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Ernst Ising ( German: [ˈiːzɪŋ] ; May 10, 1900 – May 11, 1998) was a German physicist , who is best remembered for the development of the Ising model . He was a professor of physics at Bradley University until his retirement in 1976.

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73-542: Ising is a surname. Notable people with the surname include: Ernst Ising (1900–1998), German physicist Gustav Ising (1883–1960), Swedish accelerator physicist Jane Ising (1902-2012), German-American economist Rudolf Ising , animator for MGM , together with Hugh Harman often credited as: Harman-Ising See also [ edit ] Ising model , mathematical model of ferromagnetism in statistical mechanics [REDACTED] Surname list This page lists people with

146-496: A graph , where each node has exactly one spin, and each edge connects two spins with a nonzero value of J. If all the Js are equal, it is convenient to measure energy in units of J. Then a model is completely specified by the graph and the sign of J. The antiferromagnetic one-dimensional Ising model has the energy function: where i runs over all the integers. This links each pair of nearest neighbors. In his 1924 PhD thesis, Ising solved

219-469: A knowledge economy , with the founding of the University of Luxembourg and a national space program . Luxembourg is described as a " full democracy ", with a parliamentary democracy headed by a constitutional monarch . Executive power is exercised by the grand duke and the cabinet, which consists of several members with the titles of minister, minister delegate or secretary of state, who are headed by

292-663: A soviet worker's republic similar to the ones emerging in Germany , but these attempts lasted only 2 days. In November 1918, a motion in the Chamber of Deputies demanding the abolition of the monarchy was defeated narrowly by 21 votes to 19 (with three abstentions). France questioned the Luxembourgish government's, and especially Marie-Adélaïde's, neutrality during the war, and calls for an annexation of Luxembourg to either France or Belgium grew louder in both countries. In January 1919,

365-705: A Habsburg possession, the Duchy of Luxembourg became, like many countries in Europe at the time, heavily involved in the many conflicts for dominance of Europe between the Habsburg-held countries and the Kingdom of France . In 1542, the King of France , François I , invaded Luxembourg twice, but the Habsburgs under Charles V managed to reconquer the Duchy each time. Luxembourg became part of

438-407: A Ph.D. in physics from the University of Hamburg in 1924 when he published his doctoral thesis (an excerpt or a summary of his doctoral thesis was published as an article in a scientific journal in 1925 and this has led many to believe that he published his full thesis in 1925, see, ). His doctoral thesis studied a problem suggested by his teacher, Wilhelm Lenz. He investigated the special case of

511-414: A Prime Minister. The current Constitution of Luxembourg , the supreme law of Luxembourg, was originally adopted on 17 October 1868. The Constitution was last updated on 1 July 2023. The grand duke has the power to dissolve the legislature , in which case new elections must be held within three months. But since 1919, sovereignty has resided with the nation, exercised by the grand duke in accordance with

584-513: A company of the Luxembourgish Army rebelled, declaring itself to be the army of the new republic, but French troops intervened and put an end to the rebellion. Nonetheless, the disloyalty shown by her own armed forces was too much for Marie-Adélaïde, who abdicated in favor of her sister Charlotte 5 days later. The same year, in a popular referendum , 77.8% of the Luxembourgish population declared in favor of maintaining monarchy and rejected

657-415: A consciousness of independence and a national awakening took place in the 19th century. The people of Luxembourg began referring to themselves as Luxembourgers , rather than being part of one of the larger surrounding nations. This consciousness of Mir wëlle bleiwe wat mir sinn (" We want to remain what we are ") culminated in 1890, when the last step towards full independence was finally taken: due to

730-459: A discrete collection of variables called spins , which can take on the value 1 or −1. The spins S i {\displaystyle S_{i}} interact in pairs, with energy that has one value when the two spins are the same, and a second value when the two spins are different. The energy of the Ising model is defined to be: where the sum counts each pair of spins only once. Notice that

803-703: A founding member of the United Nations in 1945. Luxembourg's neutral status under the constitution formally ended in 1948, and in April 1949 it also became a founding member of NATO . During the Cold War , Luxembourg continued its involvements on the side of the Western Bloc . In the early fifties a small contingent of troops fought in the Korean War . Luxembourg troops have also deployed to Afghanistan, to support ISAF . In

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876-442: A linear chain of magnetic moments , which are only able to take two positions, "up" and "down," and which are coupled by interactions between nearest neighbors. Mainly through following studies by Rudolf Peierls , Hendrik Kramers , Gregory Wannier and Lars Onsager the model proved to be successful explaining phase transitions between ferromagnetic and paramagnetic states. After earning his doctorate, Ernst Ising worked for

949-661: A position, first as a teacher and then as headmaster, at a Jewish school in Caputh near Potsdam for Jewish students who had been thrown out of public schools. Ernst and his wife Dr. Johanna Ising, née Ehmer, lived in Caputh near the famous summer residence of the Einstein family. In 1938, the school in Caputh was destroyed by the Nazis, and in 1939 the Isings fled to Luxembourg , where Ising earned money as

1022-1018: A pro- NATO , pro-US foreign policy. Luxembourg is considered a European capital, and is the site of the Court of Justice of the European Union , the European Court of Auditors , the European Investment Bank , the Statistical Office of the European Union ( Eurostat ) and other vital EU organs. The Secretariat of the European Parliament is located in Luxembourg, but the Parliament usually meets in Brussels and sometimes in Strasbourg . Luxembourg

1095-468: A profound effect on the historical memory of the country and its citizens. However, many republican ideas of this era continue to have a lasting effect on Luxembourg; one of the many examples features the implementation of the Napoleonic Code Civil which was introduced in 1804 and is still valid today. After the defeat of Napoleon in 1815, the Duchy of Luxembourg was restored. However, as

1168-619: A shepherd and railroad worker. After the German Wehrmacht occupied Luxembourg, Ernst Ising was forced to work for the army. In 1947, the Ising family emigrated to the United States. Though he became Professor of Physics at Bradley University in Peoria, Illinois , he never published again. Ising died at his home in Peoria in 1998, just one day after his 98th birthday. The Ising model is defined on

1241-470: A short time in business before becoming a teacher, in Salem , Strausberg and Crossen , among other places. In 1930, he married the economist Dr. Johanna Ehmer (February 2, 1902 – February 2, 2012; later known as Jane Ising and just barely becoming a supercentenarian ). As a young German–Jewish scientist, Ising was barred from teaching and researching when Hitler came to power in 1933. In 1934, he found

1314-569: A succession crisis the Dutch monarchy ceased to hold the title Grand-Duke of Luxembourg. Beginning with Adolph of Nassau-Weilburg , the Grand-Duchy would have their own monarchy , thus reaffirming its full independence. In August 1914, during World War I , Imperial Germany violated Luxembourg's neutrality by invading it in order to defeat France. Nevertheless, despite the German occupation , Luxembourg

1387-615: A succession crisis, precipitated by the lack of a male heir to assume the throne. Since Sigismund and Elizabeth of Görlitz were both heirless, all possessions of the Luxembourg Dynasty were redistributed among the European aristocracy. The Duchy of Luxembourg become a possession of Philip the Good , Duke of Burgundy . As the House of Luxembourg had become extinct and Luxembourg now became part of

1460-603: Is also known for having founded the Schueberfouer in 1340 and for his heroic death at the Battle of Crécy in 1346. John the Blind is considered a national hero in Luxembourg. In the 14th and early 15th centuries, three more members of the House of Luxembourg reigned as Holy Roman Emperors and Bohemian Kings: John's descendants Charles IV , Sigismund (who also was King of Hungary and Croatia ), and Wenceslaus IV . Charles IV created

1533-853: Is also site of the EFTA Court , which is responsible for the three EFTA members who are part of the European Single Market through the EEA Agreement . The Luxembourgish army is mostly based in its casern, the Centre militaire Caserne Grand-Duc Jean on the Härebierg in Diekirch. The general staff is based in the capital, the État-Major . The army is under civilian control , with the grand duke as Commander-in-Chief . The Minister for Defense , Yuriko Backes , oversees army operations. The professional head of

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1606-456: Is different from Wikidata All set index articles Ernst Ising Ernst Ising was born in Cologne in 1900. Ernst Ising's parents were the merchant Gustav Ising and his wife Thekla Löwe. After school, he studied physics and mathematics at the University of Göttingen and University of Hamburg . In 1922, he began researching ferromagnetism under the guidance of Wilhelm Lenz . He earned

1679-534: Is generally known as the Second Partition of Luxembourg . To compensate the Duchy for this loss, it was decided to elevate the Duchy to a Grand-Duchy , thus giving the Dutch monarchs the additional title of Grand-Duke of Luxembourg . After Belgium became an independent country following the victorious Belgian Revolution of 1830–1831 , it claimed the entire Grand-Duchy of Luxembourg as being part of Belgium, however,

1752-661: The Abbey of Echternach in AD 698, and is celebrated annually in the dancing procession of Echternach . For a few centuries the abbey would become one of northern Europe's most influential abbeys. The Codex Aureus of Echternach , an important surviving codex written entirely in gold ink, was produced here in the 11th century. The so-called Emperor's Bible and the Golden Gospels of Henry III were also produced in Echternach at this time. When

1825-554: The Bock rock by Siegfried, Count of the Ardennes , in 963 through an exchange act with St. Maximin's Abbey, Trier . Around this fort , a town gradually developed, which became the center of a state of great strategic value within the Duchy of Lorraine. Over the years, the fortress was extended by Siegfried's descendants and by 1083, one of them, Conrad I , was the first to call himself a " Count of Luxembourg ", and with it effectively creating

1898-675: The Burgundian Netherlands , this would mark the start of nearly 400 years of foreign rule over Luxembourg. In 1482, Philip the Handsome inherited all of what became then known as the Habsburg Netherlands , and with it the Duchy of Luxembourg. For nearly 320 years Luxembourg would remain a possession of the mighty House of Habsburg, at first under Austrian rule (1506–1556), then under Spanish rule (1556–1714) , before going back again to Austrian rule (1714–1794) . With having become

1971-685: The Carolingian Empire was divided many times starting with the Treaty of Verdun in 843, today's Luxembourgish territory became successively part of the Kingdom of Middle Francia (843–855), the Kingdom of Lotharingia (855–959) and finally of the Duchy of Lorraine (959–1059), which itself had become a state of the Holy Roman Empire . The recorded history of Luxembourg begins with the acquisition of Lucilinburhuc (today Luxembourg Castle ) situated on

2044-628: The Court of Justice of the European Union , the highest judicial authority. Luxembourg's culture , people, and languages are greatly influenced by its much larger neighbors France and Germany; for example, Luxembourgish , a Germanic language, is the only national language of the Luxembourgish people and of the Grand Duchy of Luxembourg, French is the only language for legislation, and all three – Luxembourgish, German and French – are used for administrative matters in

2117-828: The Dalheim Ricciacum and the Vichten mosaic , on display at the National Museum of History and Art in Luxembourg City. The territory was infiltrated by the Germanic Franks from the 4th century, and was abandoned by Rome in AD 406, after which it became part of the Kingdom of the Franks . The Salian Franks who settled in the area are often described as the ones having brought the Germanic language to present-day Luxembourg, since

2190-522: The Grand Duchy of Luxembourg , is a small landlocked country in Western Europe . It is bordered by Belgium to the west and north, Germany to the east, and France to the south. Its capital and most populous city, Luxembourg City , is one of the four institutional seats of the European Union (together with Brussels , Frankfurt , and Strasbourg ) and the seat of several EU institutions, notably

2263-510: The King of the Netherlands , would become, in personal union , the head of state. To satisfy Prussia, it was decided that not only the Fortress of Luxembourg be manned by Prussian troops , but also that large parts of Luxembourgish territory (mainly the areas around Bitburg and St. Vith) become Prussian possessions. This marked the second time that the Duchy of Luxembourg was reduced in size, and

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2336-768: The Nine Years' War in 1684, France invaded Luxembourg again , conquering and occupying the Duchy until 1697 when it was returned to the Spanish in order to garner support for the Bourbon cause during the prelude to the War of the Spanish Succession . When the war broke out in 1701 Luxembourg and the Spanish Netherlands were administered by the pro-French faction under the governor Maximilian II Emanuel, Elector of Bavaria and sided with

2409-701: The Paleolithic Age , about 35,000 years ago. From the 2nd century BC, Celtic tribes settled in the region between the rivers Rhine and Meuse . Six centuries later the Romans named the Celtic tribes inhabiting these exact regions collectively as the Treveri . Many examples of archaeological evidence proving their existence in Luxembourg have been discovered, the most famous being the Oppidum of Titelberg . In around 58 to 51 BC,

2482-450: The Red Lands ' rich iron-ore grounds in the beginning of the 20th century drove Luxembourg's industrialization. After the decline of the steel industry in the 1970s, the country focused on establishing itself as a global financial center and developed into the banking hub it is reputed to be. Since the beginning of the 21st century, its governments have focused on developing the country into

2555-581: The Spanish Netherlands in 1556, and when France and Spain went to war in 1635 it resulted in the Treaty of the Pyrenees , in which the first partition of Luxembourg was decided. Under the Treaty, Spain ceded the Luxembourgish fortresses of Stenay , Thionville, and Montmédy , and the surrounding territory to France, effectively reducing the size of Luxembourg for the first time in centuries. In context of

2628-501: The old Frankish language spoken by them is considered by linguists to be a direct forerunner of the Moselle Franconian dialect , which later evolved into, among others, the modern-day Luxembourgish language . The Christianization of Luxembourg is usually dated back to the end of the 7th century. The most famous figure in this context is Willibrord , a Northumbrian missionary saint, who together with other monks established

2701-407: The surname Ising . If an internal link intending to refer to a specific person led you to this page, you may wish to change that link by adding the person's given name (s) to the link. Retrieved from " https://en.wikipedia.org/w/index.php?title=Ising&oldid=1222591267 " Category : Surnames Hidden categories: Articles with short description Short description

2774-409: The 1950s, Luxembourg became one of the six founding countries of the European Communities , following the 1952 establishment of the European Coal and Steel Community , and subsequent 1958 creations of the European Economic Community and European Atomic Energy Community . In 1993, the former two of these were incorporated into the European Union . With Robert Schuman (one of the founding fathers of

2847-419: The Battle of Worringen helped the Counts of Luxembourg to achieve military glory, which they had previously lacked, as they had mostly enlarged their territory by means of inheritances, marriages and fiefdoms. The ascension of the Counts of Luxembourg culminated when Henry VII became King of the Romans , King of Italy and finally, in 1312, Holy Roman Emperor . With the ascension of Henry VII as Emperor,

2920-495: The Bourbons. The duchy was subsequently occupied by the pro-Austrian allied forces during the conflict and was awarded to Austria at its conclusion in 1714. As the Duchy of Luxembourg repeatedly passed back and forth from Spanish and Austrian to French rule, each of the conquering nations contributed to strengthening and expanding the Fortress that the Castle of Luxembourg had become over the years. One example of this includes French military engineer Marquis de Vauban who advanced

2993-475: The Constitution and the law. Legislative power is vested in the Chamber of Deputies , a unicameral legislature of sixty members, who are directly elected to five-year terms from four constituencies . A second body, the Council of State ( Conseil d'État ), composed of 21 ordinary citizens appointed by the grand duke, advises the Chamber of Deputies in the drafting of legislation. Luxembourg has three lower tribunals ( justices de paix ; in Esch-sur-Alzette ,

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3066-421: The Dutch King who was also Grand Duke of Luxembourg, as well as Prussia, did not want to lose their grip on the mighty fortress of Luxembourg and did not agree with the Belgian claims. The dispute would be solved at the 1839 Treaty of London where the decision of the Third Partition of Luxembourg was taken. This time the territory was reduced by more than half, as the predominantly francophone western part of

3139-455: The EU), Pierre Werner (considered the father of the Euro ), Gaston Thorn , Jacques Santer and Jean-Claude Juncker (all former Presidents of the European Commission ), Luxembourgish politicians contributed substantially to the EU's formation and establishment. In 1999, Luxembourg joined the eurozone . Thereafter, the country was elected non-permanent member of the United Nations Security Council (2013–14). The steel industry exploiting

3212-464: The First World War, under the German occupation of Luxembourg during World War II , the country was treated as German territory and informally annexed to the adjacent province of Nazi Germany , Gau Moselland . This time, Luxembourg did not remain neutral as Luxembourg's government in exile based in London supported the Allies , sending a small group of volunteers who participated in the Normandy invasion , and multiple resistance groups formed inside

3285-433: The French army. Their sacrifices have been commemorated at the Gëlle Fra . After the war, Grand-Duchess Marie-Adélaïde was seen by many people (including the French and Belgian governments) as having collaborated with the Germans and calls for her abdication and the establishment of a Republic became louder. After the retreat of the German army , communists in Luxembourg City and Esch-sur-Alzette tried to establish

3358-402: The Holy Roman Empire and elsewhere, Wenceslaus, annexed the County of Chiny in 1364, and with it, the territories of the new Duchy of Luxembourg reached its greatest extent. During these 130 years, the House of Luxembourg was contending with the House of Habsburg for supremacy within the Holy Roman Empire and Central Europe. It all came to end in 1443, when the House of Luxembourg suffered

3431-493: The Romans invaded the country when Julius Caesar conquered Gaul and part of Germania up to the Rhine border, thus the area of what is now Luxembourg became part of the Roman Empire for the next 450 years, living in relative peace under the Pax Romana . Similar to those in Gaul, the Celts of Luxembourg adopted Roman culture, language, morals and a way of life, effectively becoming what historians later described as Gallo-Roman civilization . Evidence from that period includes

3504-450: The city of Luxembourg , and Diekirch ), two district tribunals (Luxembourg and Diekirch), and a Superior Court of Justice (Luxembourg), which includes the Court of Appeal and the Court of Cassation. There is also an Administrative Tribunal and an Administrative Court, as well as a Constitutional Court, all of which are located in the capital. Luxembourg is divided into 12 cantons , which are further divided into 100 communes . Twelve of

3577-425: The city's nickname the Gibraltar of the North . Luxembourg was annexed by France, becoming the département des forêts (department of forests), and the incorporation of the former Duchy as a département into France was formalised at the Treaty of Campo Formio in 1797. From the start of the occupation the new French officials in Luxembourg, who spoke only French, implemented many republican reforms, among them

3650-404: The communes have city status ; the city of Luxembourg is the largest. Luxembourg has long been a prominent supporter of European political and economic integration . In 1921, Luxembourg and Belgium formed the Belgium–Luxembourg Economic Union (BLEU) to create a regime of inter-exchangeable currency and a common customs . Luxembourg is a member of the Benelux Economic Union and was one of

3723-402: The competing County of Vianden (and with it St Vith and Schleiden ) had either been incorporated directly or become vassal states to the County of Luxembourg. The only major setback during their rise in power came in 1288, when Henry VI and his three brothers died at the Battle of Worringen while trying unsuccessfully to add the Duchy of Limburg to their realm. But despite the defeat,

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3796-416: The country (but also the then Luxembourgish-speaking part of Arelerland ) was transferred to the new state of Belgium, thereby giving Luxembourg its modern-day borders. The treaty of 1839 also established full independence of the remaining Germanic-speaking Grand-Duchy of Luxembourg. In 1842, Luxembourg joined the German Customs Union ( Zollverein ). This resulted in the opening of the German market,

3869-451: The country. With an area of 2,586 square kilometers (998 sq mi), Luxembourg is Europe's seventh-smallest country . In 2024, it had a population of 672,050, which makes it one of the least-populated countries in Europe , albeit with the highest population growth rate ; foreigners account for nearly half the population. Luxembourg is a representative democracy headed by a constitutional monarch , Grand Duke Henri , making it

3942-406: The defeat of Napoleon , it regained independence in 1867 after the Luxembourg Crisis . Luxembourg is a developed country with an advanced economy , and has one of the world's highest GDP (PPP) per capita as per IMF and World Bank estimates. The nation's levels of human development and LGBT equality are ranked among the highest in Europe. The historic city including its fortification

4015-407: The development of Luxembourg's steel industry , and expansion of Luxembourg's railway network from 1855 to 1875. After the Luxembourg Crisis of 1866 nearly led to war between Prussia and France, as both were unwilling to see the other taking influence over Luxembourg and its mighty fortress, the Grand Duchy's independence and neutrality were reaffirmed by the Second Treaty of London and Prussia

4088-437: The dynasty of the House of Luxembourg not only began to rule the Holy Roman Empire , but rapidly began to exercise growing influence over other parts of Central Europe as well. Henry's son, John the Blind , in addition to being Count of Luxembourg, also became King of Bohemia . He remains a major figure in Luxembourgish history and folklore and is considered by many historians the epitome of chivalry in medieval times. He

4161-425: The establishment of a republic. During this time, Belgium pushed for an annexation of Luxembourg. However, all such claims were ultimately dismissed at the Paris Peace Conference , thus securing Luxembourg's independence. In 1940, after the outbreak of World War II , Luxembourg's neutrality was violated again when Nazi Germany 's Wehrmacht entered the country , "entirely without justification". In contrast to

4234-444: The fortifications around and on the heights of the city, fortification walls that are still visible today. During the War of the First Coalition , Revolutionary France invaded the Austrian Netherlands, and with it, Luxembourg. In the years 1793 and 1794 most of the Duchy was conquered relatively quickly and the French Revolutionary Army committed many atrocities and pillages against the Luxembourgish civilian population and abbeys,

4307-416: The founding members of the European Economic Community (now the European Union). It also participates in the Schengen Group (named after the Luxembourg village of Schengen where the agreements were signed). At the same time, the majority of Luxembourgers have consistently believed that European unity makes sense only in the context of a dynamic transatlantic relationship, and thus have traditionally pursued

4380-432: The independent County of Luxembourg (which was still a state within the Holy Roman Empire). By the middle of the 13th century, the counts of Luxembourg had managed to gain considerable wealth and power and had expanded their territory from the river Meuse to the Moselle . By the time of the reign of Henry V the Blonde , Bitburg , La Roche-en-Ardenne , Durbuy , Arlon , Thionville , Marville , Longwy , and in 1264

4453-409: The long-lasting Golden Bull of 1356 , a decree which fixed important aspects of the constitutional structure of the Empire. Luxembourg remained an independent fief (county) of the Holy Roman Empire, and in 1354, Charles IV elevated it to the status of a duchy with his half-brother Wenceslaus I becoming the first Duke of Luxembourg . While his kin were occupied ruling and expanding their power within

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4526-399: The model for the 1D case. In one dimension, the solution admits no phase transition . On the basis of this result, he incorrectly concluded that his model does not exhibit phase transition in any dimension. It was only in 1949 that Ising knew the importance his model attained in scientific literature, 25 years after his Ph.D. thesis. Today, each year, about 800 papers are published that use

4599-420: The model to address problems in such diverse fields as neural networks, protein folding, biological membranes and social behavior. The Ising model had significance as a historical step towards recurrent neural networks . Glauber in 1963 studied the Ising model evolving in time, as a process towards equilibrium ( Glauber dynamics ), adding in the component of time. Shun'ichi Amari in 1972 proposed to modify

4672-410: The most infamous being the massacres of Differdange and Dudelange , as well as the destruction of the abbeys of Clairefontaine , Echternach and Orval . However the Fortress of Luxembourg resisted for nearly 7 months before the Austrian forces holding it surrendered. Luxembourg's long defense led Lazare Carnot to call Luxembourg "the best fortress in the world, except Gibraltar", giving rise to

4745-417: The occupied country. With 2.45% of its prewar population killed, and a third of all buildings in Luxembourg being destroyed or heavily damaged (mainly due to the Battle of the Bulge ), Luxembourg suffered the highest such loss in Western Europe, but its commitment to the Allied war effort was never questioned. Around 1,000–2,500 of Luxembourg's Jews were murdered in the Holocaust . The Grand Duchy became

4818-475: The principle of laicism , which led to an outcry in strongly Catholic Luxembourg. Additionally French was implemented as the only official language and Luxembourgish people were barred access to all civil services. When the French Army introduced military duty for the local population, riots broke out which culminated in 1798 when Luxembourgish peasants started a rebellion. Even though the French managed to rapidly suppress this revolt called Klëppelkrich , it had

4891-518: The product of spins is either +1 if the two spins are the same ( aligned ), or −1 if they are different ( anti-aligned ). J is half the difference in energy between the two possibilities. Magnetic interactions seek to align spins relative to one another. Spins become randomized when thermal energy is greater than the strength of the interaction. For each pair, if A ferromagnetic interaction tends to align spins, and an antiferromagnetic tends to antialign them. The spins can be thought of as living on

4964-413: The territory had been part of the Holy Roman Empire as well as the Habsburgian Netherlands in the past, both the Kingdom of Prussia and the United Kingdom of the Netherlands now claimed possession of the territory. At the Congress of Vienna the great powers decided that Luxembourg would become a member state of the newly formed German Confederation , but at the same time William I of the Netherlands ,

5037-593: The weights of an Ising model by Hebbian learning rule as a model of associative memory, adding in the component of learning. This was popularized as the Hopfield network (1982). Luxembourg – in Europe  (green & dark gray) – in the European Union  (green) Luxembourg ( / ˈ l ʌ k s əm b ɜːr ɡ / LUK -səm-burg ; Luxembourgish : Lëtzebuerg [ˈlətsəbuəɕ] ; German: Luxemburg [ˈlʊksm̩bʊʁk] ; French: Luxembourg [lyksɑ̃buʁ] ), officially

5110-469: The world's only remaining sovereign grand duchy . The County of Luxembourg was established in the 11th century, as a state within the Holy Roman Empire . Its ascension culminated in its monarch, Henry VII , becoming the Holy Roman Emperor in the 14th century. Luxembourg came under Habsburg rule in the 15th century, and was annexed by France in the 18th century. Luxembourg was partitioned three times, reducing its size. Having been restored in 1815 after

5183-445: Was allowed to maintain much of its independence and political mechanisms. Unaware of the fact that Germany secretly planned to annex the Grand-Duchy in case of a German victory (the Septemberprogramm ), the Luxembourgish government continued to pursue a policy of strict neutrality. However, the Luxembourgish population did not believe Germany had good intentions, fearing that it would annex Luxembourg. Around 1,000 Luxembourgers served in

5256-599: Was declared a UNESCO World Heritage Site in 1994 due to the exceptional preservation of its vast fortifications and historic quarters. Luxembourg is a founding member of the European Union , OECD , the United Nations , NATO , and the Benelux . It served on the United Nations Security Council for the first time in 2013 and 2014. The first traces of settlement in what is now Luxembourg are dated back to

5329-539: Was finally willing to withdraw its troops from the Fortress of Luxembourg under the condition that the fortifications would be dismantled. That happened the same year. At the time of the Franco-Prussian war in 1870, Luxembourg's neutrality was respected, and neither France nor Germany invaded the country. As a result of the recurring disputes between the major European powers, the people of Luxembourg gradually developed

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