Tony Volpe

Joachim Schöpfel

Dissemination of knowledge and copyright: an historical case study

Introduction

Registration, certification, dissemination, and archiving: since the conception of Oldenburg"s Philosophical Transactions, publishers and authors share a common set of convictions on the main functions of a scientific journal.

A scientific journal should act as a public register of intellectual property, brand (or label) articles and create visibility, serve the transmission of scientific information, and help keeping a record of science (Guédon, 2001, pp. 15-18).

Of course, other features contribute to academic publishing, especially psychological drivers and human factors such as "ego, vanity, recognition, reward and immortality" (Mabe, 2009, p. 3), social and societal aspects, politics, and strategic considerations (Roosendaal et al., 2010, p. 19-36). Evaluation becomes a core function of scientific journal publishing.

Yet, the basic concept of a scientific journal remains unchanged. "The four functions of Oldenburg"s journal (…) are so fundamental to the way scientists behave and how science is carried out that all subsequent journals, even those published electronically in the 21st century, have conformed to Oldenburg"s model" (Mabe, 2009, p. 5).

These core functions of scientific communication do not necessarily co-exist in harmony. It occurred only later that there might be conflict between these functions, in particular between registration, i.e. protection of author"s rights, and broad dissemination of knowledge. Today, Internet and the open access movement challenge their importance, rank and correlation. Prevailing immediate and direct communication (dissemination) may raise quality issues (certification) and impact permanent recording. In the current crisis of copyright, this is matter of debate with regards to scientific communication via blogs, social networks, wikis, personal websites and open repositories (see for instance the proceedings in Gaster, Schweighofer and Sint, 2008).

Copyright, which means the protection of intellectual property and economic interests of publishers, authors and printers and its relationship to the creation and publication of scholarly works, is "a critical concern for the advancement of new knowledge" (Crews and van Westrienen, 2007, n.p.). Does copyright stimulate artistic and scientific creation? Does it facilitate dissemination of information and culture?

In 2010, a German economic historian roused a passionate debate about intellectual property laws, technological innovation and economic growth. Eckhard Höffner"s research on 19th century Germany suggests that a weak copyright system due to political fragmentation may have incited economic growth by making technical manuals and other practical books cheap and easily available, helping Germany catch up with Britain [START_REF] Höffner | Geschichte und Wesen des Urheberrechts[END_REF]. So, does copyright protection reduce intellectual and industrial creation? Höffner"s thesis was followed by an international and controversial debate in papers, blogs and wikis. [START_REF] Dobusch | The digital public domain: relevance and regulation[END_REF] for instance argued that "according to Höffner, this much broader availability of specialist and technical literature was then responsible for Germany"s industrial catchingup in the 19 th century" (p. 22) and concluded that "a vibrant public domain (is) relevant for economic and societal development" (p. 33). Others, as the judge and senior lecturer at the University of Chicago Law School Richard A. Posner relativized the impact of copyright, especially when compared to patent law protection [START_REF] Posner | Do patent and copyright law restrict competition and creativity excessively?[END_REF], or they stressed the role of intellectual property protection for innovation, such as a contribution from the leading Chinese business school CKSGB at Beijing [START_REF] Voyles | Promoting innovation in society[END_REF]. The only shared conviction was that a complex topic needs complex answers.

Our intention is to provide more controversial evidence to this debate based on a case study from history of science. In our case, Denys de Sallo"s Journal des Savants considered as the world's oldest scholarly journal still active today (Potts, 2011, pp. 68-69) with private rather than corporate character (Cook, 2001, p. 16) opposes Elzevier (also spelled Elzevir or Elsevier), a Dutch family of publishers, booksellers and printers of the 17 th and 18 th centuries (no link with the modern publishing house of the same name).

Our story is that Elzevier, through non authorized exploitation of a new product and without considering the editor"s legitimate interests in a context of the absence of copyright protection, largely contributed to the dissemination of the French Journal des Savants and its content, and finally to the success of scientific journals.

The story may appear old, from the past and sometimes a little bit hard to understand. Yet, the underlying questions and the search for solutions are still relevant today.

The Republic of Letters, community and communication

Beyond social barriers and political, religious or linguistic frontiers, intellectuals of the second half of the seventeenth century often saw themselves as belonging to a Republic of Letters, characterized by the search for truth and open communication among its members. The expression "Republic of Letters" designated the community of scientists and scholars who cultivated knowledge. Moreover, all those who adhered to the ideals of this great intellectual community had a duty to transmit, communicate and disseminate knowledge. They were fully aware that intellectual progress required the development of communication. Communication was at the heart of the concerns of the academic community; it was essential to its existence (see Volpe, 2008, pp. 9-13;Peiffer and Vittu, 2008, pp. 288-290).

The seventeenth century represented a fundamental moment in the history of Western thought. The forms of learned sociability grew and diversified, and the intellectual life experienced a significant development in new forms and particularly a remarkable fertility.

Rather exceptional events, journeys could not be a usual mode of transmission and dissemination of ideas that circulated most commonly through correspondence. Personal contacts were followed by the establishment of correspondences. Before the introduction of scientific journals, correspondence was the primary vehicle of scientific information, maintaining the life of the Republic of Letters and playing a leading role in the intellectual life of the seventeenth century. Some characters have distinguished themselves in their role as intermediaries. For instance, Father Marin Mersenne played an active role in forging scientific activity by continuing through his correspondence to foster research or elicit comments. Correspondences by such people like Mersenne, Peiresc, Oldenburg or Magliabechi are outstanding by their volume, the size of their corresponding networks across Europe, and the invested time and money.

Scientists not only exchanged ideas, principles and demonstrations, but also calculations, observations and experiences. The letter allowed them to express their interests and expose their activities to disseminate news about the progress of their research, and to announce the works published or in preparation to a select audience. Yet, the letter rarely remained confidential; it was released, partly or totally, among the friends of the recipient. The situation changed with the new journals.

With the advent of journals, scientific communication gradually lost its former character of personal exchange. Printed in journals, the letter reached numerous and diverse readers that it could not attain before across Europe. For scientists, periodicals became a vital tool to publish the results of their research and to register and protect the priority of a discovery or an invention. Due to the significant delay of the proceedings of academies, many scholars also submitted preprints of their papers in general journals.

The journal was a forum of expression addressed to the Republic of Letters and beyond, to an interested public taken to witness in controversy and debate. Also, those who could not afford to purchase books could nonetheless learn from the book reviews. Journals met the critical issue of access to scientific information in a time of increasing numbers of authors and printed books. With the development of scientific journals from the last decades of the seventeenth century, the ideal of communication as a powerful engine of the Republic of Letters reached its ultimate form.

Launched in 1665, the Journal des Savants inaugurated a new way of communication in the Republic of Letters (see Vittu, 2002a, p. 180;Volpe, 2008, p. 135;Beaudry, 2011, pp. 124-126). Its success could be measured in terms of its many counterfeits and also by the considerable number of journals launched in the following decades, such as the Giornale de' Letterati (Rome, 1668), Acta Eruditorum (Leipzig, 1682), Histoire des Ouvrages des Savants (Rotterdam, 1687) and the History of the Works of the Learned (London, 1699). This explosion of scholarly journals across Europe is well reported by Pierre Bayle, in his preface to the first edition of its News Republic of Letters in 1684: "It was found so convenient & pleasant to inform the public through a kind of journal of what was happening in the Republic of Letters, that no sooner had Mr Sallo, Counsellor at the Parliament of Paris, published the first tests of this project at the beginning of the year 1665, than several nations showed their joy, either by translating the Journal he was printing every week, or by publishing something similar. This emulation increased even more from that time, so that it extended not only from one nation to another, but also from one science to another science. The physicists and the chemists have published their special reports; jurisprudence and medicine have their journal; music also has his; gallant news, together with religion, war, & policy have their Mercure. After all, the initial idea of Mr. Sallo has been executed by almost all in infinite ways." [1] The "Journal des Savants" v Elzevier case

The Journal des Savants was counterfeited very early, especially in Amsterdam from 1666 on, only one year after its launch. According to a letter written by Emery Bigot to Nicolas Hensius on March 18, 1665, Daniel Elzevier had instructed his brother to bring him the first issues of the Journal (Vittu, 1988, p. 308) from Paris. This counterfeit was printed until 1680 by the Elzeviers under the name of "Pierre le Grand", then for another two years in Brussels by Fricx, who was in business with Daniel Elzevier until his death in 1680. From 1682 on, the Journal was again counterfeited in Amsterdam under the responsibility of the bookseller company Wolfgang, Van Waesberge, Boom and Van Someren. This "Amsterdam printing" was edited in a smaller size, being in duodecimo format rather than in the French quarto. [2] In 1675, the Journal des Savants took a new "impetus" under the direction of a new editor, Jean-Paul de La Roque, a real newspaper man who completely transformed the journal throughout his mandate. The development of direct or indirect correspondence by the Father de la Roque with personalities such as Oldenburg, Grew, Bayle, Huygens, Leibniz, Mencke, Bernoulli, Nicaise, Spon and Magliabechi, as well as the multiple editions of the Journal prove without any doubt the readership of the journal and its establishment in the Republic of Letters.

For the men of science, the Journal des Savants and the related meetings allowed access to the Republic of Letters as much as a vector of dissemination as a kind of "think-tank" for discussion and evaluation. Scholars, amateurs and scientists responded to the expectations of the editor and submitted regular and numerous letters, memoirs and observations. In the late 1670s, the Parisian bookseller replenished his fund by reprinting older issues of the Journal.

The postage cost per issue doubled and limited the shipping opportunities for those who lacked relationships with travellers and traders, a problem that was to facilitate the success of counterfeiting. In Amsterdam, on his own and with no intellectual property rights, Daniel Elzevier resumed his counterfeits both of the current issues and of former editions (Vittu, 2002b, p. 366). In 1684, an observation of Pierre Bayle confirmed the ultimate need of these "fakes": "If in Holland nobody printed the Journal des Savants made in Paris, it would remain forever unknown to all booksellers in the North and in Germany" [3] . Bayle"s statement was confirmed by a survey of 170 European libraries that most frequently possessed a counterfeit edition, especially in Northern Europe (Vittu, 1988, pp. 301-302 and p. 324). With the production of counterfeits, the journal"s distribution centres proliferated. The recurrence of these counterfeits corroborated the success of pirating which spread the journal across Europe. Produced in small annual volumes with different issues, the journal was no longer designed for a readership interested in cutting edge research or the latest scientific curiosities but it now satisfied readers who used it as a catalogue or a book.

The importance of the counterfeits affected the impact and visibility of the Journal des Savants in two ways. On the one hand, they increased its distribution through the creation of a second production centre. Located in Amsterdam, a crossroad of land and sea routes, it sold the journal all around in Northern and Central Europe. On the other hand, they amplified its readership community, less by lowering costs than by switching to a lower frequency that was more interesting for readers who regarded the journal rather as a book, a collection of bibliographic and erudite information. Without its counterfeits, the Journal des Savants would never have achieved such widespread dissemination throughout Europe and such success that, it must be remembered, it also generated a movement in favour of the creation of several other scientific journals.

Counterfeiting in the contextan apology

The idea of intellectual property appeared only recently in the history of mankind. Previously, writer and artist did not even imagine a literary or artistic conception might be to their benefit, subject to law. The only expected benefits of their work were the honours, awards in competitions, fame and, with it, the chance to be noticed and protected by some Maecenas, jobs and pensions from the government or some great and powerful personality.

"At the beginning of the book, there is infringement, i.e. both original sin and wonderful chance to the written word" (Moureau, 1988, p. 9). When Gutenberg "counterfeited" the manuscript with typographical lead "imitating" scriptoria practices in incunabula, he invented -with a few others -the art of the printed book, but also, from the commercial point of view, legitimized forgery. Printing is at the origin of copyright but also of counterfeit, fraud and plagiarism.

The idea of copyright began to appear in the fifteenth century with the development of the printing press. Easy copying and distribution of works by printing represented a potential source of material profits. The first traces of this right emerged across the privileges (or licences) of the city of Venice, soon followed by privileges in Italy, Spain, France, Germany and England, the first country to enact a law on copyright in 1709. Under the Ancien Régime copyright was based on a system of monopoly, i.e. a system of privileges granted by the Sovereign, in accordance with the prevailing doctrine that all right comes straight from the public authority. Denys de Sallo and his successors held a royal license for the publication of their journal. Yet, in most cases, the literary monopoly fell into hands (publishers, booksellers) other than those of the author [START_REF] Ruffini | De la protection internationale des droits sur les oeuvres littéraires et artistiques[END_REF]. Also, under the Ancien Régime, there was no international law and the library police was not very effective. In the absence of any international agreement that would protect copyright, publishers and authors were sometimes tempted to obtain privileges in different countries. In particular, these privileges were often granted by the Pope, the Emperor, the King of France, the King of Spain or the princes. For instance, in 1625 Hugo Grotius published his treatise On the Law of War and Peace in Paris; though Dutch, he had obtained in France a monopoly for a period of fifteen years.

The legal fight against counterfeit and plagiarism was relatively ineffective in the absence of international law applicable to the offense. The privilege granted to a bookseller or author according to the rules under the Ancien Régime was for a fixed term and a well-defined territory. Only books published while the privilege was still operating and on the territory covered by the privilege could be considered as counterfeits. This was the reason why printing a book protected in Paris in Avignon, City of Popes and land of the church, or distributing in Lyon a book whose Paris privilege had expired, was not fraud against law. Also, counterfeiters were often convinced that they were simply exercising a right. For example, in the words of a publisher from Liège to Marmontel of whom he published the Moral Tales without permission: "Your privileges do not extend so far. Liège is a country of franchise, we have the right to print everything that is good, and that is exactly our business." [4] Usually, the first bookseller to counterfeit a foreign work obtained de facto a sort of monopoly respected by other counterfeiters. Thus, in Holland, forgery was a priority right to the infringer as evidenced by a voluntary agreement (convention), written in Dutch and signed on March 8, 1710 by 54 booksellers: "If anyone were to infringe here any book from a foreign bookseller, the first person who gives notice or shows a printed sheet will be considered the real owner." [5] This agreement explains the great number of announcements of reprints made by Dutch booksellers in their newspapers. It was in this way that the Elzeviers had granted themselves the monopoly of French counterfeits. Many counterfeiters, in good faith, if not their right, did not hesitate to sign their forgery. When they chose not to do so, it was generally not to escape prosecution but for other, sometimes political reasons.

Finally, although each country at that time possessed its own counterfeiting site for foreign works, the different states took advantage of or, conversely, were affected by fraud in different ways. The most impacted countries could be classified into two categories: on the one hand those which, while being of the same nationality and of the same language, had the disadvantage of multiple sovereignties. This is the case of the Netherlands, divided into seven provinces, of Italy, divided into seven states, and of Germany, which had many more. On the other hand, those countries with widely used and expansive language and literature, such as England vis-à-vis America and France vis-à-vis Europe, because of the universal knowledge of the French language and the prevailing liking for French ideas. Some booksellers unable to distribute their production on their own behalf instigated or created their own "preferred counterfeiters" who disseminated in a decentralized way books in small runs that the publisher could not supply. The case of the major literary journals of the nineteenth century counterfeited in Belgium for the international market clearly shows the continuity of this labour division between an authorized publisher who keep up with demand and a "useful leech" able to meet the needs of the market.

The art of counterfeiting their own works was regular practice of authors like Voltaire for whom the multiplier and explosive effect of counterfeiting was a fundamental element of the propaganda in the Age of Enlightenment (Lumières). In the eighteenth century, the Encyclopedia still witnessed in an eloquent manner to this widespread practice and its interest: "But the main point of view from which we consider a good book is that it provides general instruction to mankind; & from this point of view, one could never spread fairly quickly or easily enough a book of this nature. Counterfeits are excellent for both goals, first because they are spread in the country where they are forged & to nearby places; & because of the price reduction, it is within reach for a very large number people who otherwise might not even have heard of the original edition. Therefore, what right do booksellers say they possess on a book written for the instruction of mankind? How dare they say that it is a property that belongs to them, and that those who counterfeit are pirating? Such remarks betray the love of gain and the ignorance of the human rights." [6] The argument that copyright was opposed to human rights, or more generally to the progress of mankind, was still put forward in the following century by many of those who continued to contest the recognition of intellectual property. Thus, in 1856, and again in 1866, the Italian politician and activist Giuseppe Mazzini wrote to a publisher in Milan: "I never believed in the law of literary property, as understood today. The author capable of really profitable ideas but without resources, should, in a well-ordered republic, be assisted and encouraged by the nation, yet the expressed thought belongs to all: it is social property. The breath of the human soul cannot be a monopoly. All have a duty to encourage, nobody has the right to impede or restrict the circulation of truth." [7]

Discussion

Telling old stories may have some distancing or alienation effect. Some familiar topics seem strange at distance. Yet, we should not be misunderstood. This is not a plaidoyer in defence of counterfeit and fraud but an invitation to debate on some specific points.

(1) Life without copyright? The first point is that Denys de Sallo"s contemporaries were not aware of anything like copyright protection. And without copyright, no infringement. But we can take this in quite a different way. Living without copyright is possible. That is historical evidence. We can imagine scientific and academic publishing without intellectual property. Or as Rupert Gatti put it: "Eliminating copyright on academic works would (…) enable more efficient and innovative mechanisms" for the delivery of academic publishing (Gatti, 2011, p. 41). At least, it is an interesting option. All those authors and scientists who put their works and data in the public domain already chose this option. Habermas described the impact of the invention of the printing technology on the development of concepts like intellectual property, public sphere and privacy [START_REF] Habermas | The structural transformation of the public sphere: an inquiry into a category of bourgeois society[END_REF]. Perhaps, the new technologies of information and communication may contribute to the vanishing of public sphere and copyright protection [START_REF] Poster | Cyberdemocracy: Internet and the public sphere[END_REF].

(2) Useful piracy? The second point is that even without copyright, there was a problem. Denys de Sallo and his successors held a royal license and never considered their Journal des Savants as a common good; neither did Elzevier. Both were conscious of the fact of the unauthorized use of printed information. Nevertheless, this illegal use contributed, through quick and broader distribution of the "Amsterdam printing", to the success of the journal and to the success of science, too. Does this mean that counterfeiting may contribute to common wealth, that "many kinds of "piracy" are useful and productive" (Lessig, 2004, p. 66)? This is a statement difficult to accept for authors, professionals and industry, especially those familiar with the continental intellectual property laws, but it is essential for the fair use concept. The everyday struggle with European copyright exceptions for non commercial educational and scientific purposes is directly related to this question.

(3) Meetings needs? Counterfeiting was facilitated if not encouraged and promoted by financial and technical barriers. High prices for printing and distribution, and long distribution processes were obstacles to the normal exploitation of the original product. Thus, counterfeit met a genuine need for scientific information, on the emerging market of early scientific communication. As described above, counterfeiting was not only or at least partly tolerated but also it was sometimes part of a conscious strategy of communication and dissemination. Looking back on more than twenty years with open repositories such as arXiv and HAL provides some concrete material for this question.

(4) Sharing for reuse? The next point is on sharing. The 17 th century scholars, amateurs and scientists were concerned with registration but above all interested in debate, controversy and exchange in their open community, the so-called Republic of Letters. Their focus was on knowledge. Translated in modern terminology, they wanted recognition of authorship but set no other conditions for reuse. One can suppose that in our time, the Journal des Savants, unlike most modern scientific journals, would have been published under Creative Commons, with a CC-BY licence only requiring attribution to the original author. Also, for a Journal des Savants published under CC-BY, counterfeiting, reprinting and reselling would not have been a real legal problem. The ongoing discussion on the best licence for reuse of data and papers (or no licence at all), on gratis or libre open access (see Suber, 2012, pp. 65-75) is just another illustration of this point.

(5) Work integrity? In the 17 th century, content was unstable, science was looked upon with distrust, and counterfeiting contributed to this problem because the different versions could be (were) different without guarantee of the original work"s integrity. This problem subsists, with different versions of academic publications being disseminated through different communication channels, but may be better controlled due to metadata and unique identifiers. Yet, the problem is still there, with preprint, postprint and publisher"s versions, on validated and on validated theses and dissertations, non peer reviewed working papers etc. being sometimes a kind of professional nightmare for librarians and scientists.

(6) Business? Law? Alternatives? Since de Sallo"s century, printing houses and publishers changed their mind on copyright and counterfeits. Yet, their function and purpose remain more or less unchanged: meet the need of the scientific community for quick, widespread and selective communication of research results through customized and addedvalue products and services, and get money out of this business. The legal context changed, the business didn"t, neither for the publishing industry nor for the Republic of Letters, i.e. the scientific community that is above all interested in recognition, debate, and sharing (see Mulligan and Mabe, 2011, p. 309). This means also, that to the extent that the mediation becomes an obstacle for open, smooth and seamless distribution of information, for instance through high pricing or excessive protection, the Republic of Letters will search for alternative options, such as direct communication. In fact, they have already started to adopt them.

Conclusion

Our paper started with the question as to whether copyright stimulates artistic and scientific creation or whether, on the contrary, copyright protection reduces dissemination of information and culture. Does a weak copyright system promote technological innovation and economic growth, as Höffner (2010) appears to suggest? Historical evidence is ambiguous and unethical. What is good today may be bad tomorrow. Often, interests, needs and business are more important than values, ethics and law. Obviously, rights infringement cannot only promote the interest and objective of the rights holder but above all, it can contribute to the emergence and development of new forms of scientific life. "Every generation welcomes the pirates from the last" (Lessig, 2004, p.66). On the other hand, the context has changed and the playersscientific community (Republic of Letters) and publishers -switched their roles. Yet, to quote Georges Santayana, "Those who cannot remember the past are condemned to repeat it." Today, in the post-Gutenberg galaxy, it appears that "copyright in the new medium will be difficult to protect" (Oppenheim, 2000, p. 295). Today, copyright battles oppose Samsung and Apple, publishers and academic libraries, and the tragic and sad hero of piracy is Aaron Swartz. Will law be the solution? Nothing is less certain. This case study tells a different story. Sometimes, when barriers to communication and debate are too strong, tolerated and calculated infringement may be an option to move the lines. This paper tells a story that is more than 300 years old. No doubt, other stories could be told, with other morals. Maybe there is no moral at all. Lessons of history lie in the eyes of the beholder. Our own story is being written right before our eyes. It is made up of open access, common goods and public domain, open data and e-Science, evaluation, innovation, sustainability, communication and sharing. We must make up our own mind to it, based on our own experiences, convictions and ethics. But we should be aware that even if our answer must adapt to the actual context, the underlying questions have been raised long time ago. Science is not private but social property. "Nobody has the right to (…) restrict the circulation of truth" (Mazzini,loc.cit.,p. 244). And let us add: nobody can, anyway.

des hommes ? Comment osent-ils dire que c'est un bien qui leur appartient, que le métier de ceux qui contrefont, est une piraterie ? Des propos pareils décelent la passion du gain & l'ignorance des droits des hommes.» In Encyclopédie ou Dictionnaire universel raisonné […], arranged by M. De Félice, Yverdon, t. XI, 1772, p. 265. [7] In G. Mazzini, Scritti editi ed ineditti, vol. IX, Roma, 1877, p. 244.


François Dubois
email: francois.dubois@math.u-psud.fr

Simulation of strong nonlinear waves with vectorial lattice Boltzmann schemes

Keywords: Hyperbolic conservation laws, entropy, shock wave, shallow water equations PACS numbers: 02.70.Ns, 05.20.Dd, 47.10.+g, 47.11.+j

We show that a hyperbolic system with a mathematical entropy can be discretized with vectorial lattice Boltzmann schemes using the methodology of kinetic representation of the dual entropy. We test this approach for the shallow water equations in one and two spatial dimensions. We obtain interesting results for a shock tube, reflection of a shock wave and non-stationary two-dimensional propagation. This contribution shows the ability of vectorial lattice Boltzmann schemes to simulate strong nonlinear waves in non-stationary situations.

Introduction

The computation of discrete shock waves with lattice Boltzmann approaches began with viscous Burgers approximations in the framework of lattice-gas automata (see Boghosian and Levermore [START_REF] Boghosian | A Cellular Automaton for Burgers's Equation[END_REF] and Elton et al. [START_REF] Elton | Convergence of Convective-Diffusive Lattice Boltzmann Methods[END_REF]). With the lattice Boltzmann methods described e.g. by Lallemand and Luo [START_REF] Lallemand | Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF], the first tentative results were proposed by d'Humières [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF] and Alexander et al. [START_REF] Alexander | A lattice Boltzmann model for compressible fluids[END_REF] among others. A D1Q2 entropic scheme for the one-dimensional viscous Burgers equation has been developed by Boghosian et al. [START_REF] Boghosian | Entropic Lattice Boltzmann Model for Burgers' Equation[END_REF]. The extension to gas-dynamic equations, and in particular to shock tube problems, is studied in the works of Philippi et al. [START_REF] Philippi | From the Boltzmann to the Lattice-Boltzmann equation: beyond BGK collision models[END_REF], Nie, Shan and Chen [START_REF] Nie | Thermal lattice Boltzmann model for gases with internal degrees of freedom[END_REF], Karlin and Asinari [START_REF] Karlin | Factorization symmetry in the lattice Boltzmann method[END_REF], and Chikatamarla and Karlin [START_REF] Chikatamarla | Lattices for the lattice Boltzmann method[END_REF].

In this contribution, we test the ability of lattice Boltzmann schemes to approach weak entropy solutions of hyperbolic equations. It is well known that a first-order hyperbolic equation exhibits shock waves. In order to enforce uniqueness, the notion of mathematical entropy has been proposed by Godunov [START_REF] Godunov | A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations[END_REF] and Friedrichs and Lax [START_REF] Friedrichs | Systems of Conservation Equations with a Convex Extension[END_REF]. A mathematical entropy is a strictly convex function of the conserved variables satisfying ad hoc differential constraints to ensure a complementary conservation law for regular solutions (see, e.g., our book with Després [START_REF] Després | Systèmes hyperboliques de lois de conservation ; Application à la dynamique des gaz[END_REF]). The gradient of the entropy defines the so-called "entropy variables." The Legendre-Fenchel-Moreau duality for convex functions allows us to define the dual of the entropy, which is a convex function of the entropy variables.

We start from the mathematical framework developed by Bouchut [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF], making the link between the finite-volume method and kinetic models in the framework of the BGK approximation. The key notion is the representation of the dual entropy with the help of convex functions associated with the discrete velocities of the lattice. If we suppose that a single distribution of particles is present, our previous contribution [START_REF] Dubois | Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF] shows that Burgers equation can be simulated in this way. We have also shown that the approach can be extended to the nonlinear wave equation but is not compatible with the system of shallow water equations.

In Section 1, we develop vectorial lattice Boltzmann schemes with a kinetic representation of the dual entropy. This framework is applied in Section 2 for the approximation of one-dimensional shallow water equations, and in Section 3 for the two-dimensional case. Stationary and non-stationary two-dimensional simulations are presented in Section 4.

1) Dual entropy vectorial lattice Boltzmann schemes

In order to treat complex physics with particle-like methods, a classical idea is to multiply the number of particle distributions, as proposed by Khobalatte and Perthame [START_REF] Khobalatte | Maximum principle on the entropy and second-order kinetic schemes[END_REF], Shan and Chen [START_REF] Shan | Lattice Boltzmann model for simulating flows with multiple phases and components[END_REF], Bouchut [START_REF] Bouchut | Construction of BGK models with a family of kinetic entropies for a given system of conservation laws[END_REF], Dellar [START_REF] Dellar | Lattice Kinetic Schemes for Magnetohydrodynamics[END_REF], and Wang et al. [START_REF] Wang | Lattice Boltzmann simulations of thermal convective flows in two dimensions[END_REF]. We follow here the idea of a dual entropy decomposition with vectorial particle distributions, as proposed by Bouchut [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF]. We consider a hyperbolic system composed of N conservation laws with space described by points in x ∈ R d . The unknowns are the conserved variables W ∈ R N (i.e. W k ∈ R). The nonlinear physical fluxes : F α (W ) ∈ R N (with 1 ≤ α ≤ d) are given regular  Strong nonlinear waves with vectorial lattice Boltzmann schemes functions. The system is of first-order:

(1)

∂ t W k + d α=1 ∂ α F k α (W ) = 0 , 1 ≤ k ≤ N .
We suppose that a mathematical entropy η(W ) is given, with associated entropy fluxes

ζ α (W ) for 0 ≤ α ≤ d: dζ α (W ) ≡ dη(W ) • dF α (W ) .
The entropy variables ϕ k ≡ ∂η(W ) ∂W k are defined as the jacobian of the entropy:

dη(W ) ≡ N k=1 ϕ k dW k .
The dual entropy η * (ϕ) and the so-called "dual entropy fluxes"

ζ * α (ϕ) satisfy (2) η * (ϕ) = ϕ • W -η(W ) , ζ * α (ϕ) ≡ ϕ • F α (W ) -ζ α (W ) .
They can be differentiated without difficulty (see e.g. [START_REF] Després | Systèmes hyperboliques de lois de conservation ; Application à la dynamique des gaz[END_REF]):

dη * (ϕ) ≡ k dϕ k W k , dζ * α (ϕ) ≡ k dϕ k F k α (W ) .
• With Bouchut [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF], we introduce N particle distributions f k j (for 1 ≤ k ≤ N ) and q velocities (0 ≤ j ≤ q -1). The conserved moments W k are simply the first discrete integrals of these distributions:

(3)

W k = q-1 j=0 f k j , 1 ≤ k ≤ N .
We suppose that the particle distributions f k j are solutions of the Boltzmann equations with discrete velocities:

∂ t f k j + v α j ∂ α f k j = Q k j , 0 ≤ j ≤ q -1 , 1 ≤ k ≤ N
We suppose j Q k j = 0 in order to enforce the conservation laws [START_REF] Alexander | A lattice Boltzmann model for compressible fluids[END_REF]. The non-equilibrium fluxes take the natural form Φ k α ≡ j v α j f k j and we have a system of N conservation laws:

∂ t W k + α ∂ α Φ k α = 0 , 1 ≤ k ≤ N .
In the following, we use the term "Perthame-Bouchut hypothesis" to refer to the fact that the dual mathematical entropy η * (ϕ) can be is decomposed into q scalar potentials, h * j . The potentials h * j are supposed to be regular convex functions of the entropy variables ϕ, and satisfy the two identities (4)

q-1 j=0 h * j ϕ ≡ η * (ϕ) , q-1 j=0 v α j h * j ϕ ≡ ζ * α (ϕ) , ∀ϕ .
The equilibrium fluxes (f eq ) k j are easy to derive from the potentials h * j :

f eq k j = ∂h * j ∂ϕ k , q-1 j=0 f eq k j = W k , 1 ≤ k ≤ N .



• We introduce the Legendre dual of the convex potentials h * j :

h j (f j , f 2 j , . . . , f N j ) ≡ sup ϕ N k=1 ϕ k f k j -h * j (ϕ) , 0 ≤ j ≤ q -1 .
We observe that each function h j (•) is a convex function of N variables. The so-called "microscopic entropy" H(f ) can now be defined according to

H(f ) ≡ q-1 j=0 h j (f 1 j , f 2 j , . . . , f N j ) .
This is a convex function in the domain where the h j 's are convex.

• We can establish a "H-theorem" for the continuous dynamics relative to time and space in a way similar to the maximal entropy approach developed by Karlin and his co-workers [START_REF] Karlin | Maximum Entropy Principle for Lattice Kinetic Equations[END_REF]. Under a BGK-type hypothesis

Q k j ≡ 1 τ f eq k j -f k j we have ∂ t H(f ) + α ∂ α j v α j h j (f 1 j , f 2 j , . . . , f N j ) ≤ 0 .
To establish this result, we derive the microscopic entropy relative to time:

∂H ∂t = jk ∂h j ∂f k j ∂f k j ∂t = jk ∂h j ∂f k j Q k j - jk ∂h j ∂f k j v α j ∂ α f k j = jk ∂h j ∂f k j Q k j -∂ α q-1 j=0 v α j h j . Then ∂H ∂t + ∂ α j v α j h j = 1 τ jk ∂h j ∂f k f k j f eq k j -f k j ≤ 1 τ jk ∂h j ∂f k j f eq j f eq k j -f k j by convexity of the potentials h j .
This last expression is equal to

1 τ jk ϕ k f eq k j -f k j
due to Legendre duality:

∂h j ∂f k j f eq = ϕ k . In consequence, ∂H ∂t + ∂ α j v α j h j ≤ k ϕ k j f eq k j -f k j = 0
by construction of the values f eq in equilibrium. The H-theorem is thereby proven.

2) "D1Q3Q2" lattice Boltzmann scheme for shallow water

We apply the previous ideas to the shallow-water equations in one spatial dimension

∂ t ρ + ∂ x q = 0 , ∂ t q + ∂ x q 2 ρ + p 0 ρ γ 0 ρ γ = 0 ,
Velocity u, pressure p and sound velocity c > 0 are given by the expressions:

u ≡ q ρ , p ≡ p 0 ρ γ 0 ρ γ , c 2 ≡ γ p ρ = γ p 0 ρ γ 0 ρ γ-1 .


Strong nonlinear waves with vectorial lattice Boltzmann schemes

The entropy η and the entropy flux ζ can be determined explicitly without difficulty (see e.g. [START_REF] Dubois | Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF]):

η = 1 2 ρ u 2 + p γ -1 , ζ = η u + p u
Then the entropy variables ϕ = (θ ≡ ∂ ρ η, β ≡ ∂ q η) can be related to the usual ones:

θ = c 2 γ -1 - u 2 2 , β = u .
Thanks to (2), the dual entropy η * and the dual entropy flux ζ * can be worked out explicitly: η * = p and ζ * = p u. We observe that

η * = K θ + β 2 2 2 = p , ζ * = K β θ + β 2 2 2 = p u , with K = k γ -1 γ k γ γ-1 .
• We model this system with a kinetic approach and a D1Q3 stencil. We have to find the particle components of the entropy variables, id est the (still unknown) convex functions h * j satisfing the Perthame-Bouchut hypothesis (4), that now can be written in the form:

(5) h * + (θ, β) + h * 0 (θ, β) + h * -(θ, β) = p , λ h * + (θ, β) -λ h * -(θ, β) = p u , where λ ≡ ∆x
∆t is the numerical velocity of the mesh. We use a simple quadratic function as in our previous contribution [START_REF] Dubois | Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF]. We suggest that when γ = 2:

(6) h * 0 = h * 0 (θ) = a 2 K θ 2 ,
with the introduction of a parameter a that has to be made precise for real numerical computations. With this choice [START_REF] Chikatamarla | Lattices for the lattice Boltzmann method[END_REF], the resolution of the system (5) with unknowns h * ± is straightforward, resulting in

(7) h * ± (θ, β) = K 2 θ + β 2 2 2 1 ± β λ - a K 4 θ 2 .
• From the previous potentials, ( 6) and [START_REF] Dellar | Lattice Kinetic Schemes for Magnetohydrodynamics[END_REF], it is possible to derive the entire distribution at equilibrium. Observe first that with a vectorial lattice Boltzmann scheme, it is necessary to use two families, f and g, of particle distributions, one for mass conservation and the other for momentum conservation. We have in this case

f eq j = ∂h * j ∂θ , g eq j = ∂h * j ∂β .
With (6), the function h * 0 is indepedent of β. Then g 0 = ∂h * 0 ∂β is unnecessary for the computation. With a very basic D1Q3 stencil, we define a "D1Q3Q2" lattice Boltzmann scheme. The equilibrium distribution is obtained by differentiation of the relations ( 6) and [START_REF] Dellar | Lattice Kinetic Schemes for Magnetohydrodynamics[END_REF]:

               f eq 0 = ∂h * 0 ∂θ = a K θ = a ρ 0 2 c 2 0 c 2 - u 2 2 = a 2 ρ - ρ 0 u 2 2 c 2 0 f eq ± = ∂h * ± ∂θ = ρ 2 1 ± u λ - a 4 ρ - ρ 0 u 2 2 c 2 0 g eq ± = ∂h * ± ∂β = ρ u 2 ± ρ 2 u 2 λ + c 2 2 λ . 

François Dubois

From these equilibria, we implement the lattice Boltzmann method within the multiplerelaxation-time (MRT) framework. The conserved moments follow the general paradigm introduced in (3): ρ = f 0 + f + + f -, q = g + + g -. The non-conserved moments are chosen in the usual way:

J ρ = λ f + -f -, ε ρ = λ 2 f + + f --2 f 0 , J q = λ g + -g -.
The relaxation step of the scheme is particularly simple when all the relaxation parameters are equal to a constant value τ as proposed in the BGK hypothesis. When a general MRT scheme is used, we follow the rule [START_REF] Lallemand | Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF] of the moments m * after relaxation: [START_REF] Després | Systèmes hyperboliques de lois de conservation ; Application à la dynamique des gaz[END_REF] m * = m + s m eq -m .

• We have tested the previous ideas for a Riemann problem for a shock tube. We have chosen the following numerical data and parameters:

γ = 2 , ρ ρ 0 = 2 , ρ r ρ 0 = 0.5 , q = q r = 0 , λ c 0 = 8 , a = 0.15 , s j ≡ 1.8 .
The numerical results are displayed in Fig. 1. The rarefaction wave (on the left) and the shock wave (on the right) are correctly captured. Riemann problem for shallow water equations. Density (blue, top) and velocity (pink, bottom) fields were computed with the D1Q3Q2 lattice Boltzmann scheme with 80 mesh points and compared to the exact solution.



3) "D2Q5Q4Q4" vectorial lattice Boltzmann scheme

We study now the two-dimensional shallow-water equations ( 9)

         ∂ t ρ + ∂ x ρ u + ∂ y ρ v = 0 ∂ t ρ u + ∂ x ρ u 2 + p 0 ρ 2 0 ρ 2 + ∂ y ρ u v = 0 ∂ t ρ v + ∂ x ρ u v + ∂ y ρ v 2 + p 0 ρ 2 0 ρ 2 = 0 .
We have three conservation laws in two spatial dimensions. We extend the previous D1Q3Q2 vectorial lattice Boltzmann scheme into a D2Q5Q4Q4 scheme. The D2Q5 stencil is associated with the following velocities:

(10) v 0 = (0, 0) , v 1 = (λ, 0) , v 2 = (0, λ) , v 3 = (-λ, 0) , v 4 = (0, -λ) .
We now have three particle distributions: f ∈ D2Q5, g x ∈ D2Q4 and g y ∈ D2Q4. The natural question is to find an intrinsic method to determine the equilibrium values f eq j for 0 ≤ j ≤ 4 and g eq xj , g eq yj for 1 ≤ j ≤ 4. As in the one-dimensional case, a key point is to be able to explicitly determine the dual entropy. In this two-dimensional case, the entropy variables ϕ ∈ R 3 can be written as

ϕ = (θ , u , v) , θ = ∂η ∂ρ = c 2 γ -1 - u 2 + v 2 2 η * (θ, u, v) ≡ p ≡ ρ 0 2 c 2 0 θ + 1 2 u 2 + v 2 2 .
In order to determine the equilibrium distributions, we search for convex functions h * j (θ, u, v) for 0 ≤ j ≤ 4 , such that the first set of Perthame-Bouchut conditions (4) are satisfied:

(11)

4 j=0 h * j (θ, u, v) ≡ η * (θ, u, v) .
Then

f eq j = ∂h * j ∂θ , g eq xj = ∂h * j ∂u , g eq yj =
∂h * j ∂v . We also have to take into account the dual entropy fluxes ζ α in order to correctly represent the first-order terms of the model, ( 1) or ( 9) in our case. With the second set of Perthame-Bouchut conditions (4), we have:

(12) 4 j=0 v 1 j h * j (θ, u, v) ≡ η * u , 4 j=0 v 2 j h * j (θ, u, v) ≡ η * v .
For the D2Q5 stencil, the conditions of (11) (12) take the form

(13) h * 0 + h * 1 + h * 2 + h * 3 + h * 3 ≡ p , λ h * 1 -h * 3 ≡ p u , λ h * 2 -h * 4 ≡ p v .
We mimic for shallow water in two spatial dimensions what we have done for the onedimensional case (6), and we suggest here to set as previously

h * 0 (θ) = a 2 K θ 2 .
Because this function h * 0 does not depend explicitly on the variables u and v, we are not defining a D1Q5Q5Q5 scheme, but rather simply a D1Q5Q4Q4 vectorial lattice Boltzmann  scheme. The positive parameter a still has to be fixed. Nevertheless, we still have many degrees of freedom. We suggest moreover to break into two parts the first relation of ( 13):

(14) h * 1 + h * 3 = 1 2 p -h * 0 , h * 2 + h * 4 = 1 2 p -h * 0 .
We have now a set of five independent equations ( 6), ( 13) and ( 14) with 5 unknowns h * j . The end of the algebraic determination of the system (6), ( 13) and ( 14) is then completely elementary.

• When the potentials h * j are known, the computation of the equilibrium values is straightforward. With the 5 + 4 + 4 = 13 particle distributions, we can construct 13 moments for the D2Q5Q4Q4 lattice Boltzmann scheme. We suggest the following five moments associated with the distribution f j :

ρ = f 0 + f 1 + f 2 + f 3 + f 4 , J x, ρ = λ f 1 -f 3 , J y, ρ = λ f 2 -f 4 , ε ρ = f 1 + f 2 + f 3 + f 4 -4 f 0 , XX ρ = f 1 -f 2 + f 3 -f 4 .
For the eight moments relative to the distributions g xj and g yj , we have chosen

q x = g x1 + g x2 + g x3 + g x4 , f xx = λ g x1 -g x3 , f xy = λ g x2 -g x4 , XX u = g x1 -g x2 + g x3 -g x4 and q y = g y1 + g y2 + g y3 + g y4 , f yx = λ g y1 -g y3 , f yy = λ g y2 -g y4 , XX v = g y1 -g y2 + g y3 -g y4 .
• The value at equilibrium of the previous moments can be determined, taking into account that the three moments ρ, q x and q y are at equilibrium. We have:

   J eq
x, ρ = ρ u = q x , J eq y, ρ = ρ v = q y , ε eq ρ = 1 -

5 a 2 ρ + 5 4 ρ 0 (u 2 + v 2 ) c 2 0 , XX eq ρ = 0
We have also

f eq xx = ρ u 2 + p , f eq xy = ρ u v , XX eq u = 0 f eq yx = ρ u v , f yy = λ g y2 -g y4 , XX eq v = 0
. The MRT algorithm can be implemented without difficulty. It is just necessary to write a relation of the type (8) for the 10 moments that are not at equilibrium. Our present choice is the BGK variant of the scheme, with all parameters s set equal. The boundary conditions of wall constraint, supersonic inflow or supersonic outflow are treated with an easy adaptation of the usual methods of bounce-back and "anti-bounce-back".

4) First test cases

We propose two bi-dimensional test cases for the shallow-water equations: A stationary shock reflection and a classical non-stationary forward-facing step, first proposed by Emery [START_REF] Emery | An evaluation of several differencing methods for inviscid fluid flow problems[END_REF] for gas dynamics. The first test case is a the reflection of an incident shock wave of angle -π/4 issued from a "left" state into a new shock of angle atan(4/3) due  Strong nonlinear waves with vectorial lattice Boltzmann schemes to the physical nature of the "top" state (in green on the left picture of Fig. 2) and the "right" state (in indigo). The exact solution is determined through the use of the Rankine-Hugoniot relations. We have chosen

   ρ = 1 , u = 1.59497132403753 , v = 0 , ρ t = 1.17150636388320 , u t = 1.47822089880855 , v t = -0.116750425228984 , ρ r = 1.38196199044604 , u r = 1.33228286727232 , v r = 0 .
The stationary result of the vectorial lattice Boltzmann scheme for this first test case can be compared with the pure finite-volume approach with the Godunov [START_REF] Godunov | A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations[END_REF] scheme, solving a discontinuity at each interface at each time step. We have used three meshes of 35 × 20, 70 × 40 and 140 × 80 grid points. The contours of constant density are presented in Fig. 2. The numerical results are similar. Shock reflection, mesh 140 × 80. Exact solution (left), Lattice Boltzmann scheme D2Q5Q4Q4 (middle) and Godunov scheme (right).

• The second test case (Emery [START_REF] Emery | An evaluation of several differencing methods for inviscid fluid flow problems[END_REF]) is purely non-stationary. At time zero a small step is created inside a flow at Froude number equal to 3. A strong shock wave separates from the wall and various nonlinear waves are generated which mutually interact. Our present experiment (Figs. 3 and4) shows the ability of a vectorial lattice Boltzmann scheme to approach such a flow. We have refined the mesh, using three families of meshes: 120 × 40, 240 × 80 and 480 × 120. We have used λ = 80 , a = 0.05 , s j = 1.8 ∀j to achieve experimental stability. The time step is very small (due to the high value of λ = ∆x ∆t ), and in consequence the computation is relatively slow.

• We present our results for the finer mesh, at a dimensionalized time equal to 1/2 (Fig. 3) and 4 (Fig. 4). The results show the ability of the vectorial scheme based on the decomposition of the dual entropy to capture such flows. Nevertheless, the Godunov scheme, well known to be only order one, gives better non-stationary results compared to the new approach. Emery test case for the shallow water equations, mesh 480 × 120, t = 4, density profile, D2Q5Q4Q4 vectorial lattice Boltzmann scheme (top) and Godunov scheme (bottom).

Conclusion

We have extended the methodology of kinetic decomposition of the dual entropy previously studied for one-dimensional problems into a general framework of vectorial lattice Boltzmann schemes for systems of conservation laws in several spatial dimensions, in the spirit of Bouchut [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF]. The key point is to decompose the dual entropy of the system into convex potentials satisfying the Perthame-Bouchut hypothesis. Our first choices show that the system of shallow-water equations can be solved numerically without major difficulty. Nevertheless, our first numerical experiments show that the resulting scheme contains high numerical viscosity. Future work is necessary to reduce this effect.

Figure 1 .

 1 Figure 1.Riemann problem for shallow water equations. Density (blue, top) and velocity (pink, bottom) fields were computed with the D1Q3Q2 lattice Boltzmann scheme with 80 mesh points and compared to the exact solution.

Figure 2 .

 2 Figure 2. Shock reflection, mesh 140 × 80. Exact solution (left), Lattice Boltzmann scheme D2Q5Q4Q4 (middle) and Godunov scheme (right).

Figure 3 .

 3 Figure 3.Emery test case for the shallow-water equations, mesh 480 × 120, t = 1/2, density profile, D2Q5Q4Q4 vectorial lattice Boltzmann scheme (top) and Godunov scheme (bottom).

Figure 4 .

 4 Figure 4.Emery test case for the shallow water equations, mesh 480 × 120, t = 4, density profile, D2Q5Q4Q4 vectorial lattice Boltzmann scheme (top) and Godunov scheme (bottom).

Acknowledgments

The author thanks François Bouchut for enlightening discussions during the elaboration of this work and Bruce Boghosian for his help with the English language.


Olivier Maurice
email: olivier.maurice@gerac.com

Alain Reineix
email: alain.reineix@xlim.fr

Philippe Durand
email: philippe.durand@cnam.fr

François Dubois
email: francois.dubois@math.u-psud.fr

Kron's method and cell complexes for magnetomotive and electromotive forces

Keywords: EMC, Kron's formalism, MKME, tensorial analysis of networks AMS Classification : 55U10, 55U15, 78M34

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1) Introduction

Gabriel Kron (1900Kron ( -1968) has transfered the tensorial analysis developed in the framework of general relativity to the world of applied electromagnetism [START_REF] Kron | Tensorial Analysis of Networks[END_REF][START_REF] Balasubramanian | Differential Forms on Electromagnetic Networks[END_REF]. He had felt the straight relations between Kirchhoff's laws and topology. To replace his reflexions at this time, we try to give a brief history of these works, from Kron to nowdays. As many words exist to call the various elements available on graphs, we propose some of them that we use in our paper. Our first purpose is to find again these straight relations as simplest as possible. Making this exercise and under the hypothesis that no spaning tree current sources are involved, only electrostatic and magnetostatic phenomenons are considered, it appears that the current can be seen as a fundamental element of the faces boundary space. Through this vision, the electromotive force belongs to the faces space. The current seen as a boundary of faces, becomes equivalent to magnetomotive forces. By a physical understanding, a relation between magnetomotive forces and electromotive forces must exists. A problem appears because both doesn't belong to the same differential form dimension. Thanks to the Hodge operator, we make this link, following previous existing works. But if the electromotive force, which give the network its energy, belongs to the faces space, self inductance reaction must belongs to the same space and mutual inductance interaction translates the relation of Hodge. These components, as sensed by Kron, represent the metric that we take to compute these interactions. This leads to the Lagrangian expression of the whole graph and to the "chords" elements that symbolize these interactions. This Lagrangian must then be increased taking into account the spaning tree sources, added to the faces ones. This new space give us the complete base to take into account far field interactions. Lamellar fields create current sources, rotationnal fields create electromotive forces. This continuous fields are connected to our topology using moment space. This space is the frontier between our first bounded manifolds which generate the graph and continuous not bounded manifolds which are radiated and propagated fields. The first topological discussion gives all the base to include this new interaction through a generalized definition of the chords. At each step, we first give a topological approach before to "translate" it in expressions more usually given by physicists.

2) Tensorial Analysis of Networks (TAN) history

Gabriel Kron has written is famous "tensorial analysis of networks" in 1939. Before this work, he has written in 1931 a first remarkable paper "Non-Riemannian Dynamics of Rotating Electrical Machinery" [START_REF] Kron | Non-Riemannian dynamics of rotating electrical machinery[END_REF]. For this work he had the Montefiore price of the university of Liege in 1933 and the M.I.T. journal of mathematics and physics publishes the entire paper in the May 1934 [START_REF] Rottman | Gabriel Kron et la formulation d'une technique de résolution des systèmes complexes sur la base de la théorie du circuit électrique[END_REF]. This paper instantly produces wide-spread discussion and contreversy [START_REF] Happ | Gabriel Kron and System Theory[END_REF]. Kron uses its own notation without regarding established ones. This leads to some mathematicians contempt. But some of them were clear enought to understand and study Kron's work. Hoffmann [START_REF] Hoffmann | Kron's non Riemannian electrodynamics[END_REF], Roth [START_REF] Roth | The validity of Kron's method of tearing[END_REF] make links between Kron's concept and topology ones. Physicists like Branin, Happ, and in France Denis-Papin and Kaufmann [START_REF] Denis-Papin | Cours de calcul tensoriel[END_REF] promote Kron's work for electrical engineers. Many studies were done after around the concept of Diakoptic initiated by Kron [START_REF] Happ | Diakoptics And Networks[END_REF]. But to focus on topology, less references are available. Major lecture was made by Balasubramanian, Lynn and Sen Gupta [START_REF] Balasubramanian | Differential Forms on Electromagnetic Networks[END_REF]. Recently, there is the work done by Gross and Kotiuga [START_REF] Gross | Electromagnetic Theory and Computation[END_REF], following first one of Bossavit [START_REF] Bossavit | Electromagnétisme en vue de la modélisation[END_REF]. These last two works was made more particularly for finite element method. But they give fundamental bases through algebraic topology, following previous works of Roth and others, clarified using benefit of years passing. In this paper we try to take benefit of all this story and to present as clear as we can our understanding on these concept.

3) Notations

We note N the set of integers, R the set of real numbers, C for complex, T for cells, etc. We work in a complex cellular T ∞ made of vertexes s ∈ T 0 , edges a ∈ T 1 and faces f ∈ T 2 , etc. T is the whole set of these geometrical or chain objects. Low indices refer to chains, high ones to the geometric objects. Geometric objects are classical forms. Chains are abstract objects embedding properties added to the geometric objects in order to represent symbolically a real thing. A set of currents runing in a system can be linked with a set of chains associated with edges. This set of currents are components of a unique current vector. The current vector constitute a chain, image of some real currents on an electronic system. More generally, vectors can be associated with each of these geometric objects and their bases: |s for vertexes, |a for edges, |f for faces, etc. We note T j the vectorial space created by the geometric vectors |x of T D(x) (D(x) = 0 if x similar to s, etc.). T j can be developped as:

(1)

T j = σ∈T j α σ |σ , α σ ∈ R, C , j ∈ N
In this definition, we see here a generalized formulation of the classical writing of a vector, using the mute index notation (each time an index is repeated, the summation symbol on the index can be omitted) in [START_REF] Peikari | Fundamentals of network analysis and synthesis[END_REF]:

f = f σ u σ .
Here f is a vector developed on the base u σ of components f σ .

4) Boundary operator

We now introduce the boundary operator. It translates the intuitive understanding of object boundary. The boundary of a segment is a pair of two points, the one of a surface is a closed line, and so on. The boundary operator is the base of all Whitney's concepts [START_REF] Whitney | Geometric Integration Theory[END_REF]. To define integration through bounded objects, anyone needs boundaries. Once more, this boundary concept is natural. It is an application from T to T , ∂ : T → T , and more precisely, ∂ is an operator from T j to T j-1 . Its self composition leads to zero: (2)

∀θ ∈ T D(θ) , ∂ • ∂ |θ = 0.
∂ |a = s∈T 0 B s a |a .
Take a look to the graph figure 1. We can easily find its incidence B making relations between the vertices s and the edges a:

(3) B =   1 -1 -1 0 0 -1 1 0 1 1 0 0 1 -1 -1   .
Each row is linked to an vertex and each column to an edge.

5) Seeing electrical current as a 1-chain and the potential as a 0-cochain

In the following we consider the electrical current i as an element |i of the space T 1 : on each edge k, the current has a component i k which is a real number:

(4) |i = k∈T 1 i k |k .
In the tensorial analysis of networks [START_REF] Kron | Tensorial Analysis of Networks[END_REF] (as previously in classical nodal techniques [START_REF] Peikari | Fundamentals of network analysis and synthesis[END_REF]), the boundary operator applied to edges is called the "incidence". Using the sign rule saying that a current entering a vertex is affected of a plus sign and a current leaving a vertex is affected of a minus sign, it is a matrix that gives the relations between vertices and edges. It is possible to create for each vertex s a linear form s| acting on all the vertices: s|σ = 0 if s and σ differ, s|σ = 1 if s = σ. This form belongs to T * 0 : the dual space of T 0 composed by the 0-cochains. Moreover to each vertex s ∈ T 0 is associated a potential value V s . With this set of numbers we construct the potential V such that:

(5) V | = s∈T 0 V s s| .

6) A topological form of Kirchhoff's laws

We propose here to write the Kirchhoff's laws in a single abstract form as:

(6) V |∂i = 0, ∀V ∈ T * 0 , ∀i ∈ T 1 .



This relation applied to electricity is usual for physicists. If we retain one particular node (for example node 1 on figure 1), the algebraic sum of the currents i k for k an edge that contains the vertex number 1 is equal to zero; in this case i 1 -i 2 + i 3 = 0. To interpret relation [START_REF] Hoffmann | Kron's non Riemannian electrodynamics[END_REF] as the second Kirchhoff's law relative to the mesh law, we need the mathematical notion of co-boundary.

From the chains T j , we introduce the space T * j of co-chains of degree j: we have defined the duality product s|σ for two vertices s and σ of the cellular complex. We do the same for each geometrical object of dimension j. For α ∈ T j , the dual form α| belongs to T * j and is defined for each θ ∈ T j by α|θ = 0 if α and θ differ, α|θ = 1 if α = θ. The co-boundary operator ∂ o is the polar operator of the boundary operator ∂. By definition for ϕ ∈ T * j and θ ∈ T j+1 we have:

(7) ∂ o ϕ|θ ≡ ϕ|∂θ , ∀ϕ ∈ T * j , θ ∈ T j+1 .
The co-boundary operator ∂ o is defined from each T * j and takes its values in the space T * j+1 . The boundary operator makes decreasing the dimension of the chains while the co-boundary operator makes it increasing. The co-boundary operator is a good tool to express the second Kirchhoff's law. We re-express the fundamental property [START_REF] Hoffmann | Kron's non Riemannian electrodynamics[END_REF] in terms of the co-boundary operator:

(8) ∂ o V |i = 0, ∀V ∈ T * 0 , ∀i ∈ T 1 .
For each edge a we introduce the potential differences in term of the potential values V s for each vertex s and the incidence matrix B as introduced in (2):

(9) U a = s∈T 0 B s a V s .
So we have:

∂ o V = a∈T 1 U a a|.
We introduce a closed circuit γ. We test the relation (8

) for i = i 0 a∈γ |a . Then ∂ o V |i >= i 0 a∈T 1 b∈γ U a a|b = i 0 a∈T 1 U a = 0.
The mesh Kirchhoff's law express that the sum of the potential differences along a closed circuit is identically equal to zero. For example if we take a look on circuit 2-3-4 figure 1, we have:

U 2 -U 4 -U 3 = 0.

7) Spanning tree for pair of nodes currents

We assume now that the network T is connected. To fix the ideas we suppose more precisely that this network is simply connected i.e. does not contain any hole. When this hypothesis is not satisfied (a torus to fix the ideas) we refer to the contribution of Rapetti et al. [START_REF] Rapetti | Discrete vector potentials for non-simply connected three-dimensional domains[END_REF]. A spanning tree A is a subgraph of the set of edges, doesn't contain any cycle, and is composed with a number of edges equal to the number of vertices minus one; if we add to this spanning tree an edge a which doesn't belongs to A we obtain a cycle γ composed by a and edges of the spanning tree. We refer for a precise definition to the book of Berge [START_REF] Berge | Theory of graphs and its applications[END_REF].

Once a spanning tree A is fixed, any 1-chain can be decomposed in terms of boundary of faces plus a term associated to the spanning tree. In particular, each current can be decomposed in the previous form:

(10) |i = f ∈T 2 β f ∂ |f + α∈A θ α |α .
The first term f ∈T 2 β f ∂ |f corresponds to the meshes currents in Kron's terminology, and the second one α∈A θ α |α to the nodes-pairs currents. The formula [START_REF] Gross | Electromagnetic Theory and Computation[END_REF] describes the direct sum of these two spaces. It is denoted as the "complete space" in Kron's approach.

We have the following theorem: if the current i satisfies the Kirchhoff's law ( 6) then the nodes pair currents is reduced to zero. We have in relation [START_REF] Gross | Electromagnetic Theory and Computation[END_REF]: θ α = 0 for each edge α of the spanning tree A. The proof can be conducted as follows. Consider an arbitrary edge α ∈ A. We construct a potential V as the one explicited on the figure 1. We have ∂ o V α = α| plus a sum related to edges that does not belong to the spanning tree A. Then we have the following calculus:

0 = V α |∂i = V α |∂ f ∈T 2 β f ∂|f + V α |∂ β∈A θ β |β = V α |∂ β∈A θ β |β = ∂ o V α | β∈A θ β |β = α| β∈A θ β |β = θ α .
The property is established. When the Kirchhoff's law are satisfied, the electrical branches currents can be represented by the meshes currents |i = f ∈T 2 β f ∂ |f . In the general case, when charges are injected to nodes, the Kirchhoff's laws [START_REF] Hoffmann | Kron's non Riemannian electrodynamics[END_REF] are no more satisfied. The node pair currents α∈A θ α |α is not equal to zero and represents these charges variations. In Maxwell's equations, the charge conservation has two terms: "divJ" which is represented by the mesh currents and "∂ t ρ" represented by the node pair currents.



8) The fundamental space of faces

In electrodynamic we want to make a relation between the meshes currents i and some quantity coming from the flux Φ. It translates the general relation of electrodynamic between currents and electromotive forces. This flux Φ is in relation with a magnetomotive force through i. The induced electrical current described in the previous section can be linked with the meshes currents through |i = f ∈T 2 β f ∂ |f . The boundary operator being linear, we can write: |i = ∂ f ∈T 2 β f |f . This makes appear clearly the magnetic flux Φ given by:

(11) Φ = f ∈T 2 β f |f .
The magnetic flux Φ ∈ T 2 is associated with the faces in the complex cellular T . The meshes current i being under this view a boundary current. We define a dissipation operator W :

T 2 × T 2 (Φ, Φ ) -→ W (Φ, Φ ) ∈ R.
It creates a positive defined quadratic form: W (Φ, Φ) ≥ 0 and W (Φ, Φ) = 0 ⇒ Φ = 0. This quadratic form W generates a linear application:

T 2 Φ -→ ζ Φ ∈ T * 2 so that: (12) ζ Φ|Ψ = W (Φ, Ψ) ∀Φ, Ψ ∈ T 2 .
This linear application is nothing else than the impedance operator. On the faces base of T 2 it creates the impedance matrix Z:

(13) W f ∈T 2 α f |f , g∈T 2 β g |g ≡ f,g∈T 2 Z f g α f β g .
The energy source for the mesh space is given by the electromotive force e: for Φ ∈ T 2 , e ∈ T * 2 the dual product e|Φ is well defined and points out e as the dual source for faces as well as θ for the current source for nodes. The natural space for the electromotive force is cochain of degree 2.

Equilibrating sources and dissipation (in the general sense, i.e. used energy, losses or stored ones) we have:

W (Φ, Ψ) = e|Ψ , ∀Ψ ∈ T 2 which means (14) ζ Φ = e .
Then the current can be obtained through:

i = ∂ ζ -1 e 
Olivier Maurice, Alain Reineix, Philippe Durand, François Dubois which is the topological expression for the Kron's tensorial equation

i µ = y µν e ν .
9) Resolution of networks in complete spaces using the Kron's method

The method starts with a graph. This graph is a engineer view of a real system. Through homotopy, homology, surgery, the problem is projected on a graph [START_REF] Hatcher | Algebraic Topology[END_REF]. In this operation, we start by finding a ST passing through the various remarkable points of the structure. This tree can be drawn on a sheet. Each point has its own connection with the original 3D space attached to the structure. Figure 2 shows such simple graph obtained from a filter. On this graph, the ST is repaired by bold lines while closing edges are repaired by thin lines. Meshes are in doted and blue lines. Each edge current can be described depending on ST edges and meshes. For example, current of the edge 1 depends on ST edge 1 and mesh 2. The sign of this dependance is positive, as all currents flows in the same direction. When the meshes are constructed through closing edges from the ST, all the sign of the dependances are positive. This is a remarkable property linked with this construction method. The edge currents projection on ST edges and meshes is synthesized in a connectivity matrix C. If we choose to number the edges firstly as functions of closed edges belonging to the ST, then to meshes, the C matrix has a particular organization:

(15) C = Q L 0 I
As edges belonging to the ST are firstly numbered, they depend on both closing edges and meshes -that's why the submatrixes Q and L make links between these edges and  both current sources and meshes. On the other hand, edges obtained by closing paths depend only on meshes. So a unity matrix links these edges with the meshes, and a zero matrix shows that there's no links between them and the closing paths.

Each edge (a) has its own intrinsic property represented by an operator z aa . This operator can depends on the current value on the edge (non linear one). This operator is a metric component which we talk about in the next paragraph. To present the Kron's method, we accept that this metric has the form:

(16) z = A B E D
We note i t the ST edges, i c the closing path edges, J the current sources belonging to the ST, k the mesh currents. The connectivity is:

(17) i t i c = Q L 0 1 J k
The Kirchhoff's law for any edges can be written [START_REF] Peikari | Fundamentals of network analysis and synthesis[END_REF] (it can be used for any physics):

(18) 0 S = V 0 + z i t i c = V 0 + z Q L 0 1 J k
S are the mesh sources on which we come back later, V the ST potential differences. By multiplying on the left by C T (index T here is for transpose operation) we make appearing a bilinear transformation C T zC. Noting A , B , E , D the components of this triple products (they are the component of the metric in the complete space: ST plus meshes), we obtain finally:

(19) 0 S = QV LV + A B E D J k
As previously demonstrated LV = 0. The first equation resolved is:

S = E J + D k ⇒ k = (D ) -1 [S -E J].
Then, knowing k, the edges voltages of the ST current sources can be obtained:

QV = -(A J + B k).

10) About the metric

Usually, a metric is a matrix that can be half-positive, singular [START_REF] Penrose | A la découverte des lois de l'univers[END_REF]. In Kron's theory, as we want to manipulate complex operator through this tensor, we accept non strictly positive matrixes as metric and non symetric. Many mathematicians can object that we are finally far from a metric? But the notion stills very relevant as it describes a distance under preferential paths for currents in the topology, and it leads to the generalized power of the network studied. As said in [START_REF] Denis-Papin | Cours de calcul tensoriel[END_REF] page 309 (this is a translation from French): A general theory for linear networks, symetrics or not, should be based on a metric space definition  with fundamental tensor.... The connectivity C can be seen as a group transformation that belongs to SO ( [START_REF] Penrose | A la découverte des lois de l'univers[END_REF] page 271). We accept from now to call metric the fundamental tensor obtained from µ. We mean by this that to create the Hodge's relation between the magnetic flux and the electromotive force (emf) we need to give ourselves a metric. This metric is like a rule to make a correspondance between the vectorial surface flux and the scalar emf. The metric makes the link between the first space of mesh currents and the dual quantity of emf obtained from an Hodge process.

From years, electronicians use the mutual inductance to translate cross talk between both isolated circuits of a transformer. As we said before, emf and so self-inductances are for us deeply linked with the mesh space. Figure 3 shows the graph for a transformer cross talking two simple circuits. On the classical schematic presented fig. 3, we have two simple circuits having each of them a self-inductance and cross talked through a mutual inductance u. For each faces of each circuit, we can, following Maxwell's laws, compute the electric field circulation [START_REF] Maurice | La compatibilité électromagnétique des systèmes complexes[END_REF]:

(20) Γ≡∂f E • dΓ = A∈Γ dΓ J σ + B∈Γ dΓ t dt J -hE 0
A and B begin part of the boundary ∂f of the face f, hE 0 begin the source electrical work giving energy to the circuit. Each kind of energy (dissipative through σ, potential through ) can belong to a single edge in the edge space. At the beginning of the circuit description, we have separate information for edges. Always for one circuit, one could be dedicate for dissipative energy (they are the classical resistors in electrical circuits) and another for potential energy (capacitances). Each elementary circuit can be for example a RC one, one edge A being a resistor, and another edge B being the capacitor. When we construct both circuits, we first begin to associate couple of edges in a single mesh. Using the mesh connectivity: L T = 1 1 , we obtain a first expression of the metric in the mesh space given for one circuit g n by:

(21)

g n = L T zL = 1 1 R n 0 0 1 Cnp 1 1



(p is the Laplace's operator, and R n and C n the resistance and capacitance of the circuit n). The metric for the two circuits comes from the union of all the metrics of separate circuits involved: g = n g n . For us it gives :

(22) g = R 1 + 1 C 1 p 0 0 R 2 + 1 C 2 p
As inductances belong only to the mesh space, we must add them to the previous matrix through a new one µ, one including the inductance parts L n :

(23) g = g + µ, µ = L 1 p 0 0 L 2 p
Now the first circuit has its own energy source hE 0 -we don't care from where it comes, the second circuit has no self energy generator. But a cross talk creates in its mesh an emf e. This emf comes from the mmf F of the first circuit. Using the previous relations we can write finally a function between the mesh current of the first circuit k 1 and the emf of the second onde e 2 : e 2 = -upk 1 . The cross talk is symetric, it means that the current in the second circuit k 2 creates an emf in the first one: e 1 = -upk 2 . Finally, the complete magnetic energy tensor added to the one obtain in the edges space becomes :

(24) µ L 1 p -up -up L 2 p
We compute the emf added to the edge first circuit in a second step, after transformation of the edge metric in the mesh space and based on a function depending on the mesh current of another circuit. The mutual inductance appears here to be directly an application of the reluctance physic [START_REF] Lorrain | Champs et ondes électromagnétiques, Armand Collin, French version[END_REF]. A reluctance network and the associated graph can be construct where edges are tubes and vertex are meshes. This graph is the "chord" that is associated with the function u of mutual coupling. In next paragraph we will generalized this approach for Maxwell's fields. Before we obtain the extradiagonal component of our metric by another way. Another graph gives the same metric that the one of figure 3. Two spaces can be construct with these topologies. An isometric bijection exists between them that doesn't preserve the graph structure, but keep their common metric [START_REF]Encyclopedia universalis, Dictionnaire des mathématiques[END_REF]. We consider the graph presented figure 4. Always without ST sources, the current can be expressed as:

I = β 1 ∂ |f 1 + β 2 ∂ |f 2 .
The boundaries of the faces can be developed on the edges, depending on the directions chosen:

∂ |f 1 = |a 1 + |a 12 , ∂ |f 2 = |a 2 -|a 12
where a 1 is edge 1 on the graph, a 12 edge 2 and a 3 edge 3. f 1 and f 2 are meshes 1 and 2 (doted blue lines fig. 5). If z 1 , z 2 and z 3 are the three components of the metric for the three edges 1, 2 and 3, the metric tensor in the mesh space (L m : z → g) becomes:

(25)

g = z 1 + z 2 -z 2 -z 2 z 2 + z 3
Comparing with previous one, we see that:

z 1 = R 1 + (C 1 p) -1 + L 1 p -up, z 2 = R 2 + (C 2 p) -1 + L 2 p -up and z 2 = up.
The natural energy distribution leads to the lagrangian operator obtained in the mesh space using the Kron's description of the networks [START_REF] Gabillard | Vibrations et phénomènes de propagation[END_REF]. Graphs from figure 3 and4 have not the same topology, but they do have the same metric, that's why we call isometric the transformation from one to the other.

11) Extension of chords to radiative fields

All our previous discussion is based on graphs to describe a topology. When linked with real objects, these objects are reduced to edges through some algebraic topology operations and concepts like homotopy, homology, etc. Representation of reality is made through agencement of edges as pieces. The whole object modeled is a group of R connex networks parts of a global graph. Each of these networks is a set of B edges, joined by N vertexes. The major characteristic of the object is its number of meshes M by respect of the equivalent Poincaré's law for complex cellular: M = B -N + R. In each connex network transmission of energy is perfectly controled because it goes from one vertex to another. By this way, the mathematical concepts can be applied to Maxwell's fields quite easily, under the hypothesis that the field behave like a bounded volume able to be represented by an edge. This is true for near and evanescent fields -for which macroscopic modelling are resistors, capacitances, inductances, reluctances, but this is not true for the free radiative field that cannot radiate in a bounded volume independently of the distance.

We recall the fundamental difference between these two kinds of fields, and then we apply our previous method to these fields.

a) Radiated versus evanescent fields

Basic demonstration for photon starts from the potential vector in the Coulomb's gauge [START_REF] Feynman | Quantum Electro-Dynamics[END_REF]. Under this gauge, there is a formal separation of the transverse part of the field with the other components. Transverse part of the field leads to the photon concept of quantum mechanics. All the evanescent parts are the longitudinal components of the potential and scalar vectors. They can be modeled using inductances, capacitances, etc., that are properties (and components of the metric) of edges or meshes. Strictly speaking, evanescent modes of the field are virtual photon [START_REF] Stahlhofen | Evanescent modes are virtual photons[END_REF]. The big difference for us is that the free field cannot be enclosed in a bounded edge (or a cycle of bounded edge, i.e. a mesh). It radiates in the infinity space. Another remarkable property of the far field is the radiation resistance. On the edge radiating, the property is increased of this radiation resistance. It can be shown that the radiation resistance is intrinsic to the radiated edge.

When enclosed in a shielded room, the same edge has its metric modified by the metallic walls of the room. Reflecting the radiated field, the radiation resistance disapears due to the back induction coming from the walls. This process allows to desmontrate that the radiation resistance is natively existing and is modified by the environment [START_REF] Leighton | Le cours de physique de Feynman[END_REF] [25].

b) Emf and chords process for free radiated fields

From years, engineers use antennas very simply: an input impedance gives the antenna equivalent circuit seen from the electronics. A radiation diagram describes the radiation of the antenna in all the free space. A gain, gives the relation between the free field and the power received. This very efficient modelling can be translated in our topology description. Basic principles are well explosed in [START_REF] Kraus | Antennas[END_REF]. We present these principles before to translate them in topology.

c) Antennas principles

Depending on the antennas gain, the receiving antenna can integrate the energy coming from the emitting antenna. The total radiated power S r in free space (whatever the distance r from the emitter) is:

(26) S r = P t G t 4πr 2 e -τ p P t is the power delivered to the emitter, G t is its own antenna gain. The last term makes sure of the causality. The available power in reception is given by: P r = S r A r . A r is the effective surface of reception of the antenna. The relation between gain and effective surface is:

(27) G x (θ, φ) = 4π λ 2 A x (θ, φ)
Both gain or effective aperture (the other name for the effective surface of reception or emission) are functions of the 3D space. Using all the previous relations we find what is the Friss' equation:

(28) P r P t = A r (θ, φ) e -τ p λ 2 r 2 A t (θ , φ )
Now we just have to make the links between the power and the topology. For that, fundamental concepts previously stated should be used.

d) From power to topology

The total power of the radiated field is obtain through the radiative resistance on an edge. This component represents all the losses due to the radiation. As inductance and capacitance are linked with evanescent and lamellar fields, the radiated and transverse one which leaves the circuit (we consider no losses due to Joule effect in the wires) is linked with a resistance. For example we have: P t = R 11 k 1 and P r = (e 2 ) 2 R 22 . These relations give us the fundamental coupling impedance for far field:

(29) z 21 = e 2 k 1 = R 11 R 22 A r (θ, φ) [G θ,φ,θ ,φ ] A t (θ , φ )
where G θ,φ,θ ,φ is a Green's kernel for the Friis formula. Once more we see that e 2 derive from an on a face that belongs to T 2 . This kind of interaction can be generalized, whatever the modes of the field propagated. Take a look to figure 5. We see four simple meshes in interactions. For each of them we can identify the emf and mesh currents, and both emitted and received power. Going from the source of dynamic field k 1 to the induced emf e 4 we can describe the path: e 4 = z 43 y 33 z 32 y 22 z 21 k 1 . The z ij are the coupling metric components and y ij the intrinsic inverse metric of meshes 2 and 3. A t and A r are properties attached to the graphs 1 and 4, i.e. to faces that belong to each graph of these circuits. In fact e 2 is the emf linked with A t : it's a face resulting from a co-boundary applied to the cycle k 1 . Each point on this face can be linked with a wave vector. The set of these wave vectors generate the flux S t (corresponding to k 2 ): the cotangent manifold of R 4 generated by k 1 using an application φ. The component of this set are transformed by the propagation operator yzy to have its image through S r (k 3 and φ -1 ) where this time, z is not a metric attached to the cellular complex T ∞ but to the 4D space-time fields propagation R 4 . The interface with the receiver is covered by the scalar product with A r → k 3 . Then the emf e 4 is created by k 3 . Figure 6 shows the general process involved in the connection between the cell complex T ∞ and the 4D space

12) Extension to functors

Previous function φ is a correspondance between both topological spaces T 2 and R 4 : φ : T 2 → R 4 . We define as morphism of graphs, functions that preserve the graph structure, i.e. keep the numbering of vertices, edges, etc. Cell complexes can be used to project any physical phenomenon on graphs. It was already usual to employ electroanalogy for various physics like thermal [START_REF] Ouin | Transferts thermiques[END_REF], mechanics [START_REF] Van Santen | Vibrations mécaniques[END_REF], quantum mechanics [START_REF] Sproull | Eléments de physique moderne[END_REF] and even biological information [START_REF] Grimnes | Bioimpedance and bioelectricity basics[END_REF][START_REF] Poudret | Exploring Topological Modelling to Discriminate Models of Golgi Apparatus Dynamics[END_REF]. In this last case, the author introduces the two categories of continuous and discrete spaces. This theory seems to be relevant for us, as it gives an algebraic approach for the method of chords [START_REF] Maurice | On mathematical definition of chords between networks[END_REF]. A possible extension of his approach could be to generalize our previous work to functors between these two categories. By the way the method of chords replaces a large spatial domain where free electromagnetic energy propagates by a discrete link between topological objects. This kind of model could be extended to other physical configurations.

13) A cavity problem as example

Considering a cavity with an aperture and an impeging parasitic wave, the problem is to compute the internal field induced by the external wave (figure 7). In the following, only the vertical polarization of the incident field will be considered. This problem can be solved using the formalism developed above. Thanks to the analogy between respectively the electrical field and the voltage and then the magnetic field and the current, it is possible to define electrical equivalent scheme of the problem. In order to make such a representation, we will begin by the definition of different topological domain, corresponding to different regions that will be modeled separatly and then connected together to reconstruct the entire cavity. Three object have to be modeled:

• the incident field that can be assimilated to a voltage source in series with a 377 ohms impedance. This model represents the electrical field propagating in the free space that is equivalent to the mathematical representation in a plane wave shape of the incident field arriving with a normal incidence.

• the aperture, that will be seen from its middle point and that can be represented as two short circuited half transmission lines. The impedance seen from the center has already been extensively studied by different authors. The famous formulation given by Gupta ( [START_REF] Balasubramanian | Differential Forms on Electromagnetic Networks[END_REF]) has proved to efficiently represent the aperture impedance by a simple formula:

(30) Z a = 120π 2 [ln(2 1 + 4 1 -(w e /b) 2 1 -4 1 -(w e /b) 2 )] -1
• the cavity that can be seen as a transmission line along the main axis with some particular terminal conditions. On the side containing the aperture, the line will be directly connected in parallel with the aperture impedance model.The opposite side will be short circuited. In the present case, only the fundamental TE propagating mode will be considered. If we want to have results in a more large frequency bandwidth, we will have to consider each mode as an individual transmission line and to connect them in parallel. In order to complete our model, we will add the possibility to realize a measurement in one point in the internal cavity. As in a real experiment, we will introduce a sensor that is modeled here as a resistor having a high value in order to avoid field perturbations inside the cavity. This can be made by cutting the line into two half transmission lines located on each side of the transmission line. After having introduced the topology of our system, it is now important to give the Kron's transmission line model. In fact the more easier way to represent a transmission line is to give a quadrupole model. This quadrupole can be represented by two branches coupled by driven voltage sources. In fact, we can notice that such a representation is nothing else a circuit model of the impedance matrix of the line seen from its two extremities:

(31)    V 1 = Z 11 i 1 + Z 12 i 2 V 2 = Z 21 i 1 + Z 22 i 2
For example, from the branch 1, the first equation shows two terms: the first one represents the influence of the current from this branch and the second one the effect of the current on the first branch. This coupling can be represented by a voltage source, the equation becomes: V 1 = Z 11 I 1 + V c . As a conclusion, the Kron's model of the cavity is given figure 8. In such an application, the important parameter that is usually modeled is the shielding effectiveness. This quantity was compared with the one obtained with a FDTD code. A very good agreement can be observed figure 9. Other examples between many others are given in [START_REF] Durand | Generalized Interaction Principle Implemented in the Kron's Method[END_REF], [START_REF] Leman | Kron's method applied to the study of electromagnetic interference occuring in aerospace system[END_REF], [START_REF] Poiré | SMPS tools for EMI filter optimization[END_REF].

14) Antenna modelling

This second example shows the use of the formalism to compute the interaction between an antenna and a metallic wall. The objective of this experience was to understand the effect of a reflexion of energy on the radiation impedance of an antenna. One horn antenna is powered through an amplifier that delivers a amplitude modulated waveform at 10 GHz. Moreover, 50 cm in front of the antenna, there is a metalic wall or absorbers. The set-up of the experiments is shown figure 10. e 2 = (V 1 + Z c i 1 ) e -τ p V 1 and V 2 are voltages respectively at the left and right of the line. i 1 and i 2 are the currents at the same extremities. Replacing V 1 and V 2 by their expressions depending on the loads and currents, Looking at the circuit figure 12, we obtain: Any expression involving forms like e i /I j can be replaced by an impedance interaction z ij . Any line or guided wave structure can be replaced by an impedance tensor as: In the graph figure 11, two of these structures were used. The last edge of the graph represents the emitting horn antenna. It is a radiation resistance of fifty ohms. After a delayed time, when the field is reflected by the metallic wall in front of the antenna, a reflected field wave comes back in the horn and creates an electromotive force given by: e 5 = Gλ (4π2R) -1 σ √ R r i 4 (R is the distance to the wall, σ the reflection coefficient on the wall, G the antenna gain and R r the radiation resistance). This creates an interaction given by e 5 /i 4 . The complete experience is detailed in [START_REF] Maurice | Expérience pour mettre en évidence la variation d'impédance d'entrée en fonction des interactions avec l'environnement[END_REF]. Various measurements were made. Some with the wall equipped of a metallic plate. Some others with absorbers. The variation of radio frequency signal envelop at 10 GHz shows both cases of short circuit on the metallic wall or free space radiation on the absorbers. Difference between the signal compute under the Kron's formalism and measurements is of 1,2%. This performance like others [START_REF] Alaeldine | EMC-oriented analysis of electric near-field in high frequency[END_REF] was obtain using the method which allows to couple various accurate equations taken from different previous works.

(33)    V 1 = E 0 -R 0 i 1 V 2 = RLi 2 .

15) Conclusion

From fundamental definitions of discrete topology like the boundary operator and the notion of duality, we propose a mathematical model to formalize major results expressed by Gabriel Kron in his "Tensorial Analysis of Networks". One remarkable fact is that the chords introduced in a previous work appear as links between electromotive forces and meshes currents. We present an application of this result to an electromagnetic cavity without the help of three dimensional Maxwell's solver. Another application shows free radiated interaction between an antenna and a wall. This algebraic concept is flexible and the extension is under work for the modeling of multidisciplinary systems with networks.

Figure 1 .

 1 Figure 1. Spanning tree.

Figure 2 .

 2 Figure 2. Graph of a filter.

Figure 3 .

 3 Figure 3. Graph of a transformer.

Figure 4 .

 4 Figure 4. Isometric graph.

Figure 5 .

 5 Figure 5. General free field process.

R 4 Figure 6 .

 46 Figure 6. Sequence of operation for free field.

Figure 7 .

 7 Figure 7. Electric field on cavity with an aperture.

Figure 8 .

 8 Figure 8. Kron's model for a cavity.

Figure 9 .

 9 Figure 9. Comparison with a Finite Difference Time Domain computer code.

Figure 10 .

 10 Figure 10. Experiment set-up.

Figure 11 .

 11 Figure 11. Experiment equivalent graph.The splitter is made of three resistors of 17 ohms. Cables are simulated using Branin's model. For a cable of electrical length τ , characteristic impedance Z c , the Branin's model consists in two equations defining the electromotive force at each extremity of the line:

Figure 12 . 1

 121 Figure 12. Experiment equivalent graph.So, by replacement in[START_REF] Maurice | On mathematical definition of chords between networks[END_REF] we understand that:

e 2 -

 2 E 0 e -τ p = (Z c -R 0 ) e -τ p i 1

R

  O + Z c (RL -Z c ) e -τ p (Z c -R 0 ) e -τ p RL + Z c  

  As we will see in next paragraph, this operator is linked to various connectivities in tensorial algebra. A face boundary is a cycle (a closed line). When we have an edge |a , we can consider its boundary ∂ |a which is a couple of vertices. The boundaries can be developed on the zero chain of vertices :

For example, ∂ • ∂ |a = 0: the boundary of an edge is a vertex and the boundary of a vertex is null, or ∂ • ∂ |f = 0: the boundary of a face is a closed line, and the boundary of this closed line is null (remember ∂ • ∂ = 0).