Marine Louis

Régis Guillot

Rémi Métivier

Clémence Allain

β-Diketone derivatives: influence of the chelating group on the photophysical and mechanofluorochromic properties

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Introduction

The development of compounds fluorescent in the solid state is currently a topic of high interest, both for elucidating the mechanisms responsible for the fluorescence changes observed when going from the solution to the solid-state, and for various applications, such as sensors or emissive displays. Among the solid-state fluorescent materials, mechanofluorochromic materials are those for which fluorescence emission changes upon application of a mechanical stress (pressure, shearing). 1 The study of these mechanofluorochromic materials is rapidly developing, with potential applications as force sensors or in security technologies, and several mechanofluorochromic systems have been described during the past five years including coordination complexes, 2 organic dyes, [START_REF] Sagara | Mechanoresponsive Luminescent Molecular Assemblies:An Emerging Class of Materials[END_REF] and polymers. [START_REF] Calvino | Approaches to polymeric mechanochromic materials[END_REF] Since the first description of their mechanofluorochromism by Fraser et al. in 2010, [START_REF] Zhang | Polymorphism and reversible mechanochromic luminescence for solid state difluoroboron avobenzone[END_REF] difluoroboron ßdiketonate (DFB) complexes have been widely studied since they possess appealing properties such as versatile synthesis, high fluorescence quantum yields in the solid state, large fluorescence emission shifts upon mechanical stimulation, and a substituent-dependent backward reaction to the original fluorescence emission after mechanical stimulation. [START_REF] Sagawa | Substituent-dependent backward reaction in mechanofluorochromism of dibenzoylmethanatoboron difluoride derivatives[END_REF] As a consequence, numerous derivatives belonging to this family have been described, such as dibenzoylmethane systems with different substituents on the phenyl rings, [START_REF] Sakai | New Fluorescence Domain "Excited Multimer" Formed upon Photoexcitation of Continuously Stacked Diaroylmethanatoboron Difluoride Molecules with Fused pi-Orbitals in Crystals[END_REF][START_REF] Galer | Crystal structures and emission properties of the BF(2) complex 1-phenyl-3-(3,5-dimethoxyphenyl)-propane-1,3-dione: multiple chromisms, aggregation-or crystallization-induced emission, and the self-assembly effect[END_REF][START_REF] Morris | Mechanochromic luminescence of halide-substituted difluoroboron ß-diketonate dyes[END_REF] or diketones substituted with larger aromatic rings [START_REF] Xu | Aromatic difluoroboron beta-diketonate complexes: effects of pi-conjugation and media on optical properties[END_REF][START_REF] Butler | Mechanochromic Luminescence and Aggregation Induced Emission of Dinaphthoylmethane beta-Diketones and Their Boronated Counterparts[END_REF] or heterocycles. [START_REF] Morris | Stimuli Responsive Furan and Thiophene Substituted Difluoroboron beta-Diketonate Materials[END_REF] However, the studies aiming at deciphering the influence of the boron chelating group on the photophysical properties of these complexes are more scarce. Comparison between free diketones and their boron difluoride complexes has been made on dinaphthoylmethane diketones, in which the complexation induces an enhancement of the fluorescence emission both in solution and in the solid state [START_REF] Butler | Mechanochromic Luminescence and Aggregation Induced Emission of Dinaphthoylmethane beta-Diketones and Their Boronated Counterparts[END_REF] and on azepane substituted dibenzoyl ß-diketones in which the complexation induces a loss of fluorescence emission in solution. [START_REF] Wang | Multi-stimuli responsive luminescent azepanesubstituted ß-diketones and difluoroboron complexes[END_REF] The ß-diketones described in the references cited above display Aggregation Induced Emission (AIE) or Aggregation Induced Enhanced Emission (AIEE) and are in most cases mechanofluorochromic. Interestingly, a dibenzoyl ß-diketone bearing a dimethylamino group in the para position of one of the phenyl rings has been studied separately as a free ligand, [START_REF] Butler | Controlling solid-state optical properties of stimuli responsive di-methylamino-substituted dibenzoylmethane materials[END_REF] as difluoroboron and di-( pentafluorophenyl) complexes [START_REF] Hu | A simple approach to aggregation-induced emission in difluoroboron dibenzoylmethane derivatives[END_REF] and as a diphenyl-boron complex. [START_REF] Wang | Luminescent chromism of boron diketonate crystals: distinct responses to different stresses[END_REF] It is worth noting that this diphenyl-boron complex is described as mechanofluorochromic, with a distinct response to distinct forces, which is promising for the development of force sensors.

Fig. 1 Structures of ß-diketone P-Ester studied, together with those of its boron complexes DPB-Ester and DFB-Ester.

Based on these data, modifying the boron chelating groupof ß-diketonate complexes appears as an interesting and straightforward strategy from the point of view of organic synthesis leading to new mechanofluorochromic compounds with distinct properties. We decided to prepare an original diphenyl-boron-ß-diketone complex substituted with methoxy and ester groups (DPB-Ester, Fig. 1) and compare its photophysical properties, in solution and in the solid state, with those of its corresponding metal-free diketone precursor (P-Ester) and its corresponding difluoroboron complex (DFB-Ester). 17

Experimental

Synthetic procedures

All chemicals were of reagent grade and used without further purification. Anhydrous solvents were dried and purified using alumina columns. All reactions were monitored by thin-layer chromatography (TLC) using a Merck TCL silica gel 60 F254. NMR spectra were recorded on a JEOL JMS ECS 400 MHz spectrometer (100 MHz for 13 C and 128.2 MHz for 11 B). The reference used for [START_REF] Butler | Mechanochromic Luminescence and Aggregation Induced Emission of Dinaphthoylmethane beta-Diketones and Their Boronated Counterparts[END_REF] B was a solution of BF 3 OEt 2 15% in CDCl 3 contained in a coaxial insert, with the reference peak at 0 ppm for [START_REF] Butler | Mechanochromic Luminescence and Aggregation Induced Emission of Dinaphthoylmethane beta-Diketones and Their Boronated Counterparts[END_REF] B. High resolution mass spectroscopy was performed using the CNRS Imaging platform. The melting points were measured using a Stuart SMP10 apparatus. Syntheses of P-Ester and DFB-Ester compounds have been described in a previous article. [START_REF] Louis | Polymorphism, Mechanofluorochromism, and Photophysical Characterization of a Carbonyl Substituted Difluoroboron-ß-Diketone Derivative[END_REF] DPB-Ester synthesis. Triphenylborane BPh 3 (0.29 g, 1.20 mmol, 1.10 eq.) was added to a solution of diketone (0.340 g, 1.1 mmol, 1.0 eq.) in anhydrous CH2Cl2 (C = 0.12 mol.L -1 ) under argon. The mixture was refluxed overnight, and then the solvent was removed in vacuo. The crude product was purified by column chromatography on silica (eluent CHCl 3 ). DPB-Ester was isolated as a yellow solid (398 mg, 0.84 mmol, 76% yield).

Rf (50/50 petroleum ether/EtOAc) = 0.71.

X-Ray crystallography

X-ray diffraction data for compound P-Ester were collected by using a VENTURE PHOTON100 CMOS Bruker diffractometer with Micro-focus IuS source Mo Ka radiation. Crystals were mounted on a CryoLoop (Hampton Research) with Paratone-N (Hampton Research) as a cryoprotectant and then flashfrozen in a nitrogen-gas stream at 100 K. The temperature of the crystal was maintained at the selected value by means of an N-Helix series Cryostream cooling device within an accuracy of ±1 K. The data were corrected for Lorentz polarization, and absorption effects. The structures were solved by direct methods using SHELXS-97 [START_REF] Sheldrick | SHELXS-97, Program for Crystal Structure Solution[END_REF] and refined against F2 by fullmatrix least-squares techniques using SHELXL-2017 [START_REF] Sheldrick | [END_REF] with anisotropic displacement parameters for all non-hydrogen atoms. Hydrogen atoms were located on a difference Fourier map and introduced into the calculations as a riding model with isotropic thermal parameters. All calculations were performed by using the Crystal Structure crystallographic software package WINGX. 20 The crystal data collection and refinement parameters are given in Table S1. CCDC 1823426 contains the supplementary crystallographic data for this paper.

Calculations

Calculations were performed using the hybrid B3LYP functional, as implemented in Gaussian 09 software package. [START_REF] Frisch | Gaussian 09[END_REF] For geometry optimization, a 6-31G(d) basis set was used. All minima were verified via a calculation of vibrational frequencies, ensuring that no imaginary frequencies were present. TD-DFT calculations were performed using a 6-311+G(d,p) basis set and solvent effects were modelled using the IEF-PCM model.

Spectroscopic measurements

Absorption spectra were recorded on a double beam CARY 100 spectrophotometer. Steady-state emission spectra in solution were obtained on a Fluoromax-3 or Fluorolog-3-221 spectrofluorimeter from Horiba Jobin-Yvon. The signal was collected at 90° with respect to the excitation beam. Fluorescence spectra of a thin film deposited on paper were recorded on the Fluorolog-3-221 spectrofluorimeter, and the signal was collected at 22.5° with respect to the excitation beam (frontface mode). Fluorescence quantum yields in a diluted solution were determined using quinine sulfate as a reference, dissolved in a solution of 0.5 M H2SO4 (literature  = 0.54). All spectra were recorded on aerated solutions, unless specified.

Fluorescence quantum yields in the solid state were obtained using a Fluorolog-3-221 integrating sphere, with a Horiba Jobin-Yvon acquisition and analysis software. Fluorescence decay curves were obtained by the time-correlated single photon counting (TCSPC) method with femtosecond laser excitation composed of a Titanium-Sapphire laser (Tsunami, Spectra-Physics) pumped by a doubled Nd:YVO laser (Millennia Xs, Spectra-Physics). Light pulses at 780 nm from the oscillator were selected by an acousto-optic crystal at a repetition rate of 4 MHz, and then doubled at 390 nm by a nonlinear crystal. Fluorescence photons were detected at 90° through a monochromator and a polarizer at the magic angle by means of a Hamamatsu MCP R3809U photomultiplier, connected to a SPC-630 TCSPC module from Becker &Hickl. The fluorescence data were analyzed using the Globals software package developed at the Laboratory for Fluorescence Dynamics at the University of Illinois at Urbana-Champaign, which includes reconvolution analysis and global nonlinear least-squares minimization method.

Results and discussion

Synthesis and photophysical properties in solution

The target DPB compound was prepared by the reaction of the diketone precursors, whose synthesis is described elsewhere, [START_REF] Louis | Polymorphism, Mechanofluorochromism, and Photophysical Characterization of a Carbonyl Substituted Difluoroboron-ß-Diketone Derivative[END_REF] with triphenylborane in refluxing dichloromethane, following an established procedure [START_REF] Wang | Luminescent chromism of boron diketonate crystals: distinct responses to different stresses[END_REF] (Scheme 1). It was isolated with 76% yield as a yellow solid after purification by chromatography on silica gel.

Scheme 1. Synthesis of the DPB-Ester compound

The photophysical properties of the P-Ester compound and its diphenyl-boron complex DPB-Ester were studied in a THF solution (Fig. 2) and compared with those of the corresponding difluoro-boron complex DFB-Ester (Table 1). Compared to the free diketone P-Ester, complexation by a boron moiety induces a red-shift of the absorption spectrum which is more important for DPB-Ester (3330 cm -1 ) than for DFB-Ester (2270 cm -1 ). The molar absorption coefficient of DPB-Ester (21 000 L mol -1 cm -1 ) is smaller than that of free diketone P-Ester (31 000 L mol -1 cm -1 ) which in turn is smaller than that of the DFB-Ester complex (46 000 L mol -1 cm -1 ).

Complexation also induces a red-shift of the fluorescence emission spectra. The fluorescence emission maximum of DPB-Ester is very similar to that of DFB-Ester, implying a limited Stokes shift and a significant overlap between the absorption and emission bands for DPB-Ester. This smaller Stokes shift for DPB-Ester suggests that in this compound, the excited state has a lower charge transfer character than the P-Ester and DFB-Ester compounds. The fluorescence quantum yield of the P-Ester compound is very low ( <2×10 -3 , associated with very short decay <> = 0.19 ns) and only slightly increases upon complexation by a boron-diphenyl moiety ( = 0.01 for the DPB-Ester, with an average fluorescence lifetime <t> = 0.92 ns) indicating that not only the presence but also the nature of the chelating boron moiety has a crucial influence on the fluorescence properties of these diketonate complexes. The bathochromic shifts in absorption and emission as well as the fluorescence enhancement upon complexation by a borondifluoride moiety are fully consistent with a previous report by Fraser et al. on dinaphthoylmethane ß-diketones. [START_REF] Butler | Mechanochromic Luminescence and Aggregation Induced Emission of Dinaphthoylmethane beta-Diketones and Their Boronated Counterparts[END_REF] In contrast, the weak fluorescence emission of the DPB-Ester molecule strongly differs from the bright emission described by Zhang et al. for a related diphenylboron diketonate complex. [START_REF] Wang | Red emissive diarylboron diketonate crystals: aggregation-induced color change and amplified spontaneous emission[END_REF] DFT calculations were performed on the P-Ester and the DPB-Ester compounds and compared with the calculations on DFB-Ester. [START_REF] Louis | Polymorphism, Mechanofluorochromism, and Photophysical Characterization of a Carbonyl Substituted Difluoroboron-ß-Diketone Derivative[END_REF] Geometry optimization shows that the P-Ester compound adopts a slightly twisted geometry (Fig. 3): the methoxyphenyl and the p-methylbenzoate rings form dihedral angles of 9.6° and 15.8° respectively with the dioxaborinine ring. Complexation of the diketone by a boron diphenyl group does not significantly change the conformation of the diketone, in a similar fashion to that observed for DFB-Ester, with a dihedral angle of -1.5° (resp. 17.9°) between the meth oxyphenyl (resp. the p-methylbenzoate) ring and the dioxaborinine ring in the DPB-Ester compound.

The geometry around the boron atom is tetrahedral.

Fig. 3 Geometry optimization (DFT B3LYP/6-31G(d)) for compounds P-Ester (top) and DPB-Ester (bottom).

TD-DFT calculations were then performed, using an implicit IEF-PCM solvent model. The position of the first Franck-Condon transition was compared with the maximum of the first absorption band in the experimental spectra. For both DPB-Ester and P-Ester good agreement was obtained (Table 2).

For DPB-Ester the first vertical transition has a weak oscillator strength and is followed by a more intense transition (Table 2), while for P-Ester and DFB-Ester the first vertical transition shows a high oscillator strength, corresponding to the first absorption band. For P-Ester, the first transition only implies a HOMO → LUMO transition which is of the  → * type with a charge transfer character.

The second transition implies a HOMO-2 → LUMO, a HOMO-4 → LUMO and a HOMO-6 → LUMO transition: Natural Transition Orbitals (NTO) [START_REF] Martin | Natural transition orbitals[END_REF] were computed (Table 3) and it was revealed that for this transition the hole is mainly located on a non-bonding orbital while the electron is located on a *-type orbital. For DPB-Ester, the first transition involves only the HOMO-1 and LUMO orbitals, which are spatially separated, HOMO-1 being localized on the diphenyl-boron moiety and the LUMO being of *-type, on the diketone. ‡ The second transition implies a HOMO → LUMO transition and a HOMO-5 → LUMO transition. Again, NTO analysis demonstrates that this transition is of the p → p* type with a weaker charge transfer character than in the case of P-Ester. This weaker charge transfer character is in good agreement with the smaller Stokes shift observed experimentally. By comparison, the first transition computed for DFB-Ester is also of the p → p* type with a charge transfer character which only implies a HOMO → LUMO transition, and is very similar to the first transition observed in P-Ester. The analysis of the orbitals involved in the first Frank-Condon transition suggests the existence of a low-energy excited-state for DPB-Ester that can be reached from the ground-state by a weakly allowed transition (low oscillator strength). This excited-state could be involved in non-radiative deactivation processes that could partly explain the low fluorescence quantum yield observed in solution. In contrast, for the brightly fluorescent DFB-Ester, only one allowed transition is computed in the main absorption band. For P-Ester, the first transition computed is also allowed; however free rotations within the diketone moiety can explain fast nonradiative deactivation and low fluorescence quantum yield in solution. Chujo et al. performed DFT calculations on a related diphenylboron ß-diketonate complex and its di-( pentafluorophenyl)boron analog. [START_REF] Nagai | Highly intense fluorescent diarylboron diketonate[END_REF] They arrived at similar conclusions from the analysis of the orbitals: a -type HOMO localized on the diketone seems to be correlated with a bright fluorescence in solution and may be favored by electron withdrawing substituents located on the boron chelating group, such as fluorine or pentafluorophenyl. a These transitions involve only one orbital for the hole and one for the electron; consequently, the orbitals involved are directly represented, without the need for NTO computation.

Table 3. Natural transition orbitals computed for the first and second vertical absorption transitions of P-Ester, DFB-Ester and DPB-Ester

compounds ‡ Despite the spatially separated orbitals, the fluorescence emission spectra are identical for aerated and degassed solutions, which indicates the absence of delayed fluorescence in this compound.

Structure determination

Crystallization tests were performed for the three compounds. Single crystals of the P-Ester compound suitable for X-ray diffraction were isolated by slow evaporation of a mixture of ethyl acetate and petroleum ether (1 : 1). Unfortunately, suitable crystals could not be obtained for DPB-Ester, although DSC analysis, which shows a fusion endotherm at 220 °C (see the ESI), indicates that the powder isolated from the synthesis is partially crystalline. As described previously, [START_REF] Louis | Polymorphism, Mechanofluorochromism, and Photophysical Characterization of a Carbonyl Substituted Difluoroboron-ß-Diketone Derivative[END_REF] two different polymorphs were isolated for DFB-Ester. P-Ester crystallizes in a face-centered cubic system, composed of 8 molecules per cells. The precursor is slightly distorted, with both phenyl moieties, namely p-methoxyphenyl and p-methylbenzoate, forming angles of 15.79° and 10.36° with the dioxaborinine ring, respectively (Fig. 4). This corresponds to a geometry similar to the one computed in the gas phase. Molecules are associated in pairs. In each pair the molecules are parallel but the substituents (methoxy and methyl) are on opposite sides (Fig. 4B andC). The molecules within one pair are separated by 3.693 Å (distance measured between the centroids) with a partial overlap of the phenyl moieties.

The pair system was also found for DFB-Ester but, in contrast to P-Ester, the molecules were always antiparallel. DFB-Ester also displayed a polymorphic behaviour which was not revealed for the precursor. Furthermore, for the two polymorphs of DFB-Ester, distances between centroids within the pairs were shorter (3.41 Å in the green-emitting crystal and 3.52 Å in the yellow-emitting crystal respectively) [START_REF] Louis | Polymorphism, Mechanofluorochromism, and Photophysical Characterization of a Carbonyl Substituted Difluoroboron-ß-Diketone Derivative[END_REF] which indicates stronger intermolecular interactions in the DFB-Ester crystals.

Fig. 4 ORTEP diagram of compound P-Ester (A), and molecular packing in crystal: view along the b axis (B) and along the c axis (C).

Solid-state fluorescence and mechanofluorochromism

In the case of P-Ester the transition from the THF solution to the micro-crystalline solid-state comes along with a modest bathochromic shift of the emission (from 431 nm to 450 nm; 1080 cm -1 ) and an enhancement of the fluorescence quantum yield (from 0.002 to 0.025), together with longer fluorescence decay (<t> = 2.41 ns at 450 nm; see the ESI †). In the solid state, the deactivation processes by rotational motion are limited compared to that in solution, which explains the fluorescence enhancement. The isolated powder displays a weak blue emission. For DPB-Ester the transition to the solid-state does not bring any modification of the fluorescence quantum yield which stays around 0.01. Here, unfortunately, the AIEE phenomenon was not observed, which is consistent with the fact that the molecule was already rigidified by complexation with the boron diphenyl moiety. Nevertheless, a strong bathochromic shift of 82 nm (3430 cm -1 ) can be underlined with a new maximum of the fluorescence centred at 532 nm. By comparison, for DFB-Ester going from the THF solution to the solid-state also induces a bathochromic shift of the emission (from 452 nm for the solution to 494 nm for a crystalline thin film; 1880 cm -1 ) but in this case a partial quenching of the fluorescence emission is observed, with quantum yields decreasing from 0.85 for the THF solution to 0.32 for the crystalline thin film. [START_REF] Louis | Polymorphism, Mechanofluorochromism, and Photophysical Characterization of a Carbonyl Substituted Difluoroboron-ß-Diketone Derivative[END_REF] Overall, the complexation of the diketonate by the boron chelating complex seems to lead to an increase of the bathochromic shift when going from the solution to the solid-state.

Fig. 5 Solid-state emission spectra of the crystalline powders isolated from synthesis (as synthesized -AS) and the powders after grinding in a mortar (smeared -SM).

The mechanofluorochromic behaviour of the three different compounds was then compared (Fig. 5). The powders, isolated from synthesis, were smeared for 30 seconds in an agate mortar. Similar to DFB-Ester, the precursor P-Ester revealed itself to be mechanofluorochromic with a 47 nm bathochromic shift (from 450 nm to 497 nm; 2100 cm -1 ) together with a significant broadening of the emission spectrum. Excitation spectra revealed comparable shapes for the different phases (see the ESI †). In the case of DPB-Ester, no mechano-induced change of emission was observed. At first sight, these results seem to differ greatly from the system described by Zhang et al. where the diphenylboron complex was not only mechanofluorochromic but also showed a distinct fluorescence response to distinct types of mechanical stimuli (grinding or hydrostatic pressure). [START_REF] Wang | Luminescent chromism of boron diketonate crystals: distinct responses to different stresses[END_REF] However, it is worth noting that Zhang et al. describe a sensitivity to hydrostatic pressure in the GPa range, as well as the apparition of an hypsochromic shift ofthe fluorescence when crystals are "smashed" but that the fluorescence of their diphenylboron complex does not change significantly upon grinding the crystals in a mortar. In the literature, the mechanofluorochromic response of the difluoroboron ßdiketonate complexes is attributed to easier excimer formation in the amorphous phase, allowed by a higher conformational freedom. [START_REF] Zhang | Reversible solid-state mechanochromic fluorescence from a boron lipid dye[END_REF] Our data, both on the solid-state fluorescence properties and on the geometry optimization by DFT, together with a careful examination of the data published by Zhang et al. suggest that the steric hindrance induced by the bulkydiphenylboron group may prevent this excimer formation, thus suppressing the sensitivity of this class of compounds to the shearing stress induced upon grinding.

Conclusions

In conclusion a diphenyl-boron ß-diketonate complex DPB-Ester was synthesized and its photophysical properties were compared in solution and in the solid state to those of its parent free diketone P-Ester and to those of the corresponding difluoro-boron ß-diketonate complex DFB-Ester. This series of compounds show clearly distinct properties. DFB-Ester is strongly fluorescent in solution, and remains fluorescent in the solid-state despite a lower quantum yield. P-Ester is very weakly fluorescent in solution and displays significant AIEE in the crystalline state, which can be explained by the rigidification of the molecule in the crystal lattice. DPB-Ester exhibits a weak fluorescence, both in solution and in the solid-state. TD-DFT calculations reveal that DPB-Ester possesses a lowlying excited state that could play a role in the efficient nonradiative deactivation processes occurring in these two compounds.

For both P-Ester and DFB-Ester, a mechanofluorochromic response was observed upon grinding the crystalline powder in a mortar, whereas for DPB-Ester no fluorescence emission change is observed. The steric hindrance induced by the bulky diphenyl-boron chelating group may prevent excimer formation in the amorphous phase for the DPB-Ester whereas this has been shown to occur at the origin of the mechanofluorochromic shift for DFB-Ester [START_REF] Louis | Polymorphism, Mechanofluorochromism, and Photophysical Characterization of a Carbonyl Substituted Difluoroboron-ß-Diketone Derivative[END_REF] and for related compounds. [START_REF] Zhang | Polymorphism and reversible mechanochromic luminescence for solid state difluoroboron avobenzone[END_REF][START_REF] Zhang | Reversible solid-state mechanochromic fluorescence from a boron lipid dye[END_REF] Thus, comparison of our results with the literature indicates that diphenyl-boron ß-diketonate complexes could be interesting to obtain compounds sensitive to hydrostatic pressure and not to shearing stress (obtained by grinding), provided they are highly fluorescent. Complexation of more electron-rich ßdiketones with a diphenyl-boron group may be a promising way to design such fluorescent complexes.

Fig. 2

 2 Fig. 2 Absorption (in black) and fluorescence emission (in coloured lines), of the P-Ester, DFB-Ester and DPB-Ester compounds in THF solution.  exc= 390 nm. λmax(abs) / nm λmax(em) / nm ε / L.mol -1 .cm -1

  transition 1 (386 nm) a P-Ester transition 2 (333 nm) DFB-Ester, transition 1 (399 nm) a DPB-Ester, transition 1 (415 nm) a DPB-Ester, transition 2 (406 nm)

Table 1

 1

Photophysical properties of the P-Ester molecule and its boron complexes DPB-Ester and DFB-Ester in THF DFT calculations

Table 2 .

 2 Experimental and computed absorption wavelengths.

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 715757 MECHANO-FLUO to C. A.) and from the IDEX ParisSaclay (Ph. D. fellowship to M. L.).

Notes and references

1 J. Xu and Z. Chi, Mechanochromic Fluorescent Materials: Phenomena, Materials and Applications, The Royal Society of Chemistry, 2014.

Conflicts of interest

There are no conflicts to declare.


Text Mining Methods Applied to Mathematical Texts

Yannis Haralambous 1

DECIDE -Lab-STICC -Télécom Bretagne

Abstract. Up to now, flexiform mathematical text has mainly been processed with the intention of formalizing mathematical knowledge so that proof engines can be applied to it. This approach can be compared with the symbolic approach to natural language processing, where methods of logic and knowledge representation are used to analyze linguistic phenomena. In the last two decades, a new approach to natural language processing has emerged, based on statistical methods and, in particular, data mining. This method, called text mining, aims to process large text corpora, in order to detect tendencies, to extract information, to classify documents, etc. In this paper we present math mining, namely the potential applications of text mining to mathematical texts. After reviewing some existing works heading in that direction, we formulate and describe several roadmap suggestions for the use and applications of statistical methods to mathematical text processing: (1) using terms instead of words as the basic unit of text processing, (2) using topics instead of subjects ("topics" in the sense of "topic models" in natural language processing, and "subjects" in the sense of various mathematical subject classifications), (3) using and correlating various graphs extracted from mathematical corpora, (4) use paraphrastic redundancy, etc. The purpose of this talk is to give a glimpse on potential applications of the math mining approach on large mathematical corpora, such as arXiv.org.

Mathematics and natural language

The goal of "Data mining" (or "Knowledge discovery in databases") is to extract potentially useful and previously unknown knowledge from huge amounts of data. According to [START_REF] Ananiadou | Text Mining for Biology and Biomedecine[END_REF], [START_REF] Hearst | Untangling Text Data Mining[END_REF] makes a distinction between text mining, which is the process of discovering and extracting knowledge from unstructured data, and data mining, which discovers knowledge from structured data. Under this view, text mining comprises three major activities: information retrieval, to gather relevant texts; information extraction, to identify and extract a range of specific types of information from texts of interest; and the usual mining algorithms, to find associations among the pieces of information extracted from many different texts.

But what is "unstructured data"? It is data with more-or-less hidden structure that has to be extracted, to be useful. Also there are so many different ways of extracting structure that it can be merely considered as an interpretation among several others, and the way your interprete your data depends on the final application you have in mind. A typical example of unstructured data is text in natural language.

And how about mathematical text? Consider the following six statements:

1. "There is no triplet of positive nonzero integers for which the sum of the cubes of the first two is equal to the cube of the third." 2. "The equation a 3 + b 3 = c 3 has no solution in the set of positive nonzero integers."

3. "(a, b, c) ∈ A ⊂ N 3 , a 3 + b 3 = c 3 ⇒ A = {(0, 0, 0)}." 4. "∀p ∀q ∀r (sum(cube(p), cube(q)) = cube(r)) ∧ inN(p) ∧ inN(q) ∧ inN(r) |= (p = 0 ∧ q = 0 ∧ r = 0)." 5
. "Fermat's Last Theorem is true for n = 3." 6. "Le dernier théorème de Fermat est vrai en degré 3." They carry more-or-less the same knowledge and natural language is used at various degrees and in different ways: in the first case the statement is given in natural language only, the second is a mixture of a formula and text (actually a textual statement on the solutions of the equation in the formula), the third is written in standard mathematical notation (assuming knowledge of notation N for the set natural numbers), the fourth is a representation of statement in FOL (assuming predicates "sum," "cube," and "inN," the latter meaning "belongs to N"). The last two refer to a known result, namely Fermat's last theorem: note that in English the capitalization of words "Last Theorem" shows the fact that they constitute a named entity.

Various degrees of language formality

[15] defines the word flexiform as an adjective to describe the fact that a representation is of flexible formality, i.e., can contain both informal (i.e., appealing to a human reader), and formal (i.e., supporting syntax-driven reasoning processes) means. According to [START_REF] Lange | Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration[END_REF], there are many steps between "informal" and "formal," Informality does not necessarily contradict rigorous style, and symbolic notation is not necessarily formal." and also "Rigorous natural language, often called "mathematical vernacular," has the potential to be understood by a machine."

Besides flexiform and "rigorous" language there is also "controlled" (a language between formal and natural, with specific constraints, like the simplified versions of English used for technical documentation, which can be automatically translated to other language without information loss, cd. 1.2) and "specialized" (= using the jargon of a given scientific or technical discipline).

But what about mathematical text? Note that in a sentence like "A is abelian," the symbol A acts as a noun and has the syntactic role of a NP. It is denoting a mathematical object (probably a group), given earlier in the text. We deduce that (flexiform) mathematical text can be analyzed by traditional NLP methods: morphology, syntax, semantics, pragmatics. There have been two important works in this area: [START_REF] Baur | Syntax und Semantik mathematischer Texte[END_REF] and [START_REF] Zinn | Understanding Informal Mathematical Discourse[END_REF]:

1. [START_REF] Baur | Syntax und Semantik mathematischer Texte[END_REF] uses the HPSG (head-driven phrase structure grammar) approach for describing syntax and λ-DRT (λ-discourse representation theory) for semantics. 2. [START_REF] Zinn | Understanding Informal Mathematical Discourse[END_REF] mentions a "parser module," for POS tagging and syntactic analysis, and then also uses DRT for semantics.

Both [START_REF] Baur | Syntax und Semantik mathematischer Texte[END_REF] and [START_REF] Zinn | Understanding Informal Mathematical Discourse[END_REF] aim to rigorously analyze mathematical text in order to use theorem provers subsequently. This corresponds to the "symbolic" approach to NLP. Unfortunately they provide only limited results: Let us recall the differences between symbolic and statistical NLP methods (from [START_REF] Liddy | Natural Language Processing[END_REF]):

-[
-"Symbolic approaches to NLP perform deep analysis of linguistic phenomena and are based on explicit representation of facts about language through wellunderstood knowledge representation schemes and associated algorithms." ". . . A good example of symbolic approaches is seen in logic-or rule-based systems. In logic-based systems, the symbolic structure is usually in the form of logic propositions. Manipulations of such structures are defined by inference procedures that are generally truth preserving. Rule-based systems usually consist of a set of rules, an inference engine, and a workspace or working memory. Knowledge is represented as facts or rules in the rule-base. The inference engine repeatedly selects a rule whose condition is satisfied and executes the rule." -"Statistical approaches employ various mathematical techniques and often use large text corpora to develop approximate generalized models of linguistic phenomena based on actual examples of these phenomena provided by the text corpora without adding significant linguistic or world knowledge. In contrast to symbolic approaches, statistical approaches use observable data as the primary source of evidence."

At this point we may ask how to deal with mathematical text. Shall we aim for an "approximative generalized model" of mathematical text? Are we going to "approximate Bourbaki"? Isn't that heresy?

Let us (re)view possible strategies.

Possible strategies for flexiform mathematical text

-Strategy #1 (for the brave): Use a controlled language from the very beginning. -Strategy #2: Use XML markup to structure as much as possible.

-Strategy #3: Use a visual language to structure as much as possible.

-Strategy #4: Use statistical methods.

Strategy #1: Use a controlled language According to Wikipedia, "Controlled natural languages (CNLs) are subsets of natural languages, obtained by restricting the grammar and vocabulary in order to reduce or eliminate ambiguity and complexity. . . . [Some of them] have a formal logical basis, i.e., they have a formal syntax and semantics, and can be mapped to an existing formal language, such as first-order logic. Thus, those languages can be used as knowledge-representation languages, and writing of those languages is supported by fully automatic consistency and redundancy checks, query answering, etc." This is, for example, the case of Mizar [START_REF] Rudnicki | An overview of the Mizar Project[END_REF] and the Journal of Formalized Mathematics, which is a special, esthetically beautiful way of writing mathematics, even though it is not (yet) the way to write a paper or a thesis, for most of us. Citing [START_REF] Gow | Closing the Gap Between Formal and Digital Libraries of Mathematics[END_REF], "Most users of mathematics are not versed in formal mathematics and, even if they were, it is not yet clear that it could support their activities adequately."

Strategy #2: use XML markup This is the OMDOC approach [START_REF] Kohlhase | An Open Markup Format for Mathematical Documents[END_REF]. A mathematical document is an XML file. The mathematical formulas are represented in OpenMath (which is content-based, as opposed to formal systems that are semantic-based and allow proof-checking), but there is also provision for formal proofs in the system. L A T E X to OMDOC translation can be done (almost) automatically. It is an ongoing project for the arXiv corpus [START_REF] Ginev | An Architecture for Linguistic and Semantic Analysis on the arXMLiv Corpus[END_REF]. An IDE for OMDOC, Sentido [START_REF] González-Palomo | Sentido: an authoring environment for OMDOC[END_REF], allows visualization of formula structure.

Strategy #3: use a visual language This is the MathLang [START_REF] Kamareddine | Restoring Natural Language as a Computerised Mathematics Input Method[END_REF][START_REF] Kamareddine | Narrative Structure of Mathematical Texts[END_REF] approach: in T E Xmacs, the user places boxes around parts of his text and formulas. Syntactical roles of text blocks or of formula parts are treated in the same way: a + 0 equals a is equivalent to a + 0 = a . The author of the text has to manually annotate (= put boxes around) his text and formulas.

Strategy #4: use statistical methods Many NLP methods exist to process ordinary text.

A morphological analyzer (or "POS-tagger") will parse, for example, the sentence The world faces a financial crisis as The/DT world/NN faces/VBZ a/DT financial/JJ crisis/NN but, doing this, will also detect, that "faces" can be the plural of noun "face" or the 2nd person of singular of present of verb "to face," with given probabilities.

A syntax parser will produce the most likely tree A semantic annotator will match the meaning of words in semantic resources, for example in WordNet, with given probabilities... There are also several other levels of processing: pragmatics, requiring knowledge of the world; discourse processing; anaphora resolution, etc.

Many mathematical structures has been used to modelize these phenomena (starting from finite state automata and probabilistic models and going all the way to fractals, quantum mechanics and high energy physics...).

To handle flexiform mathematical text, various strategies could be used:

1. Adapt existing NLP tools. 2. Adapt corpora, i.e., convert formulas (or geometric constructions, etc.) into natural language text. For formulas, this can be done, for example, by converting to L A T E X, and then using ASTER [START_REF] Raman | An audio view of L A T E X documents[END_REF] or some similar system. 3. Use a hybrid approach (adapt both corpus and tools).

Here is an example of existing work in that direction: [START_REF] Watt | Mathematical Document Classification via Symbol Frequency Analysis[END_REF] uses frequencies of mathematical symbols as features for document classification. This is the bag-of-words approach, restricted to mathematical symbols. Why not combine features such as symbols, words or even, as we will see, terms? Advantages of this approach: no knowledge resource needed, symbols are easy to detect and count. Disadvantages: semantic ambiguity (which could be easily avoided by looking into surrounding terms), working on document level only.

In the remaining part of this paper we will give suggestions about how to process mathematical text using NLP methods.

Suggestion 1: Use terms instead of words

First of all, let us ask the fundamental question: "What is a term?"

In OMDOC [START_REF] Kohlhase | An Open Markup Format for Mathematical Documents[END_REF], a (technical) term is "a phrase representing a concept for which a declaration exists in a content dictionary." Terms in OMDOC are tagged by the user, termhood property is binary.

For [START_REF] Frantzi | The C-value/NC-value Method of Automatic Recognition for Multi-word Terms[END_REF] also, "terms are linguistic representations of concepts," but, nevrtheless, one can define a termhood property which is a positive scalar. Compare for example: mobile phone / mobile / red mobile phone / red telephone automorphism / canonical automorphism / trivial automorphism "Mobile phone" is the original name of a wireless telephone for outside use. This is a complex term. For brevity, the term "mobile" has progressively become an alternative version of "mobile phone," in other words: the "phone" part has become redundant. When we add the adjective "red" we do not create a new term since red color is simply a variable property of mobile phones, and red color does not change the technical specifications of the device. Nevertheless "red telephone" is a term when we mean the specific telephone line between Washington and Moscow.

In the same way, a "canonical automorphism" is a special kind of automorphism and hence is clearly a term, but "trivial automorphism" can be a specific automorphism (for example, the identity map) or simply an automorphism whose description is very simple. Therefore we can state that the termhood property of "trivial automorphism" is less than the one of "canonical automorphism."

The phenomenon when combining two terms produces a term with a completely different meaning is called a "red herring" (which is neither red, nor a herring). There are very few red herrings in mathematics, an example: child drawings, which are not (only) drawings, and certainly not drawn by children.

We see that termhood depends on the term and on the context. There are several ways of calculating it, here is how.

C-Value

For [START_REF] Frantzi | The C-value/NC-value Method of Automatic Recognition for Multi-word Terms[END_REF], a term is a string of words of the following form:

((Adj|Noun)+ | ((Adj|Noun)*(NounPrep)?)(Adj|Noun)*)Noun
A term can be nested in longer terms, take for example "graded algebra" which is contained in "differential graded algebra," contained in "commutative differential graded algebra," etc.

The C-value for termhood is defined as follows:

C-value(a) = log 2 (|a| + 1) • f (a) if a not nested, log 2 (|a| + 1) • f (a) -1 #(Ta) b∈Ta f (b) otherwise,
where f is frequency in the corpus, | . | length (in words), T a the set of terms containing a.

NC-Value

As defined above, C-value depends strictly on frequencies in the corpus. Another characteristic of terms is that they are compatible with specific modifiers and not with others. For example, in math, multi-word units starting with "easy" have lower chances of being terms, those starting with "simplicial" have strong chances of being terms. The NC-value is defined as follows in [START_REF] Frantzi | The C-value/NC-value Method of Automatic Recognition for Multi-word Terms[END_REF]: First, the weight of a word w (adj, noun or verb, preceding or following a term) is defined as Weight(w) = t(w) n where t(w) is the number of terms w appears with, and n the number of terms considered.

Then NC-value(a) = 0.8 C-value(a) + 0.

2 b∈Ca f a (b) • Weight(b)
where C a is the set of context words of a and f a (b) the frequency of b as context word of a.

Term variation

While representing the same concept, terms can have several variants. Term variation can be:

orthographic: "pull-back" vs. "pullback," "neighbor" vs. "neighbour," morphological: singular/plural, "Riemann space" vs. "Riemannian space," lexical: "Abelian" vs. "commutative," "epimorphism" vs. "surjective morphism," structural: for example, possessive usage of nouns using prepositions, like in "isomorphism of groups" vs. "group isomorphism," abbreviational: "log" vs. "logarithm" (even outside formulas), compositional: "Wahscheinlichkeitstheorie" vs. "Theorie der Wahrscheinlichkeit," acronymic: "GCD", "CW-complex" vs. "closure-finite weak-topology complex," naming: "Banach space" vs. "complete normed vector space," reductive: "complex conjugate function" vs. "complex conjugate" (substantification), etc.

Some of these need extra resources, others can be detected by simple transformation rules. [START_REF] Nenadić | Enhancing automatic term recognition through recognition of variation[END_REF] handles term variation as follows:

1. acronym acquisition 2. inflectional normalization 3. structural normalization 4. orthographic normalization and then uses the usual method for (N)C-value calculation, after having merged variations into a standard form (called canonical representative).

Term and symbol interaction

In math, very often terms are used to describe symbols, or, inversely, symbols are used to denote objects represented by terms. [START_REF] Wolska | Using Discourse Context to Interpret Object-Denoting Mathematical Expressions[END_REF] uses word-sense disambiguation methods to map symbols (in fact, simple expressions) to terms. Instead of recognizing terms, they are taken from a pre-existing taxonomy. "Simple expressions" are atomic identifiers with optional superscripts and/or subscripts. As we see, context can help both for calculating termhood as for finding relations between terms and symbols denoting them.

Word-sense disambiguation methods as in [START_REF] Wolska | Using Discourse Context to Interpret Object-Denoting Mathematical Expressions[END_REF] can also profit from the fact that there are (domain-dependent) notational conventions so that we can calculate probabilities of term/symbol matches and inject them into the learning algorithm.

Notational conventions depend on domain but also on language: "Sei K un Körper," "Let F be a field," "Soit C un corps," . . . Also, some notations become domain-independent (R, C, ...), others not (P can denote "projective" or "probability").

This raises the question: Should we use placeholders for symbols? If yes, we loose knowledge on notational conventions. If no, we have less chances of identifying formulas. We suggest using a hybrid approach. One can build the distribution of denotations given a symbol, and then attach it to placeholders.

Suggestions about dealing with context

Up to now, the context was simply a set of words neighboring a term, or a scientific domain. But terms (and symbols) belong to documents. And documents are events in the real world : as every event they have a cause (the authors), an intention (to spread knowledge, to become rich and famous), a timeline (they are created, modified), a (physical or virtual) support (a journal), etc. Knowledge of 1, 3 and 4 can contribute to disambiguate a term or a symbol.

Authors are objects belonging to various graphs: co-authorship, citations, social networks. . . Timeline is also important since terms have an organic life span: they are born, they grow by getting used, they interact, sometimes even procreate, and sometimes die. Finally, journals sometimes alter notations or terminology.

Therefore we suggest not to consider a mathematical text as an abstract piece of pure wisdom, living in some Platonic world. After all, mathematical texts are written by people. People have their own styles, and tend to reuse the same notations and terminologies. People communicate with other people. They share ideas and knowledge. As ideas and knowledge are communicated by notation and terminology, these are (often) shared as well.

When extracting terms and finding term↔symbol matchings, a to-do list would be:

find out who wrote the text, find and analyze his/her other writings, find his/her "friends," co-authors, the people he/she cites, find and analyze their writings (especially those cited) check the timeline of all documents considered.

3 Suggestion 2: Use topics instead of subjects

Keywords, subjects

Most math papers contain a list of keywords and a subject classification, in some scheme (for ex. AMS MSC 2010, see also [START_REF] Lange | Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web Integration[END_REF]). For MSC, AMS suggests using one primary classification and several secondary ones. Keywords are freely chosen by the author. They express the author's opinion about the important concepts in his/her paper. The advantages of this approach are: the author is best suited to know the important keywords and subjects of his/her paper. And the disadvantages: they are global despite the fact that different parts of the paper may require specific metadata; there is subjectivity involved in their choice (difference between what our paper is about and what we would like it to be about).

Topics

In the text mining world, a topic is a statistical model for discovering themes occurring in a document collection. Not only can we have several topics per document, or per document section, but we can also study their interrelations, and construct topic hierarchies or graphs. Topic models are very popular because they apply not only to the document paradigm, but also to form recognition in images, bioinformatics and other fields.

Were are some statistical models using topics:

Mixture of unigrams The simplest topic model is the mixture of unigrams [START_REF] Blei | Latent Dirichlet Allocation[END_REF].

For each document (among M ) choose a topic z using some distribution p(z), and then generate N words w independently from the conditional multinomial p(w | z). The joint probability of document w and topic z is

p(w, z) = p(z) N n=1 p(w n | z).
In this model we use only one topic per document.

Latent Dirichlet Allocation A more elaborate model is Latent Dirichlet Allocation [START_REF] Blei | Latent Dirichlet Allocation[END_REF]. For each document (among M ) choose a Dirichlet distribution of K topics (θ * ) based on parameters (α * ). Then for n = 1, . . . , N choose:

a topic z n from a multinomial distribution with parameters (θ i ), a word w n from p(w n | z n , β), a multinomial distribution conditioned on z n , with parameters (β * ).

The joint probability of document w (N words), set of N topics z and topic mixture θ is

p(w, z, θ | α, β) = p(θ | α) N n=1 p(z n | θ)p(w n | z n , β).
In this model we have many topics per word and per document, but there is no correlation between topics.

Pachinko Allocation Model

The Pachinko Allocation Model [START_REF] Li | Pachinko Allocation: DAG-Structured Mixture Models of Topic Correlations[END_REF] is an enhanced version of LDA. A directed acyclic graph is constructed, whose edges are topics at various levels and words. Vertices represent dependance.

r … … S S … … … S … … V V (a) Dirichlet Multinomial … … (b) LDA (c) Four-Level PAM V (d) Arbitrary PAM
The Pachinko model is generated as follows: Suppose we have S topics. For each document, sample θ t1 , . . . θ ts from Dirichlet distributions of parameters α 1 , . . . , α s . Each θ ti is a multinomial distribution of topic t i overs its children on the graph. For each word w in the document:

sample a topic path z w of length L w . Each z w,i is a child of z w,i-1 and is sampled according to the multinomial θ zw,i-1 , sample a word w from θ z w,Lw .

The joint probability of a document d, the topic assignments z and the multinomials θ is

p(d, z, θ | α) = s i=1 p(θ ti | α i ) w Lw i=2 p(z w,i | θ zw,i-1 )p(w | θ z w,Lw ) .
Joint topic models for text and citations [START_REF] Nallapati | Joint Latent Topic Models for Text and Citations[END_REF] introduce a model (called Link-PLSA-LDA) that combines text topics and citations. The idea is that citations can contribute in capturing topicality of the document. In this figure, citation probabilities have to be multiplied by 0.0015. Intra-topic citation probability is high. [START_REF] Boyd-Graber | Syntactic Topic Models[END_REF] introduce a model that discovers topics that are both semantically and syntactically coherent. This model uses LDA for the semantic part and FTIC (Finite tree with independent children [START_REF] Finkel | The infinite tree[END_REF]) for the syntactic part. For the Syntactic Topic Model, the observed data are documents, each of which is a collection of dependency parse trees.

Syntactic topic models

Suggestions

Using topic models, one could classify math documents by topics, calculated upon terms.

calculate topic also on the section level, and combine global and local results.

use syntactic topic model to capture dependencies between symbols and terms denoting them, etc. use joint topic model for text and citation, to include information on cited documents (particularly important in mathematics). investigate correlation between metadata introduced by authors and topics obtained.

use Pachinko topic model and investigate correlation between mathematical domains (in an Algebraic Topology paper, will we find topics "algebra" and "topology," and how will they be interrelated?). create the hierarchical (dynamic) graph of mathematical topics.

And here are some potential applications:

-One can consider a paper as a vector in the space of topics. In this space we semantic/topical similarity. -An author would be the sum of papers he/she wrote (potentially weighted by "importance"). Finding people working in the same domain would be projecting their vectors upon topics of the domain and calculating their cosine.

-Social networks and research teams are graphs of people. They become graphs in the space of topics. One could investigate correlations between graphical measures and topical ones. -Journals and conferences can be considered as the sums of vectors of papers they publish. One could investigate their evolution over time, the correlation between their papers, etc.

"Gehirnsturm und Drang," open questions

Is there a way to infer quality of a paper using statistical methods? And how about readability? Can we build a service which will recommend papers for reading, based not only on topical and temporal proximity, but also on potential fertilization?

Could we keep track of everything you read and write (incl. drafts, emails, chats, etc.), of everything your friends/colleagues read and write, etc., to increase the quality of this recommendation?

Mathematics are used in other disciplines. Topics could allow us to detect parts of mathematics that have been applied elsewhere as well as those that haven't yet. Surveying them could give ideas for future work, PhDs. . . Can topics complement the Mathematician's global intuition on the field?

4 Suggestion 3: Extract graphs from corpora 4.1 The structure of mathematical corpora Some blocks of mathematical text are mainly intradocumental : proofs, lemmas, corollaries, acknowledgments. . . and others are of interdocumental nature: conjectures, definitions, theorems. They can be new statements, in which case they may obtain the names of their authors. Or they may be taken from other documents, in which case, besides their "names" they should contain citations. Or they may simply be mentioned more-or-less explicitly. . . One can build a digraph of these statements, oriented by temporality or citation direction.

The graph of interdocumental statements is only one of the many graphs one can build out of mathematical corpora:

graph of co-authors, institution sharing (use also the Mathematics Genealogy Project) graph of citation sharing graph of topic/subject/keyword sharing graph of denotation sharing (same symbol for same meaning) graph of acknowledgment sharing, etc.

mathematical ontologies. . . All of them can be naturally oriented (either by time or by intrinsic properties), and derived graphs can be built: authors, institutions, topics, symbols, funded projects, etc.

There is a research area called graph mining. It allows us to define measures and calculate distances between graphs. One can find vertices or subgraphs with specific good properties (for example: communities in social networks), one can also find frequent substructures. One can learn graph grammars using machine learning methods.

Under this view, a document is not an isolated node anymore, but the center of a neighborhood in the various graphs we mentioned. And the same goes for authors, institutions, topics, etc.

Not to mention that even if mathematics are eternal, mathematical corpora/communities evolve, and this evolution can be represented by successive versions of graphs.

Typical interaction between graphs and natural language

Geometric constructions (cf. [START_REF] Quaresma | A XML-Format for Conjectures in Geometry[END_REF]) can be modelled as knowledge base graphs:

points, lines, etc. are instances of concepts in a geometry ontology, relations between them are instances (for example RDF triplets) of relations in that ontology.

These graphs can be compared, patterns found, etc. They can be converted into natural language (with a lot of redundancy); Redundancy can be avoided by considering inference relations. Both natural language and graphs can be used for querying.

Suggestion 4: Use paraphrastic redundancy

On Fig. ?? one can see the same Chinese sentence translated in many different ways into English. Statistical NLP methods can used the various translations of the same sentence to improve accuracy of knowledge extraction. Traditionally a sentence "having the same meaning" as a give one, is called a paraphrase, this is why we call information included in various translations of the same sentence, "paraphrastic redundancy." How can this be applied to mathematics? Stating a theorem is like translating a mathematical fact into flexiform text. There are dozens of books on the same topic (Algebra, Analysis, Topology, etc.). We could improve knowledge on specific theorems or parts of theories, by comparing the various versions of the same statement in several documents, using the corresponding NLP methods.

Conclusion

While waiting for rigorous systems to analyze in depth hundreds of thousands of mathematical texts, a lot can be done by processing mathematical text using natural language tools. The results will always be "statistical," ambiguous, approximative, but "approximative" does not mean "unreliable" nor "useless." For some applications, like ("fuzzy") searching, classifying, recommending, surveying, detecting tendencies or plagiarism, etc., this may be perfectly sufficient. Statistical Mathematical Language Processing, or Math Mining, may be a useful research area for the near future.

  morphological analysis (if other trees are possible, they will be give as well, with given probabilities).

Fig. 1 .

 1 Fig. 1. Taken from [11].


Amar Abane

Paul Mühlethaler

Samia Bouzefrane

Mehammed Daoui

Paul Muhlethaler

Abdella Battou

Towards evaluating Named Data Networking for the IoT: A framework for OMNeT++

come   L'archive ouverte pluridisciplinaire

Introduction

Current IoT systems are deployed on the IP protocol suite that has been used to build the worldwide Internet decades ago. However, by involving constrained physical devices, the IoT puts the IP model to the test and highlights its limitations. For example, security is still focused on communication channels when the data itself needs to be secured. Moreover, IoT systems need efficient support for resource naming and discovery, which is not easy to deploy with IP in constrained infrastructures. Although dedicated adaptations such as CoAP and 6LoWPAN have been developed, under the hood, matching the IoT vision is still a challenge for IP [START_REF] Shang | Challenges in IoT networking via TCP/IP architecture[END_REF].

While adapting IP for the IoT might be seen as cutting corners, alternative architectures based on the Information Centric Networking (ICN) paradigm promise to natively satisfy emerging Internet applications. One of these architectures is Named Data Networking (NDN) [START_REF] Zhang | Named Data Networking[END_REF]. The NDN project was funded by NSF 1 under the Future Internet Architecture (FIA) program. The main entity in NDN is the content. Networking operations are performed on names, and hosts (without logical addresses) request named-content directly from the network. Native features come along with this principle, such as: communication without establishing end-to-end connections and name-to-address resolution. Moreover, no consumer-provider path or session needs to be maintained, which provides a native support of connection disruption resulting from mobility.

In recent years, many studies investigated the suitability of NDN for the IoT, making NDN more and more powerful. However, important challenges need to be addressed to build a strong NDN-IoT duo. Specifically, integrating NDN in low-end devices, over low-power wireless standards is mandatory due to the degree of freedom allowed by NDN in terms of unbounded name length, packet structure and semantics. NDN-based solutions for the IoT need accurate evaluation, and relative results should be provided further to convince industrials, investors and IP-enthusiasts. Examples of scenarios in which such evaluation can be useful are: NDN-based networking for robots [START_REF] Dauphin | NDN-based IoT Robotics[END_REF], NDN for public safety and tactical networks, NDN for WSN [START_REF] Amadeo | Named data networking: A natural design for data collection in wireless sensor networks[END_REF] and NDN over IEEE802.15.4 [START_REF] Amar Abane | Ndn-over-zigbee: A zigbee support for named data networking[END_REF].

Although some of these scenarios can be evaluated in testbeds, implementations are commonly based on realistic needs, and sometimes do not support certain features or "outlandish" scenarios. However, a pure network simulator does not allow to model internal interactions of a system. Therefore, experimental NDN-IoT design requires the best of both worlds: a tool capable to simulate network protocols and system-level interactions. We introduce in this paper our implementation of an NDN framework for the OMNeT++ [START_REF]OMNeT++[END_REF] simulator 2 . OMNeT++ can be used to simulate wired and wireless networks as well as on-chip networks, and so on. We designed the NDN framework with the following objectives in mind: (i) Evaluating how a design affects NDN entities in a given topology/scenario. (ii) Evaluating packet processing approaches (e.g. compression). (iii) Providing a good visualization of NDN communications at network and system levels for testing and teaching purposes. (iv) Providing an easy-to-use framework to quickly experiment initial NDN parameters without additional implementation, particularly for wireless networks and constrained devices.

The rest of this paper is organized as follows: The first part presents an overview of NDN (Section 2) and our framework design (Section 3). In the second part, the NDN framework is used to study a key aspect of NDN in IoT (Section 4). Section 5 gives a summary with future directions.

Named Data Networking

In NDN, the content is named independently from the producer location. Names are hierarchical, URI-like and formed by a sequence of components separated by slashes. Applications are free to design their own naming scheme as routers do not interpret the whole name. A producer can append components to the initial name to provide more information about the content (e.g. timestamp, sequence number, etc.).

The NDN architecture uses two types of packets: Interest and Data. Each NDN node has three data structures to process packets: (i) a Forwarding Information Base (FIB): associates content names to Face(s) through which the content can be retrieved, (ii) a Pending Interest TLV encoding represents an NDN packet as a collection of sub-TLVs. A TLV block is a sequence of bytes starting with a marker t, followed by its length l and a sequence of l bytes that represent the value.

Our framework design

OMNeT++ is not a native network simulator. However, ready-to-use domain-specific functionalities are provided by frameworks such as INET [START_REF]INET Framework[END_REF] which contains models for the IP protocol stack, link layer protocols, mobility, etc. Therefore, our framework is based on INET.

Overview

The NDN core module is designed as a network layer (NdnL3 ) that implements the network layer interface of INET (INetworkLayer ). Within an NDN host, NdnL3 is connected to the upper layer; expected to be the application layer, and the lower layer consisting on wired/wireless network interfaces. However, the NDN network layer can run on top of (or beside) IP with minimal adaptations. Following the modular approach of OMNeT++, we designed the NDN entities as independent modules included in the NdnL3 module (Figure 2).

OMNeT++ modules use message passing through connected Gates to communicate. The notion of Gate provides a native abstraction of the Face concept; making the NdnL3 module interact with upper (i.e. application) and lower (i.e. link) layers in a transparent way.

Due to space limitations, we report only the main components we implemented and a description of the packet representation we used.

Host and Application Modules

NDN hosts: To represent NDN-based IoT devices, a base NDN wireless host (NdnWire-lessHostBase) is implemented. It includes the typical wireless host components and the NDN layer (NdnL3 ). This module is used directly as a relay node since it does not include any application (Figure 3). By extending NdnWirelessHostBase, a typical IoT end-device is created (NdnWirelessHost) with consumer and/or producer applications. Figure 4 shows a typical NDN end-device with applications (NdnWirelessHost).

NDN layer Modules

Pending Interest Table : IPit is an abstraction of the PIT. It includes a typical entry which stores the following information: (i) A copy of the forwarded Interest, (ii) Incoming Face(s) of the Interest, (iii) Face(s) to which the Interest is forwarded, (iv) Expire time of the entry and, (v) Source MAC. NDN-to-MAC mapping is discussed in the next Section.

IPit provides the following functions: (i) Lookup, (ii) Create, (iii) Remove, (iv) Update, and (v) Print.

On Interest timeout, IPit implementations can emit a signal with a notification that includes the Interest, its incoming Face(s) and source MAC address(es).

A base implementation (PitBase) of IPit is provided with the following parameters: (i) interestLifetime. If no value is provided, the lifetime value of the Interest is used instead. (ii) maxSize. When the PIT is full, no Interest can be forwarded.

Forwarding Information Base: IFib is an abstraction of the FIB. It includes a typical entry which keeps the following information: (i) Name prefix, (ii) Face(s), (iii) Destination MAC address, (iv) Expire time of the entry (optional).

IFib provides the following functions: (i) Lookup, (ii) Create, (iii) Remove, (iv) Update, (v) Register prefix; used to create an entry to the local application and, (vi) Print.

When a prefix expires, IFib implementations can emit a signal with a notification that includes the prefix and its corresponding Face(s) and MAC address(es).

A base implementation of IFib (FibBase) is provided with the following user parameters: (i) entryLifetime and, (ii) maxSize.

Content Store: ICs is an abstraction of the CS. It includes a typical entry which stores the following information: (i) A copy of the Data packet and, (ii) A stale flag to manage the freshness of the Data [START_REF] Ndn Team | NDN Packet Format Specification[END_REF].

ICs provides the following functions: (i) Lookup, (ii) Add, (iii) Remove, (iv) Update freshness and, (v) Print.

A base implementation of ICs (CsBase) is provided. It uses the FIFO replacing policy and the cache size can be defined (maxSize).

Experimental unit: Some internal NDN processes that can be imagined to improve efficiency cannot be assimilated to the forwarding strategy, and are not related to native NDN entities (i.e. PIT, FIB, CS). Packet compression and Fuzzy logic NDN forwarding [START_REF] Mastorakis | Experimentation with fuzzy interest forwarding in named data networking[END_REF] can be cited as examples of such processes. To provide a clean and flexible way to implement such experimental design without disturbing the NDN base implementation, an eXperimental Unit module (XU) is included.

This module is designed to evaluate future AI-based operations such as semantics extraction from names, intelligent routing, etc.

IXu is an abstraction of the eXperimental Unit. The following signals can be emitted by IXu implementations to notify on processing progress: (i) Packet received, (ii) Packet processing begin, (iii) Packet processing end, (iv) Packet processing error and, (v) Packet processing success.

Forwarding: The forwarding module is the main component of our design and it is connected to PIT, FIB, CS and XU.

The forwarding process is abstracted in the IForwarding interface which provides the following functions: (i) processLLInterest; process Interest coming from lower layer. (ii) processLL-Data; process Data coming from lower layer. (iii) processHLInterest; process Interest coming from higher layer (i.e. application layer). (iv) processHLData; process Data coming from higher layer. (v) The communication with PIT, FIB, CS and XU can be either through message passing or direct module access as provided by OMNeT++. However, the message passing communication allows to model time processing of each module separately and independently. The forwarding module implementation is intended to subscribe to PIT, FIB and XU signals in order to handle NdnL3 events.

The current framework implementation includes a base forwarding strategy (Forwarding-Base). It supports the following parameters: (i) ndnMacMapping; code of the NDN-to-MAC mapping to use. (ii) cacheUnsolicited ; whether to cache unsolicited Data packets or not. (iii) forwarding; whether to forward Interests (router) or not (end-device).

Messages and packets

To represent NDN packets, we use the OMNeT++ message representation in order to provide an easy way to create packets and access their fields. However, for packet-related evaluation, a set of tool functions are provided to generate the TLV representation and to compute the actual size of a packet from the OMNeT++ packet representation.

For evaluation purposes, Interest and Data have a common superclass (NdnPacket) which includes non-NDN fields. These fields are of two types: (i) inherited from OMNeT++ Packet class [START_REF] Ltd | OMNeT++ Simulation Library[END_REF] and, (ii) additional fields that include a sequence number, a type (i.e. Interest or Data), a hop count and other fields that help for statistic or experimental implementations.

Use case: NDN over low-rate wireless technology

In this section, we use our framework to study an aspect of integrating NDN over a low-power low-rate wireless technology. Considering the IEEE802.15.4 standard, one of the enablers of the IoT, fundamental questions need to be explored and revisited for NDN as described in the following.

NDN wireless forwarding strategies

In one hand, wireless forwarding strategies are generally based on a flood-and-learn mechanism [START_REF] Amadeo | Forwarding strategies in named data wireless ad hoc networks: Design and evaluation[END_REF][START_REF] Michael | Listen first, broadcast later: Topology-agnostic forwarding under high dynamics[END_REF][START_REF] Wang | Rapid traffic information dissemination using named data[END_REF]. Although lightweight mechanisms for constrained devices are rare in literature, a simple one has been proposed in [START_REF] Baccelli | Information centric networking in the iot: Experiments with ndn in the wild[END_REF]: Reactive Optimistic Name-based Routing (RONR) and detailed further.

On the other hand, different approaches may be used to map learned information, which are in form of NDN names, to MAC addresses. This question has been investigated in [START_REF] Peter Kietzmann | The Need for a Name to MAC Address Mapping in NDN: Towards Quantifying the Resource Gain[END_REF] where different mappings are discussed: (i) broadcast for Interest and Data (IBDB), (ii) broadcast for Interest and unicast for Data (IBDU), (iii) unicast for Interest and broadcast for Data (IUDB), (iv) unicast for Interest and Data (IUDU). The study is particularly focused on CPU usage and system wakeup comparison.

To show a typical use of our framework, we simulate a wireless forwarding mechanism to enhance the NDN-to-MAC exploration cited above, by studying both network performance and device state in static and mobile scenarios, under different parameters. Therefore, we combined RONR with a related solution for wireless NDN forwarding [START_REF] Amadeo | Forwarding strategies in named data wireless ad hoc networks: Design and evaluation[END_REF]. The implemented mechanism (ForwardingBase) works as follows:

1. When a consumer issues an Interest for unknown content, an Interest flooding is performed by the relay-nodes until reaching the producer. Interest retransmissions are randomly delayed to minimize the risk of collisions.

2. After the first Data is sent back by the producer, each relay-node that solicited the Data creates a temporary FIB entry with the corresponding prefix.

3. Subsequent Interests are forwarded only by the nodes that have the corresponding entry in the FIB.

4. The Interest flooding phase is performed again when an entry is expired or after an Interest timeout (a flood flag is used by the consumer).

Simulation and results

We configured a local wireless network composed of 9 fixed routers, 5 mobile producers and a mobile consumer, in an area of 100m x 100m. Each producer has a unique content of 30 pieces that are randomly requested by the consumer. Figure 5 shows the OMNeT++ visualization of the simulated network. Caching is disabled in order to study the behavior in a complete consumer-producer path. The rest of the simulation parameters are reported in However, contrary to what we might expect, IUDU does not cause the lowest number of collisions, for both mobile and static scenarios as shown in Figure 9. The lowest number of collisions is achieved by the IUDB mapping, followed by IUDU. This can be explained by the fact that Data broadcast prevents some nodes from unnecessary Interest retransmission. During the flooding phase, nodes can overhear Data broadcast, while Interest unicast phase reduces the number of potential forwarders in consumer-producer direction. IUDB can be considered as a good compromise when flood-and-learn techniques are used, especially in mobile scenarios.

We observe that changing from one mapping to another has the same effect in mobile and static scenarios in terms of Interest duplication, PIT size, PIT lookups and collision number. However, performances are slightly higher in the static case, as we can expect (Figure 10 and11).

IUDU achieves the best performance in the static scenario (Figure 10 and11). This indicates that the Interest broadcast only causes extra computation without improving the performance. The study in [START_REF] Peter Kietzmann | The Need for a Name to MAC Address Mapping in NDN: Towards Quantifying the Resource Gain[END_REF] also reports that IUDU mapping provides the highest Interest satisfaction rate in static scenario. However, in the mobility scenario, IUDU only achieves the second better performance, while the best one is achieved by the IBDU mapping. This is certainly due to the exploratory nature of broadcasting Interests. Although this feature is not so helpful in static configurations, it helps to find the content producer when the hosts are mobile. The proposed NDN framework is intended to be used to evaluate NDN for low-end IoT solutions, by focusing on both local wireless networks and system state of constrained devices. Although for some evaluations, a simulation cannot serve as a suitable substitute for a realworld testbed, the proposed framework can quickly provide accurate results for different aspects of the solution. For example, it can be used to evaluate the following aspects: (i) packet fragmentation, encoding/decoding and other NDN packet processing, (ii) secured and trust-based wireless forwarding, (iii) hybrid NDN-IP gateways and, (iv) impact of NDN names to MAC addresses mapping approaches.

The current implementation does not support all NDN features; the Interest forwarding implemented is based on names only, while hop limit and lifetime should be considered also. As a future work, we plan to improve the framework with two main features: (i) A fullycustomizable forwarding strategy that take the most of OMNeT++ module parameters. (ii) A memory usage model for NDN entities, as well as accurate packet processing time measurement, in addition to the mobility and energy consumption models provided by INET.

( 3 )

 3 When the producer receives the Interest, it sends back a Data packet containing the requested content, or a part of it. The Data packet follows the reverse path of the Interest following traces left in the PIT of each router. (4) When the Data packet reaches a router, it is forwarded to the Faces from which the corresponding Interests were received. After that, the router removes the entry from the PIT, and stores the recent Data packet in its CS.Figure1summarizes the Interest and Data processing steps in a typical NDN node.

Figure 1 :

 1 Figure 1: NDN forwarding process

Figure 2 :

 2 Figure 2: NDN network layer (NdnL3)

Figure 3 :

 3 Figure 3: NDN relay node (NdnWirelessHostBase)

Figure 4 :

 4 Figure 4: NDN end-device (NdnWirelessHost)

Each simulation is executed with 7

 7 random seeds and the average results are reported. The following metrics are measured: (i) number of duplicated Interests received by relay-nodes, (ii) mean/max PIT size and PIT lookup in relay-nodes, (ii) number of collisions observed by all nodes, (iv) Interest satisfaction rate and Interest-Data RTT measured by the consumer.

Figure 5 :

 5 Figure 5: Simulated scenario

Figure 6 :

 6 Figure 6: Interest duplication

Figure 7 :

 7 Figure 7: PIT size

Figure 8 :

 8 Figure 8: PIT lookups

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure 9: Collision number

Table (

 ( The consumer requests a content by sending an Interest carrying the name of the content (e.g. home/room1/temperature). (2) A node that receives the Interest checks if matching Data already exists in its local CS: if the corresponding Data is found, it is sent back. Otherwise, the router checks if an Interest with the same name is already in the PIT; if so, the new Interest is not forwarded and only the incoming Face is added to the existing PIT entry.

	following steps: (1)
	PIT): stores each forwarded Interest associated to its incoming Face(s), while waiting to
	get the corresponding Data packet. The notion of Face represents a network device interface
	or a local application in an NDN node. (iii) A third optional structure, Content Store (CS) is
	used to store forwarded Data packets to satisfy similar future Interests; which provides native
	in-network caching in NDN.
	Typically, a communication is initiated by the consumer, and operates according to the

If no similar Interest is found in the PIT, the new Interest is forwarded according to the FIB information, and stored in the PIT.

  forwardInterestToRemote; forward Interest to remote Face (e.g. radio). (vi) forward-DataToRemote; forward Data to remote Face. (vii) forwardInterestToLocal ; forward Interest to local application. (viii) forwardDataToLocal ; forward Data to local application. (ix) map-ToMAC ; encapsulate Interest or Data considering the given unicast or broadcast MAC address. (x) onInterestTimeout; when receiving an Interest timeout signal. (xi) onPrefixExpired ; when receiving a prefix expired signal.

Table 1 .

 1

	Parameter	Value
	MAC layer	CSMA with 250kbps data rate
	Communication range	30m
	Interests number	100
	Max Interest retransmissions	3
	Interest send interval	1s
	Interest size	30 kb
	Data size	100 kb
	Mobility model	Random Waypoint

Table 1 :

 1 Simulation parameters

Source code available at https://github.com/amar-ox/NDNOMNeT