Amine Ould-Hamouda

Antonio Iazzolino

Hiroko Tokoro

Shin-Ichi Ohkoshi

Eric Freysz
email: eric.freysz@u-bordeaux.fr

Second harmonic and terahertz generation in a Prussian-blue-analog

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Coordination polymers have recently lot of attention due to their interesting nonlinear optical properties. [1][2][3] Among these compounds, it has been shown that Prussian-blue-analogues offer many prospects in nonlinear optics. [4][5][6][7][8] For instance, the cyano-bridged coordination polymer of RbxMn[Fe(CN)6](x+2)/3 z H2O switches from a centrosymmetric to a non-centrosymmetric and piezoelectric structure when one increases the Rb content above x=0.7. [6] The non-centrosymmetric structure of the compound gives rise to second-order nonlinear optical properties that were revealed by second-harmonic generation (SHG) experiments. These SHG experiments were performed using Nd:YAG nanosecond laser pulses centered at 1.064 µm. Above x=0.7, it was demonstrated that the second-order nonlinear coefficient steadily increases with the Rb content. [6] For x=0.94, as the temperature is increased from T=240 K to T=320 K, a phase transition between a tetragonal low temperature (LT) structure to a cubic noncentrosymmetric high temperature (HT) structure is recorded. [START_REF] Ōtsu | Chemical, Biological, and Nanophotonic Technologies for Nano-Optical Devices and Systems[END_REF] During this phase transition, which is associated to a broad thermal hysteresis loop spanning in between 250 K and 310 K, the color of the sample turns from light brown to dark brown. Since both phases are non-centrosymmetric, SHG signal can be recorded. It is interesting to mention that, if the Kleinman's symmetry conditions hold, this material exhibits the same non-null nonlinear coefficient in both the HT and LT phase. [START_REF] Boyd | Nonlinear Optics[END_REF] This makes it possible to directly compare the value the second-order nonlinear properties of this compound in the HT and LT phase. The modulation of SHG signal recorded during the structural change of this compound provides an original mean to record the thermal hysteresis loop. [6] However, the recorded modulation of the SHG signal may be either due to the modification of the amplitude of or of the index of refraction in the LT and HT phase. [START_REF] Ohkoshi | [END_REF] In the HT phase, exciting the sample with nanosecond pulses centered at =1.064 µm, was found to be ~0.032 pm.V -1 . However, due to the high absorption of the SHG signal at =532 nm, such measurements have not been performed in the LT phase. Hence, it is difficult to directly compare the second-order nonlinear optical properties of this material in the LT and HT phases. Hereafter, we will demonstrate that, thanks to a procedure we will describe, we have been able to determine of this compound in the LT and HT phases over a spectral range ranging from 1.1 µm to 1.8 µm. In the LT phase and at 1.1 µm, the measured second-order optical susceptibility increases about twoorders of magnitude reaching a value =2.7±0.3 pm.V -1 . We will show that this enhancement is related to an absorption band of this material whose maximum shifts from 410 nm to 460 nm as the compound switches from the HT to the LT phase. Within the hysteresis loop, we illustrate how one can optically modulate the nonlinear optical properties, namely its , by photo-switching it from the low temperature to the high temperature phase. We have also evaluated the capability of this material to generate terahertz (THz) waves. Indeed, like ZnTe a crystal widely used to generate THz pulses, [12][13][14] this material belongs to the space group. Accordingly, one may wonder if THz generation cannot be recorded in Rb0.94Mn[Fe(CN)6]0.98.0.3H2O. Performing THz time domain spectroscopy (TDS), we have first measured the absorption and the index of refraction of this compound in the THz spectral range. Our data indicates this material exhibits a transparency window ranging from 2.5 to 7 THz. Then we have used it to generate THz radiations. We have easily been able to record a non-phase-matched THz signal. In the THz frequency range, the value of the second-order optical susceptibility of this material is found to be very large and comparable with ZnTe crystal.

Results and Discussion

SHG

The evolution of the SHG signal in the LT and HT phases versus the power of the excitation pulse centered at 1200 nm is displayed in Fig. 1. In both the LT and HT phases, the intensity of the SHG signal exhibits quadratic power dependence. Moreover, regardless of the excitation intensity, the spectrum of the SHG signal is found to be in very good agreement with the spectrum of the excitation pulses. These measurements were performed repeatedly, tuning the central wavelength of the exciting pulse from 1.8 µm to 1.1 µm. At each wavelength, we found the SHG signal exhibits quadratic power dependence in both the LT and HT phase. To determine the secondorder optical susceptibility of our sample, we calibrate the SHG against a 1 mm thick ZnSe polycrystalline optical window whose microcrystallites size distribution is centered around 50-70 μm. This crystal also adopts a structure and its second-order optical susceptibility is well known ( = 53.8 pm.V -1 ). This window was inserted almost exactly at the position of our sample. To define the position of the sample, we inserted two small irises at the back and in the front of the lenses L2 and L3, respectively. These two irises fix precisely the position of the sample. [7] Then the sample was removed and the ZnSe window was installed in such a way that the excitation reflected by its front surface is passing through the center of these two irises. Moreover, to ensure we detected only the SHG generated at the surface of a ZnSe window, a razor blade placed on the top surface of the ZnSe window was used to block the SHG signal reflected by the back surface of the latter. Hence, for a fixed intensity of the exciting pulse and whatever its central wavelength, we were able to measure and to quantitatively compare the SHG signals generated by our sample and by the ZnSe window. In Figure 2, we have plotted the evolution of the ratio of the SHG signal recorded in our sample against the SHG signal recorded in ZnSe. These measurements have been performed fixing the intensity of the excitation pulse to ~5 µJ.mm -2 . The SHG signal in the HT phase is almost always larger than the SHG in the LT phase. In the HT phase, one can notice the occurrence of a sharp peak of the SHG signal centered at1.47 µm. In the LT phase the amplitude of this peak is decreased and it position is shifted to 1.52 µm. We have repeated this experiment many times and we always recorded the occurrence of these SHG peaks. The SHG signal generated by a microcrystal of size r of our sample writes: [15] (1)

Where I(), ,  k are the intensity of the exciting pulse, the second-order optical susceptibility of our sample, the absorption coefficient of SHG signal and the phase mismatch, respectively. For SHG the phase mismatch writes k=2(n-n)/cwhere n and naretheindices of the crystallite at fundamental  and second harmonic  pulsations, respectively. Equation 1 holds when r</k and r<<1. From an ensemble of micro crystallites randomly oriented within a single layer, the SHG signal writes NRbMnFe Iv(2,r) where NRbMnFe is the number of crystallite per unit surface. [16] Hereafter, we will consider the sample can be modeled as a set of layers of crystallites. Accordingly it can be shown that the SHG signal reflected by an infinite number of layers of crystallites writes (see supporting information):

(2) With * (3)
where R(2), T() and T(2) are the reflection of the SHG frequency and the transmission of the fundamental and SHG frequency by crystallite interfaces, respectively. The latter expression is slightly different from the one we previously derived. [7] It now takes into account the multiple reflection of the SHG signal by the different interfaces. Equations 1-3 stress that the SHG signal strongly depends on the indices and absorption of the studied material at the second harmonic wavelength (). The evolution of the absorption and refractive index of this material in the 0.3 µm-0.8 µm spectral range has been previously reported. [START_REF] Ohkoshi | [END_REF] With a polynomial fit we extended these data in the 1.1 µm to 1.8 µm spectral range. This approximation is valid if the sample is weakly absorbing in this spectral range. To ensure this latter approximation holds, we performed a FTIR spectrum of a thin pellet of our sample. This measurement indicates the absorption is indeed weak in the latter spectral range. Hence, we were able to compute the evolution of the coherence length Lc=/k for our sample in both the LT and HT phase. The evolution of  and Lc are displayed in Fig. 3a-b. In agreement with Equation 1, the size of the crystallites r is always smaller than the coherence length. Figure 3b indicates that in the LT phase the sample exhibits an intense absorption peak centered at ~460 nm, likely due to the intervalence transfer band from the Mn III -NC-Fe II to Mn II -NC-Fe III and Jahn-Teller distorted Mn III . Knowing the size, the index of refraction and the absorption of the crystallite and considering the second-order susceptibility of our sample is similar in the LT and HT phases, in Fig. 4 we have plotted the theoretical evolution of the ) and ), as one tunes the central frequency of the exciting laser pulse. In the HT phase, the coherence length is large compared to the size r of the crystallites, hence HT) is ~1. Similarly HT), which replicates the SHG reflection coefficient of the powder, is ~1.7 and it decreases as one tunes the excitation wavelength towards 1 µm. As displayed in Figure 3, in the LT phase the coherence length is small. As one decreases the excitation wavelength, the latter further decreases and it becomes comparable to the size r of the crystallite. Hence, LT) <1 and it starts to decrease above ~1.6 µm. The larger absorption of the sample in the LT phase also impacts the parameter LT) which, in the studied spectral range, is always smaller than 0.2. Furthermore, as one tunes the excitation wavelength below 1.6 µm, it drops abruptly and it becomes smaller than 10 -4 at =1.05 µm As already mentioned, we calibrated the SHG signal produced by our sample against a ZnSe polycrystalline window. The SHG signal produced in reflection by the air-ZnSe interface writes: [START_REF] Boyd | Nonlinear Optics[END_REF] (4)

Where I(), , Lc, NZnSe are the intensity of the exciting pulse, the second-order optical susceptibility of ZnSe, the coherence length and the number of crystallite per unit surface, respectively. For SHG in reflection, the phase mismatch writes and where n and nareindices of the ZnSe at fundamental () and second harmonic wavelength (), respectively. To compute the coherence length Lc for SHG in reflection in ZnSe, we used the dispersion formula proposed by Marple. [17] Its evolution versus the excitation wavelength is plotted in Fig. 5a. It is worth mentioning that in the studied spectral range, Lc is very small and about 20.

Accordingly, the attenuation of the SHG signal over such a small length is negligible. Figure 5a displays the surface SHG signal produced considering the intensity of the exciting pulse is constant and the dispersion of the susceptibility is negligible. [18] The coherence length and consequently the SHG signal are found to decrease as we tuned the central wavelength of the exciting pulse towards 1 µm. Our experimental set-up makes it possible to directly compare the SHG signal produced by either our sample or the ZnSe window. According to Equation 2 and 4, it can be easily shown that:

(

The latter equation indicates that keeping the same experimental conditions, one can evaluate the optical susceptibility measuring the SHG signal produced by our sample and ZnSe window provided that we know , Lc, r, , , NZnSe/NRbMnFe. Hereafter, we will consider that NZnSe/NRbMnFe~1. Note that in comparison to ZnSe, the concentration of our very slightly compacted RbMnFe sample is likely less. Hence according to Equation 5, we are underestimating the value of the . Figure 6 displays the evolution of versus the wavelength in the LT and HT phases. One can notice that the increase of the SHG signal recorded at~1.5 µm in the LT and HT phases do not noticeably impact the spectral dispersion of . In the HT phase and over the scanned spectral range we measured ~0.03±0.01 pm.V -1 . This value is very close to the one previously published for this sample and it indicates that two-photon absorption which has been considered by previous authors in second and third harmonic generation experiments are likely negligible in our sample. [7,19,20] However in the LT phase, presents a clear dispersion. Below 1.2 µm, it rapidly increases, and at 1.1 µm it is ~2.7±0.3 pm.V -1 . This increase is directly linked to the increase of the absorption of the SHG signal by the sample in the LT phase (dotted blue line in Fig. 6) and it is well known to give rise to resonant enhancement of the second-order nonlinear optical susceptibility [START_REF] Boyd | Nonlinear Optics[END_REF] . In both the LT and HT phase the sample exhibits two strong absorption bands centered at 270 µm and 54 µm, respectively. These peaks, likely related to far infrared lattice modes of the sample, induce a large absorption of the THz signal. However one can notice that they are spectral windows where the transmission of the signal remains large. For instance when >400 µm and for ~100 µm the absorption of our sample is strongly reduced. This indicates that in these spectral domains, a THz signal generated within our 300 µm thick sample may be efficiently transmitted. One can also notice that in the far infrared (>500 µm) the index of refraction the sample can be as large as 5. It rapidly decreases as the wavelength decreases and it is ~1.9 below =200 µm. Over all the spectral range displayed in Figure 8b, the phase matching condition for THz generation is not satisfied. It requires that the index nTHz of the sample in the THz spectral range equals the group index of the exciting pulse. The latter writes ng=n-dn/d where n and  are the index and the wavelength in the optical range. For our sample, the group index at the excitation wavelength in the LT (ng~1.42) and HT (ng~1.32) phases is always smaller than the index nTHz of the sample the THz spectral range. However, it is important to indicate that the coherence length Lc=THz/[2(nTHz-ng)] for THz generation is always larger than the size (~2.1 µm) of the crystallites used to prepare the pellet. The THz signal produced by exciting the sample with the 160 fs pulses centered at =1.1 µm and detected by the Golay's cell is displayed in Figure 10. THz generation is a second-order nonlinear process. Hence, the THz signal displays quadratic evolution versus the excitation intensity. In Figure 9, one can also notice that the THz signal in the LT phase is slightly larger than in the HT phase. When the energy of the excitation pulse is ~2.2 µJ, the calibration of our Golay's cell indicated that the energy of the THz pulse is ~0.11 nJ. This corresponds to a conversion efficiency of 5.10 -5 . This value has to be compared to conversion efficiency (~10 -5 ) achieved under the phase matching condition when one excite a 1 mm thick <110> cut ZnTe crystal with 120 fs pulses having a similar peak power centered at =800 nm. Hence, this experiment indicates that the nonlinear coefficient of this material in the THz range in both the LT and HT phases is, at least, larger than the nonlinear coefficient of ZnTe crystal ( >100 pm/V). This latter value is very large compared to the nonlinear coefficient we measured for SHG. Here again, this enhancement may results from the very large absorption of our sample in the THz spectral range.

THz spectroscopy and THz generation

Optical modulation of the second-order nonlinear optical properties.

One of the interesting properties of the cyano-bridged coordination polymer RbxMn[Fe(CN)6](x+2)/3 z H2O is that it can be photo-switched from the LT phase to the HT phase within the thermal hysteresis loop. We have recently shown this makes it possible to optically photoswitch its third-order nonlinear optical properties. [7] Since the secondorder nonlinear properties of these compounds are different in the LT and HT phase, it should be also possible to record a modulation of their second-order nonlinear properties as the sample is switched from the LT phase to the HT phase. To demonstrate this phenomenon we performed an experiment very similar to the one we realized to demonstrate the photo-switching of the third-order nonlinear optical properties of Rb0.94Mn[Fe(CN)6]0.98.0.3H2O. [7] The sample was initially set in the LT phase at 285 K. The SHG signal was then recorded, exciting the sample at λ=1400 nm by varying the energy of the exciting pulse in between 0.1 µJ and 2.0 µJ. Afterwards, we increase the energy of the pulse to ~10 µJ. After several seconds, when a sequence of these more powerful pulses is used, the color of the area excited by these pulses turned from dark brown to light brown (inset Fig. 10). This indicates the sample, likely heated by the laser beam, is switched from the LT phase to the HT phase. To confirm that the sample was indeed in the HT phase, we reduced the energy of the laser pulses from 10 µJ to 0.1-1.5 µJ and recorded the SHG signals for these lower energies. These SHG signal were then compared to the SHG signals we previously recorded under the same experimental condition when the sample was set in the HT phase. Figure 10, indicates the amplitude of these two SHG signals recorded at T=285 K are almost similar. This indicates that indeed we have switched the sample from the LT to the HT phase. Then, cooling the sample with a nitrogen jet, we could bring it back in the LT phase where the sample recovered its dark brown color. The amplitude of the SHG signal is then almost identical to the one we recorded at the beginning of this experiment. Repeating this experiment many times, we could reproduce these results. This experimental proof of concept indicates that one is indeed able to photo-switch back and forth the second-order nonlinear optical properties of this sample from the LT phase to the HT phase.

Conclusions

The ensemble of results presented in this manuscript indicates that Rb0.94Mn[Fe(CN)6]0.98.0.3H2O cyano-bridged coordination polymer exhibits original linear and second-order nonlinear optical properties. Within the broad thermal hysteresis loop of this compound around room temperature, we have characterized its linear properties in the THz frequency range using THz-TDS and determined its second-order nonlinear properties in the near IR and THz ranges performing SHG and THz generation, respectively. We have shown that, in the THz frequency range this compound as a high index of refraction that decreases almost steadily from ~5 to ~2 as the wavelength decreases from 700 µm down to 50 µm. We have shown it also exhibits a high and broad absorption peak centered at ~300 µm and a useful transparency range at higher and lower wavelengths. In the HT phase and in the near IR spectral range the amplitude of the second-order susceptibility is low and ~0.03±0.01 pm.V -1 . Its amplitude increases in the LT phase and it is ~2.7±0.3 pm.V -1 at 1.1 µm. This latter phenomenon is associated to the resonant enhancement of the close to a broad and intense absorption peak centered at ~460 nm, likely due to the intervalence transfer band from the Mn III -NC-Fe II to Mn II -NC-Fe III and Jahn-Teller distorted Mn III . We have also demonstrated that in the THz frequency range this compound exhibits a high second-order optical nonlinearity comparable to ZnTe that makes possible to record THz generation when this material is excited with femtosecond laser pulses. Finally, we have also evidenced that one can photo-switch the linear and nonlinear optical properties of this compound within its thermal hysteresis loop. This opens the possibility to modulate the amplitude of the nonlinear properties of this material.

Experimental Section

Sample preparation and characterization

The preparation and characterization of Rb0.94Mn[Fe(CN)6]0.98.0.3H2O has previously been described elsewhere. [6] The sample we used consists in powder of cubic micro-crystals with a size of 2.1±0.7 µm. For the SHG measurements, the powder was inserted in between two microscope slides, gently pressed, and heated at 330 K. For the THz spectroscopy and THz generation, we pressed the powder using a pressure of 4 kN to prepare a 300 µm thick pellet. After preparation, the pellet was also heated at 330 K. As displayed in Fig. 11, when the temperature is decreased from 330 K to 200 K, the color of our sample changes from light brown to dark brown. As the temperature of the sample is decreased, the sample experiences a structural phase transition from the HT phase (cubic ) to the LT phase (tetragonal ). Figure 11 also shows that this structural transition is associated to a broad thermal hysteresis loop which extends in between T~310 K and T~210 K. The change of color of the sample is associated to the shift of a broad absorption band of the compound in the visible spectral range. In the HT phase, a broad and intense absorption peak centered at ~460 nm, likely due to the intervalence transfer band from the Mn III -NC-Fe II to Mn II -NC-Fe III and Jahn-Teller distorted Mn III is recorded. [START_REF] Ohkoshi | [END_REF] In the LT phase, this absorption peak is shifted toward ~410 nm and its amplitude decreases. The latter absorption has been assigned to the ligand-to-metal charge transfer (LMCT) transition of [Fe III (CN)6]. [START_REF] Ohkoshi | [END_REF] The index of refraction is also sensible to this structural phase change. In the near infrared spectral range, the index of refraction which is ~1.32 in the HT phase increases to ~1.42 in the LT phase. [7,[START_REF] Ohkoshi | [END_REF] As shown in this manuscript, the changes of the absorption coefficient and the index of refraction impact both SHG and THz generation.

Experimental set-up

The experimental set-up we used to perform our SHG experiments has been presented elsewhere. [7] The sample was excited by 160 femtosecond (fs) pulses yielded by an optical parametric amplifier (OPA), which has a central wavelength tunable from 1100-2400 nm. At the exit of the OPA, an infrared (IR) glass filter is inserted. It transmits the IR pulses produced by the OPA and blocks the visible light produced within this system. The laser pulse is then slightly focused by a lens L1 onto the sample over a beam waist of 235 µm, and its maximal energy is 2 µJ. Due to the thickness of our sample, no SHG signal was detected in transmission. The SHG signal generated at the surface of the sample was collimated by the lens L2 and focused by the lens L3 on the entrance slit of a spectrometer in front of which we inserted a colored glass filter F2. The latter transmits the visible light and blocks the infrared excitation pulse. The spectrum transmitted by the spectrometer was recorded by a thermally cooled charge coupled detector (CCD).

The THz set-up we used to perform the THz time domain spectroscopy (TDS) has been described elsewhere. [21] In short, a THz beam consisting in a singlecycle THz wave is emitted in air ionized by the fundamental and the second harmonic of 50 fs pulses centered at =800 nm. After transmission through the sample, this wave is measured by electro-optical (EO) sampling in a 200 µmthick <110> GaP crystal. This THz detection system records the temporal profile of the electric field of the THz pulses which makes it possible to recover the complex index of refraction of the studied sample on a 0.3-7 THz frequency range. However, this method implies to perform two measurements. We first record a reference waveform ER(t) without the sample and then a signal waveform ES LS,HS with the sample in the LS and HS state, respectively. The Fourier components of the two signals are obtained through a Fourier transform and define the complex transmission function of the sample, equal to the ratio of the output amplitude spectrum to the input of the reference spectrum. The latter is directly related to the complex refractive index of the sample or to the frequency-dependent complex dielectric constant of the sample. It also makes it possible to retrieve the absorption spectrum of the sample. [21,22] For the THz generation experiment, a 300 µm thick pellet was excited by the 160 fs pulses centered at =1.1 µm provided by the OPA. The THz wave was generated without focusing the exciting pulses. It was collected and focused on the detector (a Golay's cell) by two 20 cm focal length, gold-coated, off-axis parabolic mirrors. To prevent any signal due to diffusion of the exciting beam, a silicon wafer and Teflon sheet were inserted in front of this detector. The THz wave generated by the sample was modulated by a chopper and the detector was connected to a lock-in amplifier. We checked that without the sample no signal was detected.

To run the SHG experiment the sample was held in a thermally controlled cold finger in which the temperature is set at T=290 K. To set the sample in the HT phase, it was heated up to 330 K. It was then slowly cooled down to T=290 K. As the sample switches from the LT phase to the HT phase its color turns from dark brown to light brown. To bring the sample back in LT phase, it is cooled for several minutes by means of a nitrogen jet and slowly heated back to T=290 K. During this process, the color of the sample switches from light brown to dark brown.

Figure 1 .

 1 Figure 1. Evolution of the SHG signal versus the excitation intensity. The solid lines are fits considering a square power law of the experimental data recorded in the LT (blue open circles) and HT (red open dots) phases. The inset displays the evolution of the SHG spectra in the HT phase.

Figure 3 :

 3 Figure 3: Evolution of the coherence length (a) and the absorption (b) of the sample in the LT (blue line) and HT (red line) phases

Figure 4 :

 4 Figure 4: Theoretical evolution of (a) ) and (b) ) coefficients of the sample in the LT (blue line) and HT (red line) phases.

Figure 5 :

 5 Figure 5: Theoretical evolution of (a) the SHG signal produced in reflection and (b) the coherence length in ZnSe polycrystalline window.

Figure 6 :

 6 Figure 6: Spectral evolution of the amplitude of in the LT (blue dot) and HT (red dot) phases. The dotted line displays the evolution of the absorption of the SHG signal in the LT (blue line) and HT (red line) phases.

Figure 7 :Figure 8 :

 78 Figure 7: Transmitted THz pulse recorded without the sample (black line) and with the sample in the LT (blue line) and HT (red line) phases.

Figure 9 :

 9 Figure 9: Evolution of the THz signal versus the excitation intensity. The solid lines are fits considering a square power law of the experimental data recorded in the LT (blue open circle) and HT (red open circle) phases.

Figure 10 :

 10 Figure 10 : Solid lines evolution of the SHG signal versus the energy of the exciting pulse. The dots corresponds to the SHG signal before (blue open dots) and after (red open dots) photo-switching from the LT to the HT phase. Cooling the sample in the LT phase, we were able to reproduce this experiment more than ten times without significant evolution of the SHG signal

Figure 11 .

 11 Figure 11. Hysteresis loop of the sample revealed by the MT-T plots. The insets show that in the LT and HT phase the color of the sample is dark brown and light brown, respectively.

1.1 1.2 1.3 1.4 1.5 1.6 0.000 0.002 0.004 0.006 0.008 0.010 0.012 HT LT I RbMnFe I ZnSe   (µm) Figure 2. Spectral

  evolution of the SHG signal in the LT (blue open circle) and HT (red open circles) phases. The lines are just a guide for the eyes. The signals have been calibrated against SHG signal recorded in a ZnSe polycrystalline window.


Dating nodes in a phylogeny is an important problem in evolution and is typically performed by using molecular clocks and fossil age estimates [START_REF] Donoghue | Telling the evolutionary time[END_REF]. The manuscript by Chauve et al. [START_REF] Chauve | MaxTiC: Fast ranking of a phylogenetic tree by Maximum Time Consistency with lateral gene transfers[END_REF] reports a novel method, which uses lateral gene transfers to help ordering nodes in a species tree. The idea is that a lateral gene transfer can only occur between two species living at the same time, which indirectly informs on node relative ages in a phylogeny: the donor species cannot be more recent than the recipient species. Horizontal gene transfers are increasingly recognized as frequent, even in eukaryotes, and especially in microorganisms that have little fossil records [START_REF] Ropars | Adaptive horizontal gene transfers between multiple cheese-associated fungi[END_REF][START_REF] Novo | Eukaryote-to-eukaryote gene transfer events revealed by the genome sequence of the wine yeast Saccharomyces cerevisiae EC1118[END_REF][START_REF] Naranjo-Ortíz | Widespread inter-and intradomain horizontal gene transfer of d-amino acid metabolism enzymes in Eukaryotes[END_REF][START_REF] Alexander | Horizontally acquired genes in early-diverging pathogenic fungi enable the use of host nucleosides and nucleotides[END_REF][START_REF] Marcet-Houben | Acquisition of prokaryotic genes by fungal genomes[END_REF]. Yet, such an important source of information has been very rarely used so far for inferring relative node ages in phylogenies. In this context, the method by Chauve et al. [START_REF] Chauve | MaxTiC: Fast ranking of a phylogenetic tree by Maximum Time Consistency with lateral gene transfers[END_REF] represents an innovative and original approach to a difficult problem. An obvious limitation of the approach is that it relies on inferences of horizontal transfers, which detection is in itself a difficult problem. Incomplete taxon sampling, or the extinction of the true donor lineage may render patterns difficult to interpret in a temporary fashion. Yet, for clades with no fossils this may be the only piece of information

Open Access

Published: 7th November 2017

Copyright: This work is licensed under the Creative Commons Attribution-NoDerivatives 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licen ses/by-nd/4.0/ we have at hand, and the growing amount of sequence data is likely to minimize issues derived from incomplete sampling.

The developed method, MaxTiC (for Maximal Time Consistency) [START_REF] Chauve | MaxTiC: Fast ranking of a phylogenetic tree by Maximum Time Consistency with lateral gene transfers[END_REF], represents a very nice application of theoretical developments on the well-known « Feedback Arc Set » computer science problem to the evolutionary question of ordering nodes in a phylogeny. MaxTiC uses as input a species tree and a set of time constraints based on lateral gene transfers inferred using other softwares, and minimizes conflicts between node ordering and these time constraints. The application of MaxTiC on simulated datasets indicated that node ordering was fairly accurate [START_REF] Chauve | MaxTiC: Fast ranking of a phylogenetic tree by Maximum Time Consistency with lateral gene transfers[END_REF]. MaxTiC is implemented in a freely available software, which represents original and relevant contribution to the field of evolutionary biology.

  Dating nodes in a phylogeny using inferred horizontal gene transfers Tatiana Giraud 1 and Toni Gabaldón 2 1 Laboratoire Ecologie, Systématique et Evolution, Univ. Paris Sud, CNRS, Université Paris-Saclay -91400 Orsay, France 2 Centre for Genomic Regulation (CRG), Barcelona Institute of Science and Technology --Barcelona, Spain Correspondence to Tatiana Giraud (tatiana.giraud@u-psud.fr) doi: 10.24072/pci.evolbiol.100037 Cite as: Giraud T. and Gabaldón T. 2017. Dating nodes in a phylogeny using inferred horizontal gene transfers. Peer Community in Evolutionary Biology, 100037. doi: 10.24072/pci.evolbiol.100037 A recommendation -based on reviews by Mukul Bansal, Alexandros Stamatakis and an anonymous reviewer -of Chauve C, Rafiey A, Davin AA, Scornavacca C, Veber P, Boussau B, Szöllősi GJ, Daubin V, Tannier E. 2017. MaxTiC: Fast ranking of a phylogenetic tree by Maximum Time Consistency with lateral gene transfers. bioRxiv, 127548, ver. 6 of 6 th November 2017. doi: 10.1101/127548

Appendix

Reviews by Mukul Bansal, Alexandros Stamatakis and an anonymous referee: http://dx.doi.org/10.24072/pci.evolbiol.100037


Andrzej M J Skulimowski

Inez Badecka

Software Innovation Dynamics in CMSs and its Impact on Enterprise Information Systems Development

Keywords: Software evolution, ICT foresight, ERP-CMS, vector autoregression, EIS scenarios, open-source software

This paper reports the results of a prospective study on information system development trends. It is based on the observation that SMEs seek opportunities to endow their CMS-based IS with CRM, e-commerce and ERP/ERM functionalities. Using publicly-available release data, we investigated the dynamics of subsequent versions and new functionalities of the most popular open-source CMSs -Drupal, Joomla!, and WordPress. Special attention was paid to software innovations that make possible the use of CMS-based applications for typical EIS purposes. The software technology race was modelled by a system of quasi-linear stochastic equations with state variables describing the upgrade generation time. Two such models have been built and compared for the above CMSs. Trend extrapolation with vector autoregression allowed us to predict ERP-related functionality development prospects until 2025. We maintain that the deployment of CMS-based ERP/ERM may have a relevant impact on business models and strategic ICT alignment in SMEs.

Introduction

The deployment order and intensity for different information and communication technologies (ICTs) in enterprises is an important issue from the software market point of view. It is also a research question relevant to understanding the technological evolution of Enterprise Information Systems (EISs) and their market development trends. ICT deployment trajectories in large enterprises have been extensively studied by many researchers cf. e.g. [START_REF] Castellina | SaaS and Cloud ERP trends, observations, and performances[END_REF], [START_REF] Hailu | Evaluation of Key Success Factors Influencing ERP Implementation Success[END_REF], [START_REF] Hallikainen | Evaluating strategic IT investments: an assessment of investment alternatives for a web content management system[END_REF], [START_REF] Leina | Development and Implementation of ERP/CRM System Based on Open Source Software to Small and Medium-sized Enterprise in China[END_REF], who have often referred to them as strategic ICT alignment [START_REF] Cataldo | Comparing Strategic IT Alignment Versus Process IT Alignment in SMEs[END_REF]. However, the ICT investment behaviour of the small and medium sized enterprises (SMEs) is known in far less detail due to the great diversity of SMEs structures and business models and a greater flexibility of SME managerial decisions that often depend on individual preferences. Nevertheless, SMEs are expected to be the fastest growing market for corporate ICT solutions over the next decade [START_REF]Trends and Development Scenarios of Selected Information Society Technologies[END_REF]. Software market research indicates that the most important ICT functionalities for SMEs are those offered by Web 2.0 CMS technologies. The research presented in [START_REF] Skulimowski | New trends in the technological development of CMS-based enterprise software for SMEs[END_REF] provided clues that enterprise resource planning (ERP) systems based on the collaboration of developer groups, supported by web browsers and typical CMSs, could become a significant future trend among SMEs. This conjecture has been confirmed by a needs analysis performed among enterprises during the initial phase of a foresight project [START_REF]Trends and Development Scenarios of Selected Information Society Technologies[END_REF]. In contrast to big companies, which use large and complex multi-module ERP applications such as SAP, SMEs widely use in-company online solutions [START_REF] Devos | Using bricolage to facilitate emergent collectives in SMEs[END_REF] based on open-source CMSs, for cost reasons. Further studies conducted within the above project [START_REF]Trends and Development Scenarios of Selected Information Society Technologies[END_REF] showed that small and micro enterprises in particular intend to further invest in the development of such systems. This trend can also be regarded as a manifestation of the business model alignment to software evolution in the context of syntactic integration of different enterprise software systems [START_REF] Liu | Manufacturing Perspective of Enterprise Application Integration: the State of the Art Review[END_REF].

As a sample data set, this paper uses the CMS versions and the mutual relations of ERP and CMS development trends (ERP-CMS) gathered in the project SCETIST [START_REF]Trends and Development Scenarios of Selected Information Society Technologies[END_REF]. Partial results concerning ERP-CMS have been presented in [START_REF] Skulimowski | Competition Boosts Innovativeness: A Quantitative Software Evolution Model[END_REF] and [START_REF] Skulimowski | New trends in the technological development of CMS-based enterprise software for SMEs[END_REF]. A general technology race model for the most common open-source CMSs Drupal, Joomla! and WordPress was proposed in the form of a quasi-linear system of stochastic equations describing software evolution trajectories. The state variables are time lapses between generating two subsequent software releases of each CMS under consideration. We will construct, analyse and compare two specific models of upgrade generation processes for the three CMSs. These models correspond to software development scenarios that differ on factors taken into account when making a decision to produce a new release that brings an essential software improvement.

Both models take into account the emergence of essential new functionalities in the new releases. The first model refers to a situation where the innovation leader is fully independent in shaping its development strategy, without taking into account the progress made by its main competitors. The form of this model is justified by historical data concerning the CMS innovation leader Drupal which point out that the novelties offered in new releases of WordPress and Joomla! have not influenced the main line of Drupal development so far. The second model assumes a symmetric impact of innovation among all software providers.

The model coefficients are calibrated based on historical data published by the CMS providers on their websites. However, the coefficients of the second model that describe the future dependence of the current innovation leader on the other systems after it loses leadership could not be estimated as such a situation has never occurred before. Instead, these coefficients are assumed to be similar to the reactions of other systems on the competitor's advantages. They are then used in calculation of future release generation forecasts together with estimated coefficients describing other CMSs. Thus, the main goal of this paper is to apply the above foresight model composed of two long-term software development scenarios to provide clues to SME managers as regards the viability of the open source software they use as a seed information technology [START_REF] Skulimowski | Competition Boosts Innovativeness: A Quantitative Software Evolution Model[END_REF] in their enterprises.

Trend extrapolation with vector autoregression has made it possible to construct two scenarios of ERP-related functionality development prospects with open source CMSs until 2025 and beyond, related to the above models. The technological trajectories thus generated show surprising future behaviour, indicating changes in innovation leadership. According to these forecasts, Drupal, the current innovation leader, may lose its market position if it does not take into account the innovation potential of its competitors. In the final section, we will discuss the related basic business scenarios of SMEs as well as the synergy with other EIS development trends. We conclude that the open-source CMSs are viable enough to capture a considerable portion of the ERP for SMEs market over the next decade. Consequently, vendors and authorized resellers of popular ERP systems may lose some of their revenue to CMS developers.

ERP based on CMS and web portal technologies (in this paper referred to as CMS ERP, cf. also [START_REF] Kapur | Generalized discrete time modelfor multi generational technological products[END_REF], [START_REF] Zykov | Integrating Enterprise Software Applications with Web Portal Technology[END_REF]) is an application that contains an administrative panel endowed with a complex hierarchy of user authorisation levels. It can be used to exchange information within the enterprise and externally, and manage its resources. The ERP-CMS emerged as a result of including software development tools in advanced CMSs, making possible the implementation of resource-management processes.

To conclude this section, in Box 1 below we recall a few basic notions related to EISs used in further parts of this paper.

Box 1. Basic notions relevant to Enterprise Information System development trends.

EIS: Enterprise Information System is a key notion used in this paper. EIS is a broad class of business software facilitating information integration within an enterprise and its exchange with external agents (clents, suppliers, business partners, authorities etc.). EAI: Enterprise Application Integration (EAI) is the process of sharing and linking different information and business processes in an organization [START_REF] Cataldo | Comparing Strategic IT Alignment Versus Process IT Alignment in SMEs[END_REF], [START_REF] Lee | Enterprise Integration with ERP and EAI[END_REF]. As a result of EAI, most ICTsupported business processes are entirely controlled by a unified application. The same application ensures access to all or most enterprise databases and an appropriate common data management system. The interconnection of all the organisational data forms a basis for implementing efficient decision support systems [START_REF] Asprey | Integrated document management for decision support[END_REF]. ERP: Enterprise Resource Planning is an ICT-based integration of business processes in an organisation. Usually, ERP is implemented as an information system with a modular architecture, where each module manages a specific area of activity. The same shortcut, ERP, or 'ERP system', also refers to an integrated software solution that implements the above process [START_REF] Castellina | SaaS and Cloud ERP trends, observations, and performances[END_REF]. ERM: Enterprise Resource Management is a term which refers to ERP in a more general setting and gradually replaces the ERP term. It reflects the extension of enterprise application integration beyond resource planning. The notion of resources is more general and covers any information managed by the company and its intangible assets. One of the ultimate goals of ERM is to provide decision support to enterprise management [START_REF] Castellina | SaaS and Cloud ERP trends, observations, and performances[END_REF], [START_REF] Jiang | A Generalized Norton-Bass Model for Multigeneration Diffusion[END_REF]. CMS: A Content Management System is a web application (or a set of applications) that facilitates the development of a website, and other forms of web-based information systems by non-technical content editors. CMSs are increasingly including modules and functionalities that support enterprise management [START_REF] Liu | Manufacturing Perspective of Enterprise Application Integration: the State of the Art Review[END_REF].

Basic Properties and a Review of Popular Open-Source CMSs

The main idea of a typical CMS is to allow a non-expert user to edit the content of a website and modify its design via an easy-to-use user interface, called an administration panel. The main task of CMS platforms is to separate the information content of the website from the technical aspects of its appearance. CMSs generate web pages automatically or semi-automatically. The information entered by an authorised editor is stored in a database. The CMS generates a dynamic website based on the content of this database and on a selected template. This allows for more flexible and convenient content management than with static HTML files. Thanks to this approach, the publishing of web pages has become much simpler than in the past.

A review of most popular CMS

Popularity scores of CMSs are provided by different web application market watchers [2], where the number of installations varying in time can be found, with the demarcation of specific industries or sectors of use. The three systems selected for detailed analysis in this paper cover most of the CMS market. They are briefly outlined below.

WordPress [START_REF]WordPress web site[END_REF] has become the most popular content management system during the current decade. It is written in PHP and uses a MySQL database. WordPress is distributed under the GNU -General Public License. As the official successor of the b2/cafélog blog system, WordPress is the most popular and user-friendly system for blogging. Among the many CMSs, WordPress appears as the easiest to install, operate and configure. Due to its growing range of functionalities and ease of use, WordPress is increasingly popular in the business sector, as well as with entertainment and social networking sites. With its recent enterprise-oriented premium spin-out, WordPress Vip, the platform fulfils the early predictions concerning the CMS development trends contained in [START_REF] Skulimowski | New trends in the technological development of CMS-based enterprise software for SMEs[END_REF], cf. also https://vip.wordpress.com/2014/05/22/ enterprisewordpress-content-trends.

The Joomla! project was founded in August 2005 by the team who developed Mambo, a predecessor of Joomla [START_REF]Joomla! web site[END_REF]. After the development of Mambo was abandoned, future versions have appeared, such as Aliro (http://www.aliro.org), Lanius CMS (http://www.laniuscms.org), Elxis (http://www.elxis.org) etc. Joomla has a modular structure, which means that each new feature of the system is added as an additional module. This allows enterprise users to easily extend the usability of the system into the e-commerce, frequently starting from product catalogues, as well as to use its CRM and ERP/ERM functionalities. Drupal [START_REF]Drupal web site[END_REF] is a CMS that allows its users to easily publish, manage and organise web content. It is equipped with functionalities that include environments for collaborative work on projects, file exchange and much more. An important feature of Drupal is its system of modules and taxonomy. The latter allows the users to organise web contents according to predefined categories. Drupal plays an important role in this paper due to its two exceptional features:

• In recent years, Drupal has led the CMS field in terms of implementing innovations in its releases,

• Drupal is the leader in offering ERP-related modules [START_REF]Drupal web site[END_REF].

Overall, Drupal is equipped with many special tools that are useful in the business sector. The platform is scalable and can be used to build enterprise information systems of any size. Moreover, Drupal strives to simplify its use via 'Drupal Open Enterprise' (https://www.drupal.org/project/openenterprise).

The technological evolution of CMSs

The Fig. 1 summarises, in the form of a timeline, the history of versions, milestones, and functionalities of the most popular CMSs covered in the preceding section: Wordpress, Joomla and Drupal. The data has been gathered in [START_REF]Trends and Development Scenarios of Selected Information Society Technologies[END_REF] and [START_REF] Skulimowski | New trends in the technological development of CMS-based enterprise software for SMEs[END_REF] and verified based on the recent information provided by the software suppliers.

The historical data characterising the relative innovativeness of the selected CMSs represented in Fig. 1 and used to build the models presented in the next section is based on [START_REF] Kapur | Generalized discrete time modelfor multi generational technological products[END_REF]. The identification of relevant functionalities and milestones illustrated in the above timeline, together with the version data, forms the basis on which to build both technological evolution models.

Two Scenarios of Technological Evolution

There are various models which describe the generation of software innovation, however, none of them is widely-accepted. General discrete-time innovation diffusion models usually refer to the well-known Norton-Bass model [START_REF] Jiang | A Generalized Norton-Bass Model for Multigeneration Diffusion[END_REF], [START_REF] Kapur | Generalized discrete time modelfor multi generational technological products[END_REF]. When applied to open-source software, in the absence of price equilibria, the major role played by essential technological progress must be taken into account. This is partly included in the famous eight Lehman software evolution laws, cf. e.g. [START_REF] Tripathy | Software evolution and maintenance: a practitioner's approach[END_REF], pp.49-53, 58-61, although they do not sufficiently refer to the emerging oligopolistic structure of open source software supply. Therefore, in [START_REF] Skulimowski | Competition Boosts Innovativeness: A Quantitative Software Evolution Model[END_REF], [START_REF] Skulimowski | New trends in the technological development of CMS-based enterprise software for SMEs[END_REF] we proposed a new class of stochastic models, suitable for information systems evolution modelling, including the open source CMS. The dependent variable is the time of market release of a relevant innovation by the developers. Explanatory variables are time lags between consecutive releases of the same as well as of other systems. In [START_REF] Skulimowski | Competition Boosts Innovativeness: A Quantitative Software Evolution Model[END_REF] we compared models based on major version releases and taking into account selected essential functionalities only.

Following the research reported in [START_REF] Skulimowski | Competition Boosts Innovativeness: A Quantitative Software Evolution Model[END_REF], this paper extends the scope of modelling to include models based on analogies, to be applied in case where no or too few observations are available to use statistical fitting techniques. This allows us to construct a realistic forecasting model for a situation where the hitherto market leader may lose its leadership during the forecast period.

In both models presented in the following subsections 3.1 and 3.2 we take into account only the releases of each software system that bring considerable innovation of relevance to the user community. Specifically, for a given software system S we assume that the time lags from recent releases of other systems are less important to explain the evolution of this kind of software than the dates of the introduction of relevant functionalities, which system S does not yet possess. Based on empirical evidence, in the first model we have additionally assumed that the technological leader (Drupal) develops its system autonomously, i.e. without taking into account the innovations introduced by systems that have been less advanced so far. Another development scenario, which does not admit this assumption and treats all systems as dependent on the innovations introduced by other developer teams, yields the second model. These models will generate two CMS and ERP-CMS development and deployment scenarios, termed Scenario A and Scenario B, respectively. Both models and the corresponding scenarios are investigated are compared in Section 4.

As already mentioned, the coeffcients and their goodness of fit of the second model (Scenario B) could not be determined properly because the situation where the mutual dependence of the leading system on the others did not occur in the past. However, simulations of such behavior based on analogy to other developer teams are possible and are crucial in building the second model. The goodness of fit of a simulated Model 2 may be worse than in the case of Model 1 since the estimation of missing interdependence parameters is based on analogies, and not on the extrapolation. However, the study of the second model has an exploratory character while its lower accuracy will be compensated by a better coverage of possible states of the future.

In the following sections we will present the assumptions that led to the formulation of both models, the forecasts that they generate, the impact on the development of CMS technology and the resulting conclusions.

Innovation forecasts in Scenario A

Let x(i), y(i), and z(i) denote time intervals between next (i-th) essential system improvement, for WordPress, Joomla! and Drupal, respectively. We can then examine the relationship between the time of the next improvement introduction by a particular development team and the frequency of similar innovations in the past, created by the developer teams of all above CMSs.

We assume that the time interval of the subsequent system improvement depends linearly on the n-1 previous time intervals of system improvements as well as on the frequency of the improvements emerging in other systems. Motivated by a timeline chart analysis (Fig. 1), which showed Drupal leading in the implementation of all previous innovations, the first scenario will additionally assume that the Drupal development team operates autonomously, without taking into account subsequent versions of WordPress and Joomla. The innovations introduced by the latter two are already present in Drupal. These innovations are not influenced by a competitive pressure on the Drupal development team (such pressure may, however, result from introducing marketing or organisational innovations, which are not necessarily linked with the technological innovation introduced by the successive versions of software). These assumptions lead to the formulation of the following model of innovation creation in the three analysed systems [START_REF] Asprey | Integrated document management for decision support[END_REF], which is an extension of that proposed in [START_REF] Skulimowski | Competition Boosts Innovativeness: A Quantitative Software Evolution Model[END_REF]:

x(k+1)=a 1,1 x(k)+a 1,2 x(k-1)+…+a 1n x(k-n+1)+b 1,2 v 1,2 (k) + b 1,3 v 1,3 (k) +c 1 (1a) y(k+1)=a 2,1 y(k)+a 2,2 y(k-1)+…+a 2n y(k-n+1)+b 2,1 v 2,1 (k) + b 2,3 v 2,3 (k)+c 2 (1b)
z(k+1)=a 31 z(k)+a 32 z(k-1)+…+a 3n z(k-n+1) + c 3 (1c) where:

v 12 (k) -average frequency of introducing a new functionality of Joomla calculated on the basis of P 1,2 (k) time intervals between essential releases of this system directly preceding the k-th essential functionality of WordPress; v 21 (k) -average frequency of introducing of a new functionality of WordPress calculated on the basis of P 2,1 (k) time intervals between essential releases of this system directly preceding the k-th essential functionality of Joomla v j,3 (k), for j=1,2 -average frequency of introducing of a new version of Drupal calculated on the basis of P j,3 (k) time intervals between essential releases of this system directly preceding the k-th essential functionality of WordPress -for j=1, and Joomla -for j=2.

Based on numerical experiments, in the above model we will admit a further simplifying assumption, namely as P i,j (k), we will take the maximum value P i,j (k)=3, for i=1,2, j=1,2,3, i≠j which, however, will allow us to obtain a sufficient statistical significance of the model. Furthermore, we will take into account only positive time lags between launching essential functionalities, i.e. we assume that exclusively the technological arrears play a motivating role for the developer teams.

Let a j1 ,a j2 ,…,a jn denote the direction coefficients of the regression equation for the dependence of the time interval of the j-th system new functionality appearance on the previous frequencies of innovation introduced in the same system, i.e. WordPress, Joomla and Drupal, respectively, for j=1,2,3. The b i,j for i=1,2, j=1,2,3, and i≠j, denote the coefficients of the linear regression functions that describe the following dependencies:

-for i=1, j=2,3, b i,j is the coefficient of multivariate linear regression which explains x(k) with v 1,j (k), -for i=2, j=1, b i,j is the coefficient of multivariate linear regression which explains y(k) with v 2,1 (k) and for i=2, j=3, b i,j explains y(k) with v 1,3 (k).

After finding the coefficients of (1) with the least squares method (LSM), we will get the regression function relating the expected time of a new innovation release by each system -as dependent variables -to the average time lags between the introduction of technological innovations in all systems. The lags are calculated to the releases directly preceding the latest innovation in the i-th system, prior to the k-th improvement of the system described by the equations (1a) or (1b). The trend drift coefficients c i , i=1,2,3 should vanish after the Model (1) for autoregresssion purposes was integrated sufficiently many times to yield a stationary time series for each system. However, they re-appear when calculating the forecasts for the original time series.

The equations (1a), (1b) take into account the pressure of competition. According to the initial assumption, equation (1c) describes time lags between subsequent Drupal improvements only.

It turned out impossible to find a statistically significant model where the variables were simple time lags between innovations. However, a stationary time series and, at the same time, significant regression functions could be found when averaging the variables three times. From the observation that, by the definition of variables as time lags, they possess the property (2),

x

(t)-x(t-n) =[(x(t)-x(t-1)] + [x(t-1)-x(t-2)]+…+[x(t-n+1)-x(t-n)],

(2) it follows that the above averaging operation is equivalent to the integration of the original time series.

The statistical significance of both models was investigated with the F (Fisher-Snedecor, cf. e.g. [START_REF] Snedecor | Statistical Methods[END_REF]) and goodness of fit tests with the determination coefficient R 2 . Tests confirmed the significance of the calculation results for triple averaged data, which corresponds to the case where all variables characterising the frequency of innovations are calculated as an average of three time lags between essential new functionalities in all systems modeled.

Finally, the forecasting model in Scenario A is provided in eq. ( 3) below.

x(k+1)= 0.1521 x(k) + 0.6284 v 1,2 (k) + 2.9762 v 1,3 (k) +8.2919 (3a)

y(k+1)= -0.3679 y(k)+2.9762 v 2,1 (k) -2.0860 v 2,3 (k)-18.3726 (3b) z(k+1)= 1.1662 z(k) -0.6892 z(k-1)+ 4.6565 (3c)
As the significance of the regression function does not guarantee that the coefficients are significant, we calculated the confidence intervals for each of them. Then, to complete the usual statistical analysis of forecasted values, we calculated the confidence intervals for the number of innovations. The forecasting procedure was pursued until the last release of the most slowly evolving system reached the year 2025, i.e. the foresight horizon of the project SCETIST [START_REF]Trends and Development Scenarios of Selected Information Society Technologies[END_REF]. The forecasting results for the horizons 2025 and 2030 are provided in Table 1, while the forecasts of the number of implemented innovations are visualised in Fig. 2. The chart shows that Drupal, which was assumed to develop autonomously, will temporarily lose its leadership twice. Joomla will catch up with Drupal within a few years and the development of both systems will follow the same trajectory until 2025. Both systems will overtake Wordpress. In about 10 years from now all systems will converge, i.e. the number of implemented innovations in WordPress, Joomla and Drupal will be practically equal. This may ultimately result in mergers or otherwise elimination of some systems from the market as the community of developers will not differentiate them anymore. The simulation points out that this situation may change until 2030. However, the confidence intervals of the forecasted quantities will already be rather large in 2025 and grow further beyond this time horizon (cf. Table 1) while the longer-term forecasts may be disturbed by new factors and phenomena that did not appear by now and were not included in model [START_REF] Asprey | Integrated document management for decision support[END_REF].

Innovation trajectories in Scenario B

Let us recall that the Scenario B differs from the Scenario A on the assumption that all systems, including the current innovation leader, take into account the new functionalities introduced in the software releases of their competitors. Their decisions concerning the time of new version releases are modified quasi-linearly according to the time elapsed since the new functionalities were launched by other systems. To build a CMS forecasting model with this assumption, we will use the same essential functional improvements that have been identified for the Model 1 and Scenario A, cf. eq. ( 1), Fig. 2. The interdependence coefficients between competing systems are calculated based on the time lags (in months) between subsequent (i-th) essential system improvements. The assumptions concerning the variables are the same as in Model 1, in particular only positive lags between the introduction of new functionalities are taken into account. This led us to the formulation of an extended model of essential innovation generation in WordPress, Joomla, and Drupal, described by eq. ( 4).

x(k+1)=a 1,1 x(k)+a 1,2 x(k-1)+…+a 1n x(k-n+1)+b 1,2 v 1,2 (k) + b 1,3 v 1,3 (k) +c 1 (4a) y(k+1)=a 2,1 y(k)+a 2,2 y(k-1)+…+a 2n y(k-n+1)+b 2,1 v 2,1 (k) + b 2,3 v 2,3 (k)+c 2 (4b) z(k+1)=a 31 z(k)+a 32 z(k-1)+…+a 3n z(k-n+1) + b 3,1 v 3,1 (k) + b 3,2 v 3,2 (k)+ c 3 (4c)
The notation is similar to that used in Scenario A and Model [START_REF] Asprey | Integrated document management for decision support[END_REF]. The new coefficients of the Model 2, v 3,j (k), for j=1,2, denote the average frequency of introducing a new version of Wordpress (j=1) or Joomla (j=2) calculated on the basis of P 3,j (k) time intervals between essential releases of the corresponding system directly preceding the k-th essential functionality of Drupal. The other difference consists in the fact that the coefficients b 3,1 and b 3,2 could not be estimated with multivariate regression because during the observation period (2004-2012) Drupal was always a leader in launching new functionalities, so the variables v 3,1 (k) and v 3,2 (k) were identically equal to 0. To simulate the reaction of the Drupal's team on a potential different situation that may occur in the future, b 3,1 and b 3,2 were assumed equal to 1.5 and 0.7, respectively. These values resulted from the averaging of the reactions of WordPress and Joomla teams on the past Drupal's advantages, taking into account the mean values of the variables x, y, and z. The subjectivity of this assumption illustrates the difference between the forecasting and foresight methodology: foresight approaches make it possible to explore future in the situation where no extrapolation of the past is possible, based on heuristic observations and assumptions. Analogy-based reasoning is the only way to investigate the phenomena that have never occurred before.

The triple averaging of input data yielded again the best outcomes. The coefficients of the second model are provided in eq. ( 5) below.

x(k+1)= 0.1521 x(k) + 0.6284 v 1,2 (k) + 2.9762 v 1,3 (k) +8.2919 (5a) y(k+1)= -0.3679 y(k)+2.9762 v 2,1 (k) -2.0860 v 2,3 (k) -18.3726 (5b) z(k+1)= 1.1662 z(k) -0.6892 z(k-1)+ 1.5 v 3,1 (k) + 0.7 v 3,2 (k) + 4.6565 (5c)
The significance of Model ( 5) is the same as of Model (1) during the observation period and it is undefined during the forecasting period. The occurrence of any event that causes non-zero values of v 3,1 (k) and v 3,2 (k) would make it possible to use supervisory learning techniques to update the coefficients b 3,1 and b 3,2 . The results of the CMS innovation implementation forecasts in Scenario B are shown in Fig. 3. The ability to observe the other systems and react gives the Drupal team a development impulse every time it is surpassed by Joomla. In this scenario WordPress stays initially behind, but at the end of the forecasting period the functionalities of all systems differ only slightly. Similarly as in Scenario A, it gives again clues as regards potential mergers or transformations of main open source CMSs during the next decade. The evolution of Internet, the emergence of new business models, and new information technologies may additionally contribute to an end of CMS era.

The confidence intervals for the quantities of implemented innovations by the two forecasting horizons 2025 and 2030 are presented in Table 2 below. The assumed coefficients b 3,1 and b 3,2 are regarded deterministic and do not influence the stochastic properties of Model (5).

A Comparison of Scenarios A and B

A comparison of both CMS evolution scenarios is shown in Table 3 below. It turns out that there is no considerable difference in CMS innovative behavior between the scenarios, with some more activity in Scenario B. As the only exception, WordPress performs worse on about 4% in Scenario B in 2030. The ability to consider technological and market signals from the other two systems allowed Drupal to generate slightly more (about 5% in 2030) new essential functionalities in its releases. Without this ability Drupal will loose its leadership to WordPress until 2030. More active Drupal in Scenario B boosts the innovativeness of Joomla -its number of essential innovation rises on 5 (or on 12%) until 2030. A better performance of Joomla is accompanied by a loss of the innovateveness of WordPress between 2025 and 2030.

The convergence trend is more salient in Scenario B, where the number of functionalities of all systems in 2025 and 2030 is almost the same. Another comparison is provided in Table 4 which shows the value of the indicator "CMS innovativeness growth index" defined in [START_REF] Skulimowski | Competition Boosts Innovativeness: A Quantitative Software Evolution Model[END_REF]. This indicator (5 th row in Table 4) is calculated as the ratio of the forecasted yearly average number of innovations during the period of 2012-2025 (4 th row in Table 4) and the actual average value observed during the period of 2005-2012 (3 rd row). The above results show that the assumption concerning the market leader behavior that differentiate both scenarios may have a remarkable influence on the innovativeness of all the developer teams concerned. It turns out that a 'weaker' leader that follows its own development strategy and does not react to its competitors' achievements suppresses the innovative activity of all market players. On the contrary, its activity may increase the market strength of all open source CMS suppliers and reduces the differences between their products and innovativeness of developer teams.

Future features and functionalities of CMSs

An important issue that has not been explained yet is the transition between quantitative innovation characteristics and a concrete sequence of innovations that will be created by each of the CMS development teams. A partial answer can be derived from development plans that, however, are announced for a relatively short period of for one or two years only [START_REF]Drupal web site[END_REF]. Other clues are given by the version and functionality analysis performed in Sec. 2.2 and by both innovation development models presented in Secs.

3.1, 3.2 and 4. It follows that Drupal plays the role of leading innovator, while the other systems follow its functionalities with some delay. Taking into account that Drupal has already entered the ERP/ERM market and plans further expansion in enterprise software sector, it can be concluded that the other systems will follow suit during the next decade.

In addition, all the above-analysed CMS developers pay much attention to research, which allows them to quickly implement new technologies. Below is a (nonexhaustive) list of ideas, functionalities, and new modules that are either in the process of being tested or are planned for implementation during the next few years (cf. the CMS web sites [START_REF]Drupal web site[END_REF], [START_REF]Joomla! web site[END_REF], [START_REF]WordPress web site[END_REF]). Some of them will support ERP functionalities or meet general enterprise needs. A few new ERP-related features, namely innovation support modules as well as HR management systems, have been pointed out as relevant by the respondents of the Delphi survey performed within the project SCETIST [START_REF]Trends and Development Scenarios of Selected Information Society Technologies[END_REF]. They are also included in the list below.

• Cognitive user models allowing the system to learn users interact with the website and the applications installed there. These models will be further developed towards intelligent recommendation systems. • General enterprise managerial decision support modules using integrated CMS-supported databases containing data provided by other subsystems [START_REF] Asprey | Integrated document management for decision support[END_REF]. • HR management.

• Quality management modules -different versions according to the sector needs and ISO, EMAS, or other quality norms implemented. • Template builder -easy creation of templates from the administration panel.

• In-company innovation support systems. The approach applied in this paper yields models that put emphasis on the overall number of functionalities implemented in software systems. They do not yet allow us to determine the sequence of implementation of specific features. However, the latter may be estimated based on development roadmaps published on CMS websites, on research of market expectations and scenarios of software use, and on general software evolution trends. A more detailed study of technological trajectories of ERP-CMS and enterprise decision support systems will be a subject of further research.

Conclusions

In this study, we described the foundations of an important software development trend, namely enterprise application integration (EAI) based on the gradual expansion of the scope of applications covered by a company's CMS functionalities. We conducted an overview and comparison of the most popular open source CMSs used primarily, but not exclusively, by SMEs. We maintain that this trend will bring new technological opportunities to SMEs in particular. Using open-source software modules, SMEs will be able to cover more areas of their commercial activity with modern ICT solutions. This trend will interfere with the "going mobile" and "moving to the cloud" trends that are influencing all enterprises, but are of particular importance to small and micro companies. Specifically, web-based ERP/ERM applications are in a better position to include mobile technologies than offline systems. This is the case of applications built with CMS technology, both open-source and commercial.

While CMS-based systems will expand into the cloud and mobile worlds, according to the Delphi survey performed in [START_REF]Trends and Development Scenarios of Selected Information Society Technologies[END_REF], the accounting software used in SMEs in the perspective of 2025 will still resist full integration in cloud-based systems. This is partly due to the traditional vigilance and concern that entrepreneurs have as regards revealing their financial data in the public space, despite the fact that ICT security has been steadily improving. This contradicts the widely-held belief that a vast majority of enterprises will gradually implement professional ERP software built around accounting, inventory and sales modules. Thus, accounting and personal record files may remain the core of traditional ERP systems. However, this could vary from country to country [START_REF] Leina | Development and Implementation of ERP/CRM System Based on Open Source Software to Small and Medium-sized Enterprise in China[END_REF] and may depend on the economic sector.

Finally, let us observe that the methodology presented in Secs. 2, 3 and 4 can be used to investigate the technology race between companies that regularly release new versions of non-CMS software products. A necessary prerequisite is that the market for the product under study is oligopolistic, with a reasonably small number of suppliers (say, less than 20) that enables building and constructively analyzing a model of type [START_REF] Asprey | Integrated document management for decision support[END_REF]. Some hints regarding sample-based generalization strategies of software-related models are provided in [START_REF] Wieringa | Six strategies for generalizing software engineering theories[END_REF]. While many open source applications and programming environments fulfill this assumption well, the approach presented in this paper is not restricted to software and may be used to forecast the development of electronic components, cars, and other products and technologies.

Fig. 1 .

 1 Fig. 1. A timeline of the development of the three most frequently-used open source CMSs. The coloured lines point out the release of main functionalities of these CMSs on the time scale.

Fig. 2 .

 2 Fig. 2. Forecasted number of implemented innovations by each of the systems in Scenario A.

Fig. 3 .

 3 Fig. 3. Forecasted number of innovations for the three CMS in Model 2 -Scenario B

Table 1 .

 1 The forecasts of implemented innovations in system releases and their confidence intervals (p=0.95) for the Scenario A

		Drupal			WordPress			Joomla!
	Forecast	Number of	Confidence	Forecast	Number of	Confidence	Forecast	Number of	Confidence
	horizon	innovations	interval	horizon	innovations	interval	horizon	innovations	interval
	2025-01	34	(27,42)	2025-06	33	(28,39)	2025-12	34	(27,42)
	2030-05	42	(36,52)	2030-01	45	(38,52)	2030-07	39	(32,49)

Table 2 .

 2 The forecasts of implemented innovations in system releases and their confidence intervals (p=0.95) for the Scenario B

		Drupal			WordPress			Joomla!
	Forecast	Number of	Confidence	Forecast	Number of	Confidence	Forecast	Number of	Confidence
	horizon	innovations	interval	horizon	innovations	interval	horizon	innovations	interval
	2025-05	35	(29,41)	2025-08	36	(32,41)	2025-11	36	(29,42)
	2030-07	44	(36,53)	2030-03	43	(37,50)	2031-11	44	(36,54)

Table 3 .

 3 Expected number of essential innovations in the CMS releases until 2030 -a scenario comparison

		Drupal	WordPress	Joomla!
	Forecast horizon	Scenario A	Scenario B	Scenario A	Scenario B	Scenario A	Scenario B
	2020	28	30	26	27	27	29
	2025	34	35	33	36	34	36
	2030	42	44	45	43	39	44

Table 4 .

 4 Number of innovations in CMS releases until 2025 -Scenarios A and B compared

	Drupal	WordPress	Joomla!

Scenario A Scenario B Scenario A Scenario B Scenario A Scenario B No. of innovations in 2005-2012

		10	9	10	10	14	14
	No. of innovations in 2012-2025	17	20	17	21	18	22
	Mean innovations no. in 2005-2012	1.43	1.29	1.43	1.43	2	2
	Mean innovations no. in 2012-2025	1.31	1.53	1.53	1.62	1.38	1.69
	Average innovation growth	0.92	1.20	1.07	1.13	0.69	0,85

Acknowledgement. This research was supported by the research project "Scenarios and Development Trends of Selected Information Society Technologies until 2025" (SCETIST) co-financed by the ERDF, Contract No. WND-POIG.01.01.01-00-021/09.