From the private bank to the joint-stock bank:

the case of France (second half of the 19th century) by Nicolas Stoskopf, CRESAT, University of Haute-Alsace, F-Mulhouse

The political revolution of 1848 resulting in the fall of the July Monarchy and the advent of the Second Republic may be regarded as the starting point of the banking revolution in France. Initially, the credit crisis which accompanied these events underlined the failings of a banking system which, until then, had been highly unstable and deficient. The upper echelons of the system admittedly stood up reasonably well, even though operations were reduced: the Banque de France continued to fulfil its role as bank of issue and discount house but under very restrictive conditions; the major banking-house system (haute banque), comprising private banks -which stood apart by way of age, respectability and mutual recognitionmanaged the accounts of a wealthy clientele and were responsible for major business (State loans and mining and railway investments). At the lower echelons, however, the lower-ranked regional and local banks -inadequate for the needs of small and medium businesses as well as of ordinary clients even before 1848 -were unable to withstand the crisis and left a gap which was unfavourable to recovery of business 1 .

It was in this context that the Comptoir national d'escompte de Paris was established, as a matter of urgency, in March 1848. It was the starting point of a banking revolution which flourished under the Second Empire (1852 -1870) with the establishment of Crédit foncier (1852) and the Pereire brothers' Crédit mobilier (1852), followed by the foundation of the major deposit banks, the Crédit industriel et commercial (or CIC, 1859), Crédit lyonnais (1863) and Société générale (1864). This revolution continued at the beginning of the Third Republic with the creation of the investment banks -Paribas (1872) and the Banque de 1 On the financial crisis of 1848 see A.J. TUDESQ, « La crise de 1847 vue par les milieux d'affaires parisiens », Aspects de la crise et de la dépression de l'économie française au milieu du XIX e siècle (1846-1851), Bibliothèque de la révolution de 1848 , 1956, vol l'Indochine (1874) 2 . The new banking system which was then set up, was very different from the old. The latter comprised simple or limited partnerships whose directors invested their own capital and were liable for their acts to the extent of their entire fortunes; the former were limited liability companies run by managers who were accountable to their shareholders and sometimes to the State, should they have been appointed by the latter. The banking revolution therefore took place at two levels: it renewed the banking offer by allowing banks to use clients' deposits on a large scale and diversify the distribution of credit and it drastically modified banking management practices by appointing salaried directors to head the new establishments.

In his famous article, David Landes pointed out that this divergence should not be exaggerated 3 ; despite appearances, in particular the conflict between the Rothschilds and the Pereires, the new banking system was not created to compete with the old, but to exist alongside it. The heads of the major banking houses (Fould, Bischoffsheim, Blount, d'Eichthal, Mirabaud, Seillière etc.) made a considerable contribution to the foundation of the new establishments. The prosopographic study on the patrons of the Second Empire 4 , applied to Parisian financiers and bankers 5 , enables us to study this question and compare the directors of the old and the new banks, or, if one prefers, of the private 6 and joint-stock banks. Do they differ in their cultural and sociological characteristics, their professional attitudes or in their banking practices?

I. Two types of banker…

The first fact to emerge from the study is that each of these banking categories had its own distinct type of banker:

-The private Parisian bank was multi-confessional and cosmopolitan but fairly homogenous in its social origins. There were three main branches: the first, native French, comprised Catholics (Davillier, Laffitte, Perier, Seillière), Protestants (André, Neuflize) and Jews (Fould); the second, was made up of a sizeable contingent of Swiss bankers of reformed religion who began to establish themselves in France at the beginning of the 18 th century (Mallet) and continued up until the Restoration (1815-1830); most came from Geneva and its surrounding districts (Dassier, Delessert, Hentsch, Mirabaud, Vernes), with a few from Zurich (Hottinguer) or Bern (Marcuard). Finally, the most recent branch, who settled in France between the beginning of the century and the war of 1870, were German Jewish bankers from the Rhineland (Bischoffsheim, Cahen d'Anvers, Erlanger, Rothschild, Stern) or Bavaria (d'Eichthal, Koenigswarter). A few personalities such as the English Catholic Edward Blount or the Savoyard Frederic Pillet-Will completed this heterogeneous picture. This remarkable cultural diversity was partly attenuated by mutual professional experience; these bankers usually belonged to families who had been engaged in business trading, and often even in banking7 , for so long that at the time of the Second Empire they represented the second, third or -as in the case of Adolphe d'Eichthal, a fifth generation banker -even later generations in banking.

The new managers' profile was exactly the reverse, culturally homogenous but sociologically diverse; with the exception of the Italian Cernuschi, co-founder of the Banque de Paris in 1869, they were French and mainly Catholic. Yet, almost all of them were new men on the scene, not only to banking but also to commerce, which was not usually part of their family background. In this group we find a significant contingent of noblemen who thus accomplished a spectacular aggiornamento to the capitalist economy. Their professional careers were equally varied; those who had experience of banking were a minority; they included Emile Pereire, who was employed by the Rothschilds for a long time and Pinard, a partner at a small banking house since 1840. Some started out as lawyers, others as industrialists and still others acquired considerable experience sitting on Boards of Directors before becoming directors. Yet the most frequent and remarkable career path -at the origin of a particularly French tradition -was via the civil service. At the outset, the government became obsessed with the dilution of responsibility in limited companies; this obsession led the Conseil d'Etat to refuse all requests for authorization from limited companies until 1848 and to accept them in dribs and drabs thereafter, coupling them with restrictive conditions, ostensibly for the protection of shareholders, until the enactment of liberal laws which ended the control procedures for companies with a capital of less than twenty million francs in 1863, and then for all companies in 1867. These restrictions included the government's right to appoint the directors of credit establishments or at least to approve their appointment. The presence of senior civil servants at the head of the major credit establishments was justified by the need to establish confidence, no longer based on the solvency of a director who was liable to the full extent of his fortune but on recognized competence and a residual authority inherited from the exercise of public responsibility. Thus, when France's first deposit bank, the CIC, was created in 1859, the State insisted on appointing the 73-year-old Marquis d'Audiffret, Grand Officier of the Légion d'Honneur, Chairman of the Cour des Comptes, senator and French specialist in public finance. Many civil servants, and in particular the finance inspectors, were attracted by the notion of pantouflage8 ('crossing the floor' from the public to the private sector): Ploeüc, for example, tried to enter Crédit mobilier in 1853 and gave up because the salary was insufficient, but he 'crossed the floor' in 1863, when he accepted the directorship of the Imperial Ottoman Bank. Several of his colleagues, including Adrien Delahante, the first director of the Société générale, forsook the service of the State and the Prince to serve the shareholder and the client.

The careers of these new bankers bore no resemblance to those of a Rothschild, a

Hottinguer or a Seillière. There was Louis Frémy (1805 -1891), born in the little commune of the Yonne department where his father had been mayor, and orphaned at the age of two; nevertheless he was able to study law, to begin his career as a lawyer and then to enter the Conseil d'Etat as an auditor. He became sous-préfet of Domfront in 1835 and of Gien in 1837 and was appointed to the Railways Management Commission, where he established relations with many of the investors who shared the railway concessions. He also had political connections, among others with Léon Faucher who took him as departmental head at the Ministry of the Interior in 1848, with Persigny, a faithful ally of Louis-Napoleon Bonaparte and Baron Haussmann, future préfet of Paris under the Second Empire. These multiple connections enabled him to start a banking career in 1857 at the age of fifty-two, when he was appointed Governor of Crédit foncier. When he resigned in 1877, he was managing credits of the order of two billion francs, significantly greater than those of any other major banking house.

The French language makes no distinction between the various categories of financial professionals, who are all included in the generic term 'banker', but there are doubtless several historical reasons for this lack of precision.

II. Of those who mixed the genres…

When we mention the passages of arms between James de Rothschild and Emile Pereire we usually forget to specify that the latter was not a banker in the strict sense of the term: under the Second Empire he chaired the Compagnie des chemins de fer du Midi, the Compagnie immobilière de Paris, the Compagnie générale Transatlantique and several other insurance or service companies, but he always sufficed himself with a simple administrative position in the board of the Crédit mobilier, headed by his brother Isaac after Benoît Fould resigned in December 1854. However, in the eyes of his contemporaries and of historians, Crédit mobilier was very similar to a Pereire bank and almost the equal of a major banking house. Admittedly, the Pereire and Fould families subscribed 28.6% of the capital in November 1852 and therefore held a majority of the original capital9 ; they brought men such as Ernest André, Adolphe d'Eichthal, the Duke de Galliera, Charles Mallet and Achille Seillière onto the Board of Directors -these men, imbued with major banking, brought the caution of the old bank to the new. When Crédit mobilier was in difficulties in 1867, the Banque de France was only prepared to help on condition that the Pereire brothers resigned and personally deposited ten million francs by way of guarantee10 . The personalization of the management of limited companies may come as no surprise to an observer at the beginning of the twenty-first century, but contemporaries such as the economist Alfred Neymarck found it much more striking; in 1880 he wrote 'In France we have a very fine group of first rank credit institutions (which are) huge associations, vast aggregates of interest, which are limited companies (sociétés anonymes) according to the law and their articles of association. Well, well! In reality they are far from 'limited' (anonymes). What one sees in them, and sometimes above them, is the name of a talented founder, an experienced administrator, a competent and approved director, a noted and well-known organizer11 ».

Stable management was probably a significant factor in this identification. Biesta and Pinard remained at the head of the Comptoir d'escompte from 1848 to 1870-1871, Frémy and Soubeyran at Crédit foncier from 1860 to 1877-1878, Henri Germain at Crédit lyonnais from its foundation in 1863 until his death in 1905. From here to regarding the company as personal property was just one step that some people give the impression of having take. At any rate they felt that they were omnipotent and untouchable. This development was particularly noticeable at the Comptoir d'escompte de Paris and at Crédit foncier after 1860, the starting point of a liberal revolution inaugurated by the Franco-British commerce treaty.

-Though the Comptoir d'escompte had been created to meet commercial needs in Paris, it was under governmental pressure to finance external business, particularly overseas and in the colonies 12 . This new orientation was orchestrated by its sub-director, Alphonse Pinard, who was closely associated with Chasseloup-Laubat, the Minister for the Colonies: he was so committed to it that in October 1860 he personally subscribed 4.7% of the capital of Crédit colonial, established under the patronage of the Comptoir d'escompte, of which he became Chairman 13 . He formed an alliance with Louis Bischoffsheim and Edouard Hentsch, two major bank heavyweights, to found the Banque de dépôt et de crédit des Pays-Bas, the forerunner of Paribas, in March 1863 and the Société générale in May of the following year. When the shares were subscribed he took 16 000 for the Comptoir and 7 450 for his own account -a capital investment of 3 725 000 francs 14 . The Comptoir specialized in French foreign loan business from 1864 onward and Pinard 'gambled' recklessly with the company's money and with his own. Rumour had it that the Comptoir had 15 million francs' worth of its director's discounted bills in its portfolio! Pinard was under attack -from shareholders at general meetings, in lampoons and in the press. He was accused of betraying the Comptoir's original mission, of scarcifying money in France, of using the company for his own personal ends, of getting rich at the shareholders' expense and on the backs of the investors who subscribed to the Comptoir's 'rotten' loans 15 . All of this was probably true, but Pinard could count on his political protectors and take refuge behind the excellent results of his establishment which paid out a dividend in excess of 10% from 1864 to 1868.

-At Crédit foncier 16 , Frémy and Soubeyran did not draw a very clear line between their public function and their private affairs either: Fremy, for example, chaired a number of companies whose activities would clash with those of the establishment of which he was Governor. When Crédit foncier was in difficulties following the Egyptian bankruptcy of April 1875, Soubeyran behaved like a major banking house patron: he placed a governors' guarantee of sixteen million francs on the table, fifteen million of which he provided himself. When the Government revoked his appointment in August 1878, to the amazement of his fellow colleagues, who had stood in admiration of his competence, Soubeyran at least had the satisfaction of seeing the Egyptian operation -which could have been disastrous -finally settled, with a gross profit of thirty-eight million for Crédit Foncier 17 ; however, he was never able to recover his fifteen-million-franc guarantee deposit and the major part of his will, drafted in January 1890, was devoted to enjoining his heirs to continue fighting for its restitution 18 . Several times he had occasion to explain why he had acted in such a manner; he believed that bankers, even in limited liability companies, should put their personal fortune on the line in order to inspire confidence in their partners and their clients, their wealth being the ultimate proof of their competence. As far as his opponents were concerned (among them the Pereires and the Republican left), his success was proof of his dishonesty.

Private bankers, on the other hand, took the opposite path, as we have seen in the case of the Pereires and the Comptoir d'Escompte: they agreed to commit part of their fortune to the 'vast interest aggregates' and to lend their name and reputation to the new companies. Frédéric Pillet-Will was an exception; he distanced himself from all the financial groups of his time and constantly demonstrated his hostility to the new joint-stock banks. Among his fellow bankers, the prospect of profit took precedence over respect for outmoded principles: David Landes is correct in declaring that "the joint-stock banks promised to be profitable businesses which, far from clashing with the established framework of their activities, seemed to be in perfect harmony with it". 19 The first to understand this and to declare himself in favour was Benoît Fould, co-founder and short-term Chairman of Crédit mobilier, before attacks of insanity forced him to resign. He had many imitators, including Louis In every case, Donon and his personal associates in Donon, Aubry, Gautier & Co, and the other companies in the group subscribed a significant part of the new company's capital. In the end Donon controlled a large group of companies, built around a private bank of which the partners, their families and their friends in the political and financial worlds were directors of many limited companies.

The absence of a definite dividing line between private and public money, private and public business or between business and politics is altogether typical of the climate of political racketeering (affairisme) which reigned under the Second Empire and at the beginning of the Third Republic. It has often been denounced as immoral. Its economic effectiveness remains to be seen.

III. For a brief period of euphoria

If we may judge by the international spread of French capitalism in the third quarter of the nineteenth century, it must be recognised that the banking system established from 1850 onward was particularly effective 21 . It was as if French savings were a dormant nest egg which only needed to be called upon in order to reveal its full potential. The phenomenon was explosive, particularly under the aegis of the Pereire brothers and Credit mobilier, who multiplied their major international operations between 1853 and 1856; they included the cofoundation of the Darmstädter Bank (1853), financing the Mines and foundries of the Vieille Montagne in Belgium (1853), founding the Austrian State Railway (1854), the Crédit mobilier espagnol (1855), the Russian Railway Great Company (1856) [START_REF] Cameron | The Credit Mobilier and the Economic Development of Europe[END_REF] , etc. Rivalry with the Rothschilds was a further stimulant to this expansion in which other French groups took part, particularly after 1860: French capital was dominant in Spain, Italy, Austria, the Ottoman Empire, Egypt and Tunisia etc. When Bismarck imposed an indemnity of five billion francs on France under the Treaty of Frankfurt, everyone was amazed at the ease with which the territorial liberation loans were raised in 1872-1873. We may wonder whether this remarkable efficiency is to be attributed to political racketeering -if, in other words, the mixture of genres observed in the bankers' behaviour also extended to banking practices.

To answer this question we should start with the private Parisian bank, and in particular its elite, the major banking houses. Unlike the London exchange, with its specialized establishments, each with very specific functions 23 , the major banking houses were nonspecific: the same houses undertook classic banking operations (exchange, discount, recoveries), financial operations (issuing loans and company shares), as well as international trade and direct investment (railways, mines, heavy industry, real estate) [START_REF] Landes | Vieille banque…[END_REF] . Admittedly, we can discern certain orientations which reveal very precise strategies; thus the Seillière bank was very active in financing the European steel industry, granting sizeable advances to Schneider, Wendel and Krupp; the Hottinguers had a solid position in transatlantic commerce and the Marcuard-Andrés main interests were in the Mediterranean. Yet this kind of specialization did not lead to a sharing of markets or to a real division of labour. On the contrary, the activity of a Charles Lafitte, striking by way of its sectorial and geographical eclecticism, was doubtless more representative of this all-purpose bank, a bank for all types of business, which first and foremost kept an open mind in seeking the best opportunities for making a profit, as soon as the opportunity arose: under the Second Empire, Lafitte was chairman of the Victor-Emmanuel Railway Company (Piedmont-Sardinia), obtained the concession for Cordova-Seville, speculated in the Bois de Boulogne land at the gates of Paris, financed a canned-food company at Meaux, issued an Egyptian loan, took a passing interest in an Ottoman bank creation project, set up the Compagnie des lits militaires and administered the Maubeuge blast furnaces.

The banking revolution should have established some order in these practices by setting up a specialized, hierarchic banking system. This was also the main concern of the Pereire brothers who, as good disciples of the Count de Saint-Simon, wanted to end the anarchy of liberal capitalism, damaging both to entrepreneurs and to workers, and to replace it with harmonious organisation of production 25 . The Pereire brothers were already very much involved in the creation of the Comptoir d'escompte and Crédit foncier and they believed they could achieve their objectives through Crédit mobilier: the basic idea was to centralise the financing of major enterprises by issuing debentures which would attract dormant savings and replace the multitude of certificates on the market with a single bond -an 'omnium' bondwhich would distribute credit throughout the French economy. This centralisation would be the prelude to a progressive merging of all enterprises in the same sector into a single company. The process knew no apparent limits: in 1855, Isaac Pereire not only explained the need to create similar establishments abroad, but also envisaged the issue of a European credit bond which would eliminate the differences in interest and exchange rates in the various financial markets. The Pereire brothers were visionaries who came up against powerful opponents: the Rothschilds, who denounced their hegemonic ambitions, but above all the State, which prohibited Crédit mobilier from increasing its resources by issuing short-or long-term debentures; they were never able to realize their ambitions and their failure to gain acceptance for their ideas was at least as great as the failure of their business enterprise.

Whilst the interventionism of the Bonapartist State initially got the better of the school of Saint-Simon, the liberal turning point of the 1860s was the heyday of 'capitalist anarchy'. The apparent specialisation of the major credit establishments such as Crédit foncier or the Comptoir d'escompte was progressively diminished either by changes in the articles of association or by the creation of subsidiaries with more liberal articles. Thus in 1860, Frémy, the Governor of Crédit foncier, founded the Crédit agricole and managed to ensure that the new establishment could operate in ways which were prohibited at Crédit foncier. His resources were partly diverted through his Marseilles branch to highly remunerative unsecured short-term investments in Turkey, Russia, and Tunisia and above all in Egypt 26 . As Chairman, in a private capacity, of the Société générale Algérienne, a company which could compete with Crédit foncier when necessary, Frémy also distanced himself from his obligations under the treaty with the State, which was signed in May 1865: instead of undertaking works in Algeria on behalf of the State and supplying the necessary funding, he turned the company's resources to more directly and rapidly profitable operations such as the territorial liberation loans of 1872 -1873, advances to the Spanish Treasury and the Spanish Mortgage Bank and other foreign loans. At the beginning of the Third Republic, it was common knowledge that the Hôtel d'Evreux, the residence of the Governor of Crédit foncier, was the place where all major business was shared out among the Parisian bankers. The example of Frémy and Crédit foncier was indicative of a marked trend in the French banking system, i.e. the progressive slide of specialist establishments towards mixed banking, the all-purpose bank which was undifferentiated and modelled upon the private bank. They were joined by the deposit banks established in the 1860s (Crédit lyonnais, Société générale), even when this was not their original intention: Armand Donon, for example, championed the notion, in France, of the distribution of banking tasks and a deposit bank along English lines, in which the assets were as liquid as their liabilities, concentrating solely on short-term credit 27 . Yet he did not strictly adhere to his own beliefs when he created a multifunctional group; although it was made up of separate companies, these were financially interconnected for better or -as was finally the case -for worse. The new managers not only aligned themselves to the behaviour of traditional bankers but they also constantly competed with them on their own territory and shared the best sources of profit with them. The scope of the banking revolution was thereby greatly reduced; it applied to obtaining resources but it made few innovations in respect of utilisation of same. The new bank did not strike out in new directions, but extended the old system, of which it was the culmination. Its effect was quantitative rather than qualitative: it was content to merely deliver a solid boost to French capitalism.

However, the euphoria was of short duration and the aftermath was like the painful hangover which follows one drink too many. The Great Depression provided the necessary toxic agent and the effect was maximised when Republican politicians settled a few accounts with bankers who were regarded as creatures from the Second Empire. Most of these new managers, as we have already seen, had generally allowed their private and public interests to overlap and were sanctioned one after another by the government or by the market:

-At Crédit foncier, as we have seen, Frémy was obliged to resign in January 1877 following the Egyptian bankruptcy and Soubeyran was dismissed in August 1878. .

-1882 was marked by the most famous crash of the era, that of the Union générale, a Catholic bank, founded in 1877 and directed by Eugene Bontoux, who cannot be omitted 27 He defended these positions at the Study into the principles and general facts governing fiduciary and monetary circulation (Enquête sur les principes et les faits généraux qui régissent la circulation monétaire et fiduciaire), 1867-1869, t. I.

from this chain of disasters, though his was a particular case28 .

-Guillaume Denière, however, as Chairman of the Société générale from 1868 and Regent Banque de France, was both an eminent player in the banking revolution and a pure product of Second-Empire racketeering: he received, for example, rent from the Société Générale branch which occupied premises in the Parisian building he owned. In 1866 he was dismissed after revelations about bribes which he had demanded for bringing the Société générale into the Peruvian guano business and accusations that he had placed his private interests before those of the company.

-In 1889, Edouard Hentsch, Chairman of the Comptoir d'escompte, paid dearly for his lack of vigilance regarding the imprudence of his Director, Eugène Denfert-Rochereau, the son-in-law of Alphonse Pinard, who involved the Comptoir in a disastrous speculation in copper 29 . It proved fatal for the Comptoir d'escompte, the private bank Hentsch Frères & Co and for the Hentsch family30 .

-In 1891, the depositors of the Société de dépôts et de comptes courants (chaired by Armand Donon) made a run on the company which duly succumbed as it was unable to realise its assets. The company was liquidated, with heavy losses for the shareholders, for some of the administrators, who paid out 20 million francs to cover part of the debt, and for Armand Donon, who served six months in prison and went totally bankrupt.

-Finally, Soubeyran, who had set up a powerful financial group combining a investment bank, mortgage company and a real estate and insurance company, was forced into bankruptcy in 1894.

These repeated failures marked the death throes of the mixed bank in France and, with it, that of the conduct inherited from the Second Empire. The joint-stock bank, arising from the ashes of a totally defective banking system in 1848 and carried by new men, never really succeeded in establishing its autonomy. It remained immature and was caught up by the old system which eventually imposed its own conduct and methods. The new managers acted as if they owned the capital and took a great many liberties with their shareholders and the articles of association which they were supposed to respect. While the economic and political environment remained favourable they met with financial success but the spectre of failure darkened the horizon at the end of the 1870s. The few that managed to escape disaster were either those who had already died or those who had been more far-sighted and above all more prudent. As Jean Bouvier demonstrated, Henri Germain, Chairman of Crédit lyonnais, felt the wind change and in the 1870s he adapted his establishment to the new economic and political

  Bischoffsheim, Chairman of the Banque franco-égyptienne (1870-1873), Edouard Hentsch, Chairman of the Comptoir d'escompte (1873-1889) and Edward Blount, Chairman of the Société générale (1886-1901), etc. The latter was an exception, having liquidated his personal businesses, but the rule was to combine the management of a private bank and one or two joint-stock banks. In this respect Armand Donon and his associates Maurice Aubry and Jules Gautier were typical: their private bank Donon, Aubry, Gautier & Co, founded in 1851, immediately took its place among the leaders in the market, acting as a small-scale Crédit mobilier and investing the deposits of several large clients such as the Count de Morny, the Emperor's half-brother, in France and overseas. Donon joined up with William Gladstone -of the London bank J. Thomson, T. Bonar & Co. -a cousin of the English statesman of the same name, in promoting joint-stock banking on the English model in France. Their efforts resulted in the foundation in 1859 of the CIC -of which they subsequently lost control 20 -and of the Société de dépôts et de comptes courants in 1863, of which Donon was appointed Chairman. Subsequently, they linked up with the CIC to found similar establishments in the provinces in Lyon, Lille, Marseille and Bordeaux, with more specialized real estate and investment banking companies in Paris (Société financière de Paris) or even popular banks (Caisse centrale populaire).

  . XIX, p. 29 ; A. ROWLEY, « Deux crises économiques modernes : 1846 et 1848 ? », 1848, révolution et mutations au XIX e siècle, n° 2, 1986, p. 86 ; N. STOSKOPF, « La fondation du Comptoir d'escompte de Paris, banque révolutionnaire », Histoire, Economie, Société, sept. 2002.

  They included the Count de Germiny, governor of Crédit foncier from 1854 to 1857 and of the Banque de France from 1857 to 1863 and of Crédit mobilier, which he liquidated; the Marquis d'Audiffret, Chairman of the CIC from 1859 until his death in 1878; the Marquis de Ploeüc, Managing Director of the Imperial Ottoman Bank from 1863 to 1867 and then Sub-Governor of the Banque de France from 1867 to 1878 and Baron de Soubeyran, Sub-Governor of

	Crédit foncier from 1860 to 1878. Their colleagues were drawn from a wide social spectrum,
	ranging from the Delahante brothers, born of a general farming family (fermiers généraux)
	notorious for their wealth and their role as tax collectors under the Ancien Régime, and thus
	part of the financial upper middle class, to Alphonse Pinard, Sub-director of the Comptoir
	d'escompte de Paris from 1848 to 1870, the son of a secondary school teacher.

Only the most important establishments are mentioned here. For the full list see P. DUPONT-FERRIER, Le marché financier de Paris sous le Second Empire,, Paris, 1925, p. 245 ; R. BIGO, Les banques françaises au cours du XIX e siècle, Paris, 1947,

p. 3 D. S. LANDES, « Vieille banque et banque nouvelle: la révolution financière du XIX e siècle », Revue d'histoire moderne et contemporaine,, III, 1956, p. 204-222 4 Les patrons du Second Empire, Paris, Picard-Cenomane, 6 volumes published: 1. Normandy, Maine, Anjou; 2. Burgundy; 3. Franche-Comté ; 4. Alsace ; 5. Marseille ; 6. Bordeaux. 5 It covers 93 bankers or financiers. Cf. N. STOSKOPF, les patrons du Second Empire, 7, , Banquiers et financiers parisiens, Paris, Picard, 2002.We refer to this work for bibliographic references on bankers' biographies; with a few exceptions these cannot be printed in the context of this article, given their extreme dispersal.6 The expression "private bank" is not altogether satisfactory since the joint-stock bank is also privately owned; however, the term is used here for lack of a better expression and in accordance with common practice.

It is for this reason that D. Landes prefers to refer to them by the English term merchant banker.

The term pantouflage, which means leaving public service for employment in the private sector, appeared in 1878, according to A. REY, Dictionnaire historique de la langue française, Paris, Le Robert, 1992.

Bulletin des lois, partie supplémentaire, 1852, t. 2.

Crédit mobilier general meetings, Centre des archives du monde du travail (Working World Archive Centre) (CAMT, Roubaix), 65 AQ A 949-1.

Cited by C. JANNET, Le capital, la spéculation et la finance au XIXe siècle, 1880, p. 177.

Articles of Association and General Meeting Minutes, Comptoir d'escompte de Paris, BNF 4 Wz 3159 et 4324.

D. LANDES, op. cit., p. 217.

G. BEAUJOUAN and E. LEBÉE, «La fondation du Crédit industriel et commercial », Histoire des entreprises, 6, November 1960, p. 5-40.

B. GILLE, La banque en France au XIX e siècle,, Geneva-Paris, 1970, pp. 15-143 and 163-183.

A. GUESLIN, Les origines du Crédit agricole, Nancy, 1978, p. 110-115.

Cf. J. BOUVIER, Le krach de l'Union générale, 1878-1885, 1960, 308 p.

E. KAUFMANN, op. cit. , pp 243-247; C. Jannet, op. cit., p 320-331.

R. HENTSCH, Hentsch, banquiers à Genève et à Paris au XIXe siècle,, Paris, 1996, p. 286-303.


On the application of the Schoenberg quasi-interpolant for complexity reduction in trajectory generation

Vincent Marguet 1 Florin Stoican 2 Ionela Prodan 1

1 Univ. Grenoble Alpes, Grenoble INP † , LCIS, F-26000, Valence, France † Institute of Engineering and Management Univ. Grenoble Alpes Email:{vincent.marguet, ionela.prodan}@lcis.grenoble-inp.fr 2 Faculty of Automatic Control and Computers, Politehnica University of Bucharest, ACSE, Bucharest, Romania, Email:florin.stoican@upb.ro

Highlights

Employ the Schoenberg operator to approximate nonlinear functions (which stem from cost and/or constraints representation in an optimisation-based trajectory generation problem); Present a motion planning algorithm which efficiently provides feasible trajectories via Schoenberg-based approximations.

B-spline curves

We consider the parametrization of a trajectory z(t) with B-spline curves as a linear combination of control points and B-spline basis functions:

z(t) = n-1 k=0 P k B k,p,ξ (t) = PB p,ξ (t), ∀t ∈ [0, T ], (1)
with P = P 0 . . . P n-1 the matrix which gathers the control points, B p,ξ (t) = B 0,p,ξ (t) . . . B n-1,p,ξ (t) ⊤ the basis vector, ξ = {τ 0 ≤ τ 1 ≤ ... ≤ τ m } a knot sequence starting at 0 and ending at T . If m ≥ p + 2, we can define B-splines of order up to p over the knot sequence.

Schoenberg operator

The Schoenberg operator [START_REF] Lyche | Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement[END_REF]] approximates a function, denoted as f (•), by interpolating it as weighted sum of B-splines basis functions and the function's values in the Greville points of the B-spline basis:

f (t) = ñ-1 k=0 f ( tk ) Bk,p, ξ(t), (2)
where the Greville points are tk = τk+1 + ... + τk+p-1 p -1 .

(3)

Trajectory generation for a fixed-wing UAV

Let us consider a 2D 3-DOF model of a fixed wing UAV, [START_REF] Beard | Small unmanned aircraft: Theory and practice[END_REF], where the position (x(t), y(t)) and the heading (yaw) angle ψ(t) ∈ [0; 2π] rad are the state variables. Their dynamics are described by the following equations, where V a (t) and ϕ(t) are the inputs, the airspeed velocity and the roll angle, respectively:

x y X Y ψ ẋ(t) = V a (t) cos ψ(t), (4a) ẏ(t) = V a (t) sin ψ(t), (4b) ψ(t) = g tan ϕ(t) V a (t) . ( 4c
)
We consider as flat output (for more details on flatness and the UAV's flat representation see [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF], [START_REF] Prodan | Receding horizon flight control for trajectory tracking of autonomous aerial vehicles[END_REF]) the position of the aircraft

z(t) = z 1 (t) z 2 (t) ⊤ = x(t) y(t) ⊤ .
Thus, we are able to express all the state variables and inputs via the flat output and its first and second derivatives:

x(t) = z 1 (t), (5a) y(t) = z 2 (t), (5b)
ψ(t) = arctan ż2 (t) ż1 (t) , ( 5c
) V a (t) = ż2 1 (t) + ż2 2 (t), (5d)
ϕ(t) = arctan    1 g z2 (t) ż1 (t) -ż2 (t)z 1 (t) ż2 1 (t) + ż2 2 (t)    . (5e)
This will allow to reformulate the various costs/constraints in terms of the flat output z(t) which will be further parameterized as in (1).

Schoenberg-based approximation of the optimization problem

Motion planning often reduces to solving a constrained optimization problem of the form min

z(t) T 0 C(z(t))dt (6a) s.t. g i (z(t)) ≤ 0, 0 ≤ t ≤ T, (6b) h j (z(t)) = 0, 0 ≤ t ≤ T. ( 6c
)
The goal is to find a trajectory z(t) which minimizes the cost C(•) from (6a) while simultaneously respecting the inequality (6b) and equality constraints (6c) over the time interval [0, T ].

To better illustrate the subsequent ideas let us particularize ( 6) by taking z(t) as in (5). Then, we have that: the cost C(•) minimizes the trajectory length along the time horizon [0, T ]; the bounds on the velocity's magnitude are denoted by v and v, respectively; the bounds on the roll angle are denoted by ϕ and ϕ, respectively; the way-points passing (through p j at t j ) conditions. Thus, ( 6) is instantiated to the particular form:

min z(t) T 0 ∥ ż(t)∥dt (7a) s.t. v ≤ ∥ ż(t)∥ ≤ v, ∀t ∈ [0, T ] (7b) ϕ ≤ ϕ(t) ≤ ϕ, ∀t ∈ [0, T ] (7c) z(t j ) = p j , ∀j. (7d)
Parameterizing z(t) as in (1) allows to reformulate (7) into

min P k ,B k (t) T 0 f c (t)dt, (8a) s.t. v 2 ≤ f v (t) ≤ v 2 , (8b) tan(ϕ) ≤ f ϕ (t) ≤ tan(ϕ), (8c) n-1 k=0 P k B k,p (t j ) = p j , ∀j, (8d)
with the notations

f c (t) = n-2 k 1 ,k 2 =0 P ′ k 1 ⊤ P ′ k 2 B k 1 ,p-1 (t)B k 2 ,p-1 (t), (9a)
f v (t) = n-2 k 1 ,k 2 =0 P ′ k 1 ⊤ P ′ k 2 B k 1 ,p-1 (t)B k 2 ,p-1 (t), (9b)
f ϕ (t) = n-3 k 1 =0 n-2 k 2 =0 P ′′ k 1 ⊤ QP ′ k 2 B k 1 ,p-2 (t)B k 2 ,p-1 (t) g n-2 k 1 ,k 2 =0 P ′ k 1 ⊤ P ′ k 2 B k 1 ,p-1 (t)B k 2 ,p-1 (t) , ( 9c
)
where Q = 0 -1 1 0 and P ′ k , P ′′ k are used as short-hands for the control points associated to the first and second order derivatives of the B-spline curve. Using again the illustrative example from (8) we apply (2) thrice, once to approximate the nonlinear cost (8a), once to approximate the velocity in (8b) and once to approximate the roll angle in (8c). Correspondingly, the approximations are

f{c,v,ϕ} (t) = ñ-1 k=0 f {c,v,ϕ} ( tk )B k,p, ξ(t), (10)
which allow to approximate:

1 the cost from (8a) as

T 0 fc (t)dt = T 0 ñ-1 k=0 f c ( tk )B k,p, ξ(t) = ñ-1 k=0 f c ( tk ) T 0 B k,p, ξ(t)dt (11) 2 the velocity inequalities v 2 ≤ f c (t) ≤ v 2 from (8b) by v 2 ≤ ñ-1 k=0 f v ( tk )B k,p, ξ(t) ≤ v 2 , ( 12
)
3 the roll angle inequalities tan(ϕ) ≤ f ϕ (t) ≤ tan(ϕ) from (8c) by tan(ϕ) ≤ ñ-1 k=0 f ϕ ( tk )B k,p, ξ(t) ≤ tan(ϕ). (13)
Introducing approximations (11)-( 13) and keeping the already linear equality (8d) leads to the approximated constrained-optimization problem [START_REF] Marguet | On the application of the Schoenberg quasi-interpolant for complexity reduction in trajectory generation[END_REF]]

min P k ϵ 2 (14a) -ϵ ≤ ñ-1 k=0 f c ( tk ) T 0 B k,p (t)dt ≤ ϵ, (14b) s.t. v 2 ≤ f v ( tk ) ≤ v 2 , 0 ≤ k ≤ ñ -1, (14c) tan(ϕ) ≤ f ϕ ( tk ) ≤ tan(ϕ), 0 ≤ k ≤ ñ -1, (14d)
p j = n-1 k=0 P k B k,p (t j ), ∀j. (14e)

Simulation results

Simulation data

Future work

Extend this algorithm to multiple UAVs, taking into account "inter-UAV" constraints like communication range among them and collision avoidance Apply this algorithm to other UAV models: Quadcopter Crazyflie that we are using in the Esisarium platform.

Study the approximation error caused by the Schoenberg operator in order to obtain bounds to know a priori the constraints tightening.

  Numerical Value Gravitational acceleration g 9.81 m/s 2 Velocity control input V a ∈ [18 25] m/s Bank control input ϕ ∈ [-0.43 0.43] rad Altitude 150 m Simulation time T 150 s Way-point list P={(1000,1000,150),(1100, 250,150),(400,0,150),(-250, 250,150), (400,500,150)} Number of control points n 10 Order of the B-spline curves p 4 Number of greville points ñ 20

Table 1 .

 1 Numerical values of the simulationsDue to the approximation error, the velocity constraint was tightened from 40%, centered in the middle of its interval.

	Simulation Results

Table 2 .

 2 Computational time analysis for the approximated curves


Giorgio Palma
email: giorgio.palma@uniroma3.it

Stefano Meloni
email: stefano.meloni@unitus.it

Roberto Camussi
email: roberto.camussi@uniroma3.it

Umberto Iemma
email: umberto.iemma@uniroma3.it

Christophe Bogey
email: christophe.bogey@ec-lyon.fr.

Subsonic installed Jet noise scattering prediction using BEM and near-field-data-optimized Wave-Packet model

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. Introduction

Aviation noise has been widely identified as a driver of several negative stress-mediated health effects, from sleep disorders to cardiovascular issues [START_REF] Baudin | Aircraft Noise and Psychological Ill-Health: The Results of a Cross-Sectional Study in France[END_REF][START_REF] Basner | WHO Environmental Noise Guidelines for the European Region: A Systematic Review on Environmental Noise and Effects on Sleep[END_REF], which incidence is increased in the exposed population. The operation and expansion of airports are nowadays limited by strict regulations aiming at controlling and limiting the exposure of the surrounding community to aircraft noise and the number of people affected by it. Forecasts of the (international regulation authorities) ICAO, published in the last report on Global Environment Trends, indicates that this situation is the most likely scenario in the future, at least for most regions of the world.

The research on noise reduction devices is nowadays very active in all the aircraft areas, involving relatively mature technologies for quieter high lift devices [START_REF] Burghignoli | Evaluation of the noise impact of flap-tip fences installed on laminar wings[END_REF], chevrons for jet exhaust [START_REF] Jawahar | Jet noise sources for chevron nozzles in under-expanded condition[END_REF][START_REF] Meloni | A Wavelet-Based Time-Frequency Analysis on the Supersonic Jet Noise Features with Chevrons[END_REF], the evolution of acoustic liners [START_REF] Iemma | Optimization of metasurfaces for the design of noise trapping metadevices[END_REF][START_REF] Palma | On the integration of acoustic phase-gradient metasurfaces in aeronautics[END_REF][START_REF] Palma | Innovative Acoustic Treatments of Nacelle Intakes Based on Optimised Metamaterials[END_REF][START_REF] Jones | Evaluation of Variable-Depth Liners with Slotted Cores[END_REF][START_REF] Dodge | An Acoustic Liner with a Multilayered Active Facesheet[END_REF][START_REF] Palani | Experimental and numerical assessment of novel acoustic liners for aero-engine applications[END_REF] for turbofans ducts, and also more innovative treatments with lower Technology Readiness Level [START_REF] Palma | Optimized metamaterials for enhanced noise shielding of innovative aircraft configurations[END_REF]. Jet noise has always been a dominant noise source for turbojets and turbofans especially during take-off operations. So far, it has been mainly tackled by reducing jet velocity, increasing the thrust portion provided by the fan, reaching higher bypass ratios, and improving the propulsive system efficiency. This led to engines with larger relative sizes compared to aircraft, which integration into the airframe is becoming more and more critical, with the nacelle and the jet coming in the extreme proximity of the wing and high-lift devices. The aerodynamic interference between the jet shear layer and the flaps generates sources of noise that can easily dominate the generic free jet noise.

Projecting the research to the mid-and long-term future, groundbreaking solutions are also being developed, aiming at overcoming the saturation trend in noise reduction that characterizes mature technologies. Innovative configurations such as Blended and Hybrid Wing Body (BWB and HWB) aircraft are probably the most promising alternative to the well-known tube-and-wing configuration in terms of aerodynamic efficiency and community noise reduction [START_REF] Liebeck | Design of the Blended Wing Body Subsonic Transport[END_REF][START_REF] Centracchio | Noise shielding models for the conceptual design of unconventional aircraft[END_REF][START_REF] Burghignoli | Multi-objective optimization of BWB aircraft for noise shielding improvement[END_REF][START_REF] Xie | Integrated Sizing and Optimization of Hybrid Wing Body Aircraft in Conceptual Design[END_REF]. The most popular interpretation of these innovative configurations involves the upper installation of the propulsion system on top of the large center body surface, offering interesting acoustic shielding capability to be exploited for engine-related community noise reduction [START_REF] Palma | Optimized metamaterials for enhanced noise shielding of innovative aircraft configurations[END_REF][START_REF] Papamoschou | Jet noise shielding for advanced hybrid wing-body configuration[END_REF][START_REF] Denisov | Investigation of noise-shielding efficiency with the method of sequences of maximum length in application to the problems of aviation acoustics[END_REF]. The propulsion-airframe acoustic interaction is an aspect of growing research interest for future aircraft and should be accounted for since the beginning of the design process.

However, the simulation of the scattering and shielding from large surfaces in the audible range of frequencies can be computationally very expensive, requiring accurate solutions up to extremely high Helmholtz number 𝐻𝑒 = 𝑘𝑙 (where 𝑘 is the wave-number and 𝑙 is the characteristic length of the scattering object). The resources required for direct simulation with high-fidelity CFD or CAA methods make them unfeasible for extensive usage in the conceptual design phase and design optimization processes.

There is hence a strong need for low-and mid-fidelity models and solvers able to catch the fundamental feature of installed jet noise, avoiding the solution of the complete set of equations holding the dynamic of the complex fluid structures involved. Adaptive metamodeling techniques have been recently applied to this class of problems [START_REF] Burghignoli | Adaptive RBF with hyperparameter optimisation for aeroacoustic applications[END_REF][START_REF] Centracchio | Noise shielding surrogate models using dynamic artificial neural networks[END_REF] to reduce the computational effort required in determining the optimal position of the propulsion system that minimizes the noise directed towards the ground and community. Artificial Neural Networks, trained with experimental data in both near and far field, have been used as a non-linear surrogate model to predict the noise emitted by a single-stream jet in under-expanded conditions [START_REF] Centracchio | Under-expanded jet noise prediction using surrogate models based on artificial neural networks[END_REF], showing good agreement for a wide range of Mach number. Other jet noise models used distributions of acoustic monopoles as noise sources. However, it has been demonstrated that such a simple source is not able to satisfactorily capture the jet noise features [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF]. In some low-order models, the noise sources of a single circular jet were represented by a set of uncorrelated quadrupoles [START_REF] Kopiev | Simulation of azimuthal characteristics of turbulent jet noise by correlation model of quadrupole noise sources[END_REF]. However, it was shown that the directivity at shallow angles was not well reproduced [START_REF] Denisov | Using the Correlation Model of Random Quadrupoles of Sources to Calculate the Efficiency of Turbulent Jet Noise Screening with Geometric Diffraction Theory[END_REF]. The perspective on jet noise changed when the presence of coherent structures in jets and their importance in the aft-angle radiation for high subsonic and supersonic jets was demonstrated, providing a basis for the introduction of wave-packets as modeling approach [START_REF] Jordan | Wave Packets and Turbulent Jet Noise[END_REF][START_REF] Koenig | Farfield filtering and source imaging of subsonic jet noise[END_REF]. The wave-packet model has been widely used as a low-order model for the jet noise source. In Papamoschou [START_REF] Huang | Numerical Study of Noise Shielding by Airframe Structures[END_REF], a virtual cylindrical surface hosting the wave-packet is assumed to surround the jet region and radiate the pressure perturbation. The parameters such as envelope amplitude, wavelength, position, and convection velocity were typically estimated from far-field measurements, optimizing their values and maximizing the agreement on experimental data on a training set [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF][START_REF] Papamoschou | Wavepacket Modeling of the Jet Noise Source[END_REF][START_REF] Papamoschou | Wavepacket modeling of the jet noise source[END_REF].

A large body of the literature demonstrated that the low-frequency amplification of the jet noise in the classical jet-wing architecture could be ascribed to the scattering of the jet hydrodynamic field [START_REF] Meloni | An experimental investigation into model-scale installed jet-pylon-wing noise[END_REF][START_REF] Meloni | Wall-Pressure Fluctuations Induced by a Compressible Jet in Installed Configuration[END_REF][START_REF] Jordan | Jet-flap interaction tones[END_REF], and the same mechanism will hold for innovative turbofan powered BWB and HWB. The jet-surface interaction is expected to happen at a few nozzle diameters from the jet axis, both for classic and innovative aircraft configurations, implying that the wave-packet should be calibrated with near-field data to maximize its prediction reliability where the scattering takes place. For the prediction of the acoustic scattering and shielding, source models able to capture the near-field characteristics of the impinging jet noise are needed to accurately address the effects of the wing and/or airframe. Recently, Palma et. al. [START_REF] Palma | A Multi-Objective Optimization of a Wave-Packet Model Using Near-Field Subsonic Jet Data[END_REF][START_REF] Palma | Data-Driven Multiobjective Optimization of Wave-Packets for Near-Field Subsonic Jet Noise[END_REF] followed the approach introduced by Papamoschou [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF][START_REF] Huang | Numerical Study of Noise Shielding by Airframe Structures[END_REF][START_REF] Papamoschou | Wavepacket Modeling of the Jet Noise Source[END_REF][START_REF] Papamoschou | Wavepacket modeling of the jet noise source[END_REF], calibrating the model parameters on near-field large eddy simulations (LES) data of a high-speed subsonic isothermal jet. The mentioned paper presents a multi-Strouhal number analysis optimizing the wave-packet source model separately for each value in the set 𝑆𝑡 𝐷 = 0.25, 0.5, 0.75 and 1 and using pressure data from the numerical database for the dominant axisymmetric azimuthal mode. It has been shown that optimizing the model parameters with pressure data at multiple distances in the near field provides a noise source that also preserves adherence to the reference data for radial positions outside the training set, improving the reliability of its prediction.

This article aims to extend the work started in [START_REF] Palma | A Multi-Objective Optimization of a Wave-Packet Model Using Near-Field Subsonic Jet Data[END_REF], training the wave-packet model with information from both the near and far fields using the same multi-objective approach. Data from a high-fidelity LES are used in the optimization defining the reference solutions the optimized wave-packet has to reproduce. An extra far-field line is added to improve the fidelity of the prediction of the optimized source far from the jet axis, providing additional information on the radiating acoustic component of the wave-packet; the near-field lines, on the other hand, contain information also on the hydrodynamic part of the acoustic field. Combining the features of the two classes guides the optimization towards a prediction valid in a wider range of distances. The optimized wave-packet is then coupled with a Boundary Element Method (BEM) solver for jet noise scattering predictions. The aim is to develop an open-source mid-fidelity tool for predicting installed jet noise scattering and shielding effects.

The paper is organized as follows: the wave-packet model and the BEM theory are described in sections II and III, respectively; Results from the optimization and the BEM simulations are reported in section V. Final remarks can be found in section VI.

II. The wave-packet model

A wave-packet model is used as a source for reproducing the noise produced by a subsonic jet. The one here used has been introduced by Morris [START_REF] Morris | Jet noise prediction: Past, present and future[END_REF][START_REF] Morris | A Note on Noise Generation by Large Scale Turbulent Structures in Subsonic and Supersonic Jets[END_REF] and by Tam and Burton [START_REF] Tam | Sound generated by instability waves of supersonic flows. Part 2. Axisymmetric jets[END_REF], Crighton and Huerre [START_REF] Crighton | Shear-layer pressure fluctuations and superdirective acoustic sources[END_REF], and Avital et al. [START_REF] Avital | Mach Wave Radiation by Mixing Layers. Part I: Analysis of the Sound Field[END_REF]. It has been developed by Papamoschou [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF][START_REF] Huang | Numerical Study of Noise Shielding by Airframe Structures[END_REF][START_REF] Papamoschou | Wavepacket Modeling of the Jet Noise Source[END_REF][START_REF] Papamoschou | Wavepacket modeling of the jet noise source[END_REF], which derived the formulation adopted in this paper. The model is based on the fundamental assumption stating that the peak noise radiation from the jet in the aft region is related to the large-scale coherent structures in the jet flow that can be modeled as instability waves at its boundary, growing and then decaying along the axial distance [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF].

The present formulation introduces a cylindrical virtual surface surrounding the original jet. The surface radiates the pressure perturbation imposed on it, representing and substituting the jet from the acoustic point of view. Applying the wave-packet ansatz, the pressure on the cylindrical surface at 𝑟 0 surrounding the jet is prescribed as:

𝑝 𝑤 (𝑚, 𝑟 0 , 𝑥, 𝜃, 𝑡) = 𝑝 0 (𝑥)𝑒 -𝑖 𝜔𝑡+𝑖𝑚𝜃 , ( 1
)
where m is the azimuthal mode number, x denotes the axial coordinate, 𝜃 is the azimuthal angle, 𝜔 = 2𝜋 𝑓 is the pulsation. The wave-packet axial shape 𝑝 0 (𝑥) is given in the form [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF]:

𝑝 0 (𝑥) = tanh (𝑥 -𝑥 0 ) 𝑝 1 𝑏 𝑝 1 1 1 -tanh (𝑥 -𝑥 0 ) 𝑝 2 𝑏 𝑝 2 2 𝑒 𝑖 𝛼( 𝑥-𝑥 0 ) . (2)
The parameter 𝑟 0 determines the radial distance of the virtual surface from the jet axis, which is taken as 𝑟 0 = 𝐷. The coordinate 𝑥 0 is used to locate the relative position between the origin of the wave-packet function and the nozzle exit.

The two are considered to be coincident in this work, i.e., 𝑥 0 = 0. The signal growth is controlled by the parameters 𝑏 1 and 𝑝 1 , while 𝑏 2 and 𝑝 2 define its decaying rate. Following Morris [START_REF] Morris | A Note on Noise Generation by Large Scale Turbulent Structures in Subsonic and Supersonic Jets[END_REF], and Papamoschou [START_REF] Papamoschou | Prediction of Jet Noise Shielding[END_REF], the solution in the linear regime (i.e., solution for the 3D wave equation in cylindrical polar coordinates) for an arbitrary radial distance 𝑟 ≥ 𝑟 0 can be evaluated as

𝑝 𝑤 (𝑚, 𝑟, 𝑥, 𝜃, 𝑡) = 1 2𝜋 𝑒 -𝑖 𝜔𝑡+𝑖𝑚𝜙 ∫ ∞ -∞ p0 (𝑘)
𝐻 (1) 𝑚 (𝜆𝑟) 𝐻 (1) 𝑚 (𝜆𝑟 0 )

𝑒 𝑖𝑘 𝑥 𝑑𝑘 with 𝜆 = 𝜔 𝑐 ∞ 2 -𝑘 2 1/2 , - 𝜋 2 < 𝑎𝑟𝑔(𝜆) < 𝜋 2 , (3)
where p0 (𝑘) is the Fourier transform of 𝑝 0 (𝑥), 𝑐 ∞ is the speed of sound of the unperturbed flow, and 𝐻 (1) 𝑚 is the Hankel function of the first kind and order 𝑚. In the radiation process, particular care must be taken to the spatial length of the wave-packet from the numerical point of view. A premature truncation of the waveform introduces noise and errors in the signal propagated to higher 𝑟. The phase speed can be used to distinguish among the radiative and decaying components of the pressure field generated by the wave-packet, identified respectively by supersonic (|𝜔/𝑘 | ≥ 𝑐 ∞ ) and subsonic (|𝜔/𝑘 | < 𝑐 ∞ ) values.

In this work, the method described in Palma et. al. [START_REF] Palma | A Multi-Objective Optimization of a Wave-Packet Model Using Near-Field Subsonic Jet Data[END_REF][START_REF] Palma | Data-Driven Multiobjective Optimization of Wave-Packets for Near-Field Subsonic Jet Noise[END_REF] is followed and further extended. A wave-packet describing the pressure fluctuations by a free jet is obtained by optimizing its parameters from near-field data on co-axial lines at several radial distances from the jet axis, namely 𝑟/𝐷 = 0.5, 1, 2 and 2.5, and on a far-field polar arc, 𝑟/𝐷 = 75. The reference data used for training the model are obtained through LES simulations [START_REF] Bogey | Large-eddy simulation of the flow and acoustic fields of a Reynolds number 105 subsonic jet with tripped exit boundary layers[END_REF][START_REF] Bogey | Acoustic tones in the near-nozzle region of jets: characteristics and variations between Mach numbers 0.5 and 2[END_REF][START_REF] Camussi | On the influence of the nozzle exhaust initial conditions on the near field acoustic pressure[END_REF]. A multi-objective optimization procedure aims at matching the complete pressure fluctuation envelope from the model with the one from the numerical simulations for each of the considered lines. The wave-packet obtained is then used as an acoustic source in a Boundary Element formulation to evaluate scattering and reflection from surfaces, i.e. installed jet noise.

A. Wave-packet optimization

The wave-packet noise source introduces some parameters whose value can be adjusted to match the actual pressure fluctuation from a reference jet. In this work, this is performed involving a multi-objective optimization procedure. A generic unconstrained optimization problem consists of the research of the set of variables v that yields a minimum of the 𝑁 𝐽 objective functions: 𝐽 𝑛 (v, q) minimize/maximize [𝐽 𝑛 (v, q)] , 𝑛 = 1, ..., 𝑁 𝐽 and v ∈ D v with bounds 𝑣 𝐿 𝑚 ≤ 𝑣 𝑚 ≤ 𝑣 𝑈 𝑚 , 𝑚 = 1, ..., 𝑁 𝑣

where q is the vector of the parameters, v is the vector of the 𝑁 𝑣 design variables bounded by 𝑣 𝐿 𝑛 and 𝑣 𝑈 𝑛 in the design space D v , In the present application, v represents the vector collecting the wave-packet parameters v = 𝑝 1 , 𝑏 1 , 𝑝 2 , 𝑏 2 , 𝜔/(𝛼𝑈 𝑗 ) . The number of objective functions to be minimized at the same time is 𝑁 𝑗 = 5, four refer to the near field and one to the far field prediction, described by

𝐽 𝑛 (x, y) = √ ∫ L 𝑛 | 𝑝 𝑛 -p𝐿𝐸𝑆 𝑛 | max (| p𝐿𝐸𝑆 𝑛 |) 2 𝑑𝑠 (5)
where p𝐿𝐸𝑆 𝑛 = 𝑝 𝐿𝐸𝑆 𝑛 / p is the value of the reference pressure field on the n-th line normalized with the maximum value at the line closest to the jet axis 𝑟/𝐷 = 0.5. The objective functions represent the L2-norm of the difference between the pressure predicted by the wave-packet source model and the reference pressure from the LES. The integral is defined over the axial extension, from 0 up to 𝑥/𝐷 = 20, of lines having a constant 𝑟 𝑛 for the four radial distances in the near field, (𝑟 1 = 0.5𝐷, 𝐷, 2𝐷, 2.5𝐷), defining 𝐽 1 -𝐽 4 , and over a polar arc, ranging from 15 to 165 degrees, centered on the jet axis at the nozzle exit with radius 𝑅 = 75𝐷 defining 𝐽 5 . According to Eq. 5, each objective function is normalized by the peak value from the reference pressure field on the respective line L 𝑛 .

III. The Boundary Element formulation

The Boundary Element Method (BEM) is a numerical computational method that solves linear partial differential equations formulated as integral equations. Its application is popular in acoustics for the solution of scattering and diffraction problems, exploiting its efficiency in terms of computational resources for problems with a small surface/volume ratio. The linear acoustics problem is here described by the classic D'Alambert operator applied to the pressure 𝑝, which in the Laplace domain reads:

∇ 2 𝑝(x) + 𝜅 2 𝑝(x) = q, (x) ∈ V (6)
where the • indicates Laplace transformation, whereas q represents the acoustics sources present in the domain V, and 𝜅 = 𝑠/𝑐 0 is the complex wave number, being 𝑠 = 𝛼 + 𝑗 𝜔 the Laplace variable, and 𝑐 0 the speed of sound in the reference conditions. The formulation of the problem in the Laplace domain is equivalent to the time convention 𝑒 𝑗 𝜔𝑡 . For this study, we employed an open-source solver AcouSTO [START_REF] Palma | Implementation of a dual surface regularization technique in ACA-based BEM solver[END_REF] for the corresponding Kirchhoff-Helmholtz Integral Equation (KHIE)

𝐸 (y) p(y) = ∮ S 𝐺 0 𝜕 p 𝜕𝑛 - p 𝜕𝐺 0 𝜕𝑛 + 𝑠 p𝐺 0 𝜕𝜃 𝜕𝑛 𝑒 -𝑠 𝜃 𝑑S(x) + ∫ V 𝐺 0 q𝑒 -𝑠 𝜃 𝑑V (x) (7)
where S = 𝜕𝑉, 𝑠 = 𝛼 + 𝑖𝜔 is the Laplace variable, 𝐸 (y) is the domain function (equal to 1, 1/2, or 0, if y ∈ 𝑉, y ∈ 𝜕𝑉, or y ∉ 𝑉, respectively), and 𝐺 is the free-field fundamental solution for the 3D KHIE equation defined as

𝐺 (x, y, 𝑠) = - 𝑒 -𝑠 𝜃 4𝜋𝑟 = 𝐺 0 𝑒 -𝑠 𝜃 , with 𝑟 = ||x -y|| and 𝜃 = 𝑟 𝑐 0 (8)
The integral form of the problem is derived directly from the non-homogeneous wave equation, and the acoustic potential p in Eq.7 is the total field, that includes the incident acoustic field and the part scattered by the surfaces p = p𝑖𝑛 + p𝑠𝑐 . This formulation is convenient when the primary field produced by the acoustic source in the field is easily calculable. The boundary conditions at the scattering surfaces are given in terms of normal velocity, imposed to be null modeling a rigid boundary 𝜕 p(x, 𝜅) 𝜕𝑛

= 0. ( 9
)
The boundary of the domain is partitioned into 𝑁 (quadrilateral) panels, and all the quantities are considered to be constant within each panel (0 th -order approximation). The collocation method is used by locating the collocation points at the centers of the panels and the discrete version of Eq.7 is obtained leading to a linear system of equations that reads

𝐸 p𝑛 = 𝑁 ∑ 𝑚=1 𝐵 𝑛,𝑚 𝜒 𝑚 + 𝐶 𝑛,𝑚 + 𝑠𝐷 𝑛,𝑚 p𝑚 𝑒 -𝑠 𝜃 𝑛,𝑚 + p𝐼 𝑛 , 𝑛 = 1, 𝑁 (10)
where 𝑝 𝐼 𝑛 is the value of the incident field induced by the sources at the collocation points, and the integral coefficients have the form

𝐵 𝑛,𝑚 = ∫ S 𝑚 𝐺 0 𝑑S, 𝐶 𝑛,𝑚 = ∫ S 𝑚 𝜕𝐺 0 𝜕𝑛 𝑑S, 𝐷 𝑛,𝑚 = ∫ S 𝑚 𝐺 0 𝜕𝜃 𝜕𝑛 𝑑S (11)
The open-source software has been adapted to accept an external custom incident field as input, which in this study is given by Eq. 3. Once the solution is known on the scattering surfaces, Eq. 7 can be used as a boundary integral representation to obtain the solution at arbitrary points in the field.

IV. Numerical database details

The near-field of the isothermal round free jet at a Reynolds number 𝑅𝑒 𝐷 = 10 5 used for this paper has been computed by (LES). The nozzle exhaust jet Mach number has been fixed at 𝑀 = 0.9, with the nozzle-exhaust boundary-layer thickness set at 𝛿 𝑏𝑙 = 0.15𝑟 0 and the nozzle exit turbulence intensity at 9% (see [START_REF] Bogey | Acoustic tones in the near-nozzle region of jets: characteristics and variations between Mach numbers 0.5 and 2[END_REF] for details). The LES has been carried out using an in-house solver of the three-dimensional filtered compressible Navier-Stokes equations in cylindrical coordinates (r, 𝜃, x) based on low-dissipation and low-dispersion explicit schemes. The quality of the grid for the present jet LES has been assessed in a previous work [START_REF] Bogey | Grid sensitivity of flow field and noise of high-Reynolds-number jets computed by large-eddy simulation[END_REF]. Specifically, the grid contains approximately one billion points. Pressure has been recorded at several locations spanning a large near-field domain and gaining time-resolved signals, see reference [START_REF] Bogey | A database of flow and near pressure field signals obtained for subsonic and nearly ideally expanded supersonic free jets using large-eddy simulations[END_REF] for a description of the available data. In addition, the near-pressure field of this jet has been also investigated in [START_REF] Adam | Imprint of Vortical Structures on the Near-Field Pressure of a Turbulent Jet[END_REF]. It has been propagated to the far field in [START_REF] Bogey | Acoustic tones in the near-nozzle region of jets: characteristics and variations between Mach numbers 0.5 and 2[END_REF][START_REF] Bogey | Tones in the acoustic far field of jets in the upstream direction[END_REF] using an in-house OpenMP-based solver of the isentropic linearized Euler equations in cylindrical coordinates based on the same numerical methods as the LES. Concerning the near-field domain, we consider arrays of virtual microphones parallel to the nozzle exhaust, containing 1024 probes covering a domain that spans between 𝑥=0 and 𝑥/D=20. The data have been stored at a sampling frequency corresponding to 𝑆𝑡 𝐷 = 12.8, with a total of 3221 time snapshots. A representative one is shown in Fig. 1. In the far field, we consider a polar arc of virtual microphones centred at the nozzle exit, positioned at 𝑅 = 75𝐷, from 15 to 165 degrees relative to the jet direction, with a spacing of one degree.

The original pressure signals are represented in terms of their azimuthal components through the azimuthal decomposition [START_REF] Michalke | On turbulence and noise of an axisymmetric shear flow[END_REF]. The Fourier coefficients are stored for the first four azimuthal modes that dominate the sound field for low polar angles. As aforementioned the wave-packet model presented in this paper has been carried out for the 0 th azimuthal mode, which is dominant for the noise generation at Strouhal numbers lower than 1 [START_REF] Cavalieri | Axisymmetric superdirectivity in subsonic jets[END_REF].

V. Results

A. Wave-packet optimization on free jet

The multi-objective optimization of the wave-packet produces a set of Pareto-efficient solutions constituting the Pareto front, which is a 𝑁 𝐽 -1 variety in the 𝑁 𝐽 dimensional codomain. The solutions lying on the front have equal dignity and are optimal in a Paretian sense, i.e., it is not possible, moving along the front, to improve the value of one of the objectives without worsening at least one of the other. One of the techniques that may be employed to identify the preferred solution among the optima is to identify a ranking criterion (also called Decision Maker algorithm). The criterion can be interpreted as an additional objective, allowing the sorting of the solutions on the basis of their score over it and then selecting the solution resulting as the most suitable. Any selection criterion is valid in principle and may be used reasonably, from simple subjective preferences to more complex analyses of the results.

In the present work, the optimal solutions have been sorted by their performance in reproducing the reference field on the far field line. A subset of the Pareto front satisfying 𝐽 5 ≤ 2 min(𝐽 5 ) is isolated, and the solution in this subset closest to the utopia point is taken as the preferred one. The parameters of the selected wave-packet are reported in Tab. 1. Figure 2 compares the predicted wave-packet envelopes with the original LES data at different radial distances from the jet axis and at 75𝐷 from the nozzle exit. Specifically, black continuous lines represent the original data, whilst the optimized wave-packets are in blue dashed lines. Results are presented at this early stage research only for a single 𝑆𝑡 𝐷 number. A value of 𝑆𝑡 𝐷 = 0.25 was chosen because it corresponds to the peak Strouhal number in the jet far-field in the downstream direction. Reference data refers to the 0 th axisimmetric azimuthal mode, which is predominant in the far-field for the selected 𝑆𝑡 𝐷 . The model predictions are observed to provide a quite good reconstruction of the original data. The directivity is quite well captured in the far-field for almost all the polar angles, Fig. 2f.

B. Scattering prediction

The pressure field produced by the wave-packet source represents the incident field for the scattering calculations by the BEM solver. A simple geometry, sketched in Fig. 3, has been tested to provide a preliminary characterisation of the scattered far-field noise using the BEM method. Specifically, a wedge-shaped flat plate is used as a scattering body, representing a wing interacting with the pressure perturbation emitted by the jet. The wing has been extended upstream to minimise the scattering from the leading edge and has been designed sufficiently large in a spanwise direction to avoid side edge effects. The wave-packet frame of reference is fixed on the symmetry plane of the wing, at Δ𝑥 𝑝 = 𝐷 from the plate leading edge and Δ𝑧 𝑝 = 𝐷 in the radial direction. The flat plate extends for 𝑙 𝑝 = 4𝐷 in the 𝑥 (jet-axis) direction, and its span is 𝑠 𝑝 = 10𝐷.

Fig. 3 Sketch of the BEM setup

The far-field pressure field is presented in Fig 4 comparing installed and isolated jet configurations. It is worth noting that a substantial noise increase has been predicted at higher polar angles in the installed case, according to the literature [START_REF] Meloni | An experimental investigation into model-scale installed jet-pylon-wing noise[END_REF]. This indicates that the model is able to capture quite well the jet-wing scattering noise. However, further analyses are needed to include different 𝑆𝑡 𝐷 numbers in the analysis, increasing the robustness of the model.

VI. Conclusions

A multi-objective optimization of a wave-packet, including both near-and far-field noise, is presented to predict the behaviour of the 0 th azimuthal mode. A Pareto front has been obtained as a solution for the optimization due to the 1 Wave-packet parameters concurrency between the different objectives. The preferred solution on the front is then selected using a Pareto ranking criterion method, considering the performance of the solution in reproducing both the near-and the far-field reference pressure field. The results of the optimizations are found to provide a good agreement between the reference numerical data, generated through well-resolved LES simulations, and the model in the tested Strouhal number (𝑆𝑡 𝐷 = 0.25).

The optimized wave-packet has been coupled with a BEM solver to predict the installation effects in terms of scattering noise. The noise increase in the upstream scattered field of the installed configuration shows the adherence of the simulations to the physics observed experimentally.

Further investigations that will involve multi-Strouhal analyses, including the possibility of reducing the use of flow-immersed virtual probes, which can be a limit of the model, are currently ongoing. The opportunity to use different objective functions and wave-packet shapes will also be explored.

Fig. 1

 1 Fig. 1 Snapshot in the (x,r) plane of the pressure signals. The black dashed lines represent the probe arrays in the near field.

Fig. 2

 2 Fig. 2 Comparison between reference and optimized wave-packet normalized pressure. Near-field predictions in Figures from (a) to (e), far-field prediction in (f), 𝑆𝑡 𝐷 = 0.25.

Fig. 4

 4 Fig. 4 Comparison between isolated and installed jet far-field acoustic pressures at 𝑆𝑡 = 0.25 from BEM simulations.


Gilles Dowek

The Undecidability of Typability in the Lambda-Pi-Calculus

The set of pure terms which are typable in the λΠ-calculus in a given context is not recursive. So there is no general type inference algorithm for the programming language Elf and, in some cases, some type information has to be mentioned by the programmer.

Introduction

The programming language Elf [START_REF] Pfenning | Logic Programming in the LF Logical Framework, Logical Frameworks[END_REF] is an extension of λ-Prolog in which the clauses are expressed in a λ-calculus with dependent types (λΠ-calculus [START_REF] Harper | A Framework for Defining Logics[END_REF]). Since this calculus verifies the propositions-as-types principle, a proof of a proposition is merely a term of the calculus. Using this property of the λΠ-calculus, the programmer can either express a proposition and let the machine search for a proof of this proposition (as in usual logic programming) or express both a proposition and its proof and let the machine check that this proof is correct (as in proof-verification systems). Thus Elf can be used both to express logic programs and to reason about of their properties.

A type inference algorithm for a given language is an algorithm which assigns a type to each variable of a program. Thus, when such an algorithm exists, the types of the variables do not need to be mentioned by the programmer. As an example, a type inference algorithm for the language ML is given in [START_REF] Damas | Principal Type-Scheme for Functional Programs[END_REF].

We show here that the set of pure terms which are typable in the λΠ-calculus in a given context is not recursive. So there is no general type inference algorithm for the language Elf and, in some cases, some type information has to be mentioned by the programmer.

As already remarked in [START_REF] Damas | Principal Type-Scheme for Functional Programs[END_REF], typing a term requires the solution of a unification problem. Typing a term in the simply typed λ-calculus (and in ML) requires the solution of a first order unification problem and thus typability is decidable in these languages.

Typing a term in the λΠ-calculus requires the solution of a unification problem which is also formulated in the λΠ-calculus. Unification in the λΠ-calculus has been shown to be undecidable (third order unification in [START_REF] Huet | The Undecidability of Unification in Third Order Logic[END_REF], then second order unification in [START_REF] Goldfarb | The Undecidability of the Second-Order Unification Problem[END_REF] and third order pattern matching in [START_REF] Dowek | L'Indécidabilité du Filtrage du Troisième Ordre dans les Calculs avec Types Dépendants ou Constructeurs de Types (The Undecidability of Third Order Pattern Matching in Calculi with Dependent Types or Type Constructors)[END_REF]), i.e. there is no algorithm that decides if such a unification problem has a solution. But in order to prove the undecidability of typability in the λΠ-calculus we need to prove that there is no algorithm that decides if a unification problem produced by a typing problem has a solution. Unification problems produced by typing problems are very restricted and the undecidability proofs of unification have to be adapted to this class of problems. We show here that the proof of [START_REF] Huet | The Undecidability of Unification in Third Order Logic[END_REF] can easily be adapted.

1 The Lambda-Pi-Calculus

We follow [START_REF] Barendregt | Introduction to Generalized Type Systems[END_REF] for a presentation of the λΠ-calculus. The set of terms is inductively defined by

T ::= T ype | Kind | x | (T T ) | λx : T.T | Πx : T.T
In this note, we ignore variable renaming problems. A rigorous presentation would use de Bruijn indices. The terms T ype and Kind are called sorts, the terms x variables, the terms (t t ′ ) applications, the terms λx : t.t ′ abstractions and the terms Πx : t.t ′ products. The notation t → t ′ is used for Πx : t.t ′ when x has no free occurrence in t ′ .

Let t and t ′ be terms and x a variable. We write t[x ← t ′ ] for the term obtained by substituting t ′ for x in t. We write t ∼ = t ′ when the terms t and t ′ are A term t is said to be well-typed in a context Γ if there exists a term T such that Γ ⊢ t : T .

The reduction relation on well-typed terms is strongly normalizable and confluent. Thus each well-typed term has a unique normal form and two terms are equivalent if they have the same normal form [START_REF] Harper | A Framework for Defining Logics[END_REF]

([6] [15] [2] if βη-equivalence is considered).
A term t well-typed in a context Γ has a unique type modulo equivalence.

A normal term t well-typed in a context Γ has either the form

t = λx 1 : T 1 ....λx n : T n .(x c 1 ... c n )
where x is a variable or a sort or

t = λx 1 : T 1 ....λx n : T n .Πx : P.Q
The head symbol of t is x is the first case and, by convention, the symbol Π in the second. The top variables of t are the variables x 1 , ..., x n .

2 Typability in the Lambda-Pi-Calculus

Definition 1. A term t of type T in a context Γ is said to be an object in Γ if Γ ⊢ T : T ype.
Proposition 1. If a term t is an object in a context Γ then it is either a variable, an application or an abstraction. If it is an application t = (u v) then both terms u and v are objects in Γ , if it is an abstraction t = λx : U.u then the term u is an object in the context Γ [x : U ].

Definition 2. The set of pure terms is inductively defined by

T ::= x | (T T ) | λx.T Definition 3.
Let t be an object in a context Γ , the content of t (|t|) is the pure term defined by induction over the structure of t by

• |x| = x, • |(t t ′ )| = (|t| |t ′ |), • |λx : U.t| = λx.|t|.
A pure term t is said to be typable in a context Γ if there exists a term t ′ well-typed in an extension Γ ∆ of Γ such that t ′ is an object in Γ ∆ and t = |t ′ |.

Remark 1. Typing a pure term in a context Γ is assigning a type to bound variables and to some of the free variables, while the type of the other free variables is given in the context Γ . When the context Γ is empty, then typing a term in Γ is assigning a type to both bound and free variables.

Proposition 2. Typability in the empty context is decidable in the λΠ-calculus.

Proof. Pure terms typable in the empty context in the λΠ-calculus and in the simply typed λ-calculus are the same [START_REF] Harper | A Framework for Defining Logics[END_REF] and typability is decidable in simply typed λ-calculus [START_REF] Damas | Principal Type-Scheme for Functional Programs[END_REF]. Proof. By induction on n.

Proposition 5. Let t, u 1 , ..., u n , v be normal terms such that (t u 1 ... u n ) is a well-typed term and its normal form is v. The head symbol of the t is either the head symbol of v or a top variable of t.

Proof. Let x be the head symbol of t. If x is not a top variable of t then the head symbol of the normal form of (t u 1 ... u n ) is also x, so x is the head symbol of v. for some sequence i 1 , ..., i p .

Proof. By induction on the number of variable occurrences in t.

Theorem 2. It is undecidable whether or not a pure term is typable in a given context.

Proof. Consider a Post problem {< φ 1 , ψ 1 >, ..., < φ n , ψ n >}. We construct the pure term t such that t is typable in Γ if and only if the Post problem has a solution. and the type T to all the other variables of the term t, we get a term t ′ well-typed in Γ , which is an object and such that t = |t ′ |.

Remark 2. Along the way, we have proved that in the simply typed λ-calculus, the unification problems of the form

(f t 1 ... t n ) = (f t ′ 1 ... t ′ n ) (f u 1 ... u n ) = u ′ (f v 1 ... v n ) = v ′
where t i , t ′ i , u i , u ′ , v i , v ′ are closed terms and f a third order variable are undecidable.

It is decidable if each of these equations has a solution or not (since the first one is flexible-flexible [START_REF] Huet | A Unification Algorithm for Typed λ-calculus[END_REF] [START_REF] Huet | Résolution d' Équations dans les Langages d[END_REF] and the others third order matching problems [START_REF] Dowek | Third Order Matching is Decidable[END_REF]), but it is undecidable whether or not they have a solution in common. If the variable f is second order the problems of this form are decidable since the second order matching algorithm [START_REF] Huet | Résolution d' Équations dans les Langages d[END_REF] [12] produces a finite complete set of closed solutions.

Definition 4 .4Proposition 4 .

 44 Post Correspondence Problem A Post correspondence problem is a finite set of pairs of words over the two letters alphabet {A, B} : {< φ 1 , ψ 1 >, ..., < φ n , ψ n >}. A solution to such a problem is a non empty sequence of integers i 1 , ..., i p such that φ i1 ...φ ip = ψ i1 ...ψ ip Theorem 1. (Post [14]) It is undecidable whether or not a Post problem has a solution. Undecidability of Typability in the Lambda-Pi-Calculus Let us consider the context Γ = [T : T ype; a : T → T ; b : T → T ; c : T ; d : T ; P : T → T ype; F : Πx : T.((P x) → T )] Definition 5. (Huet [9]) Let φ be a word in the two letters alphabet {A, B}, we define by induction on the length of φ the term φ well-typed in Γ and the pure term φ as follows ε = λy : T.y Âφ = λy : T.(a ( φ y)) Bφ = λy : T.(b ( φ y)) φ = | φ| Proposition 3. Let {< φ 1 , ψ 1 >, ..., < φ n , ψ n >} be a Post problem, the non empty sequence i 1 , ..., i p is a solution to this problem if and only if ( φi1 (...( φip c)...)) ∼ = ( ψi1 (...( ψip c)...)) If g is a term such that the term (g a ... a) (n symbols a) is well-typed and is an object in an extension Γ ∆ of Γ then the term g is well-typed in the context Γ ∆ and its type is equivalent to the term Πx 1 : T → T....Πx n : T → T.(β x 1 ... x n ) for some term β of type (T → T ) → ... → (T → T ) → T ype in the context Γ ∆.

Proposition 6 .

 6 Let t be a normal term of type (T → T ) → ... → (T → T ) → T in the context Γ such that the normal form of (t λy : T.y ... λy : T.y) is equal to c. Then the term t has the form t = λx 1 : T → T....λx n : T → T.(x i1 (...(x ip c)...))

t

  = λf.λg.λh.(f (g a ... a) (h (g φ1 ... φn )) (h (g ψ1 ... ψn )) (F c (g λy.y ... λy.y)) (F d (g λy.d ... λy.d))) Assume this term is typable and call α the type of g. The term (g a ... a) is well-typed and is an object in Γ ∆ so α ∼ = Πx 1 : T → T....Πx n : T → T.(β x 1 ... x n ) where β is a term of type (T → T ) → ... → (T → T ) → T ype in Γ ∆. Then all the variables y bound in the terms φi , ψi , λy.y and λy.d have type T . The term (g φ1 ... φn ) has the type (β φ1 ... φn ), so from the well-typedness of the term (h (g φ1 ... φn )) we get that the type of the variable h has the form Πx : γ.γ ′ and γ ∼ = (β φ1 ... φn ) in the same way, from the well-typedness of the term (h (g ψ1 ... ψn )) we get γ ∼ = (β ψ1 ... ψn ) so (β φ1 ... φn ) ∼ = (β ψ1 ... ψn ) From the well-typedness of the term (F c (g λy : T.y ... λy : T.y)) we get (β λy : T.y ... λy : T.y) ∼ = (P c) At last from the the well-typedness of the term (F d (g λy : T.d ... λy : T.d)) we get (β λy : T.d ... λy : T.d) ∼ = (P d) Since the term β has type (T → T ) → ... → (T → T ) → T ype, the head symbol of the normal form of the term β cannot be a top variable of β, so it is the variable P and we have β ∼ = λx 1 : T → T....λx n : T → T.(P (δ x 1 ... x n )) For some term δ of type (T → T ) → ... → (T → T ) → T . We get (δ φ1 ... φn ) ∼ = (δ ψ1 ... ψn ) (δ λy : T.y ... λy : T.y) ∼ = c (δ λy : T.d ... λy : T.d) ∼ = d The second equality shows that the normal form of the term δ has the form λx 1 : T → T....λx n : T → T.(x i1 (...(x ip c)...)) for some sequence i 1 , ..., i p . The third equality shows that p > 0 and the first one that ( φi1 (...( φip c)...)) ∼ = ( ψi1 (...( ψip c)...)) so the sequence i 1 , ..., i p is a solution to the Post problem. Conversely assume that the Post problem has a solution i 1 , ..., i p , then by giving the following types to the variables f , g and h f : (P (a (...(a c)...))) → T → T → T → T → T g : Πx 1 : T → T....Πx n : T → T.(P (x i1 (...(x ip c)...))) h : (P ( φi1 (...( φip c)...))) → T

β-equivalent (βη-equivalence can also be considered without affecting the proof given here).A context is a list of pairs < x, T > (written x : T ) where x is a variable and T a term.We define inductively two judgements: Γ is well-formed and t has type T in Γ (Γ ⊢ t : T ) where Γ is a context and t and T are terms.[ ] well-formedΓ ⊢ T : s s ∈ {T ype, Kind} Γ [x : T ] well-formed Γ well-formed Γ ⊢ T ype : Kind Γ well-formed x : T ∈ Γ Γ ⊢ x : T Γ ⊢ T : T ype Γ [x : T ] ⊢ T ′ : s s ∈ {T ype, Kind} Γ ⊢ Πx : T.T ′ : s Γ ⊢ Πx : T.T ′ : s Γ [x : T ] ⊢ t : T ′ s ∈ {T ype, Kind} Γ ⊢ λx : T.t : Πx : T.T ′ Γ ⊢ t : Πx : T.T ′ Γ ⊢ t ′ : T Γ ⊢ (t t ′ ) : T ′ [x ← t ′ ] Γ ⊢ T : s Γ ⊢ T ′ : s Γ ⊢ t : T T ∼ = T ′ s ∈ {T ype, Kind} Γ ⊢ t : T ′

Acknowledgements

The author thanks Frank Pfenning for many stimulating and helpful discussions on this problem and Pawel Urzyczyn for his careful reading of a previous draft of this paper.


Claire Girard

Anne Silvestre

Thibaut Chabrillat

Sylvain Kerros

In vitro antiprotozoal action of 7 spices, medicinal plants and essential oils against 3 stages of Eimeria tenella

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

Phytosynthese

C. GIRARD (1) , A. SILVESTRE (2) , T. CHABRILLAT (1) , S. KERROS (1).

(1) PHYTOSYNTHESE, F-63200 MOZAC, FRANCE

MATERIAL AND METHODS

• Natural products tested:

• MedP: Fabacea powder, Urticaceae extract, Punicaceae extract, Acanthaceae extract • EO: Lilaceae EO, Lamiaceae EO, Lauraceae EO

RESULTS AND DISCUSSION

The 4 MedP samples (Fabacea powder, Urticaceae extract, Punicaceae extract, Acanthaceae extract) were less cytotoxic (lethal dose 10% (LD10) than the 3 EO samples (Lilaceae EO, Lamiaceae EO, Lauraceae EO) (MedP diluted at 10 -2 , EO diluted at 10 -4 -10 -6 ). MedP samples showed direct anti-protozoal activity against E. tenella stages: sporozoite viability rates were between 76.2% and 27% (p=0.05). Lilaceae EO and Lauraceae EO exhibited 81% and 77% (p=0.05) sporozoite viability rates, respectively (Fig. 1). Acanthaceae extract and Urticaceae extract showed a sporulation inhibition of E. tenella oocysts (p=0.05) (Fig. 3). These medicinal plant extracts exhibited anti-protozoal activity against the different E. tenella stages contrary to the tested EOs which target only sporozoites. These results were in accordance with studies reporting several modes of action of medicinal plant extracts, both direct anti-protozoal activity and protective cell activity [1].

CONCLUSION

Medicinal plant extracts exhibited an anti-protozoal activity against the different E. tenella stages. EOs only target sporozoites in this study. These findings help to understand how plant bio-actives could act to reduce E. tenella pathogenicity. Further studies could be carried out to test in vitro and/or in vivo potential synergies between different natural compounds.

Growing societal concerns to replace coccidiostats in poultry farms have led to the use of alternatives such as plant bio-actives. To investigate the mode of action of natural molecules, we designed this in vitro study to screen anti-protozoal activity of 4 spices or medicinal plants described by non-volatile active molecules (MedP) and 3 essential oils (EO) against 3 different stages of E. tenella.

In vitro antiprotozoal action of 7 spices, medicinal plants and essential oils against 3 stages of Eimeria tenella

  DE TOURS, ISP, F-37380 NOUZILLY, FRANCE

Fig. 1

 1 Fig.1 DIRECT ANTI-PROTOZOAL ACTIVITY AGAINST E. TENELLA SPOROZOITES (*: p<0.05 ; **:p<0.01; ***: p<0.001)

  Fig.2 MDBK CELL INVASION BY E. TENELLA (*: p<0.05; **: p<0.01; ***: p<0.001)Punicaceae extract showed a reduction of 32% of cell invasion.

Viability rate with fluorescence microscopy as % viability = number of live parasite in tested sample/ number of live parasite in negative control*100

	2 Invasion inhibition test: 96 microplates: 1.5×10 4
	MDBK cells, 1.5×10 4 E. tenella sporozoites expressed
	enzymatic activity; substrate added for 2h at 37°C, 4
	replicates per tested sample.
	%

• Negative control: H 2 0, soybean hemicellulose, DMSO • Protozoal strain: Eimeria tenella recombined strain • 3 experimental steps after cytotoxicity assay (colorimetric MTS test, 1,5.10 4 MDBK seed, incubation 5h at 37°C): 1 Direct anti-protozoite activity test: incubation 41°C for 1h. Killed sporozoites were stained with Evans Blue Dye. Each sample tested in triplicate.

invasion = absorbance (550 nm) tested sample/ absorbance (550 nm) negative control*100 3

Sporulation rate test: 5.10 5 non-sporulated oocysts (in 2% potassium dichromate solution), incubation for 72h at 26°C; each sample tested in triplicate. %

sporulation = number of sporulated oocysts in tested sample/number of sporulated oocyst in negative control*100

• Statistical test: Mann-whitney test compared to negative control for each test (p<0.05).


Sharvani Shintre

Florent Dingli

Damarys Loew

Anne Silvestre

Najma Rachidi

Studying two complementary infection models to identify common mechanisms of intracellular parasite survival: The roles of Leishmania and Eimeria exo-kinases in subversion of host cell signalling

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

Studying two complementary infection models to identify common mechanisms of intracellular parasite survival: The roles of Leishmania and Eimeria exo-kinases in subversion of host cell signalling.

Sharvani

1

  Institut Pasteur, Université Paris Cité, INSERM U1201, Groupe signalisation et interactions hôte-parasite, Unité de Parasitologie moléculaire et Signalisation, Paris, France ; 2 Institut Pasteur, Université Paris Cité, INSERM U1201, Unité de Parasitologie moléculaire et Signalisation, Paris, France ; 3 Infectiologie et Santé Publique, INRAE, Université de Tours, Nouzilly, France ; 4 Institut Curie, Laboratoire de spectrométrie de masse protéomique, Paris, France. infection, parasites secrete effector molecules particularly kinases, that modify their host cells to create a permissive environment. For instance, Leishmania secretes casein kinase 1.2 (L-CK1.2) into the host cell via exosomes. Previous studies in the lab demonstrated that L-CK1.2 interacts with and phosphorylates more than 200 host proteins in vitro and when expressed ectopically alters the phosphorylation of 771 host proteins involved in numerous host pathways, including a few known to be modified during Leishmania infection. During Eimeria tenella infection, rhoptry kinase 1 (EtROP1) inhibits host cell apoptosis by interacting with host p53. It appears that these two exo-kinases from divergent parasites, impact similar host pathways. Our hypothesis states that there might exist convergence in the host pathways modified by Leishmania and Eimeria. To investigate this hypothesis, we will perform phosphoproteomic and transcriptomic studies. For Leishmania, we have just established the first ever phosphoproteome for L. donovani infected mouse BMDMs. We quantified about 12,000 phosphosites comparing infected versus uninfected, including 663 sites with a p-value below 0.05. In order to uncover L-CK1.2 specific effects, we also had infected cells treated or untreated with the L-CK1.2 inhibitor, D4476. Further analysis is ongoing. For E. tenella, chicken lung epithelial cells (CLEC213) were infected with either wild type or EtROP1 overexpressing parasites; the samples were collected and will soon be sent for transcriptomic analysis. Once we have the complete set, including the phosphoproteome for Eimeria-infected CLEC213 cells and the transcriptome for L. donovani-infected macrophages, we will then assess which pathways are similarly or differently modified during both infections and decipher their role by interfering with important genes of these pathways using CRISPR-Cas knockout, RNAi or overexpression. 5) Identification of host pathways affected by LdCK1.2 in infected BMDMs A) Optimization of infection conditions B) Phosphoproteomics CONCLUSIONS LdCK1.2 and EtROP1, although belonging to very different parasites, seem to target common host pathways in vitro. Some of the LdCK1.2 targets are also modulated during Leishmania infection. We established the first phospho-proteome of L. donovani infected versus non infected macrophages to confirm the in vitro results. Over 50% of the differentially phosphorylated proteins upon infection were found in the phosphoproteome of macrophages expressing LdCK1.2. We also found that the pathways enriched in LdCK1.2 expressing macrophages contained all the enriched pathways found during infection. This indicates that LdCK1.2 might play a pivotal role in host modification during infection. PERSPECTIVES • We will validate our in vitro observations for the two kinases in an infection model and compare them to find similarities or unique signatures of host modulation during infection. • Using tools such as CRISPR-Cas system, overexpression, RNAi, we will assess the essentiality of these modulations during infection • The 134 common proteins found between infected macrophages and LdCK1.2 expressing macrophages will be assessed by MOTIF analysis and for shortlisted proteins in vitro kinase assay will be performed to confirm the involvement of LdCK1.2 • We have identified a few candidates that seem to be affected by infection or are substrates of both the kinases, but this is being currently investigated. To obtain the best infection conditions (>70% infection at the chosen timepoint) for phosphoproteomic analysis, optimization of the infection conditions was performed. Bone marrow derived macrophages were plated in 24 well plates with 1.5x10 5 MØ/ well and infected with L. donovani amastigotes (MOI 6, 8 or 10) 24 hrs after plating. a c To validate the in vitro results in L. donovani infected macrophages, we established the phosphoproteome of infected BMDMs versus that of uninfected BMDMs. BMDMs were plated in 6 well plates with 3x10 6 MØ/ well (Day 0) and the workplan (fig a) was followed. 48 hrs post infection (Day 3), the lysates were sent for phosphoproteomic analysis. We identified 12000 phosphosites (663 with p<0.05) and 252 proteins with an altered phosphorylation (, BMDMs were infected with L. donovani amastigotes at MOI 10 and collected at 48h post infection. a) Experimental plan b) MOI 6 and 10 at 3h, 48h and 6 days post infection. The fixed cells were stained with Hoechst (Blue) for nuclei and hamster immune serum (Green) for parasites. The cells were imaged with the Andor camera of Zeiss Axiophot fluorescence microscope at 63X magnification. c) 100-200 MØ per condition were analysed. The percentage of cells infected per field and the number of parasites per infected cell for different timepoints post infection was calculated and the mean was plotted. Error bars = SEM. • EtROP1 is an active kinase • Protein family secreted into host cells • Interacts with many host proteins in vitro • EtROP1 inhibits apoptosis and induces cell cycle arrest in host (MA Diallo et al., 2019) Comparison of host pathways enriched during infection and in macrophages expressing LdCK1.2 We compared the phosphoproteome for MØ expressing LdCK1.2 and that of infected MØ (b) and found 134 proteins in common with modulated phosphorylation (c). We also looked at enriched pathways and found 14 common clusters (circled in red in d) such as apoptosis, cell cycle, signalling, response to stimuli, metabolism, transcription. host pathways might be targeted by LdCK1.2 and EtROP1 OMICS Burrell et al., Parasitology, 2020 Coccidiosis: Eimeria tenella (Host : poultry) Kaye et al., 2011 Visceral Leishmaniasis: L. donovani (Host: sandfly and humans) 1) Leishmaniasis and Coccidiosis • L-CK1.2 is a signalling protein kinase. • Released via extracellular vesicles into the host macrophage (Silverman et al., 2010) • Plays an essential role in intracellular parasite survival. • It is a validated drug target.Using Cytoscape, the in vitro interactome data for the two kinases was used to compare the enriched pathways. The clusters affected by both kinases were identified.