ρ ε -f ε ) , (t, x, v) ∈ R + × R n × V. (3)
Then, we perform the following Hopf-Cole transformation:

f ε (t, x, v) = M (v) e -ϕ ε (t,x,v) ε
, where we expect the potential ϕ ε to become independent of v as ε → 0. Such techniques have already been studied for a more general case of Markov process with a finite discrete set of states in [START_REF] Bressloff | On the Hamiltonian structure of large deviations in stochastic hybrid systems[END_REF] and, from a probabilistic point of view, in [START_REF] Faggionato | Averaging and large deviations principles for fully piecewise deterministic Markov Process and applications to moleular motors[END_REF]. Here, assume that the initial condition is well-prepared, i.e. it does not depend on v and ε: ϕ ε (0, x, v) = ϕ 0 (x). The equation satisfied by ϕ ε reads

∂ t ϕ ε + v • ∇ x ϕ ε = V M (v ′ ) 1 -e ϕ ε -ϕ ′ε ε dv ′ , (t, x, v) ∈ R + × R n × V. (4)
As in [START_REF] Freidlin | Random perturbations of dynamical systems[END_REF], the limit potential satisfy a Hamilton-Jacobi equation. Surprisingly enough, our Hamiltonian may lack C 1 regularity as we will show in Proposition 2.1.

Theorem 1.1 Under the previous assumptions, ϕ ε converges locally uniformly on R + × R n × V toward ϕ, where ϕ does not depend on v. Moreover, ϕ is the viscosity solution of the following Hamilton-Jacobi equation:

∂ t ϕ (t, x) + H (∇ x ϕ (t, x)) = 0, (t, x) ∈ R + × R n , (5)
where the hamiltonian H is given as follows: if p ∈ Sing (M ), then H (p) = µ (p) -1. Else, H (p) is uniquely determined by the following formula:

V M (v) 1 + H (p) -v • p dv = 1. ( 6
)
2 Identification of the hamiltonian In order to identify the limit ε → 0 of the equation ( 4) we perform the formal WKB expansion: ϕ ε (t, x, v) = ϕ (t, x) + εη (t, x, v) , where ϕ and η are to be determined. Plugging this ansatz into the kinetic formulation (4), we get, taking the formal limit ε → 0:

∂ t ϕ + v • ∇ x ϕ = 1 - V M (v ′ ) e η-η ′ dv ′ .
Let us write p = ∇ x ϕ and H = -∂ t ϕ. The equation for Q = e -η is the following spectral problem:

HQ = (v • p -1) Q + V M (v ′ ) Q (v ′ ) dv ′ . The positivity of Q yields H ≥ v • p -1 for all v ∈ V hence H ≥ µ (p) -1. Suppose H > µ (p) -1. Then, 1 + H -vp > 0 for all v ∈ V and Q (v) = V M (v ′ ) Q (v ′ ) dv ′ 1 + H -v • p . Integrating against
M with respect to v, we obtain the following problem: find

H such that V M(v) 1+H-v•p dv = 1. If p ∈ Sing (M ) c , by
monotonicity, such H exists and is unique. Equation ( 6), however, does not have a solution for p ∈ Sing (M ), so we necessarily have H = µ (p) -1. Then, a possible solution of the spectral problem is the positive measure

Q = dv µ(p)-v•p +α (p) δ w where α (p) = 1-V M(v)
µ(p)-vp dv ≥ 0 and δ w is the Dirac measure centered in w ∈ Argµ (p)∩V . Here is an example where Sing (M ) = ∅:

Example 1 Let n > 1 and M = ω -1 n .1 B(0,1)
where ω n is the Lebesgue measure of the n-dimensional unit ball. Then,

Sing (M ) = B 0, n n-1 c . Indeed, for p = |p| • e 1 , we have µ (p) = |p| and v • p = |p| v 1 hence V M (v) µ (p) -v • p dv = 1 |p| ω n B(0,1) 1 1 -v 1 dv = ω n-1 |p| ω n 1 -1 1 -v 2 1 n-1 2 1 -v 1 dv 1 = 1 |p| × n n -1
. Proof Let us first notice that µ is positively 1-homogeneous. Moreover, it is convex since it is a supremum of linear functions.

By rotational invariance, we conclude that

(i) Let p, q ∈ Sing (M ) c with p = q. Since µ is convex, we have for all τ ∈ [0, 1]

I (τ ) := V M (v) µ (p) -v • p + τ (µ (q) -µ (p) -v • (q -p)) dv ≤ V M (v) µ ((1 -τ ) p + τ q) -v • ((1 -τ ) p + τ q) dv.
Moreover, I (0) , I (1) > 1 and I is differentiable on [0, 1] with

∂ τ I (τ ) = V M (v) (µ (p) -v • p + τ (µ (q) -µ (p) -v • (q -p))) 2 (µ (p) -µ (q) -v • (p -q)) dv.
It is clear that the sign of ∂ τ I does not change hence I (τ ) > 1, which proves (i).

(ii) We refer to [START_REF] Bouin | A kinetic eikonal equation[END_REF] to prove that H is C 2 and strictly convex on Sing (M ) c and that

V M (v) (1 + H (q) -v • q) 2 (∇H (q) -v) dv = 0, ∀q ∈ Sing (M ) c . (7)
In particular, ∇H (q) ∈ Conv (V ) for all q ∈ Sing (M ) c . It is easy to see that H is continuous in the interior of Sing (M ). To show continuity of H on ∂Sing (M ), let (p m ) m converge to p ∈ ∂Sing (M ) ⊂ Sing (M ). If we can extract a subsequence (p m l ) l ⊂ Sing (M ), then

H (p m l ) = µ (p m l ) -1 -→ l→∞ µ (p) -1 = H (p). If not, then p m ∈ Sing (M ) c for m large enough and 1 = V M(v) 1+H(pm)-v•pm dv < V M(v)
µ(pm)-v•pm dv. Taking the limit, we get by dominated convergence

1 = V lim m→∞ M(v) 1+H(pm)-v•pm dv ≤ V M(v) µ(p)-v•pm dv ≤ 1 hence H (p m ) -→ m→∞ µ (p) -1 = H (p).
We now show that H is convex by proving that it is a maximum of convex functions:

H (p) = max (sup {∇H (q) • (p -q) + H (q) | q ∈ Sing (M ) c } , µ (p) -1) , ∀p ∈ R n . (8)
In Sing (M ) c , (8) holds by convexity of H and H (p) > µ (p) -1. Let p ∈ Sing (M ) and q ∈ Sing (M ) c . By convexity of Sing (M ) c ∋ 0, there exists a unique λ ∈ (0, 1] such that λp ∈ ∂Sing (M ). For all τ ∈ [0, 1], we set ω 1 (τ ) := µ (τ p) -1 = τ µ (p) -1 and ω 2 (τ ) := ∇H (q) • (τ pq) + H (q) . By continuity of H, µ (λp) -1 = H (λp) ≥ ∇H (q) (λpq) + H (q) hence ω 1 (λ) ≥ ω 2 (λ). Moreover, ω 1 and ω 2 are both differentiable and [START_REF] Bouin | A Hamilton-Jacobi approach for front propagation in kinetic equations[END_REF], which ends the proof of (ii).

∂ τ ω 1 (τ ) = µ (p) ≥ ∇H (q) • p = ∂ τ ω 2 (τ ) since ∇H (q) ∈ Conv (V ). Hence, ω 1 (1) ≥ ω 2
(iii) Suppose Sing (M ) = ∅ and H is C 1 . Since H + 1 = µ is positive homogeneous of degree 1 on Sing (M ) and since λp ∈ Sing (M ) for all λ ≥ 1 and p ∈ Sing (M ), we know that ∇H (p) • p = H (p) + 1 = µ (p) for all p ∈ Sing (M ) ⊂ Sing (M ) hence p • (∇H (p)v) ≥ 0, for all v ∈ V , the inequality being strict on a neighborhood of 0. Then,

p • V M (v) (1 + H (p) -v • p) 2 (∇H (p) -v) dv > 0, ∀p ∈ ∂Sing (M ) . ( 9
)
By continuity, equations ( 7) and ( 9) are contradictory.

3 Proof of Theorem 1.1

Let ϕ 0 ∈ W 1,∞ (R n ).
We refer to Proposition 2.1 in [START_REF] Bouin | A kinetic eikonal equation[END_REF] to prove that the Cauchy Problem (4) with initial condition ϕ 0 has a unique solution ϕ ε ∈ W 1,∞ which is locally (in t) uniformly (in ε, x and v) bounded in norm W 1,∞ . In particular, let us mention that

0 ≤ ϕ ε (t, •, •) ≤ ϕ 0 ∞ , ∇ v ϕ ε (t, •, •) ∞ ≤ ∇ x ϕ 0 ∞ . (10)
Using the Arzelá-Ascoli theorem, we extract a locally uniformly converging subsequence. We denote by ϕ the limit. The function ϕ does not depend on v since V M (v) e ϕ ε -ϕ ′ε ε dv is uniformly bounded on [0, T ]× R n × V for all T > 0. We use the perturbed test function method [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] to show that ϕ is a viscosity solution of (5). Theorem 1.1 will follow by uniqueness of the solution [START_REF] Crandall | Some properties of viscosity solutions of Hamilton-Jacobi equations[END_REF].

Subsolution procedure

Let ψ ∈ C 1 (R + × R n ) be a test function such that ϕψ has a local strict maximum at t 0 , x 0 . We want to show that ψ is a subsolution of (5). If ∇ x ψ t 0 , x 0 ∈ Sing (M ) c , then we refer to [START_REF] Bouin | A kinetic eikonal equation[END_REF].

Suppose now that ∇ x ψ t 0 , x 0 ∈ Sing (M ). Let w ∈ Argµ ∇ x ψ t 0 , x 0 ∩ V . Then, w • ∇ x ψ t 0 , x 0 = µ ∇ x ψ t 0 , x 0 . The uniform convergence of ϕ ε toward ϕ ensures that the function (t, x) → ϕ ε (t, x, w)ψ (t, x) has a local maximum at a point (t ε , x ε ) satisfying (t ε , x ε ) → t 0 , x 0 , as ε → 0. We then have:

∂ t ψ (t ε , x ε ) + w • ∇ x ψ (t ε , x ε ) = ∂ t ϕ ε (t ε , x ε ) + w • ∇ x ϕ ε (t ε , x ε ) = 1 - V M (v ′ ) e ϕ ε (t ε ,x ε ,w)-ϕ ε (t ε ,x ε ,v ′ ) ε dv ′ ≤ 1.
Passing to the limit ε → 0, we get ∂ t ψ t 0 , x 0 + µ ∇ x ψ t 0 , x 0 ≤ 1. We conclude that ϕ is a viscosity subsolution of (5).

Supersolution procedure

Let ψ ∈ C 1 (R + × R n ) be a test function such that ϕψ has a local strict minimum at t 0 , x 0 . We want to show that ψ is a supersolution of [START_REF] Crandall | Some properties of viscosity solutions of Hamilton-Jacobi equations[END_REF]. If ∇ x ψ t 0 , x 0 ∈ Sing (M )

c , then we refer to [START_REF] Bouin | A kinetic eikonal equation[END_REF].

Suppose now that ∇ x ψ t 0 , x 0 ∈ Sing (M ). Then, ∇ x ψ t 0 , x 0 = 0 because 0 ∈ Sing (M ) c . We suppose without loss of generality that the minimum of ϕ-ψ is global and that ϕ t 0 , x 0 -ψ t 0 , x 0 = 0. Let ψ ε := ψ-C tt 0 2 +εη with C > 0 yet to be determined and

η (v) := ln µ ∇ x ψ t 0 , x 0 -v • ∇ x ψ t 0 , x 0 .
Then, η is a continuous function on D (η) = V \ Argµ ∇ x ψ t 0 , x 0 and, for all w ∈ Argµ ∇ x ψ t 0 , x 0 ∩ V , we have lim v→w η (v) = -∞. Moreover, η is bounded from below on all compact sets yielding the uniform convergence

ψ ε → ψ on all compact sets of D (η). Finally, V M (v ′ ) e -η(v ′ ) dv ′ ≤ 1 since ∇ x ψ t 0 , x 0 ∈ Sing (M ).
The function ϕψ -C tt 0 2 has a global strict minimum at t 0 , x 0 . The first inequality (10) ensures that the function ϕ εψ ε has a local minimum at a point (t ε , x ε , v ε ) ∈ R + × R n × D (η). As V compact, we can extract a subsequence (v ε ) ε , without relabelling, such that v ε → v 0 , as ε → 0.

If v 0 ∈ V \ Argµ (p), then there exists a compact A ⊂ D (η) such that v 0 ∈ A and the uniform convergence of ψ ε towards ψ on A guarantees that (t ε , x ε ) → t 0 , x 0 , as ε → 0. We then get at point (t ε , x ε , v ε ),

∂ t ψ -2C t ε -t 0 + v ε • ∇ x ψ = ∂ t ψ ε + v ε • ∇ x ψ ε = ∂ t ϕ ε + v ε • ∇ x ϕ ε = 1 - V M ′ e ϕ ε -ϕ ′ε ε dv ′ ≥ 1 - V M (v ′ ) e η(v ε )-η(v ′ ) dv ′ .
We take the limit ε → 0:

∂ t ψ t 0 , x 0 + v 0 • ∇ x ψ t 0 , x 0 ≥ 1 -e η(v 0 ) V M (v ′ ) e -η(v ′ ) dv ′ ≥ 1 -e η(v 0 ) . By construction, for all v, v ′ ∈ D (η), we have e η(v) -e η(v ′ ) = (v ′ -v) • ∇ x ψ t 0 , x 0 hence, for all v ∈ D (η),we have ∂ t ψ t 0 , x 0 + v • ∇ x ψ t 0 , x 0 ≥ 1 -e η(v) . Let w ∈ V ∩ Argµ ∇ x ψ t 0 , x 0 . Since Argµ ∇ x ψ t 0 , x 0 is a null-set, V is dense in Argµ ∇ x ψ t 0 , x 0 . Taking the limit v → w, we get: ∂ t ψ t 0 , x 0 + µ ∇ x ψ t 0 , x 0 ≥ 1. If v 0 ∈ V ∩ Argµ (p)
, we still have (t ε , x ε ) -→ ε→0 t 0 , x 0 thanks to the following lemma:

Lemma 3.1 For C = 4 ϕ 0 ∞ , we have lim ε→0 εη (v ε ) = 0.
Proof of Lemma 3.1 We have ϕ ε (t, x, v)ϕ (t, x) ≥ -2 ϕ 0 ∞ by (10) and ϕ (t, x)ψ (t, x) ≥ 0 hence

ϕ ε (t, x, v) -ψ ε (t, x, v) ≥ -2 ϕ 0 ∞ + C t -t 0 2 -εη (v) , ∀ε > 0.
Moreover,

ϕ ε t 0 , x 0 , v -ψ ε t 0 , x 0 , v = ϕ ε t 0 , x 0 , v -ϕ t 0 , x 0 -εη (v) ≤ 2 ϕ 0 ∞ -εη (v) . Since C = 4 ϕ 0 ∞ , we have ϕ ε (t, x, v) -ψ ε (t, x, v) > ϕ ε t 0 , x 0
, vψ ε t 0 , x 0 , v for all t > t 0 + 1 and, thus, the minimum of ϕ εψ ε cannot be attained for t > t 0 + 1 hence t ε ≤ t 0 + 1 for all ε > 0. At point (t ε , x ε , v ε ) we have:

∇ v ϕ ε (t ε , x ε , v ε ) = ∇ v ψ ε (t ε , x ε , v ε ) = ε∇ v η (v ε ) = - ε∇ x ψ t 0 , x 0 µ (∇ x ψ (t 0 , x 0 )) -v ε • ∇ x ψ (t 0 , x 0 )
.

The second estimation (10) yields

∇ v ϕ ε (t ε , •, •) ∞ ≤ t ε ∇ x ϕ 0 ∞ ≤ t 0 + 1 ∇ x ϕ 0 ∞ hence ε (t 0 + 1) ∇ x ϕ 0 ∞
∇ x ψ t 0 , x 0 ≤ µ ∇ x ψ t 0 , x 0v ε • ∇ x ψ t 0 , x 0 , =⇒ εK ≥ εη (v ε ) ≥ εln ε (t 0 + 1) ∇ x ϕ 0 ∞ ∇ x ψ t 0 , x 0 , and εη (v ε ) → 0 as ε → 0.

Thanks to Lemma 3.1, the function (t, x) → ψ ε (t, x, v ε ) = ψ (t, x) -4 ϕ 0 ∞ tt 0 2 + εη (v ε ) converges uniformly towards (t, x) → ψ (t, x) -4 ϕ 0 ∞ tt 0 2 and has a local minimum at (t ε , x ε ) satisfying (t ε , x ε ) → t 0 , x 0 , as ε → 0. At point (t ε , x ε , v ε ), we have:

∂ t ψ ε + v ε • ∇ x ψ ε = ∂ t ϕ ε + v ε • ∇ x ϕ ε = 1 - V M (v ′ ) e ϕ ε (t ε ,x ε ,v ε )-ϕ ε (t ε ,x ε ,v ′ ) ε dv ′ .
The minimal property of (t ε , x ε , v ε ) implies at this point:

∂ t ψ (t ε , x ε ) -8 ϕ 0 ∞ t ε -t 0 + v ε • ∇ x ψ (t ε , x ε ) = ∂ t ψ ε + v ε • ∇ x ψ ε ≥ 1 - V M (v ′ ) e η(v ε )-η(v ′ ) dv ′ ≥ 1 -e η(v ε ) .
Passing to the limit ε → 0, we get ∂ t ψ t 0 , x 0 +µ ∇ x ψ t 0 , x 0 ≥ 1. We conclude that ϕ is a viscosity supersolution of (5).

1 c.

 1 Sing (M ) = B 0, n n-The Figure1gives illustrations of the hamiltonian and µ as functions of the radius of p, in the cases n = 1 and n = 3. In the cases n = 3 we can see the C 1 singularity where |p| =3 2 .

Figure 1 -Proposition 2 . 1

 121 Figure 1 -Blue plain lines : Hamiltonian for n = 1, 3 and M = ω -1 n .1 B(0,1) . Black dotted lines : |p| → µ (p) -1. Lignes pleines bleues : Hamiltonien pour n = 1, 3 et M = ω -1 n .1 B(0,1) . Lignes noires en pointillés : |p| → µ (p) -1.

Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 639638).

The author also wishes to thank Vincent Calvez and Julien Vovelle for their kind help.