Jacques Moret-Bailly
email: jmo@laposte.net

Theory of redshifts

Keywords: Quasars: absorption lines, Radiative transfer, Scattering, ISM:atoms,structure

published or not. The documents may come

Introduction: General properties of Lyman forests of quasars.

This paper uses data selected by astrophysicists to find by spectroscopy properties of the source of light.

Our interpretations are mainly different from theirs by the employment of coherence of light-matter interactions in a collisionless pure gas, as assumed by Einstein in theory which founds lasers [START_REF] Einstein | Zur Quantentheorie der Strahlung[END_REF]. Indeed, in despite of discovery of gas lasers, most astrophysicists continue to follow Menzel's point of view that light-matter interactions in low pressure gas are incoherent while they are perfectly coherent. Optical coherence of interactions between light and low pressure gas is not only a consequence of thermodynamics, but also observed, for instance, in stratosphere: incoherent scattering which produces blue sky disappears where free path time of molecules becomes larger than duration of light pulses making ordinary, temporally incoherent light.

Redshift Z = Z (ν emit ,ν obs ) of a spectral line emitted by a star is defined as ratio of emitted frequency to observed frequency, minus one. G. Burbidge [START_REF] Burbidge | The distribution of redshifts in quasi-stellar objects, nsystems and some radio and compact galaxies[END_REF] found that redshifts Z(n) of quasars he selected obey formula Z(n) = 0, 061n, where allowed values of n are 1, 3, 4, 6, 7, 9, 10, . . . . Burbidge's conclusions were careful because this result required a questionable selection of few observations. Using Rydberg's formula, Burbidge's redshifts Z(3) and Z(4) bring respectively β end γ Lyman frequencies of H atom to α frequency [START_REF] Moret-Bailly | The parametric light-matter interactions in astrophysics[END_REF]:

Z (β,α) = ν β -ν α ν α ≈ (1 -1/3 2 ) -(1 -1/2 2 ) 1 -1/2 2 ≈ 5/27 ≈ (1)
≈ 0.1852 ≈ 3 * 0.0617;

(2)

Z (γ,α) = ν γ -ν α ν α ≈ (1 -1/4 2 ) -(1 -1/2 2 ) 1 -1/2 2 = (3) = 1/4 = 0.025 = 4 * 0.0625; (4)
Absence of n=2 and n=5 in Burbidge's result is not a coincidence if redshifts result from a spectroscopic interaction involving redshifts from β or γ lines of atomic hydrogen to α line. K. G. Karlsson [START_REF] Karlsson | Possible discretization of quasar redshifts[END_REF][START_REF] Karlsson | Quasar redshifts and nearby galaxies[END_REF] introduced a periodicity 0.089 of decimal logarithms of (1 + Z) in his study of spectra of quasars. We remark that this periodicity corresponds to a multiplication of all frequencies of a spectrum by 1.23, as by a Doppler frequency shift. Multiplication of Lyman α frequency of H by 1.23 gives Lyman γ frequency.

Burbidge's and Karlsson's conclusions were criticized because they required a strong selection of neighborhood of quasars [START_REF] Kjaergaard | On the periodicity in the distribution of quasar redshifts[END_REF][START_REF] Tang | Critical examinations of qso redshift periodicities and associations with galaxies in sloan digital sky survey data[END_REF].

Studying a Lyman forest of quasar, P. Petitjean [8] assigned the name "redshifted Lyman α line of H atom" to many sharp, saturated absorption lines, and he found, at laboratory frequency, Lyman α and β lines. But he did not find any (shifted or not) Lyman γ line. In fact, it is impossible to observe an absorption line at Lyman γ frequency, because, this frequency is too hight, out of redshifted emission of quasars.

When a Lyman α line of H atom is absorbed, Lyman β and γ must be also absorbed (if there is absorbable energy at their frequencies). Applying this result to shifted or not α lines, we found other so-called alpha lines ! Using an exact superposition of sharp lines, this result is more reliable than statistical results of previous authors. Thus, three different studies lead to a common result: Spectroscopy of H atom appears implied in redshifts of quasars.

2 Conditions for an interaction of light emitted by an observed star with matter.

Spatial coherence.

So that an image of star remains sharp in despite of an interaction with matter, wave surfaces must be preserved, that is interaction of light with matter must be spatially coherent. Einstein's theory [START_REF] Einstein | Zur Quantentheorie der Strahlung[END_REF] of coherent light-matter interactions shows that interacting molecules must be in large sets of identical molecules. States of colliding molecules depend on many time-dependant parameters, so that spectra of colliding molecules, therefore Raman emissions, are all differents, while Rayleigh are not. Thus, a full coherent interaction of light with a gas requires a low pressure, with an exception for Rayleigh scatterings which produce refraction.

Model of Lyman forest of a quasar surrounded exclusively by hydrogen.

Following properties of "Ly α forest" are directly deduced from Petitjean's paper [8]: A "Lyman forest" is built by absorption of thermal emission of an extremely hot star by relatively cold (≈ 2000 -50000K), low pressure, thus non-excited, atomic hydrogen. Absorbed lines making "Lyman forests" are distinguished from usual lines by their sharpness and their saturation.

Observations of a Lyman forest shows that: -A-As lines are sharp, widening of lines by collisions must be negligible, pressure of gas must be very low.

-B-To obtain absorptions of Lyman lines at many frequencies, redshift processes of electromagnetic waves are necessary. In a first approximation, we assume that redshifts multiply all frequencies of light by a constant lower than one. It is the rule for Doppler and cosmological redshifts.

-C-A single Ly β is observed, unshifted; no unshifted Ly γ appears because it does not remain absorbable energy at high frequencies after redshifts of thermal emission profile.

Absence of an absorption line may result from : -a-Absence of emitted energy around frequency of line. It explains absence of non-shifted Ly γ absorbed line.

-b-A permanent, fast shift of light frequencies dilutes absorption, so that all absorbed (or emitted) lines have width of shift and lines are weak, not observable. Accordingly, absorption (or emission) of sharp lines requires extremely slow frequency shifts.

-c-Accurate superposition of observed lines results from the choice of redshift equations: Absence of shifted β and γ lines while so-called sharp α lines are observed, is due to a shift of frequencies of these lines to frequencies of differently shifted α lines.

In standard theory, sharpness of saturated absorbed lines requires contradictory conditions:

-Column density of gas must be large for saturation; -Absorbing gas must be thin to avoid a broadening of lines by frequency shift during absorption; -Pressure of gas must be low to avoid collisional broadening of line. Thus gas must be in filaments which are only detected on rays of observation of quasars.

Supposing that redshift is related to a physical property of gas, the condition for hiding an absorbed line under an α line is:

-Light is redshifted until an absorbed line reaches Lyman α frequency, that is until generation of 2P hydrogen requiring Lyman α absorption disappears. Rules used to build spectrum taking into account only hydrogen atoms are simple :

Building of spectrum.

-i -At start, close to star, we suppose that α, β and γ lines have been absorbed.

End of star emission

Initial absorption → First redshift --> -ii -It appears a frequency shift until an absorbed line of initial frequency ν reaches ν α frequency. Thus all absorbed frequencies have been multiplied by ν α /ν, coefficent lower than 1; this provides Karlsson's result. In pure atomic hydrogen ν may be ν β or ν γ , assuming that higher frequency lines are too weak; this provides Burbidge's result.

-iii -During stop of frequency shift, the three main lines of H could be absorbed, but there remains no energy at Ly α frequency.

-iv -During the stop which corresponds to absence of energy at α frequency, Ly β absorption excites H atoms to 3P state, weak catalyst of a very weak redshift. During this negligible redshift, gas lines are visibly absorbed. If weak redshift is able to shift absorbed frequencies off Ly α absorption line before full absorption of light at Ly β frequency, fast redshift restarts, return to -ii -for a new cycle. Else there is no more redshift, absorbed Ly β line is visible, but Ly γ line is probably not because its frequency is probably larger than the shifted high frequency limit of emission of star.

In pure H, total redshift results from several relative shifts Z β,α and Z γ,α which correspond to multiplication of light frequencies by ν α /ν β or ν α /ν γ .

Computed lower frequencies written on figure are not very good. They can be corrected by multiplication by a unique dispersion function F (ν) equal to 1 at ν α frequency. F(ν) may be measured from a quasar spectrum, from its direct observation on shifted multiplets emitted by the core of quasars, by laboratory measure or ab initio computation of dispersion of hyperfine polarizabilities of H atom.

Physical interpretation of interaction of light

with nebula close to quasar

Impulsive Stimulated Raman Scattering (ISRS).

Assume that redshifts are produced by low pressure, relatively cold hydrogen, so that atoms are free and in their ground state 1S.

To modify frequencies, interactions must involve several levels of atom. Such Raman interactions of light with matter were proposed by Jean-Claude Pecker.

If wavelengths of exciting light and Raman coherently scattered light are different, a phaseshift appears along rays of light, so that phases of scattered fields are shifted, become opposite, destroy. To avoid this destruction, ordinary and extraordinary refractions of an optically anisotropic crystal are used. This is not possible in a gas.

The trick needed in gas is cutting light into pulses: this increases line spectral profile of exciting and scattered lights, so that a common zone of equal frequencies may interfere into a single, frequency shifted beam. This is named "Impulsive Stimulated Raman Scattering" (ISRS), much used in chemistry to follow evolutions of reactions. Usually, experiments use 10 femtosecond pulses of lasers.

G. L. Lamb [START_REF] Lamb | Analytical description of ultra-short optical pulse propagation in a resonant medium[END_REF] wrote conditions for observation of ISRS: "The length of pulses must be shorter than all involved time constant". Temporally incoherent usual, thermal origin light is made of nanosecond pulses, k ≈ 10 5 times longer than pulses commonly used in laboratory ISRS. This factor k decreases ISRS in several ways:

-Collisional time of atoms must be increased by factor k, thus pressure must be decreased;

-Raman resonance period must be decreased by k, which has two consequences:

-Raman shift is decreased and -difference of populations between levels involved in Raman resonance is decreased.

Thus, compared to a lab experiment, frequency shift is roughly decreased by a factor k 3 = 10 15 : observation of ISRS with natural light requires very long paths available only in space.

Coherent Raman Effects on temporally Incoherent Light (CREIL).

An other problem is that resonant Raman levels excited by studied ISRS cannot be de-excited by too rare collisions. Happily, there are other ISRS which blueshift background thermal radiation. Excitation of hyperfine levels becomes stable, so that H atoms make a catalyst. Thus, several ISRS are bound into a "parametric interaction". This interaction is named: "Coherent Raman Effects on Incoherent Lights" (CREIL). CREIL could increase, by frequency shifts, the entropy of other systems of light beams, temperature of which being deduced by Planck's law from radiance and frequency. Variations of energy correspond to variations of frequencies.

Lamb's condition il fulfilled by resonance frequencies, periods of which are longer than one nanosecond (1ns) length of light pulses, that is for quadrupolar resonance frequencies lower than 1 GHz.

In H atom: ν= 1420 MHz (T=0.7ns, λ =21cm) resonance in ground state (1S) is too high to catalyze a CREIL; on the contrary 178 MHz (T=5.6 ns, λ =1,7 m) in state 2S 1/2 , 59 Mhz (T=17 ns, λ = 5 m) in 2P 1/2 and 24 MHz (T=42 ns, λ = 12 m) in 2P 3/2 provide a CREIL. In higher states, quadrupolar (spin) resonances have much lower frequencies, so that their CREIL are lower.

Atoms able to catalyze CREIL may be obtained by heating low pressure H over ≈ 50 000K, or pumping cold atoms 1S to 2P by radiation at Lyman α frequency. This last condition is the key of Z (β,α) (resp. Z (γ,α) ) redshifts: Assume that a continuous spectrum light propagates in cold, low pressure atomic gas. Light at Lyman α frequency pumps atoms from 1S to 2P state, so that CREIL redshifts light: energy at Lyman α frequency is renewed by redshift of all light frequencies, CREIL and redshift are permanent except if an absorbed line reaches α frequency. Where redshift stops, the spectrum of local gas is written into light. Remark that absorptions during redshift make very broad and very weak, invisible lines.

In pure hydrogen, after fundamental redshifts Z (β,α) or Z (γ,α) of spectrum previously written into light, redshift stops while new β and γ lines are absorbed, written saturated into light.

Hyperfine frequencies of levels reached by Lyman β pumping from 1S to 3P state are much lower than in 2P state, they provide a weak shift which restarts the previous cycle, except if the fall of emission by the source at high frequencies, forbids a restart of a cycle before total absorption of β line. Thus the last cycle is full.

Is the first cycle full ? Probably often, but it is difficult to find a sure answer: For instance, the start of the first cycle may appear in a region hot enough to provide excited hydrogen without Lyman α pumping. This explains the need of a selection of quasars to obtain some observed redshifts.

Lines found in figure 1 appear in real spectra, after a correction of their frequencies by the factor F (ν). But many lines have an other origin: Hydrogen is not pure in nebulae, some local molecules may have, at frequencies higher than ν α strong absorption lines able to play the role of Ly β and Ly γ , multiplying density of lines. Thus, figures much more complex than fig. 1 are drawn.

Structuring space.

Along path of light, light is sometimes redshifted by pumped H atoms, sometimes not. Thus space is divided into regions where generated 2P atoms shift light frequencies, and regions in which there is no 2P atoms and no frequency shift. As quasars are not alone in space, perturbation of generation of these regions may result from pumpings of atoms by light of other stars. Thus regions in which quantized redshifts appear must be small enough to avoid an important lighting by other stars.

To avoid different pumpings by light coming from different regions of the surface of the star, redshift must appear only at a distance much larger than size of the star, that is in a region where pressure of gas is low.

These conditions are evidently not verified for a set of relatively close stars as a galaxy.

As the quasar produces very high light frequencies, it is extremely hot. They may be accreting neutron stars, a type of stars which were never found in nebulae while they should be seen. 2. To avoid a blur of image of quasar, interaction of light with hydrogen must be coherent. Impulsive Stimulated Raman Scattering which produces frequency shifts in labs using laser femtosecond pulses is very weak using pulses which make ordinary temporally incoherent light, so that space paths are needed for its observation.

3. Around quasars, space is structured into regions in which: Either H atoms are excited by Lyman α absorption, so that light is redshifted; or light was absorbed at this frequency, spectrum of gas is written into light. This cannot work around galaxies.

4. Quasar spectra and freqency shifts are explained using only theories well verified in laboratories.

5. Coherence of interaction of light with low pressure gas of nebulae should be considered by astrophysicists. For instance superradiance, may be useful to explain bubbles.

Figure 1

 1 Figure 1 represents a canvas of atomic hydrogen spectrum:Rules used to build spectrum taking into account only hydrogen atoms are simple :-i -At start, close to star, we suppose that α, β and γ lines have been absorbed.

Figure 1 :

 1 Figure 1: Lyman hydrogen forest of a quasar of redshift Z=0.56. Arrival of an absorbed line at ν α frequency almost stops redshift. During stop, Ly β and Ly γ lines are absorbed. Lines of other local gas may be also absorbed and may later play a similar role if their frequencies are larger than ν α . Written frequencies do not take into account dispersion of hyperfine polarizability of H atom.

1 .

 1 Three independant experimental results (Burbidge's and Karlsson's periodicities of redshifts of quasars, absence of Ly β and Ly γ lines in spectrum of quasar studied by P.Petitjean) show that hyperfine resonances of H atom are involved in redshifts of quasars. These redshifts need excitation of atomic hydrogen.


Ariane Tichit
email: ariane.tichit@udamail.fr

Clément Mathonnat
email: clement.mathonnat@udamail.fr

Diego Landivar
email: diego.landivar@france-bs.com

Catherine Araujo Bonjean

Keywords: Non-banking money, Text mining, Web data, Downward hierarchical clustering, Similarity analysis. JEL codes O35, E42, C38

This paper develops a new classification of non-banking currencies projects based on a lexical analysis from web data. The advantage of this method is to derive an endogenous typology of monetary projects, based on how currencies are depicted on the internet. Therefore, it allows to by-pass issues face for now in the literature to uncover a clear classification of non-banking currencies projects from exogenous elements. Our textual corpus consists in 320 web pages, corresponding to 1210 text pages. We first applied to our data a downward hierarchical clustering, which enables us to endogenously derive five different classes, allowing us to operate distinctions not only between non-banking currencies projects, but also between these latter and the standard monetary system. Then, we resorted to a similarity analysis and according to our results; all non-banking currencies projects define themselves in relation to the standard monetary system, with the exception of Local Exchange Trading Systems (LETs).

Introduction

What is money? Or should we rather say what currencies are? It is now very difficult to give a clear answer to this question, as the current surge in various monetary forms around the world calls for a re-conceptualization of this economic and social object. In the literature, money is usually defined by the following three functions: (i) a medium of exchange, (ii) a reserve value and (iii) a unit of account. However, what is exceptional in the current historical period is the fact that these three functions are gathered in one unique national (our supranational) official currency. Indeed, in most previous historical periods, several currencies were used to serve these three functions separately [START_REF] Douthwaite | Short Circuit: Strengthening Local Economies for Security in an Unstable World[END_REF][START_REF] Greco | Money: Understanding and Creating Alternatives to Legal Tender[END_REF][START_REF] Lietaer | The future of money[END_REF][START_REF] Lietaer | Au coeur de la monnaie[END_REF].

Hence, complementary currencies are not a new phenomenon. They only resurface in a crisis time, where the lack of liquidity and the need for finding new reserve of values, call for a redefinition of money, where its three traditional functions could be embodied by different currencies.

To understand what complementary currencies are, and better analyze their issues, their impacts and better manage and support their development, it seems necessary to establish a clear classification of the latter. In this regard, the literature on complementary currencies classification, initiated by [START_REF] Kennedy | Regionalwährungen: Neue Wege zu nachhaltigem Wohlstand[END_REF] and [START_REF] Bode | Potentiale regionaler Komplementar währungen zur Förderung einer endogenen Regionalentwicklung[END_REF], currently represents a rapidly-growing research field (see, [START_REF] Blanc | Classifying "CCs": Community, complementary and local currencies' types and generations[END_REF][START_REF] Blanc | Penser la pluralité des monnaies à partir de Polanyi : Un essai de typologie[END_REF][START_REF] Schroeder | Community currency research: An analysis of the literature[END_REF][START_REF] Slay | More than money[END_REF][START_REF] Martignoni | A new approach to a typology of complementary currencies[END_REF][START_REF] Bindewald | Validating complementary and community currencies as an efficient tool for social and solidarity economy networking and development: The deployment of theory of change approach and evaluation standards for their impact assessment[END_REF][START_REF] Seyfang | Growing green money? Mapping community currencies for sustainable development[END_REF]. Because of the massive surge in complementary currencies in recent years and their puzzling diversity, researchers have realized the need for better understanding what they truly are and what are their similarities and discrepancies. Overall, authors deal mainly with questions such as, what kinds of exchange they aim at promoting? between who? for what purposes? [START_REF] Bode | Potentiale regionaler Komplementar währungen zur Förderung einer endogenen Regionalentwicklung[END_REF] suggests a classification of complementary currencies according to the following two criteria: (i) theirs compensations schemes and (ii) the type of co-contracting parties involved; and within this typology, further distinguishes between "services based complementary currencies" and "monetary based complementary currencies". Here, we notice a clear dividing line between barter's clubs and other Local Exchange and Trading systems (LETs), independent from standard moneys, and citizen's currencies, anchored to national or supranational currencies. [START_REF] Kennedy | Regionalwährungen: Neue Wege zu nachhaltigem Wohlstand[END_REF] propose a more detailed and complex typology, including technical features of complementary currencies, and define five main classification factors: (i) the objectives they served, (ii) theirs functions, (iii) their medium of exchange, (iv) their underlying process of monetary creation and (v) their cost recovery schemes. Starting from this basis, the current evolution of this literature aims at clarifying and deepening these initial classifications by accounting for a larger set of characteristics (see [START_REF] Blanc | Classifying "CCs": Community, complementary and local currencies' types and generations[END_REF][START_REF] Martignoni | A new approach to a typology of complementary currencies[END_REF][START_REF] Seyfang | Growing green money? Mapping community currencies for sustainable development[END_REF]Bindewald et al., 2013;Place & Bindewald, 2013). Indeed, relying on Derruder & Lepesant (2011) and Dittmer (2013), which mainly divide complementary currencies according to their objectives (characterized from a microeconomic point of view for the former and a meso/macroeconomic one for the latter), [START_REF] Bindewald | Validating complementary and community currencies as an efficient tool for social and solidarity economy networking and development: The deployment of theory of change approach and evaluation standards for their impact assessment[END_REF] put forward a typology of complementary currencies projects based on four categories, namely (i) politic, (ii) economic, (iii) social and (iv) environmental; which are then subdivided according to their respective scope (meta, macro, meso and micro). 1 Therefore, the current trend in this literature seems to be in search of a growing number of complementary currencies features, with a more and more complex division into sub-fields. However, we believe that these developments make the understanding of complementary currencies projects trickier, and in this perspective, classifications seem to gradually lose their goal of simplification and clarification of a given phenomenon.

In this respect, [START_REF] Blanc | Penser la pluralité des monnaies à partir de Polanyi : Un essai de typologie[END_REF], in the introduction of his paper, acknowledges the relative failure of the literature for now to define a clear-cut typology of complementary currencies, and we believe this will be even more difficult by following the current trend consisting in increasing classifications criteria. First, the problem lies in the fact that authors do not seek to classify the same object. Some of them want to account for all existing currencies, whereas some others only want to account for a limited set of these latters.

According to [START_REF] Blanc | Classifying "CCs": Community, complementary and local currencies' types and generations[END_REF], the current heterogeneity of complementary currencies is so important that it turns out to be unavoidable to resort to several classifications. Moreover, if the literature has difficulties in defining a clear and efficient typology of complementary currencies, it might come from their focus on moneys only, while emphasis should rather be placed on systems. 1 Place and Bindewald (2013) also subdivide the « politic » category between two further distinct categories, namely "culture" and "governance".

As a result, [START_REF] Blanc | Classifying "CCs": Community, complementary and local currencies' types and generations[END_REF] suggests a classification based on projects rather on objects and determines three main systems classes: (i) local currencies (territorial/geographical-based projects), (ii) community currencies (originating from preexisting communities) and (iii) complementary currencies (economic-based projects focused on production and exchange activities into markets). Going one step further by broadening his previous analysis, Blanc (2013), drawing upon Polanyi's works, proposes a typology of monetary projects according to three "ideal types": (i) public currency, (ii) profit-making currency and (iii) citizen's currency and six subcategories (state, sub-state, captative, community and trade). This taxonomy has the clear advantage to enable the characterization, qualification and classification of all types of currencies, with the only exception of crypto-currencies. According to us, it represents the most successful typology to date. [START_REF] Seyfang | Growing green money? Mapping community currencies for sustainable development[END_REF] define a classification of 3428 monetary projects from 23 countries located across 6 continents. Their sources come from the compilation of existing database and field information. In order to classify these projects, they assume that three distinct types of monetary projects appear in the literature: (i) credits services, (ii) mutual exchange systems and (iii) local currencies. They then add a fourth category to the previous ones, namely (iv) barter's clubs. According to [START_REF] Seyfang | Growing green money? Mapping community currencies for sustainable development[END_REF], the first two classes gather 91.5% of all recorded initiatives, whereas barter's clubs only account for 1.4% of monetary projects in their database. Therefore, local currencies and barter's clubs seem to account for a very limited number of complementary currencies projects over the world. 2 Consequently, in light of this brief literature review, classification and structuration of complementary currencies clearly appears as a thorny issue and has not yet succeeded in finding a clear-cut typology. For us, this relative failure might come from two main factors.

One the one hand, authors have focused on different monetary objects or projects: their functioning, the actors they involve, the types of goods and services exchanged, in what conditions, through what medium of exchange and in order to serve what goals. Yet, although these various elements are strongly interrelated, each author favors some specific features of monetary projects he believes more significant and representative, carries out his own subdivision within each element, which makes the existing classifications very hard to compare. On the other hand, database on complementary currencies are still poorly organized 2 Unless if it comes from a bias arising from the database itself, as well as the classification method used by the authors. Indeed, we believe their choice to differentiate between barter's clubs and mutual exchange systems is questionable. When we look at the definitions the authors provide for these two categories, they seem to be almost equivalent and separation between them is only based on the fact that projects call themselves "barter's clubs" or "mutual exchange system", without any significant difference in terms of functioning, structure and goals. and available. As a result, it proves to be very challenging to form an exhaustive and efficient database, which further complicates the classification of complementary currencies according to their characteristics. Since authors resort to different and partial database, they obviously find different results.

In order to circumvent these problems, we think that a relevant way to classify complementary currencies could be to neither resort to a classification based on recorded objective data, nor according to a priori factors, but instead, to endogenously categorize the largest possible set of monetary projects using web data. Indeed, internet abounds of web pages dedicated to complementary currencies, articles presenting their characteristics, blogs talking about them, which represents an invaluable source of information on these social objects, albeit these data are in a textual form. Yet, for more than two decades, we notice an important development in statistical methods for text analysis, especially regarding endogenous classification of textual corpus according to their content. A clear benefit of this methodology is to neither resort to a priori hypotheses about factors driving the typology, nor focusing on specific subsets of monetary projects. Hence we use the term "non-banking" to qualify most of the existing monetary projects.

The rest of this paper is organized as follows. Section 2 presents our lexical corpus data, as well as our statistical methodology. Section 3 displays the main results regarding the endogenous classification of non-banking currencies using a top-down hierarchical clustering.

Section 4 explores the relationships between the various estimated classes in section 3 using a similarity analysis and section 5 concludes.

Methodology

According to Gerin-Pace (1997), statistical methods for text analysis were born in the eighties and since that time they have followed two mains development paths: a first set of methods aims at analyzing writing style (texts comparisons and evolutions), while a second set deals with the analysis of the meaning of a given textual corpus. Our paper draws upon this latter set of methods.

The design of the textual corpus

We initially set 38 keywords related to complementary currencies. They were chosen in order to cover most of the actual diversity of these projects. We also decided to keep 10 results for each keyword, so as to end up with a textual corpus including around 300 web pages. Then, our raw data underwent three types of "cleaning" procedures. The first one, consisted in testing for each keyword, results obtained with the Google search engine. Here, we chose to withdraw a given keyword from our initial list if: (i) its first ten URLs results gave webpages not related to complementary currencies or (ii) its URLs results were exactly the same as some other keywords in the list (our textual corpus does not include duplication webpages).

The second "cleaning" procedure deals with the selection of data collected from the crawled webpages. To do this, we decided to focus only on webpages including informative data directly available in a textual form on each webpage we have crawled. As a result, we dropped URL corresponding to multimedia supports (for instance, videos, radio programs), those leading only to homepages without any informative content, and finally those related to translated webpages. Likewise, in order to keep our textual corpus balanced, we chose to drop URL leading to pdf files. Moreover, we also canceled URL referring to book (for example, ecommerce sites, google books and editors webpages). Therefore, our textual corpus is constituted from four main sources: (i) newspapers and magazines, (ii) blogs, (iii) free online encyclopedias and (iv) webpages from different actors of complementary currencies projects.

Finally, the third "cleaning" procedure was based on the crawled webpages themselves, and consisted in keeping only text data from each webpages (meaning, dropping terms only dedicated to web navigation such as tags, signs and pictures). We additionally removed internet user's comments, since they often reached an extremely huge size and rather consist in values judgments than in an objective description. Since our goal in this paper is to derive a classification of complementary currency through the way they are presented on the web, we believe the inclusion of such user's comments in our textual corpus irrelevant.

In the end, our textual corpus includes 1210 webpages coming from the crawling of the URLs of the first ten results in the search engine Google associated to each of the 32 final keywords used to depict non-banking currencies systems. 3

Descriptive statistics of the textual corpus

Our textual corpus is formed by 320 distinct texts. Starting from these raw data, we resorted to a corpus lemmatization. In order to reduce vocabulary diversity and so better emphasize semantic proximities within a given corpus, this method implements an identification process of forms corresponding to the different flexions of the same lemma (Beaudoin & Lahlou, 3 Table 1 in appendix gives the exhaustive list of these 32 keywords. 1993). 4 In other words, lemmatization can be viewed as a way to "undress" words from their grammatical shape, so as to gather them in one family. For instance, all conjugations of the "have" verb will be gathered in the same lemma "have". This seems to be especially relevant in our context, since we are only interested in the informative content of texts and not their forms.

Furthermore, when importing our corpus, this latter was divided in segments of 20 consecutive occurrence terms. Consequently, the final partition of our corpus is the following:

 320 Initial Context Units (ICU).
 17939 Elementary Contexts Units (ECU), also called text segments, which represents subsets of 20 successive occurrences in a given ICU.

 359223 occurrences and 22369 forms after lemmatization (i.e. distinct terms). 5 Appendix 2 gives the 50 most frequent active forms in our corpus, with their respective total frequency. 6 Since webpages were crawled with keywords including the term "currency", it seems logical that this term is the most frequent word included in our corpus, with 4733 occurrences. Then, the three other most frequent forms are "exchange" (1569 occurrences), "local" (1443 occurrences) and "system" (1314 occurrences).

Downward hierarchical clustering and similarity analysis 2.3.1. Downward hierarchical clustering

In order to implement our downward hierarchical clustering, we used the [START_REF] Reinert | Une méthode de classification descendante hiérarchique : application à l'analyse lexicale par contexte[END_REF][START_REF] Reinert | ALCESTE : Une méthodologie d'analyse des données textuelles et une application : Aurélia de Gérard de Nerval[END_REF]) ALCESTE (Analyse des Lexèmes Cooccurrents dans un Ensemble de Segmentation du Texte Etudié) method with the software IramuteQ. 7 The downward hierarchical clustering (henceforth DHC) is an algorithm which starts by assuming that all active forms initially belong to the same class. A DHC applies to a binary lexical table (0: absence; 1: presence) combining ECU in raws and active forms (after lemmatization) in columns. Then, for each algorithm iteration, we derive the two most distinct classes of words from our corpus. That is to say, we determine a partition of active forms from our corpus maximizing inter-class inertia 4 Here we considered the term form as graphical form, that is to say "any minimal concatenation of non-separator types, including information and delimited to the left and to the right by defined separators." Michel [START_REF] Demonet | Des tracts en mai 1968[END_REF], Des tracts en mai 1968, Paris Presses de Sciences Po, p.21. 5 A form may appear many times, i.e. corresponds to several occurrences. 6 Active forms are those containing most information (e.g. nouns, adjectives and verbs) and the remaining other are called supplementary forms (e.g. adverbs, articles and pronouns). 7 In English this could be translated into Analysis of Co-occurrent Lexemes in a Set of Segmentation from the Studied Text. and minimizing intra-class inertia.8 This iterative process stops when the extracted inter-class inertia is not improved by a new partition of data. From this perspective, the final number of classes is a priori undetermined, which in our case is especially relevant for deriving an endogenous classification of non-banking currencies systems without ex ante hypotheses.9

When the algorithm comes to its end, we obtain a binary lexical table depicting the repartition of active forms according to the estimated k classes. Then, once we have divided our corpus in k classes, we need to determine features related to each estimated class, which consists in analyzing active forms included in each class, and especially the contribution of each active form j to a given class k. For this purpose, we rely on a Chi-square statistic, which involves assessing the extent of connection between each active form and each class.

Afterwards, from these estimated connections, the use of a Factor Component Analysis (henceforth FCA) enables us to characterize similarities and oppositions between estimated classes by pooling them in factors delimiting theirs respective outlines. Furthermore, it is also possible to implement a DHC on segments, texts or pooled segments. On the one hand, our classifications tests based on texts (each webpage is processed as a whole) were not conclusive, since the estimated classes were somewhat dispersed and uninformative. This comes from the fact that most of the time each webpage contains several topics, so that considering it as a whole does not make any sense from a semantic point of view. Therefore, text partitioning into segments turns out to be essential in order to carry out an efficient text analysis. On the other hand, our classifications tests based on simple texts segments proved to be more significant. However, one obvious issue here is to define the length of segments to be considered. When importing corpus, IramuteQ offers different type of segmentations based on the size of occurrences (i.e. the number of successive forms needed to be considered), signs or paragraphs. In our case, a classification based on successive occurrences seems to be the most suited, given that webpages are not necessary structured in paragraphs and classification based on signs would not make any sense. We did several tests for different segments size (40, 30, 20 and 10 occurrences) and the most clear-cut results were obtained from a classification based on 20 successive occurrences, albeit few discrepancies appeared between classification based on 20 and 10 successive occurrences.

However, classification with more than 30 occurrences led to poorer results, which means that too long segments includes too much heterogeneous information and makes classification less efficient.

Lastly, we have implemented a DHC for both simple segments and pooled segments.

Classification leading to the most convincing results is the one based on pooled segments. This latter consists in a two parts classification. In a first step, we specify a given number of active forms to be pooled and then, with a first DHC, the algorithm aggregates segments in order to end up with the ex ante specified number of segments. As a result, it enables to gather similar segments in terms of included active forms. In a second step, the algorithm implements another DHC on segments, but this time by considering an aggregated number of active forms, different from the first step. Then the algorithm combines these two classifications in order to finally keep, for each class, the pooling of segments which maximizes inter-class inertia. With this method, the estimated classes are much more cohesive and significant than those obtained on simple segments, which is logical since the final classification is derived from groups of active forms generated according to their proximity, and not only according to a given size of successive occurrences (like classification on simple segments does). 10 Hence, we decided to keep classification estimates obtained through the implementation of a DHC on pooled segments of 20 occurrences.

Similarity analysis

Similarity analysis allows measuring distances between different semantic territories. In our case, this identification work is required if we want to highlight differences or similarities in the relationship between lexical representations of money. Therefore, what we seek to gauge is the distance between the k estimated classes? To do this, once we have done our DHC, we undertake an identification work of semantic categories. Indeed, despite a DHC enables to endogenously estimate a given number of classes according to relative factors, this latter is unable to depict the relationship between close semantic territories, as well as combinatory territories between several classes.

To assess distances between semantic territories, we chose to work with a method coming from applied graphs theory to lexical corpus. This method allows depicting nods and 10 For robustness purpose, we also changed these parameters: increasing and decreasing the number of successive active forms to be accounted for in each step compared with the default setting of IramuteQ. Overall, the number of derived classes increases with the decreased in the number of active forms accounted for. Nevertheless, our baseline classification is robust to changes in these parameters and only the number of classes is sometimes divided in a more precise way. It is also possible to change the number of pooled segments in the first step (the default setting being 10). Again, estimated classes are more numerous, but this does not call into question our baseline classification estimates derived with a threshold of 20. links from a modularity calculation, based on [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF] and [START_REF] Lambiotte | Laplacian Dynamics and Multiscale Modular Structure in Networks[END_REF], to determine statistically, and so endogenously, groups of nods (in this case semantic continents) gather several nods sharing commons features. This method also permits to endogenously identify central and peripheral entities. A nod is said to be central when most of possible paths connecting the graph pass through it. Here, the Betweness Centrality algorithm from Brandes (2001) allows computing the point where most of possible paths in the graph pass through. In terms of interpretation, the more nods are at the center of the graph, the more they are central. Conversely, the more nods are far away from the center of the graph, the more they are peripheral. When applying to lexical data, this centrality algorithm leads to the identification of reference classes, that is to say classes which serve as origin point (or centre of gravity) for the definition of other classes.

Results from the downward hierarchical clustering

When applying a DHC to our corpus with IramuteQ, we derived 5 classes. The following dendrogram helps to better visualize this result:

Graphic 1. Dendrogram

This dendrogram shows that our five estimated classes are well-balanced in terms of texts segments they respectively include, since each of them covers around 20% of all segments. In order to see which classes have been endogenously created, we now turn to Class 1 is associated with the traditional representation of the standard monetary system. Indeed, in this first class, we massively find vocabulary depicting monetary creation, banks, currencies and their uses. There is also terms dealing with free and private moneys, such as the "fonte" for instance. This is not surprising since this class is the result of presentation of the standard monetary system by web sources related to non-banking currencies. Therefore, some segments from this class oppose free and private moneys to standard moneys.

Class 2 refers to the recent financial and economic crisis, with a lot of terms related to its causes and consequences. This result is especially interesting since it supports previous works emphasizing the countercyclical dimension of complementary currencies [START_REF] Lietaer | Halte à la toute puissance des banques ! Paris : Odile Jacob[END_REF][START_REF] Lietaer | Au coeur de la monnaie[END_REF][START_REF] Herlin | Repenser l'économie[END_REF][START_REF] Herlin | Apple, Bitcoin, Paypal, Google La fin des banques? Comment[END_REF]. Therefore, non-banking currencies are clearly defined in our corpus as an alternative way to deal with the consequences of financial and economic crises.

For instance, recent experiments, such as in Europe (e.g. France, Germany, Japan and Belgium) and Latin America (Argentina and Brazil), and older ones, like the Swiss's WIR during the great depression of the 30's, show that surges in complementary currencies often arise in trouble financial and economic times, with increasing difficulties for people to access to liquidity. Classes 1 and 2 appear to be embedded in each other. This result seems logical since they correspond respectively to the standard monetary system and the recent financial and economic crisis. As a result, local currencies stand out from standard monetary system through factor 1, but are linked together through factor 2. Moreover, it is worthwhile to note that class 2, referring to the recent financial and economic crisis, makes the link between standard moneys and social moneys. Indeed, since most of the time, the perceived failure of the standard system in time of crises gives rise to a surge in social moneys projects; it seems relevant to see class 2 located in between classes 1 and 4, which are respectively associated to the standard monetary system and local currencies. As for, class 3 (LETs), it appears to be the most distant class from the standard system, since it differentiates from this latter through both factors 1 and 2. Then, class 5 (Bitcoin) and class 4 (Local currencies) are the two other most opposing classes. Putting together these results, it is therefore possible to derive the signification of the two factors behind our five estimated classes.

On the one hand, classes 1 and 2 with respect to class 4, as well as class 3 compared to class 5; are opposed to each other's through factor 1. As a result, factor 1 enables to distinguish local moneys from standard money, as well as between bitcoin and other cryptocurrencies from LETs and barter's clubs. Therefore factor 1 could be related to objectives and values of the existing moneys: whether (i) profit-making and speculative (negative values of factor 1), or (ii) social (positive value of factor 1). We can also note that all new non-banking currencies projects own 2 of the 3 classical functions of money, namely medium of exchange and unit account; despite they do not necessarily have a reserve value function. If we get a closer look at data distribution according to factor 1, bitcoin and standard moneys share a common feature, since they are both reserve moneys; contrary to LETs and local currencies.

Hence, we believe factor 1 to be composed of an additional characteristic related to the ability of moneys to be a value reserve. Indeed, this additional feature is found in negative values of factor 1, whereas it is absent in its positive values.

On the other hand, class 3 and class 5, compared to classes 1, 2 and 4, are opposed to each other's through factor 2. As a result, factor 2 enables to distinguish standard and local money from LETs and bitcoins. We believe this separation based on a more functional criterion, namely the anchoring to national or supranational moneys. Indeed, to our knowledge, the similarity between bitcoins and most of the LETs and barter's clubs is their independence with respect to national or supranational currencies. As a result, negative values of factor 2 can be related to independence from national or supranational currencies, i.e.

monetary creation outside any anchoring to standard money, whereas positive values corresponds to reliance on national or supranational currencies.

Consequently, by pooling results coming from the implementation of a DHC to our lexical corpus, we had endogenously derived two structural features of non-banking currencies projects than enable to classify all existing currencies projects in a rather simple way. The following table 2 gives our resulting classification of non-banking currencies. 2013). So it makes sense if they are opposing on this issue, while at the same time they are linked to standard money owing to their convertibility in national or supranational currencies, their liability to the same taxes for transactions they support and their monitoring by monetary authorities. Furthermore, local currencies are completely in opposition to cryptocurrencies, not only with regard to their respective values, but also in terms of dependence with respect to standard moneys; bitcoin being independent from the standard monetary system. By the way, cryptocurrencies were born precisely from a search for independence compared to monetary authority. However, we can see that virtual currencies share common values with standard moneys such as profit seeking, speculation and wealth accumulation.

As we have already seen with graphic 2, LETs and barter's clubs are the most opposing monetary projects to the standard ones. They share with virtual currencies the feature of being not reliant on the standard monetary system, without any relationship with this latter, neither regarding monetary creation, nor on the liability to taxes. Nevertheless, LETs and barter's clubs are distinct from cryptocurrencies when dealing with their respective values, since the former ones advocate social, mutual, ethic and environmental values, whereas the latter does not.

Finally, one additional interesting feature of our results comes from the fact that they strongly echo complementary currencies classification from [START_REF] Blanc | Penser la pluralité des monnaies à partir de Polanyi : Un essai de typologie[END_REF]. Indeed, like the author, we find the same partition between LETs and barter's clubs on one side and local currencies on the other side; and surprisingly according to the same criteria. 17 However, contrary to [START_REF] Blanc | Penser la pluralité des monnaies à partir de Polanyi : Un essai de typologie[END_REF], our typology does not enable us to differentiate between public and profit-making moneys within the standard money class. This result could stem from a lack of data regarding this class in our corpus. 18 Yet, our classification accounts for cryptocurrencies, when Blanc's (2013) classification does not.

Results from the similarity analysis

In this section, we go one step further and resort to a similarity analysis in order to better understand the outlines associated to our five previously estimated classes, their semantic similarities and discrepancies. To do this, we draw upon the lexicometrical litterature dealing with the distance measurement between lexical fields from different documents.

Graphic 3 below displays results from our similarity analysis. First of all, we notice that class 1 is at the center of this graphic, hence representing the center of gravity of the other classes. Furthermore, this statistical method allows us to identify three distinct "communities" that we will henceforth call "continents". The two first continents related to classes 3 and 4 are mono-classes. However, classes 1, 2 and 5 belong to the same continent. We can therefore interpret results from our similarity analysis in the following way: (i) banking moneys are the 17 [START_REF] Blanc | Penser la pluralité des monnaies à partir de Polanyi : Un essai de typologie[END_REF] gathers behind the ideal-type « citizen currency » all non-profit monetary projects and divides the latter according to whether they are anchored or not to national or supranational currencies. 18 On this purpose, one relevant extension of our paper would be to crawl more institutional data relative to central banks, monetary authorities and commercial banks in order to study the standard money class in a finer way and analyze to what extent we find the distinction suggested by [START_REF] Blanc | Penser la pluralité des monnaies à partir de Polanyi : Un essai de typologie[END_REF].

center of gravity, the central semantic reference to non-banking currencies. This reference might therefore justify the denominations "complementary currencies" or "non-banking currencies", since these two terms rely on a same reference norm: standard money; (ii) only

LETs currencies types seem to be not reliant on this reference to standard money, this former being the most far away from the center of the graphic; (iii) cryptocurrencies, crisis and standard money classes belong to the same semantic community, and therefore to the potentially same system of values and social representations. This latter result can also be interpreted as a semantic oppositional expression formulated by people's knowledge stemming from the local or complementary currencies projects. Indeed, the latter define themselves in reaction to behaviors and values associated to the standard monetary system. In addition, we can further extend this same conclusion to currencies related to class 3.

Graphic 3. Semantic communities' detection using similarity analysis

Conclusion

This paper offers a new classification of non-banking currencies projects based on a lexical analysis from web data. Starting from the issue that it is often difficult to access to exhaustive and factual data on complementary currencies, we decided to circumvent this drawback by using lexical web data. In light of the recent literature, the structuration of existing complementary currencies clearly appears as a thorny issue and has not yet succeeded in finding a clear-cut classification (see also [START_REF] Blanc | Penser la pluralité des monnaies à partir de Polanyi : Un essai de typologie[END_REF]. From our point of view, the existing classification are very hard to compare, because the authors focus on different monetary objects or projects, favor some specific features of projects they believe more significant and representative and carry out their own subdivision within each element.

In order to avoid these pitfalls, we built a vast lexical corpus covering the largest possible set of these new monetary objects and then resorted to an endogenous classification method, enabling to derive structural factors behind our lexical data. The corpus was created from 32 French keywords referring to complementary currencies. We kept the first 10 Urls results for each keyword with the Google search engine, and then we extracted their respective content. As a result, our corpus is formed by 320 webpages, corresponding to 1210 text pages and 342 585 words, that is to say 17 939 segments of 20 successive occurrences.

Then, in the first step we ran a downward hierarchical clustering (DHC) on text segments. This algorithm recursively finds the best way to divide data into cohesive groups and derives the optimal number of monetary projects classes. Afterwards, the implementation of a Factor Component Analysis (FCA) allows determining the latent factors behind the previously estimated classes with the DHC. This classification method enabled us to derive 5 consistent and significant classes from our lexical corpus: (i) standard moneys, (ii) the recent financial and economic crisis, (iii) local currencies, (iv) LETs and barter's clubs and (v) cryptocurrencies. One clear advantage of this method is to neither resort to a priori subjective hypotheses about factors driving the typology, nor focusing on specific subsets of monetary projects. Our results lead to a simple, clear-cut and exhaustive classification of all existing current monetary forms and uncover two fundamentals sources of differentiation between them, namely: (i) connection or dependence to national or supranational currencies and (ii) values and goals behind monetary projects. Afterwards, the implementation of a similarity analysis allowed us to better understand the outlines associated to our five estimated classes, their semantic similarities and discrepancies. Results derived from this method show that, except for LETs and barter's clubs, all new monetary forms define themselves with respect to the standard monetary system.

Therefore, we believe that this paper clarifies in a relevant and fresh way the research fields on non-banking currencies, representing an essential contribution to the current literature. Moreover, since our results strongly echo theoretical classification from Blanc (2013), our paper can be viewed, to a certain extent, as an empirical test of its non-banking currencies typology.

We acknowledge that this paper has focused only on French lexical data. Hence, one relevant extension of this work would be to apply the same approach and methodology to other languages, such as English, German and Spanish, so as to foster lexical comparisons between monetary projects according their geographic origins or the language used to describe them.

Finally, we think that this paper opens a new methodological field of research, by

showing the possibility of deriving relevant typologies from various economic or social phenomenon through the analysis of lexical data from the internet. Consequently, we hope that beyond our conclusions relative to non-banking currencies, this paper will contribute to the increased diffusion and use of textual statistics in economic and social studies.

  Class 3 is clearly related to Local Exchange and Trading systems (LETs) and barter's clubs, since it gathers all currencies representing ways of exchanging directly goods or services between people, without resorting to intermediaries, such as accorderies, LETs, REN (Reciprocal Exchanges Networks). As a result, these organizations operate outside the traditional market system. Class 4 depicts social moneys projects, located in a specific territory and based on ethical and social values. They are distinct from the LETs category since they take part in the traditional market system, albeit they share most of their values with this former.Finally, class 5 is clearly related to virtual currencies, where bitcoin is without contest the most famous representative. In this class, the most recurrent vocabulary deals with all references associated to computer monetary creation and also to the presumed bitcoin creator, namely Satoshi Nakamoto. Now, we need to go one step further and study factors behind the separation between estimated classes, in order to derive classifications criteria at the roots of non-banking currencies systems. To this end, we used a Factor Component Analysis (henceforth FCA) which enables to interpret factors giving rise to our classification with the DHC. Given that the first two factors are those contributing most to our classification (they account together for 61% of the total variance in our data), and are also the most meaningful in terms of classes division, we focus our interpretation on these two factors only. In order to get better insights on the distribution of our five estimated classes according to factors 1 and 2, as well as their respective distance to these factors, graphic 2 gives on the horizontal axis factor's 1 values and on the vertical axis factor's 2 values. Each class is depicted with a specific color and forms belonging to each class are located according to their respective coordinates with respect to these two factors.Graphic 2. Factor Component Analysis of the classified segmentsGraphic 3 clearly shows that classes 1 (in red, to the North-West) and 2 (in black, in a more central position) are closed to each other, whereas classes 4 (in blue, to the North-East), 5 (in purple, to the South-West) and 3 (in green, to the South-East) are opposed to classes 1 and 2.

Table 2 . Classification of non-banking currencies Goals/values/reserve function Profit/commercial value/ reserve function Non profit/social value/ no reserve function Connexion/taxes

 2

	Connected to standard currencies	National and supranational legal currencies	Complementary local currencies, citizen's currencies, social currencies…
	Disconnected from standard currencies	Transaction currencies like bitcoin, crypto-currencies	LETs, barter's clubs, Accorderies, Reciprocal Exchange of Knowledge Networks…
	Table 2 allows us to derive some interesting conclusions. First, local currencies seem
	mainly contest standard money values (speculation, concentration of wealth; see Blanc,

This amounts to determine a k class's partition of our data, such as maximizing the variance proportion (inertia) from all active forms in our corpus accounted by the k classes. Recall that the total inertia of our data (all active forms) = the extracted inertia (i.e. accounted by the k classes) + the residual inertia (i.e. not accounted by our k classes).

Only the maximum number of classes needs to be specified.

Ariane Tichit This work was supported by the LABEX IDGM+ (ANR-10-LABX-14-01) within the program "Investissements d'Avenir" operated by the French National Research Agency (ANR). http://www.cerdi.org/This article was prepared as part of the Research-Action program in Social Innovation financed by the Region Auvergne. Usual disclaimers apply.

We are very grateful to Jerome Blanc and Vincent Mazenod for their supports.

analyze the lexical content of each of them. To this end, the following table 1 gives the first 20 most specific terms from each estimated class (according to their Chi-square statistic). 11 11 Forms related to the same lemma and recursive segments have been pooled together.

12 Corresponding French words in parenthesis. 13 We did not translated the French word « violet » in « purple » because this term directly refers to a complementary currency used in the French town Toulouse and is called « Sol Violette». 14 In the cryptocurrencies context, mining refers to the process by which computers' calculation power is partly allocated to make easier and secure virtual moneys transactions from computer; and this service provided is paid in virtual moneys. 15 REN stands for Reciprocal Exchanges Networks. 16 In this class context, a valorimeter can be viewed as a reference enabling to give value to moneys. 20 Frequencies are displayed in brackets and French translation of each word is given in parenthesis.

Appendix


V Chabannes

C Prud'homme

M Szopos

R Tarabay

Keywords: Navier-Stokes equations, large scale simulations, computational fluid dynamics, scalable parallel preconditioners, experimental validation, Feel++

come

SimRace 2015 -Numerical methods and high performance computing for industrial fluid flows Efficient solving strategies for incompressible Navier-Stokes equations for large scale simulations using the open source software Feel++

Over the past few decades, the computational fluid dynamics has evolved to become an important tool for the description of the complex multi-physics, multi-scale phenomena characterizing blood flow. Its reliability depends both on the verification of the numerical methods and on the validation of the mathematical models.

The aim of the first part of the talk is to present a preconditioning framework for the linear system arising from the finite element discretizations and time advancing finite difference schemes of the 3D steady and unsteady Navier-Stokes equations. In particular, we are interested in preconditioners based on an algebraic factorization of the system's matrix which exploit its block structure, such as the Pressure Convection-Diffusion (PCD) preconditioner , the SIMPLE preconditioner or the LSC preconditioner, see [START_REF] Elman | Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics[END_REF]]. A comparison between the efficiency of these preconditioners is ascertain by testing them over the 3D backward facing step benchmark. The iteration counts using the PCD preconditioner are independent of mesh size and high order finite elements and mildly dependent on Reynolds numbers for steady flow problems which is not the case for the other preconditioners.

In the second part of the talk we describe a framework for the solution of flow problems relevant to biomechanics strongly supported by the aforementioned solving strategies. We assess the efficiency of this framework through experimental data for fluid flow in a nozzle model with rigid boundaries, a device designed to reproduce acceleration, deceleration and recirculation, features commonly encountered in medical devices [START_REF] Stewart | Assessment of cfd performance in simulations of an idealized medical device: Results of fda's first computational interlaboratory study[END_REF]]. The flow rates were chosen to cover laminar (Re = 500), transient (Re = 2000) and turbulent (Re = 3500) regimes.

The numerical results displayed during the presentation are obtained using the open-source library Feel++ (Finite Element method Embedded Language in C++, www.feelpp.org).

  Efficient solving strategies for incompressible Navier-Stokes equations for large scale simulations using the open source software Feel++ Vincent Chabannes, Christophe Prud'Homme, Marcela Szopos, Ranine Tarabay To cite this version: Vincent Chabannes, Christophe Prud'Homme, Marcela Szopos, Ranine Tarabay. Efficient solving strategies for incompressible Navier-Stokes equations for large scale simulations using the open source software Feel++. SimRace-2015 -Numerical methods and high performance computing for industrial fluid flows, Dec 2015, Paris, France. hal-01230451

Figure 1 :

 1 Figure 1: Computational Fluid Dynamics FDA Benchmark at Re = 2000.

Table 1 :

 1 Number of outer iterations (using PCD) at the last nonlinear iteration of the Picard algorithm and [the total number of nonlinear iterations] for the 3D step geometry