Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian mathematician who worked in the field of differential geometry . He wrote a book, Lezioni di geometria intrinseca (Naples, 1890), on this topic, in which he also describes fractal , space-filling curves , partly covered by the larger class of de Rham curves , but are still known today in his honor as Cesàro curves. He is known also for his 'averaging' method for the 'Cesàro-summation' of divergent series , known as the Cesàro mean .
57-565: After a rather disappointing start of his academic career and a journey through Europe - with the most important stop at Liège , where his older brother Giuseppe Raimondo Pio Cesàro was teaching mineralogy at the local university - Ernesto Cesàro graduated from the University of Rome in 1887, while he was already part of the Royal Science Society of Belgium for the numerous works that he had already published. The following year, he obtained
114-432: A fractal curve . After that, Cesàro also studied the "snowflake curve" of von Koch , continuous but not differentiable in any of its points. Among his other works are Introduction to the mathematical theory of infinitesimal calculus (1893), Algebraic analysis (1894), Elements of infinitesimal calculus (1897). He proposed a possible definition of a limit of divergent sequence, known today as "Cesàro's sum," given by
171-478: A martyr for his faith. To enshrine St. Lambert's relics , his successor, Hubertus (later to become St. Hubert), built a basilica near the bishop's residence which became the true nucleus of the city. A few centuries later, the city became the capital of a prince-bishopric , which lasted from 985 till 1794. The first prince-bishop , Notger , transformed the city into a major intellectual and ecclesiastical centre, which maintained its cultural importance during
228-400: A bus stop. There were six fatalities, including the attacker (who shot himself), and 123 people were injured. On 29 May 2018, two female police officers and one civilian—a 22-year-old man—were shot dead by a gunman near a café on Boulevard d'Avroy in central Liège. The attacker then began firing at the officers in an attempt to escape, injuring a number of them "around their legs", before he
285-731: A democratic spirit in Liège that survived the Middle Ages. At the end of the Liège Wars , a rebellion took place against rule from Burgundy . In 1468 Duke Charles the Bold of Burgundy, witnessed by King Louis XI of France , captured and largely destroyed the city after a bitter siege which was ended with a successful surprise attack. The rebellion figures prominently in the plot of Sir Walter Scott 's 1823 novel Quentin Durward . The Prince-Bishopric of Liège
342-520: A frequent target of armies and insurgencies over the centuries. It was fortified early on with a castle on the steep hill that overlooks the city's western side. During this medieval period, three women from the Liège region made significant contributions to Christian spirituality: Elizabeth Spaakbeek, Christina the Astonishing , and Marie of Oignies . In 1345, the citizens of Liège rebelled against Prince-Bishop Engelbert III de la Marck , their ruler at
399-546: A hard-headed but resourceful Walloon boy who lived during Charlemagne 's times. Tchantchès is remembered with a statue, a museum, and a number of puppets found all over the city. Liège hosts one of the oldest and biggest Christmas Markets in Belgium, and the oldest kermesse , the Foire de Liège held each year from 28 October. The city is well known for its very crowded folk festivals. The 15 August festival ("Le 15 août") may be
456-573: A mathematics chair at the University of Palermo , which he kept until 1891. He settled in Rome, where he stayed as a professor at the Sapienza University until his accidental death, while trying to rescue his youngest son Manlio from drowning. Cesàro's main contributions are in the field of differential geometry. Lessons of intrinsic geometry , written in 1894, explains in particular the construction of
513-422: A rampage and severely damaged the central railway station Guillemins. The unrest was so intense that "army troops had to wade through caltrops , trees, concrete blocks, car and crane wrecks to advance. Streets were dug up. Liège saw the worst fighting on 6 January 1961. In all, 75 people were injured during seven hours of street battles." On 6 December 1985, the city's courthouse was heavily damaged and one person
570-486: A total population of 749,110 on 1 January 2008. This includes a total of 52 municipalities, among others, Herstal and Seraing . Liège ranks as the third most populous urban area in Belgium, after Brussels and Antwerp , and the fourth municipality after Antwerp, Ghent and Charleroi . The city is part of the Meuse-Rhine Euroregion . The name is of Germanic origin and is reconstructible as * liudik- , from
627-585: Is a Euroregion created in 1976, with judicial status achieved in 1991. It comprises 11.000 km and has around 3.9 million inhabitants around the city-corridor of Aachen – Maastricht – Hasselt – Liège . The seat of the region has been in Eupen, Belgium since 1 January 2007. Within a wider context, the region is part of what is called the Blue Banana European urbanisation corridor. The Meuse–Rhine Euregion comprises: Though they do not officially comprise
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#1733085186045684-401: Is a large Italian community, and Italian can be heard in many places. The city has a number of football teams, most notably Standard Liège , who have won several championships and which was previously owned by Roland Duchâtelet , and R.F.C. de Liège , one of the oldest football clubs in Belgium. It is also known for being the club who refused to release Jean-Marc Bosman , a case which led to
741-503: Is going down and now is a mere shadow of its former self, the manufacture of steel goods remain important. The economy of the region is now diversified; the most important centres are: Mechanical industries ( Aircraft engine and Spacecraft propulsion ), space technology, information technology , biotechnology and the production of water , beer or chocolate . Liège has an important group of headquarters dedicated to high-technology, such as Techspace Aero , which manufactures pieces for
798-582: Is owned by the Walloon government along with some private investors. The Port of Liège , located on the river Meuse , is the 3rd largest river port in Europe. Liège also has direct links to Antwerp through the Albert Canal and to Rotterdam via the river Meuse. It stretches over a distance of 26 kilometres and comprises 32 port areas and covers 3.7 square kilometres. Liège is served by many direct rail links with
855-540: Is the Place du Marché. The "Batte" market is where most locals visit on Sundays. The outdoor market goes along the river Meuse and also attracts many visitors to Liège. The market typically runs from early morning to 2 o'clock in the afternoon every Sunday year long. Produce, clothing, and snack vendors are the main concentration of the market. Liège is home to the Opéra Royal de Wallonie (English: Royal Opera of Wallonia ) and
912-462: The Airbus A380 or the rocket Ariane 5 . Other stand-out sectors include Amós which manufactures optical components for telescopes and Drytec , which produces compressed air dryers. Liège also has many other electronic companies such as SAP , EVS , Gillam , AnB , Balteau , IP Trade . Other prominent businesses are the global leader in light armament FN Herstal , the beer company Jupiler ,
969-478: The Bosman ruling . In spring, Liège hosts the start and finish of the annual Liège–Bastogne–Liège cycling race , one of the spring classics and the oldest of the five monuments of cycling . The race starts in the centre of Liège, before heading south to Bastogne and returning north to finish in the industrial suburb of Ans . Traveling through the hilly Ardennes , it is one of the longest and most arduous races of
1026-464: The General strike against Leopold III of Belgium and "seized control over the city of Liège". The strike ultimately led to Leopold's abdication. Liège began to suffer from a relative decline of its industry, particularly the coal industry, and later the steel industry, producing high levels of unemployment and stoking social tension. During the 1960–1961 Winter General Strike , disgruntled workers went on
1083-602: The Middle Ages . Pope Clement VI recruited several musicians from Liège to perform in the Papal court at Avignon , thereby sanctioning the practice of polyphony in the religious realm. The city was renowned for its many churches, the oldest of which, St Martin's, dates from 682. Although nominally part of the Holy Roman Empire , in practice it possessed a large degree of independence. The strategic position of Liège has made it
1140-623: The Russian Far East and the fellow Francophone province of Quebec . Summers are also moderated by the maritime air, with average temperatures being similar to areas as far north as in Scandinavia . Being inland though, Liège has a relatively low winter seasonal lag . On 1 January 2013, the municipality of Liège had a total population of 197,013. The metropolitan area has about 750,000 inhabitants. Its inhabitants are predominantly French -speaking, with German and Dutch-speaking minorities. Akin to
1197-617: The Thirty Years' War , the city was besieged by Imperial forces under Johann von Werth from April to July. The army, mainly consisting of mercenaries, extensively and viciously plundered the surrounding bishopric during the siege. The Duke of Marlborough captured the city from the Bavarian prince-bishop and his French allies in 1704 during the War of the Spanish Succession . In the middle of
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#17330851860451254-644: The United Kingdom of the Netherlands . Dutch rule lasted only until 1830, when the Belgian Revolution led to the establishment of an independent, Catholic and neutral Belgium which incorporated Liège. After this, Liège developed rapidly into a major industrial city which became one of continental Europe's first large-scale steel making centres. The Walloon Jacquerie of 1886 saw a large-scale working class revolt. No less than 6,000 regular troops were called into
1311-499: The Virgin Mary . It is one of the biggest folkloric displays in the city, with a religious procession, a flea market, dances, concerts, and a series of popular games. Nowadays these celebrations start a few days earlier and last until the 16th. Some citizens open their doors to party goers, and serve "peket", the traditional local alcohol. This tradition is linked to the important folkloric character Tchantchès ( Walloon for François ),
1368-404: The limit of the mean of the sequence partial terms' sum. Li%C3%A8ge Liège ( / l i ˈ ɛ ʒ , l i ˈ eɪ ʒ / lee- EZH , lee- AYZH ; French: [ljɛʒ] ; Walloon : Lîdje [liːtʃ] ; Dutch : Luik [lœyk] ; German : Lüttich [ˈlʏtɪç] ) is a city and municipality of Wallonia , and
1425-499: The Bavarian royal house of Wittelsbach , with second sons of the Bavarian monarch ruling as prince-bishop. Beginning with the ascension of Ernest of Bavaria in 1581, Bavarian princes ruled over Cologne , Münster , and other bishoprics in the northwest of the Holy Roman Empire in addition to Liège. Ferdinand of Bavaria (bishop) ruled from 1612 to 1650, and Maximilian Henry of Bavaria ruled from 1650 to 1688. In 1636, during
1482-745: The Germanic word * liudiz "people", which is found in for example Dutch lui(den) , lieden , Polish "ludzie", Czech "lidé", German Leute , Old English lēod (English lede ), Icelandic lýður ("people"), Latvian ļaudis ("people"), Lithuanian liaudis ("people"). It is found in Ukrainian as liudy ("people"), in Russian as russian: люди, romanised: liudi ("people"), in Latin as Leodicum or Leodium , in Middle Dutch as ludic or ludeke . Until 17 September 1946,
1539-556: The Germans until the end of the war. Liège received the Légion d'Honneur for its resistance in 1914. As part of Chancellor Theobald von Bethmann Hollweg 's Septemberprogramm , Berlin planned to annexe Liege under the name Lüttich to the German Empire in any post-war peace agreement. The Germans returned in 1940 , this time taking the forts in only three days. Most Jews were saved, with
1596-828: The Limburgish and Ripuarian is that they are spoken on both sides of the border. Limburgish, although only recognised as such in The Netherlands, is also spoken in Belgian Limburg and North Rhine-Westphalia. Ripuarian is also spoken on both sides of the Dutch–German border, but with the extra trait of having the same variant spoken on both sides of the border. In daily life, one rarely makes a distinction between Limburgish and Ripuarian. The latter language hardly enjoys any recognition by people who speak it. The dialects of both linguistic groups change gradually from village to village, and
1653-666: The Meuse valley and the Ardennes en route to France. The German invasion of Belgium on 5 August 1914 soon reached Liège, which was defended by 30,000 troops under General Gérard Leman in the Battle of Liège . The forts initially held off General Alexander von Kluck 's German First Army of about 100,000 men but were pulverised into submission by a five-day bombardment by heavy artillery, including thirty-two 21 cm mortars and two German 42 cm Big Bertha howitzers . Due to faulty planning of
1710-526: The Netherlands, making Liège the first city that has hosted stages of all three cycling Grand Tours. Liège is also home to boxer Ermano Fegatilli , the current European Boxing Union Super Featherweight champion. Liège is the most important city of the Walloon region from an economic perspective. In the past, Liège was one of the most important industrial centres in Europe, particularly in steel -making. Starting in 1817, John Cockerill extensively developed
1767-596: The Orchestre Philharmonique Royal de Liège (OPRL) (English: Liège Royal Philharmonic Orchestra ). The city annually hosts a significant electro-rock festival Les Ardentes and jazz festival Jazz à Liège . Liège has active alternative cinemas, Le Churchill, Le Parc and Le Sauvenière. There are also two mainstream cinemas, the Kinepolis multiplexes. Liège also has a particular Walloon dialect, sometimes said to be one of Belgium's most distinctive. There
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1824-522: The University; for a few days before the event, students (wearing very dirty lab-coats) beg for money, mostly for drinking. Liège is renowned for its nightlife. Within the pedestrian zone behind the Opera House, there is a square city block known locally as Le Carré (the Square) with many lively pubs which are reputed to remain open until the last customer leaves (typically around 6 am). Another active area
1881-401: The best known. The population gathers in a quarter named Outre-Meuse with plenty of tiny pedestrian streets and old yards. Many people come to see the procession but also to drink alcohol (mostly peket ) and beer, eat cooked pears, boûkètes or sausages or simply enjoy the atmosphere until the early hours. The Saint Nicholas festival around 6 December is organized by and for the students of
1938-500: The capital of the province of Liège , Belgium. The city is situated in the valley of the Meuse , in the east of Belgium, not far from borders with the Netherlands ( Maastricht is about 33 km (20.5 mi) to the north) and with Germany ( Aachen is about 53 km (32.9 mi) north-east). In Liège, the Meuse meets the river Ourthe . The city is part of the sillon industriel ,
1995-418: The chocolate company Galler , and the water and soda companies Spa and Chaudfontaine . A science park south east of the city, near the University of Liège campus, houses spin-offs and high technology businesses. In 1812 there were three coal pits ( Bure ) in close proximity just outside the city gates: Bure Triquenotte, Bure de Beaujone and Bure Mamonster. The first two shafts were joined underground, but
2052-468: The city to quell the unrest, while strike spread through the whole sillon industriel . Liège's fortifications were redesigned by Henri Alexis Brialmont in the 1880s and a chain of twelve forts was constructed around the city to provide defence in depth . This presented a major obstacle to the Imperial German Army in 1914, whose Schlieffen Plan relied on being able to quickly pass through
2109-564: The city's name was written Liége , with the acute accent instead of a grave accent . In French, Liège is associated with the epithet la cité ardente ("the fervent city"). This term, which emerged around 1905, originally referred to the city's history of rebellions against Burgundian rule, but was appropriated to refer to its economic dynamism during the Industrial Revolution . Although settlements already existed in Roman times,
2166-732: The core of the Euroregion, the following areas are often associated with it through inter-regional agreements: The official languages of the three countries involved in the Euregion are Dutch (in Belgium and the Netherlands), French (in Belgium) and German (in Belgium and Germany). Regional languages are also spoken namely Limburgish (which is recognised as a regional language in Dutch Limburg), Ripuarian and Walloon . The intra-cultural aspect of
2223-690: The course of the 1794 campaigns of the French Revolution , the French Revolutionary Army took the city and imposed strongly anticlerical regime, destroying St. Lambert's Cathedral . The overthrow of the Prince-Bishopric of Liège was confirmed in 1801 by the Concordat co-signed by Napoléon Bonaparte and Pope Pius VII . France lost the city in 1815 when the Congress of Vienna awarded it to
2280-657: The eighteenth century the ideas of the French Encyclopédistes began to gain popularity in the region. Bishop François-Charles de Velbrück (1772–84), encouraged their propagation, thus prepared the way for the Liège Revolution which started in the episcopal city on 18 August 1789 and led to the creation of the Republic of Liège before it was invaded by counter-revolutionary forces of the Habsburg monarchy in 1791. In
2337-481: The first references to Liège are from 558, when it was known as Vicus Leudicus. Around 705, Saint Lambert of Maastricht is credited with completing the Christianization of the region, indicating that up to the early 8th century the religious practices of antiquity had survived in some form. Christian conversion may still not have been quite universal, since Lambert was murdered in Liège and thereafter regarded as
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2394-459: The former industrial backbone of Wallonia. It still is the principal economic and cultural centre of the region. The municipality consists of the following sub-municipalities : Angleur , Bressoux , Chênée , Glain , Grivegnée , Jupille-sur-Meuse , Liège proper, Rocourt , and Wandre . In November 2012, Liège had 198,280 inhabitants. The metropolitan area , including the outer commuter zone, covers an area of 1,879 km (725 sq mi) and had
2451-625: The help of the sympathetic population, as many Jewish children and refugees were hidden in the numerous monasteries. Liege was liberated by the British Second Army in September 1944. After the war ended, the Royal Question came to the fore, since many saw King Leopold III as collaborating with the Germans during the war. In July 1950, André Renard , leader of the Liègian FGTB launched
2508-400: The iron and steel industry. The industrial complex of Seraing was the largest in the world. It once boasted numerous blast furnaces and mills. Liège has also been an important centre for gunsmithing since the Middle Ages and the arms industry is still strong today, with the headquarters of FN Herstal and CMI Defence being located in Liège. Although from 1960 on the secondary sector
2565-458: The last one was a separate colliery. The shafts were 120 fathoms (720 ft; 220 m) deep. Water was led to a sump ( serrement ) from which it could be pumped to the surface. At 11:00 on 28 February 1812 the sump in the Beaujone mine failed and flooded the entire colliery. Of the 127 men down the mine at the time 35 escaped by the main shaft, but 74 were trapped. [These numbers are taken from
2622-456: The opening up of borders within the European Union , surging steel prices, and improved administration. Several new shopping centres have been built, and numerous repairs carried out. On 13 December 2011, there was a grenade and gun attack at Place Saint-Lambert . An attacker, later identified as Nordine Amrani, aged 33, armed with grenades and an assault rifle, attacked people waiting at
2679-513: The protection of the underground defence tunnels beneath the main citadel, one direct artillery hit caused a huge explosion, which eventually led to the surrender of the Belgian forces. The Belgian resistance was shorter than had been intended, but the twelve days of delay caused by the siege nonetheless contributed to the eventual failure of the German invasion of France. The city was subsequently occupied by
2736-522: The report, the 18 miner discrepancy is unexplained.] The trapped men attempted to dig a passageway into Mamonster. After 23 feet (7.0 m) there was a firedamp explosion and they realised that they had penetrated some old workings belonging to an abandoned mine, Martin Wery. The overseer, Monsieur Goffin, led the men to the point in Martin Wery which he judged closest to Mamonster and they commence to dig. By
2793-577: The rest of Belgium, the population of minorities has grown significantly since the 1990s. The city has become the home to large numbers of Algerian, Moroccan, Turkish, and Vietnamese immigrants. Liège also houses a significant Afro-Belgian community. The city is a major educational hub in Belgium. There are 42,000 pupils attending more than 24 schools. The University of Liège , founded in 1817, has 20,000 students. The "Le Quinze Août" celebration takes place annually on 15 August in Outremeuse and celebrates
2850-643: The rest of Western Europe. Its three principal stations are Liège-Guillemins railway station , Liège-Carré, and Liège-Saint-Lambert. The InterCity Express and Thalys call at Liège-Guillemins, providing direct connections to Cologne and Frankfurt and Paris-Nord respectively. Meuse%E2%80%93Rhine Euroregion The Meuse–Rhine Euroregion ( Dutch : Euregio Maas–Rijn [øːˈreːɣijoː ˌmaːsˈrɛin] , French : Eurorégion Meuse–Rhin [øʁɔʁeʒjɔ̃ møz ʁɛ̃] , German : Euregio Maas–Rhein [ɔʏˈʁeːɡi̯o ˌmaːsˈʁaɪn] , Limburgish : Euregio Maas–Rien [øːˈʀeːɣijoː ˌmaːsˈʀiːn] )
2907-563: The risk of a firedamp explosion. By 7pm that evening an opening was made, 511 feet (156 m) of tunnel had been dug by hand in five days. All of the 74 miners in Goffin's part survived and were brought to the surface. Liège is served by Liège Airport , located in Bierset, a few kilometres west of the city. It is the principal axis for the delivery of freight and in 2011 was the world's 33rd busiest cargo airport. Passenger services are very few. It
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#17330851860452964-502: The season. Liège is the only city that has hosted stages of all three cycling Grand Tours . It staged the start of the 1973 and 2006 Giro d'Italia ; as well as the Grand Départ of the 2004 , 2012 , and 2017 Tour de France making it the first city outside France to host the Grand Départ twice or more times. In 2009 , the Vuelta a España visited Liège after four stages in
3021-423: The second day they had run out of candles and dug the remainder of a 36 feet (11 m) gallery in darkness. On the surface the only possible rescue was held to be via Mamonster. A heading was driven towards Beaujone with all possible speed, including blasting. The trapped miners heard the rescuers and vice versa. Five days after the accident communication was possible and the rescuers worked in darkness to avoid
3078-478: The time, and defeated him in battle near the city. Shortly after, a unique political system formed in Liège, whereby the city's 32 guilds shared sole political control of the municipal government. Each person on the register of each guild was eligible to participate, and each guild's voice was equal, making it the most democratic system that the Low Countries had ever known. The system spread to Utrecht , and left
3135-744: Was killed in a bomb attack by a lawyer . Liège is also known as a traditionally socialist city. In 1991, powerful Socialist André Cools , a former Deputy Prime Minister, was gunned down in front of his girlfriend's apartment. Many suspected that the assassination was related to a corruption scandal which swept the Socialist Party , and the Belgian Federal Government in general, after Cools' death. Two men were sentenced to twenty years in jail in 2004, for involvement in Cools' murder. Liège has shown some signs of economic recovery in recent years with
3192-547: Was shot dead. Belgian broadcaster RTBF said the gunman was temporarily released from prison on 28 May where he had been serving time on drug offences. The incident is currently being treated as terrorism. In spite of its inland position Liège has a maritime climate influenced by the mildening sea winds originating from the Gulf Stream , travelling over Belgium's interior. As a result, Liège has very mild winters for its latitude and inland position, especially compared to areas in
3249-579: Was technically part of the Holy Roman Empire which, after 1477, came under the rule of the Habsburgs . The reign of prince-bishop Érard de La Marck (1506–1538) coincides with the dawn of the Renaissance . During the Counter-Reformation , the diocese of Liège was split and progressively lost its role as a regional power. By the 17th century, the bishopric of Liège became a virtual Secundogeniture of
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