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Schwyz

Schwyz (Schwytz; Svitto) is a town and the capital of the canton of Schwyz in Switzerland.

The Federal Charter of 1291 or Bundesbrief, the charter that eventually led to the foundation of Switzerland, can be seen at the Bundesbriefmuseum.

The official language of Schwyz is (the Swiss variety of) German, but the main spoken language is the local variant of the Alemannic Swiss German dialect.

Name
The earliest certain record of the name dates to 972, recorded in Medieval Latin as villa Suittes. There are a number of uncertain records dated between 924 and 960, in the form Swites (Suuites) and Switz. The name is recorded as Schwitz in the 13th century, and in the 17th to 18th century often as Schweitz. The name's etymology is uncertain. It was long presented as derived from the name of an eponymous founder in Swiss legend, one Suito or Switer, an explanation found in Swiss school textbooks until the first half of the 20th century. There is currently no consensus on the name's derivation. Isaac Wake, diplomat of King James VI and I in Bern, suggested in 1625 that the name originated in Sweden, among the Suecia, "who in the time of king [sic] Sigebert made a transmigration out of Suecia and planted themselves in this country". A Germanic etymology was suggested by Gatschet (1867), deriving the name from an Old High German verb suedan "to burn" (referring to slash-and-burn clearing of woodland for habitation). Brandstetter (1871) is critical of Gatschet's suggestion and prefers derivation from an Alemannic personal name in Svid- as it were presenting a scholarly defense of the Suito of the founding legend. The etymology proposed for the Schweizerisches Idiotikon by Hubschmied (1929) derives the name from a Gallo-Roman *(alpes) suētas, from the Gaulish or Latin word for "pig", via a Romance *suēdes "(mountain, pasture) of pigs" yielding an Alemannic Swītes. Hubschmied distanced himself from this opinion in 1961, preferring an unspecified pre-Roman (or "Etruscan") source. Sonderegger (1966) revisits Gatschet's suedan "slash-and-burn" proposal, but now claims derivation from a cognate Celtic root, sveit-, Proto-Celtic sveitos with a meaning of "clearing" or similar, giving Gaulish *Svētos (the long vowel as in Rēnos "Rhine"), Gallo-Romance *Svēdus, -is, and finally Swītes in Old High German by the 8th century.

The name Schwyz was extended to the area dominated by Schwyz (the Canton of Schwyz), and later to the entire Old Swiss Confederacy. Other cantons tended to resent this in the 15th century, but after 1499 the term Schwyzer was widely self-adopted, out of spite so to speak, since it had been employed as a term of abuse by the Swabian side during the Swabian War. Eidgenossenschaft and Schwytzerland (the origin of the English name Switzerland) could be used interchangeably as country names in the 16th century.

The Swiss German pronunciation is the same for the name of the town and that of the country (the two are distinguished only by use of the definite article for the latter,  "Schwyz",  "Switzerland"). The spelling of y for [iː] originates from the ligature ij in 15th-century handwriting.

History
While a few Roman era coins have been found in Schwyz, the earliest evidence of a settlement comes from the 8th century. The Alamanni cemetery at the parish church and the church itself are both from the first half of the 8th century. This first church was followed by a second ottonian church around 1000, which may have been destroyed by the 1117 Verona earthquake. In 1121 the third church building, a romanesque building, was consecrated. This was followed in the 15th century by the much larger fourth church which was destroyed, along with much of the village, by fire in 1642. The fifth church, an early baroque church was replaced because of serious structural defects by the current late baroque church which was dedicated in 1774.

Because Schwyz was the capital of a canton, many of the government organizations administered both the town and the canton at the same time, and the history of the town is closely tied to the history of the canton.

According to the chronicle of Johann Stumpf from 1548, the old town originally consisted of a village square, the church and its cemetery, the town hall, the inn, the archive tower and a number of scattered wooden houses. Around 1500, to distinguish it from the Canton of Schwyz, Schwyz town was often called Kilchgassen, which meant the village around the church but not the surrounding villages. The fire of 1642, which destroyed 47 buildings in the center of the village, allowed the town to be totally rebuilt. A new, larger town square with major roads radiating out was built in front of the new church and the new city hall. The houses were rebuilt as urban townhouses and a ring of about 30 large patrician farm houses grew up surrounding the village center.

Geography
Besides the town of Schwyz, the municipality includes the settlements of Ibach, Seewen and Rickenbach. To the east, the municipality includes, or borders on, the mountains of Hochstuckli, Kleiner Mythen, Grosser Mythen, Rotenflue, and Furggelenstock. The river Muota flows out of these mountains and through the municipality on its way to Lake Lucerne. The Haggenegg Pass and Holzegg Pass both cross to Alpthal, whilst the Ibergeregg Pass crosses to Oberiberg.

Schwyz has an area,, of 53.2 km2. Of this area, 46.4% is used for agricultural purposes, while 39.1% is forested. Of the rest of the land, 8.7% is settled (buildings or roads) and the remainder (5.8%) is non-productive (rivers, glaciers or mountains).

Coat of arms
The blazon of the municipal coat of arms is Gules, a Confederate cross couped in the hoist argent.

Demographics
Schwyz had a population of. , 15.6% of the population were resident foreign nationals. Over the year 2010–2011 the population reduced by 0.6%. Migration accounted for −0.9%, while births and deaths accounted for 0.0%. Most of the population speaks German (12,441 or 90.1%) as their first language, Serbo-Croatian is the second most common (378 or 2.7%) and Italian is the third (273 or 2.0%). There are 23 people who speak French and 25 people who speak Romansh.

, the population was 49.9% male and 50.1% female. The population was made up of 5,824 Swiss men (42.2% of the population), 1,058 (7.7%) non-Swiss men, 5,932 Swiss women (43.0%) and 988 (7.2%) non-Swiss women. Of the population in the municipality, 6,681 or about 48.4% were born in Schwyz and lived there in 2000. There were 2,195 or 15.9% who were born in the same canton, while 2,780 or 20.1% were born somewhere else in Switzerland, and 1,797 or 13.0% were born outside of Switzerland.

, children and teenagers (0–19 years old) make up 25.6% of the population, while adults (20–64 years old) make up 60.3% and seniors (over 64 years old) make up 14.1%.

, there were 6,314 people who were single and never married in the municipality. There were 6,305 married individuals, 722 widows or widowers and 461 individuals who are divorced.

, there were 5,250 private households in the municipality, and an average of 2.5 persons per household. There were 1,582 households that consist of only one person and 536 households with five or more people. , a total of 4,968 apartments (90.3% of the total) were permanently occupied, while 375 apartments (6.8%) were seasonally occupied and 156 apartments (2.8%) were empty. , the construction rate of new housing units was 4.8 new units per 1000 residents.

the average price to rent an average apartment in Schwyz was 1185.58 Swiss francs (CHF) per month (US$950, £530, €760 approx. exchange rate from 2003). The average rate for a one-room apartment was 543.08 CHF (US$430, £240, €350), a two-room apartment was about 904.87 CHF (US$720, £410, €580), a three-room apartment was about 1068.78 CHF (US$860, £480, €680) and a six or more room apartment cost an average of 1461.34 CHF (US$1170, £660, €940). The average apartment price in Schwyz was 106.2% of the national average of 1116 CHF. The vacancy rate for the municipality,, was 0.25%.

Historic population
The historical population is given in the following chart:

Heritage sites of national significance
The Bundesbriefmuseum (Museum of the Swiss Charters of Confederation), the Dominican nuns Convent of St. Peter am Bach, the entire medieval and early modern settlement, the Hermitage and chapel, the Forum der Schweizer Geschichte (Forum of Swiss History), the Ab Yberg im Grund House, the Bethlehem House at Reichsstrasse 9, the Ceberg im Feldli house at Theodosiusweg 20, the house at Gotthardstrasse 99 in Ibach, the Grosshus at Strehlgasse 12, the Immenfeld house, the house at Langfeldweg 14 in Kaltbach, the house at Oberschönenbuch 79 in Ibach, the Herrenhaus Waldegg, the Hettlingerhäuser, the Hofstatt Ital Reding, the Catholic Parish Church of St. Martin, the Maihof, the Palais Büeler, the Rathaus (Town council house), the Reding House, the State Archives of Schwyz and the Köplihaus house are listed as a Swiss heritage site of national significance. The entire old city of Schwyz is part of the Inventory of Swiss Heritage Sites.

Politics
In the 2007 federal election the most popular party was the SVP which received 39.91% of the vote. The next three most popular parties were the CVP (26.12%), the SPS (17.05%) and the FDP (12.72%). In the federal election, a total of 5,554 votes were cast, and the voter turnout was 57.8%.

Economy
, Schwyz had an unemployment rate of 1.8%. , there were 484 people employed in the primary economic sector and about 174 businesses involved in this sector. 2,756 people were employed in the secondary sector and there were 179 businesses in this sector. 7,099 people were employed in the tertiary sector, with 696 businesses in this sector.

the total number of full-time equivalent jobs was 8,570. The number of jobs in the primary sector was 303, of which 273 were in agriculture and 30 were in forestry or lumber production. The number of jobs in the secondary sector was 2,647 of which 1,589 or (60.0%) were in manufacturing, 8 or (0.3%) were in mining and 928 (35.1%) were in construction. The number of jobs in the tertiary sector was 5,620. In the tertiary sector; 1,357 or 24.1% were in wholesale or retail sales or the repair of motor vehicles, 306 or 5.4% were in the movement and storage of goods, 272 or 4.8% were in a hotel or restaurant, 136 or 2.4% were in the information industry, 733 or 13.0% were the insurance or financial industry, 427 or 7.6% were technical professionals or scientists, 260 or 4.6% were in education and 1,053 or 18.7% were in health care.

, there were 4,484 workers who commuted into the municipality and 2,168 workers who commuted away. The municipality is a net importer of workers, with about 2.1 workers entering the municipality for every one leaving. Of the working population, 13.6% used public transportation to get to work, and 45.3% used a private car.

Religion
From the, 11,269 or 81.6% were Roman Catholic, while 675 or 4.9% belonged to the Swiss Reformed Church. Of the rest of the population, there were 423 members of an Orthodox church (or about 3.06% of the population), there were 7 individuals (or about 0.05% of the population) who belonged to the Christian Catholic Church, and there were 155 individuals (or about 1.12% of the population) who belonged to another Christian church. There were 5 individuals (or about 0.04% of the population) who were Jewish, and 502 (or about 3.64% of the population) who were Islamic. There were 42 individuals who were Buddhist, 31 individuals who were Hindu and 7 individuals who belonged to another church. 377 (or about 2.73% of the population) belonged to no church, are agnostic or atheist, and 385 individuals (or about 2.79% of the population) did not answer the question.

Education
In Schwyz about 4,873 or (35.3%) of the population have completed non-mandatory upper secondary education, and 1,473 or (10.7%) have completed additional higher education (either university or a Fachhochschule). Of the 1,473 who completed tertiary schooling, 71.1% were Swiss men, 19.4% were Swiss women, 5.3% were non-Swiss men and 4.1% were non-Swiss women. , there were 419 students in Schwyz who came from another municipality, while 186 residents attended schools outside the municipality.

Schwyz is home to the Kantonsbibliothek Schwyz library. The library has 108,142 books or other media, and loaned out 136,064 items in the same year. It was open a total of 276 days with average of 29 hours per week during that year.

A major school in Schwyz is the Kantonsschule Kollegium Schwyz (KKS), an upper Secondary school that is a Gymnasium and a vocational or technical college. The KKS has operated for over 150 years, though it builds on several older schools. The first Latin school in Schwyz opened in 1627 in the former Capuchin monastery of St. Josef im Loo. This school remained open until the 1798 French invasion. On 25 July 1841, the Jesuits laid the cornerstone of what would become the Jesuit College on the site of the modern Kollegium. The school opened in 1844 but only remained under Jesuit control for three years. In 1847, Federal troops marched into Schwyz to suppress the Catholic Sonderbund and forced the Jesuits to flee. It was reopened in 1855 under the Capuchin Father Theodosius Florentini and in the following year began teaching students. The school continued to teach students using both religious and secular teachers until the 1970s. In 1972, the lower Secondary students moved to Pfäffikon and the school became an upper Secondary Kantonsschule.

Weather
Schwyz has an average of 149.2 days of rain per year and on average receives 1629 mm of precipitation. The wettest month is August during which time Schwyz receives an average of 199 mm of precipitation. During this month there is precipitation for an average of 13.9 days. The month with the most days of precipitation is June, with an average of 14.8, but with only 182 mm of precipitation. The driest month of the year is February with an average of 99 mm of precipitation over 13.9 days.

Transport
The A4 motorway, between Zürich and the Brunnen passes through the west of the municipality, and the town is linked to it by main roads and motorway junctions. Other main roads connect the town to Lucerne (along both banks of Lake Lucerne), to the Gotthard Pass and southern Switzerland, and to Pfaffikon and Einsiedeln in the north of the canton of Schwyz. A minor road crosses the Ibergeregg Pass to Oberiberg, providing an alternative route to Einsiedeln. Other minor roads reach to near the summits of both the Haggenegg Pass and Holzegg Pass, but only hiking trails actually cross these passes and continue to Alpthal.

Schwyz railway station, on the Gotthard railway, is located about 2 km outside the town, in the parish of Seewen. The station is served by InterRegio and S-Bahn trains.

Early plans for the Schweizerische Südostbahn included a proposal for what is now the Pfäffikon SZ–Arth-Goldau railway to terminate at Brunnen railway station instead of Arth-Goldau railway station. If that proposal had come to fruition, the Schwyz town centre would have had a railway station – initially on the Kollegi football field, and later in Steisteg.

From 6 October 1900 to 14 December 1963, the Schwyzer Strassenbahnen linked the Schwyz railway station with the town centre. On 8 May 1915, the additional section between Schwyz and Brunnen See was opened – and on the same day, the Schwyz SBB–Schwyz line was shut down. The trams were eventually replaced by the Auto AG Schwyz, which today operates 12 bus lines in the cantons of Lucerne and Schwyz.

The Rotenfluebahn, a gondola lift, links Rickenbach with the summit of the Rotenfluh mountain, which is, in summer, a popular vantage point over the Lake Lucerne region, and, in winter, a ski area.

Notable people

 * Theodor von Reding (1755 in Schwyz – 1809) a Spanish general, led Swiss and Spanish troops against Napoleon
 * Alois von Reding (1765 in Schwyz – 1818) a Swiss patriot, military officer and politician, brother of Theodor
 * Carl Elsener Sr. (1922 – 2013 in Schwyz) a Swiss entrepreneur with the Swiss Army knife
 * Xavier Koller (born 1944 in Schwyz) a Swiss film director and screenwriter
 * Gertrud Leutenegger (born 1948 in Schwyz) a Swiss poet, novelist, playwright and theatre director
 * Monika Kaelin (born 1954 in Schwyz) a model and actress

 * Sport
 * Oscar Camenzind (born 1971 in Schwyz) a former professional road racing cyclist
 * Luca Schuler (born 1998 in Schwyz) a Swiss freestyle skier, competed at the 2014 Winter Olympics

title: Schwyz


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Canton of Bern

The canton of Bern, or Berne (Kanton Bern; canton de Berne; Chantun Berna; Canton Berna), is one of the 26 cantons forming the Swiss Confederation. Its capital city, Bern, is also the de facto capital of Switzerland. The bear is the heraldic symbol of the canton, displayed on a red-yellow background.

Comprising ten districts, Bern is the second-largest canton by both surface area and population. Located in west-central Switzerland, it is surrounded by eleven cantons. It borders the cantons of Jura and Solothurn to the north. To the west lie the cantons of Neuchâtel, Fribourg, and Vaud. To the south lies the cantons of Valais. East of the canton of Bern lie the cantons of Uri, Nidwalden, Obwalden, Lucerne and Aargau. The geography of the canton includes a large share of all three natural regions of Switzerland: the Jura Mountains (the Bernese Jura), the Swiss Plateau (the Bernese Mittelland) and the Alps (the Bernese Oberland).

The canton of Bern is bilingual, officially German- and French-speaking, and has a population (as of ) of. The largest city, Bern, is also the seat of the federal government of Switzerland. Other major cities are Thun and Biel/Bienne. The canton is also renowned for its numerous Alpine resort towns, notably Interlaken and Gstaad.

Formerly part of the Holy Roman Empire, the canton of Bern entered an alliance with the Swiss Forest Cantons in 1323 and joined the Old Swiss Confederacy in 1353.

History
Bern joined the Old Swiss Confederation in 1353. Between 1803 and 1814 it was one of the six directorial cantons of the Napoleonic Swiss Confederation.

Early prehistory
The earliest traces of a human presence in the area of the modern Canton is found in three caves in the Simmental region; Schnurenloch near Oberwil, Ranggiloch above Boltigen and Chilchlihöhle above Erlenbach. These caves were used at various times during the last ice age. The first open-air settlement in the area is an upper paleolithic settlement at Moosbühl in Moosseedorf. During the warmer climate of the mesolithic period, increasing forest cover restricted the movement of hunters, fishers and gatherers. Their temporary settlements were built along lake and marsh edges, which remained free of trees due to fluctuations in water level. Important mesolithic sites in the Canton are at Pieterlenmoos and Burgäschisee lake along with alpine valleys at Diemtig and Simmental. During the neolithic period, there were a number of settlements on the shores of Lake Biel, the Toteisbecken (Lobsigensee, Moossee, Burgäschisee and Inkwilersee) and along rivers (Aare, Zihl). Several of these sites are part of the Prehistoric Pile dwellings around the Alps, a UNESCO World Heritage Site.

One of the best explored neolithic sites is at Twann (now Twann-Tüscherz). In the Twannbach delta there were about 25 Cortaillod culture and Horgen culture villages that existed between 3800 and 2950 BC. One of the oldest examples of bread from Switzerland, a sourdough from 3560 to 3530 BC, came from one of these villages. A preserved Bronze Age dugout boat has been found on the dried out lake bed, it was part of lake village near Twann, a stone-lined well had been dug near the preserved boat in 1874.

Iron Age
During the Early Iron Age changes in climate forced the Hallstatt culture (800–450 BCE) to abandon settlements along many waterways and in the valley floors and move to the plateaus and hills. With increased trade contacts across the Alps, the cultural influence of the Mediterranean region grew in the area. Evidence of this trade include a hydria which was discovered in Grächwil. Burial rituals and social classes became more developed during this time. The so-called princely graves became more common, many of the burial mounds were over 30 m in diameter and 4 m high and richly outfitted with grave goods. In a grave mound in Bützberg the first burial in the mound was followed by several later burials. Often, several grave mounds combined to become a necropolis, such as at Grossaffoltern, Ins, Bannwil, Langenthal and Bützberg. Most of the knowledge about the Hallstatt culture in the Canton comes from graves. The only discovered settlement is around Blanche Church in La Neuveville.

The grave goods show that iron was forged into swords, daggers, spearheads, knives and wagon accessories. Gold, which was probably collected from river sand, was made into diadems, rings and pendants. Thin bronze arm, leg and neck plates with geometric designs were often buried, especially in the graves at Allenlüften in Mühleberg, at Ins and at Bützberg. The jewelry that was buried included bracelets and rings which were also made of jet and lignite coal. At Münchringen, the grave pottery was both shaped by hand or thrown on a potter's wheel, and was painted with multi-colored ornamentation.

The transition to the Late Iron Age of the La Tène culture (450−1st century BCE) is indicated by a sudden change of style in the metalworking and ceramic industries. Numerous graves (from unknown settlements), along with the two oppida at Bern-Engehalbinsel and Jensberg by Studen, mark the population centers during the late Iron Age. Gold coins (from Melchnau) along with silver and bronze coins first start to appear during this era. A sword with Greek characters that said Korisios was found at the Port site. At the oppidum at Bern-Engehalbinsel, there were studios for glass and ceramic production, and iron working achieved a high level of skill, along with craftsmen who worked in wood, leather and goldsmithing. There was a nearby place of worship in the Bremgarten wood, and cemeteries at Münsingen and Bern-Engehalbinsel.

Roman era

After the Roman era victory at Battle of Bibracte in 58 BCE, the Helvetii were forced to return to their homes as foederati of the Romans. Under increasing Roman influence, the local economy and trade flourished. The main settlements lay at the foot of the Jura Mountains and on the Central Plateau.

The existing roads were expanded, especially the Aventicum-Vindonissa and the Petinesca-Augusta Raurica roads. A fourth alpine pass, the Rawil pass, was added to the traditional three; the Grimsel, Brünig and Susten passes. In the Bernese Jura the Mont Raimeux and Pierre Pertuis passes opened.

Under the Romans, many of the old fortified places were expanded and refortified. The old Helvetii oppidum at the Engehalbinsel became a Roman vicus, which was probably known as Brenodor or Brenodurum. At the foot of the Jens mountain, the fort Petinesca was built to guard the roads over the Jura Mountains. This was reinforced in the late-Roman era (368–369 CE) by a fortified bridge over the Thielle/Zihl river between Aegerten and Brügg.

A number of Roman villas were built around the Canton. At Oberwichtrach both the main building (pars urbana) and the industrial section (pars rustica) of a Roman villa have been discovered. The villas at Münsingen, Toffen and Herzogenbuchsee have richly detailed Roman mosaics that are still partly visible.

The religious practices of the local population merged with Roman beliefs and the Canton of Bern is home to a number of unique cult centers. They include the larger than life statues of gods (including enthroned Jupiter) at Petinesca, Engehalbinsel, and Thun-Allmendingen; and a number of stone inscriptions. At the beginning of the 5th century CE, Rome withdrew its troops from the Rhine garrisons, but allowed the Burgundians to settle on Helvetii lands in 443 CE.

Early Middle Ages
East of the Aare, the transition from the Gallo-Roman dominated population to a Germanic population in the Early Middle Ages happened relatively peacefully. One exception was the Battle of Wangen in 610, but elsewhere it generally was a slow process of cultural infiltration. By the 7th century, the Alamannic settlers had already taken most of the good locations southeast of the Aare and they began moving up the Aare to the regions of Lake Thun and Lake Brienz. All areas west of the Aare belonged to the romanised Burgundian kingdom, which became part of the Merovingian Frankish kingdom in the 534. During the Carolingian era, the political structure of the Franks had spread into parts of what became Switzerland. In 762/778 the County of Aargau was founded, followed in 861 by the County of Oberaargau and in 965 the County of Bargen. The Treaty of Verdun in 843 put the border between Central and Eastern Frankish Empires at the Aare, and divided the Aare region in half. The population west of the Aare generally spoke a Romance language, while those to the east spoke a Germanic language. The region between the rivers Saane and Aare became the language border.

Christianity spread slowly into the Aare valley. The dioceses of Lausanne, Basel, Sion, Chur and Constance were all established before the new faith made inroads into the valley. The Aare valley was bordered by three dioceses; Lausanne, Constance and Basel. However, the first Christian missionaries came into the valley from Alsace and other western areas. In 630 the Abbey of Luxeuil established the Abbey of Moutier-Grandval along the old transit route through the Pierre Pertuis Pass. By the 9th or 10th century, this abbey had property and influence all the way to Lake Biel and into the Balsthal valley. The first monastic cells of what would become the Abbey of Saint-Imier was also founded in 600. In the Seeland and Aare valleys, wooden churches were first built during the Merovingian period. The current churches in Kirchlindach, Oberwil bei Buren and Bleibach were all built above the ruins of these early churches. About 30 churches in the Bern and Solothurn portions of the Aare valley were created over ruins of Roman villas and subsequent burial grounds in the 7th century (including Meikirch and Oberbipp). In Mett, the church was built over a 5th-century mausoleum, which was built over a 4th-century tomb. In 700, six sarcophagi were buried on St. Peter's Island next to a Roman temple complex. A wooden monastery was built over the complex in the 8th–9th century.

Middle Ages

During the 10th century, the Aare valley came fully under the Second Kingdom of Burgundy. Under King Rudolph I and his son Rudolph II Burgundy's influence reached across most of modern Switzerland. They established royal courts at Bümpliz, Münsingen, Uetendorf, Wimmis, Kirchberg and Utzenstorf to allow them to govern the Aare valley. Later, the Aare valley moved toward closer ties with the Holy Roman Empire during the Ottonian and Salian dynasties.

The succession dispute following the death of Rudolph III in 1032 allowed the Salian kings to acquire the Kingdom of Upper Burgundy and with it the Aare valley. As a part of the Holy Roman Empire, the valley was involved when the Investiture Controversy broke out in 1056. In 1077, the Regent of Burgundy, Count Rudolf of Rheinfelden, declared himself as an anti-king against King Henry IV. The Bishops of Basel and Lausanne remained loyal to King Henry IV, and took the Rheinfelden lands in Oberaargau and the upper Aare valley. After Rudolf's death in 1090, his lands went to his son, Berchtold II of Zähringen. He and his son, Berchtold III, tried to use these lands to expand their power. In 1127 the Zähringer were appointed Rector or delegate of the king in Burgundy. Their hopes for a new, independent Burgundy were dashed in 1156, and the last Zähringen count, Berchtold V, embarked on a program of city founding. The cities of Burgdorf, Murten, Thun and Bern were all founded by Berchtold V. When he died without an heir, the Zähringen lands went to the House of Kyburg, while the offices and fiefs reverted to the empire.

During the High Middle Ages both the Aare valley and the Bernese Oberland were divided into a number of small counties, each with their own baron. Nobles from the Holy Roman Empire began to marry into the local noble families and a number of the Zähringen Ministerialis families (unfree knights in the service of a feudal overlord) moved into the Oberland. The nobles also began to found monasteries to spread their power into the Oberland. During the period between 1070 and 1150, at least ten large monasteries were founded by local nobles. These include; the Cluniac Priory of Rüeggisberg (Lords of Rümligen in 1072), Münchenwiler (Wiler family in 1080), St. Peter's Island (Count of Hochburgund-Mâcon in the late 11th century), Hettiswil (1107), Röthenbach im Emmental (Lords of Rümligen or Signau), the Benedictine monastery at St. Johannsen in Erlach (Fenis family in 1100), Trub (Lords of Lützelflüh before 1130) and Rüegsau (possibly also the Lords of Lützelflüh in first half of the 12th century), the Augustinian Collegiate church in Interlaken (Oberhofen family in 1130) and the Cistercian Frienisberg Abbey (Count Saugern around 1130). However, in 1191 the Oberland barons revolted against Berchtold V of Zähringen and many of the Oberland barons were killed in the battle of Grindelwald.

During the 13th century a number of the cities near Bern were granted the city right and appointed mayors and city councils. Bern became an imperial city. During the mid-13th century, the Empire's presence weakened in the Aare valley, forcing the local nobles to find allies to protect themselves. Bern entered into a number of treaties with its neighbors in the 13th century. In 1274, the Emperor Rudolph I of Habsburg, confirmed Bern's imperial immediacy. However, in 1285 he imposed an imperial tax which drove the city to support Rudolph's enemies. Although it withstood two sieges by the Emperor in 1288, after the defeat at Schosshalde in 1289 it had to pay taxes and a penalty.

In 1298 Bernese forces won a victory at Oberwangen in Köniz against the County of Savoy and the Habsburg Austrian nobility. In 1300, the city acquired the four surrounding parishes of Bolligen, Vechigen, Stettlen and Muri, destroyed the threatening castles of Bremgarten and Belp and gave the Baron of Montenach Bernese citizenship. After the victory of Louis IV of Bavaria over the Habsburg Frederick the Fair in the battle of Mühldorf (Bavaria) in 1322, Bern entered an alliance with the anti-Habsburg Swiss Forest Cantons in 1323.

Old Swiss Confederacy

Bern joined the Old Swiss Confederation in 1353.

14th century
In 1310 Emperor Henry VII pledged Laupen Castle, Laupen and the surrounding lands as collateral for a loan. In 1324, Bern acquired the pledged castle and lands. When the Emperor was unable to repay the loan, Laupen became the first bailiwick of Bern.

In 1322, the brothers Eberhard II of Neu-Kyburg and Hartmann II of Neu-Kyburg started fighting with each other over who would inherit the family's lands around Thun. The fighting led to the "fratricide at Thun Castle" where Eberhard killed his brother Hartmann. To avoid punishment by his Habsburg overlords, Eberhard fled to Bern. In the following year, he sold the town of Thun, its castle and the land surrounding Thun to Bern. Bern then granted the land back to Eberhard as a fief. Bern's support of Eberhard, their resulting expansion into the Oberland and their alliance with the Forest Cantons brought the city into conflict with the Habsburgs during the 14th century. The Gümmenenkrieg in 1333 between Bern and Fribourg over rights and influence in the Sense/Singine area ended without resolving anything.

In 1334, Bern fought with the Barons of Weissenburg and occupied Wimmis and Unspunnen in the Oberland. Bern's victory allowed them to bring the Oberhasli region, its capital of Meiringen and Weissenburg under their control. Bern's continued expansion was at the expense of the feudal lords in the surrounding lands. In 1339 the Habsburgs, Kyburgs and Fribourg, marched against Bern with 17,000 men and besieged the border town of Laupen. To raise the siege, Bern raised a force of 6,000, consisting of Bernese, supported by the Forest Cantons, and other allies (Simmental, Weissenbur and Oberhasli). The allied Bernese forces were victorious at the Battle of Laupen and Bern drew closer to the Swiss Forest Cantons. It entered into a permanent or eternal alliance with Uri, Schwyz and Unterwalden in 1353. This date is usually considered the date of Bern's entrance into the Swiss Confederation; however, the alliance only indirectly tied Bern to Zürich and Lucerne.

In 1358 the cash-strapped Count Peter II of Aarberg pawned the County of Aarberg to Bern. However, in 1367 he sold it, without repaying Bern, to his cousin Rudolf IV of Nidau. After Rudolf's death (1375) Bern acquired clear right to the Aarberg lands from the other heirs.

Throughout the 14th century, Bern entered into alliances and treaties with many of its neighbors, including members of the Swiss Confederation. Bern entered into a series of agreements with Biel/Bienne, which led to a conflict with Biel's ruler, Jean de Vienne, the Prince-Bishop of Basel, in 1367–68. The bishop marched south and destroyed Biel along with a number of towns in the southern Jura Mountains. During the Gugler war in 1375, there were several battles between Bernese troops and English mercenaries under Enguerrand de Coucy.

By the 1370s, the Kyburgs (which still held Thun as a fief for Bern) were deeply in debt to Bern. On 11 November 1382, Rudolf II of Neu-Kyburg made an unsuccessful raid on Solothurn. The ensuing conflict with the Old Swiss Confederacy (known as the Burgdorferkrieg or Kyburgerkrieg) allowed Bern to move against the Habsburgs in Aargau. After the Bernese laid siege to Burgdorf, Neu-Kyburg was forced to concede an unfavourable peace. Bern bought Thun and Burgdorf, the most important cities of Neu-Kyburg, and their remaining towns passed to Bern and Solothurn by 1408. The last of the Neu-Kyburgs, Berchtold, died destitute in Bern in 1417.

In 1386, the Austrians under Leopold of Habsburg invaded eastern Switzerland. When they besieged the city of Sempach, troops from Zürich, Lucerne and the Forest Cantons marched out and defeated the Austrians at the Battle of Sempach. While Bern was not involved at Sempach they took advantage of the Austrian weakness to march into the Oberland in 1386, followed by the Seeland (the region south of the Jura Mountains containing the Morat (Murten), Neuchâtel and Bienne (Biel) lakes) in 1388 and the Aargau in 1389. In the peace agreement of 1389 Bern got Unterseen and the Upper Simme valley (Simmental) from the Habsburgs. In the same year, Fribourg accepted Berns acquisition of the Iselgaus, i.e. the area between Lake Biel and the Seeland. Over the next several decades Bern continued to expand to the detriment of the Habsburgs and Fribourg. They acquired the following towns: 1391 Simmenegg, 1399 Signau, 1400 Frutigen, 1407 Wangen, 1408 Trachselwald and Huttwil, 1412 Oltigen and in 1413 (together with Solothurn) Bipp and Bechburg.

15th century
By 1400, Bern controlled the entire Bernese Oberland. Under their control, the five valleys of the Oberland enjoyed extensive rights and far-reaching autonomy in the Bäuerten (farming cooperative municipalities) and Talverbänden (rural alpine communities). However, throughout the Late Middle Ages, the Oberland, as a whole or in part, revolted several times against Bernese authority. The Evil League (Böser Bund) in 1445 fought against Bernese military service and taxes following the Old Zürich War, in 1528 the Oberland rose up in resistance to the Protestant Reformation and in 1641 Thun revolted.

In the Bernese Oberland during the 13th and the beginning of the 14th century Interlaken Monastery grew to become the largest landholder in the region. The Monastery controlled the towns of Grindelwald, Lauterbrunnen and numerous farms along Lake Brienz. However, in 1350 a period of crises and conflicts led to a decline in the number of monks and nuns and increasing debt. In 1348, the people of Grindelwald and Wilderswil joined a mutual defense league with Unterwalden. Bern responded with a military expedition to the Bernese Oberland, which ended in defeat for Unterwalden and its allies. By 1472, Bern was the patron of the Monastery. During the Protestant Reformation, the Monastery was secularized in 1528. The canons received a financial settlement and the properties were now managed by a Bernese bailiff. The tenants of the Monastery who had expected the abolition of all owed debts, responded by rioting, which was suppressed by Bern.

The lands around the northern shore of Lake Geneva and between Lake Neuchâtel were the next area into which Bern expanded. The region had originally been part of the Carolingian Empire. Then, in 1032 the Zähringens of Germany defeated the Burgundians, who were then replaced by the counts of Savoy in 1218. Under the counts of Savoy the area was given political unity, and established as the Barony of Vaud. However, as the power of the Savoys declined at the beginning of the 15th century the land was occupied by troops from Bern. By 1536 the area was completely annexed. Reformation was started by co-workers of John Calvin, including Viret, who engaged in a famous debate at the cathedral of Lausanne; but it was only decisively implemented when Bern put its full force behind it. Vaud was another French-speaking region in the mostly German-speaking canton, which caused several uprisings.

Both Château-d'Œx and Saanen were part of the county of Gruyère. Both districts had quite a bit of independence and often entered into treaties against the will and best interests of the Counts. In 1340 the Saanen valley concluded a peace treaty with the Simmental, which provided for arbitration in disputes. They entered into another treaty in 1393 with the Valais. In 1401, Count Rudolph of Gruyère entered into a treaty with Bern which included Saanen. Two years later Saanen and Château-d'Œx negotiated their own alliances with Bern. Due to the Bernese alliance, Saanen sent troops, under their own banner, to support the Bernese invasions of Aargau in 1415 and Valais in 1418. The military losses and taxes following the Old Zürich War led Saanen to support the Evil League (Böser Bund) in 1445 against Bern.

In 1475, during the Burgundian Wars, the mountain regions of Saanen and Pays-d'Enhaut, who were allied with Bern, attacked and burned the tower of Aigle Castle. They then gave Aigle town and the surrounding district to Bern in exchange for not having to pay one-third of their income to Bern. In the treaty of Fribourg from 1476, Fribourg received rights over the Aigle district, which they gave up to Bern in 1483. Bern rebuilt Aigle Castle in 1489 and made it the seat of the bailiwick of Aigle. Also in 1475, the Confederation attacked and captured Grandson Castle. In the next year, Charles the Bold retook the castle and executed the Bernese garrison. In 1476, at the Battle of Grandson, Charles' forces retreated and the Bernese retook Grandson. After Charles' defeat at the Battle of Murten, Grandson became a shared territory between Bern and Fribourg. Each city appointed a vogt for five-year periods. Aigle and Grandson were the first French speaking regions in the Canton of Bern.

During the Burgundian War in 1475, Saanen, together with troops from Château-d'Œx and the Simmental captured the Savoy district of Aigle for Bern. Saanen and the surrounding district enjoyed a great deal of independence during the 16th century. However, in 1555 the last Count of Gruyère lost both districts to Bern when his county went bankrupt. Bern took over the entire Saanen valley in the following year and introduced the Protestant Reformation. They incorporated the Pays-d'Enhaut with Château-d'Œx into the new Bernese district of Saanen.

Acquired districts
The area of the canton of Bern consists of lands acquired by the city of Bern mostly between the 14th and the 16th century during the original Swiss Confederacy period, both by conquest and purchase. Acquired districts, with dates of acquisition, include:
 * Laupen (1324)
 * Oberhasli (1334)
 * Aarberg (1375)
 * Thun and Burgdorf (1384)
 * Unterseen and the Upper Simme valley (1386)
 * Frutigen and other towns in the Bernese Oberland (1400)
 * Aargau (1415)
 * Lower Simme valley (1439–1449)
 * Aigle and Grandson (1475)
 * Interlaken, with Grindelwald, Lauterbrunnen and Brienz (1528, all the suppression of the Augustinian Canons at Interlaken Monastery)
 * Vaud (1536)
 * Saanen or Gessenay (1555)
 * the Pays d'En-Haut including Château-d'Œx (1555)
 * Köniz (1729).

Social changes in Early Modern Bern
In the Middle Ages, upwards mobility and access to public offices was relatively easy for successful traders and craftsmen, but Bernese society became ever more stratified and aristocratic as the power and wealth of the city grew.

By the 17th century in the Early Modern period, citizenship had become an inherited prerogative, all political bodies elected one another and officials were elected for life. In effect, public offices were now the exclusive prerogative of the gnädige Herren, the "merciful lords", as the small number of noble families now ruling Bern came to be called. In 1605 there were 152 families that were qualified to rule, by 1691 that number was only 104, while towards the end of the 18th century there were only 69 such families. Meanwhile, the land ruled by the town was extending over more and more territory, so that finally it governed 52 bailiwicks. These offices became very lucrative as the Bernese territories grew. Patrician Landvögte, sheriffs, ruled the politically powerless countryside, often using armed force to put down peasant revolts.

In Vaud the Bernese occupation was not popular amongst the population. In 1723, Major Abraham Davel led a revolt against Bern, in protest at what he saw as the denial of political rights of the French-speaking Vaudois by the German-speaking Bernese, and was subsequently beheaded.

Napoleonic period

Inspired by the French Revolution, the Vaudois drove out the Bernese governor in 1798 and declared the Lemanic Republic. Vaud nationalists like Frédéric-César de La Harpe had called for French intervention in liberating the area and French Revolutionary troops moved in, taking over the whole of Switzerland itself in the process and setting up the Helvetic Republic.

In 1798, with the establishment of the Helvetic Republic, Bern was divided, the canton of Oberland with Thun as its capital and the canton of Léman with Lausanne as its capital were detached from what was left of the Canton of Bern.

Within the new canton of Oberland, historic borders and traditional rights were not considered. As there had been no previous separatist feeling amongst the conservative population, there was little enthusiasm for the new order. The situation in the canton of Léman was quite different. The French-speaking Vaudois had never felt like part of the German-speaking Canton of Bern. When they joined the Swiss Confederation in 1803, it was as the Canton of Vaud.

Under the Helvetic Republic, Pays-d'Enhaut with Château-d'Œx became part of the Canton of Léman while Saanen and the rest of the district became part of the Canton of Oberland. When the Helvetic Republic collapsed in 1803, Saanen and its district became a district in the new Canton of Bern while Château-d'Oex and its district joined Vaud.

The 1801 Malmaison Constitution proposed reuniting the Oberland with Bern, but it was not until the Act of Mediation, two years later, with the abolition of the Helvetic Republic and the partial restoration of the ancien régime, that the two cantons were reunited.

Between 1803 and 1814 Bern was one of the six directorial cantons of the Napoleonic Swiss Confederation.

Modern history
With the post−Napoleonic Restoration of 1815, Bern acquired the Bernese Jura with Biel/Bienne from the bishopric of Basel, while the canton of Léman became the canton of Vaud and remained separate from Bern.

Bern still remained the largest canton of the confederacy from 1815 to 1979, when parts of the Bernese Jura broke away to form the canton of Jura. In 1994 the Laufen District was transferred to the canton of Basel-Landschaft.

Geography
The canton of Bern is mainly drained by the river Aare and its tributaries. The area of the canton is commonly divided into six regions. The most populated area is the Bernese Mittelland on the plateau north of the Alps, with the capital city of Bern. The northmost part of the canton is the Bernese Jura bordering the Canton of Jura. The Bernese Oberland is the mountainous region which lies in the south of the canton.

The area of the canton is 5959 km2. Of this area, 6.9% is occupied with houses, businesses or roads. 42.6% of the canton is agricultural land, and 31.3% is forested. Less than 19.2% is considered non-productive, which includes glaciers, cliffs and lakes.

Bernese Mittelland
The Bernese Mittelland (Bernese Midlands) is made up of the valley of the rivers Aare, the (Grosse) Emme, some of the foothills of the Bernese Alps, as well as the plain around the capital Bern, and has many small farms and hilly forested regions with small to mid-sized towns scattered throughout. It is perhaps best known by foreigners and visitors for the Emmental. The classic Swiss cheese with holes Emmentaler comes from this region's forests and pastures, of hilly and low mountainous countryside in the 1000 to 2000 m range.

Three Lakes Region and Bernese Jura
In the north of the canton lies the predominantly French-speaking Three Lakes Region (Seeland), concentrated around Lake Biel, Neuchâtel, and Murten, which rises from the plain up to the northernmost Swiss mountain chain of the Jura. This area, culminating at the Chasseral, has more relaxed geography, although still some lower mountains and some waterfalls, and large lakes.

Bernese Oberland

The Bernese Oberland (Berner Oberland) constitute the north side of the Bernese Alps and the west side of the Urner Alps within the canton of Bern. The highest mountain in the Bernese Alps is the Finsteraarhorn at 4274 m, but the best known mountains are Eiger, Mönch, and Jungfrau.

The well known hiking and ski resorts in the eastern Oberland are located around Interlaken and the Jungfrau, such as car-free Mürren and Wengen in the Lauterbrunnen valley, and Grindelwald. Further east, in the Haslital are the Aareschlucht and the town of Meiringen, famous for the fateful scene of Sherlock Holmes's 'death' at the hands of Professor Moriarty on the nearby Reichenbach Falls. In the Western Bernese Oberland there are many other resorts and small villages catering to visitors. These are accessed from the lake town of Thun, and the most notable of them are Kandersteg with the Oeschinensee and Adelboden. Further west is the Simmental with Lenk and Zweisimmen and the Saanenland with the famous resorts Gstaad and Saanen.

The whole area is very mountainous, with steep cliffs, many glaciers, and countless waterfalls. It is renowned for its scenic beauty and the charm of the small Swiss villages that dot the area. As a result of this, tourism is one of the main sources of income in the Bernese Oberland. The region also has an extensive train network as well as many cable cars and funiculars, with the highest train station in Europe at the Jungfraujoch and the longest gondola cableway in the world from Grindelwald to the Männlichen.

Mountains in the Bernese Oberland include:

Government
The Grand Council of Bern (Grosser Rat/Grand conseil) is the parliament of the canton of Bern. It consists of 160 representatives elected by proportional representation for four-year terms of office. The French-speaking part of the canton, the Bernese Jura, has 12 seats guaranteed and 3 seats are guaranteed for the French-speaking minority of the bilingual district of Biel/Bienne.

The Executive Council of Bern (Regierungsrat/Conseil-éxecutif) is the government of the canton of Bern. This seven-member collegial body is elected by the people for a period of four years. The cantonal constitution reserves one seat in the Executive Council for a French-speaking citizen from the Bernese Jura.

The canton has a two-tiered court system, consisting of district courts and a cantonal Supreme Court (Obergericht, Cour suprême). There is also an administrative court (Verwaltungsgericht; Tribunal administratif) as well as other specialised courts and judicial boards.

Political subdivisions

On 1 January 2010, the 26 districts (Amtsbezirke) were combined into 10 new districts (Verwaltungskreise):

 * Bern-Mittelland with capital Ostermundigen, made up of all or part of the former districts of Bern, Fraubrunnen, Konolfingen, Laupen, Schwarzenburg and Seftigen
 * Biel/Bienne with capital Biel/Bienne, made up of all of the former district of Biel and about half of the former district of Nidau
 * Emmental with capital Langnau im Emmental, made up of all or part of the former districts of Burgdorf, Signau and Trachselwald
 * Frutigen-Niedersimmental with capital Frutigen, made up of all or part of the former districts of Frutigen and Niedersimmental
 * Interlaken-Oberhasli with capital Interlaken, made up of all or part of the former districts of Interlaken and Oberhasli
 * Jura bernois with capital Courtelary, made up of all or part of the former districts of Courtelary, Moutier and La Neuveville
 * Oberaargau with capital Wangen an der Aare, made up of all or part of the former districts of Aarwangen and Wangen
 * Obersimmental-Saanen with capital Saanen, made up of all of the former districts of Obersimmental and Saanen
 * Seeland with capital Aarberg, made up of all or part of the former districts of Aarberg, Büren, Erlach and Nidau
 * Thun with capital Thun, made up of all of the former district of Thun

Demographics
The canton of Bern is bilingual: both German (85.1% of the population) and French (10.4%) are spoken. The German-speaking majority speaks Bernese German, a Swiss German dialect. French-speakers live in the western and northern part of the canton, the Bernese Jura. Both German and French are spoken in the bilingual district of Biel/Bienne. In the cantonal government and administration, both languages are official languages of equal standing.

In December 2020, Bern had a population of. In 2014, 15.0% of the population were resident foreign nationals. From 2010 to 2014, the population increased at a rate of 3.0% annually. In 2000, most of the population spoke German (804,190 or 84.0%) as their first language; French was the second most common (72,646 or 7.6%) native language and Italian was the third (18,908 or 2.0%). There were 688 people who speak Romansh. Using different methodology, the 2014 census found that the proportion of German native speakers to be 85.1%, that of French speakers 10.4%, and that 3.1% were Italian speakers. The number of Romansh speakers was too small to accurately estimate, but was around 0.1%. The census also reported that 2.9% of the population spoke English as their native language. Respondents could choose up to three native languages, leading to a total above 100%.

In 2008, the population was 47.5% male and 52.5% female. The population was made up of 44,032 Swiss men (35.4% of the population) and 15,092 (12.1%) non-Swiss men. There were 51,531 Swiss women (41.4%) and 13,726 (11.0%) non-Swiss women. Of the population in the canton, 292,559 or about 30.6% were born in Bern and lived there in 2000. There were 339,659 or 35.5% who were born in the same canton, while 154,709 or 16.2% were born somewhere else in Switzerland, and 129,864 or 13.6% were born outside of Switzerland.

The age distribution of the canton in 2014 was as follows: those from 0–19 years old made up 19.1% of the population, while those aged 20–64 years old were 60.9%, and those 65 and older comprised. The 2000 census had found these groups to represent 22%, 60.9% and 17.1% of the total, respectively.

In 2000, there were 397,095 people who were single and never married in the canton. There were 449,014 married individuals, 61,206 widows or widowers and 49,882 individuals who are divorced.

There were 454,000 private households in the canton in 2014, and an average of 2.2 persons per household. , the construction rate of new housing units was 4.4 new units per 1000 residents.

In 2003, the average price to rent an average apartment in Bern city was 1108.92 Swiss francs (CHF) per month (US$890, £500, €710 approx. exchange rate from 2003). The average rate for a one-room apartment was 619.82 CHF (US$500, £280, €400), a two-room apartment was about 879.36 CHF (US$700, £400, €560), a three-room apartment was about 1040.54 CHF (US$830, £470, €670) and a six or more room apartment cost an average of 2094.80 CHF (US$1680, £940, €1340). The average apartment price in Bern was 99.4% of the national average of 1116 CHF. The vacancy rate for the canton,, was 1.19%.

Historic population
The historical population is given in the following chart:

Politics
In the 2011 federal election, the most popular party was the SVP which received 29.0% of the vote. The next three most popular parties were the SP/PS (19.3%), the BDP (14.9%) and the Green Party (9.4%). The SVP received about the same percentage of the vote as they did in the 2007 Federal election (33.6% in 2007 vs 29.0% in 2011). The SPS retained about the same popularity (21.2% in 2007), the BDP moved from below fourth place in 2007 to third and the GPS moved from below fourth place in 2007 to fourth.

Federal election results

 * FDP before 2009, FDP.The Liberals after 2009
 * "*" indicates that the party was not on the ballot in this canton.
 * Part of a coalition with Separatist Party
 * Part of a coalition with the FGA
 * Party fragmented, part remained in a coalition with the FGA and the remainder formed the Green Party of Bern
 * Part of a coalition with the Green Party

Coat of arms
The blazon of the coat of arms is Gules, on a bend or, a bear passant sable, langued, armed and vilené of the field. The official blazon specifies that the tongue, claws and penis are red—and by extension it is important to always depict the bear as male.

Religion
Most Bernese are Protestant (, 67%), and most Protestants belong to the Swiss Reformed Church, which is officially recognised as a state church (Landeskirche), although it is autonomous in its governance and is organised along democratic principles. The canton is also home to a great number of small Evangelical Christian denominations unaffiliated with the state church. Bernese evangelical groups are mostly found in the Emmental and Berner Oberland, where they have a long tradition; several contemporary North American religious groups, such as the Amish and Mennonites, were founded or co-founded by Bernese emigrants to the United States. Two small Evangelical political parties are represented in the Bernese cantonal parliament.

Bern features substantial Roman Catholic (16%) and Christian Catholic minorities. These churches also have state church status, and the small Jewish community is similarly recognised by law. As everywhere in Switzerland, there are also significant religious communities of immigrants, including Sikhs (who have a prominent Gurdwara, or temple, in Langenthal), The Church of Jesus Christ of Latter-day Saints (who have built the Bern Switzerland Temple) and Muslims. , the plans to expand a backyard mosque in Langenthal with a symbolic minaret have, as elsewhere in Switzerland, caused a public stir due to vocal opposition from local conservative and evangelical leaders.

From the, 607,358 or 63.5% belonged to the Swiss Reformed Church, while 153,357 or 16.0% were Roman Catholic. Of the rest of the population, there were 9,153 members of an Orthodox church (or about 0.96% of the population), there were 1,064 individuals (or about 0.11% of the population) who belonged to the Christian Catholic Church, and there were 71,233 individuals (or about 7.44% of the population) who belonged to another Christian church. There were 807 individuals (or about 0.08% of the population) who were Jewish, and 28,377 (or about 2.96% of the population) who were Muslims. There were 2,662 individuals who were Buddhist, 5,991 individuals who were Hindu and 913 individuals who belonged to another church. 74,162 (or about 7.75% of the population) belonged to no church, are agnostic or atheist, and 37,059 individuals (or about 3.87% of the population) did not answer the question.

Economy
Tourism is the main source of income in the Bernese Oberland. Other important sectors are agriculture (especially cattle breeding), cheese making, and hydroelectric power generation. The Bernese cheese Emmentaler is known around the world. In the Bernese Midlands the lands are more fertile. Agriculture is of great importance, but this part of the canton is also the most industrialized. Small and middle-sized businesses are important employers in this part of the canton of Bern. There is a nuclear power plant at Mühleberg; however, it is no longer operational.

The area around Lake Biel is renowned for its wine production. The 3 French-speaking districts of the Bernese Jura and the bilingual district of Biel/Bienne are renowned for their watch industry and its mechanical industry (high precision machine tools, automation and machining).

, Bern had an unemployment rate of 2.58%, compared to the Swiss national average of 3.3%. , there were 34,200 people employed in the primary economic sector and about 11,563 businesses involved in this sector. About 132,800 people were employed in the secondary sector and there were 11,925 businesses in this sector. Approximately 453,800 people were employed in the tertiary sector, with 55,347 businesses in this sector. Of the working population, 31.0% used public transportation to get to work, and 51.0% used a private car.

Education
In Bern about 385,640 or (40.3%) of the population have completed non-mandatory upper secondary education, and 121,749 or (12.7%) have completed additional higher education (either university or a Fachhochschule). Of the 121,749 who completed tertiary schooling, 65.0% were Swiss men, 24.8% were Swiss women, 6.2% were non-Swiss men and 4.0% were non-Swiss women.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Canton of Bern


text:
Symplectic vector space

In mathematics, a symplectic vector space is a vector space $$V$$ over a field $$F$$ (for example the real numbers $$\mathbb{R}$$) equipped with a symplectic bilinear form.

A symplectic bilinear form is a mapping $$\omega : V \times V \mapsto F$$ that is
 * Bilinear: Linear in each argument separately;
 * Alternating: $$\omega(v, v) = 0$$ holds for all $$v \in V$$; and
 * Non-degenerate: $$\omega(v, u) = 0$$ for all $$v \in V$$ implies that $$u = 0$$.

If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa.

Working in a fixed basis, $$\omega$$ can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If $$V$$ is finite-dimensional, then its dimension must necessarily be even since every skew-symmetric, hollow matrix of odd size has determinant zero. Notice that the condition that the matrix be hollow is not redundant if the characteristic of the field is 2. A symplectic form behaves quite differently from a symmetric form, for example, the scalar product on Euclidean vector spaces.

Standard symplectic space
The standard symplectic space is $$\mathbb{R}^{2n}$$ with the symplectic form given by a nonsingular, skew-symmetric matrix. Typically $$\omega$$ is chosen to be the block matrix

 * $$\omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}$$

where In is the n × n identity matrix. In terms of basis vectors (x1, ..., xn, y1, ..., yn):

 * $$\begin{align}

\omega(x_i, y_j) = -\omega(y_j, x_i) &= \delta_{ij}, \\ \omega(x_i, x_j) = \omega(y_i, y_j) &= 0. \end{align}$$

A modified version of the Gram–Schmidt process shows that any finite-dimensional symplectic vector space has a basis such that $$\omega$$ takes this form, often called a Darboux basis or symplectic basis.

Sketch of process:

Start with an arbitrary basis $$v_1, ..., v_n$$, and represent the dual of each basis vector by the dual basis: $$\omega(v_i, \cdot) = \sum_j \omega(v_i, v_j) v_j^*$$. This gives us a $$n\times n$$ matrix with entries $$\omega(v_i, v_j)$$. Solve for its null space. Now for any $$(\lambda_1, ..., \lambda_n)$$ in the null space, we have $$\sum_i \omega(v_i, \cdot) = 0$$, so the null space gives us the degenerate subspace $$V_0$$.

Now arbitrarily pick a complementary $$W$$ such that $$V = V_0 \oplus W$$, and let $$w_1, ..., w_m$$ be a basis of $$W$$. Since $$\omega(w_1, \cdot) \neq 0$$, and $$\omega(w_1, w_1) = 0$$, WLOG $$\omega(w_1, w_2 ) \neq 0$$. Now scale $$w_2$$ so that $$\omega(w_1, w_2) =1$$. Then define $$w' = w - \omega(w, w_2) w_1 + \omega(w, w_1) w_2$$ for each of $$w = w_3, w_4, ..., w_m$$. Iterate.

Notice that this method applies for symplectic vector space over any field, not just the field of real numbers.

Case of real or complex field:

When the space is over the field of real numbers, then we can modify the modified Gram-Schmidt process as follows: Start the same way. Let $$w_1, ..., w_m$$ be an orthonormal basis (with respect to the usual inner product on $$\R^n$$) of $$W$$. Since $$\omega(w_1, \cdot) \neq 0$$, and $$\omega(w_1, w_1) = 0$$, WLOG $$\omega(w_1, w_2 ) \neq 0$$. Now multiply $$w_2$$ by a sign, so that $$\omega(w_1, w_2) \geq 0$$. Then define $$w' = w - \omega(w, w_2) w_1 + \omega(w, w_1) w_2$$ for each of $$w = w_3, w_4, ..., w_m$$, then scale each $$w'$$ so that it has norm one. Iterate.

Similarly, for the field of complex numbers, we may choose a unitary basis. This proves the spectral theory of antisymmetric matrices.

Lagrangian form
There is another way to interpret this standard symplectic form. Since the model space R2n used above carries much canonical structure which might easily lead to misinterpretation, we will use "anonymous" vector spaces instead. Let V be a real vector space of dimension n and V∗ its dual space. Now consider the direct sum W = V ⊕ V∗ of these spaces equipped with the following form:

 * $$\omega(x \oplus \eta, y \oplus \xi) = \xi(x) - \eta(y).$$

Now choose any basis (v1, ..., vn) of V and consider its dual basis

 * $$\left(v^*_1, \ldots, v^*_n\right).$$

We can interpret the basis vectors as lying in W if we write xi = (vi, 0) and yi = (0, vi∗). Taken together, these form a complete basis of W,

 * $$(x_1, \ldots, x_n, y_1, \ldots, y_n).$$

The form ω defined here can be shown to have the same properties as in the beginning of this section. On the other hand, every symplectic structure is isomorphic to one of the form V ⊕ V∗. The subspace V is not unique, and a choice of subspace V is called a polarization. The subspaces that give such an isomorphism are called Lagrangian subspaces or simply Lagrangians.

Explicitly, given a Lagrangian subspace as defined below, then a choice of basis (x1, ..., xn) defines a dual basis for a complement, by ω(xi, yj) = δij.

Analogy with complex structures
Just as every symplectic structure is isomorphic to one of the form V ⊕ V∗, every complex structure on a vector space is isomorphic to one of the form V ⊕ V. Using these structures, the tangent bundle of an n-manifold, considered as a 2n-manifold, has an almost complex structure, and the cotangent bundle of an n-manifold, considered as a 2n-manifold, has a symplectic structure: T∗(T∗M)p = Tp(M) ⊕ (Tp(M))∗.

The complex analog to a Lagrangian subspace is a real subspace, a subspace whose complexification is the whole space: W = V ⊕ J V. As can be seen from the standard symplectic form above, every symplectic form on R2n is isomorphic to the imaginary part of the standard complex (Hermitian) inner product on Cn (with the convention of the first argument being anti-linear).

Volume form
Let ω be an alternating bilinear form on an n-dimensional real vector space V, ω ∈ Λ2(V). Then ω is non-degenerate if and only if n is even and ωn/2 = ω ∧ ... ∧ ω is a volume form. A volume form on a n-dimensional vector space V is a non-zero multiple of the n-form e1∗ ∧ ... ∧ en∗ where e1, e2, ..., en is a basis of V.

For the standard basis defined in the previous section, we have

 * $$\omega^n = (-1)^\frac{n}{2} x^*_1 \wedge \dotsb \wedge x^*_n \wedge y^*_1 \wedge \dotsb \wedge y^*_n.$$

By reordering, one can write

 * $$\omega^n = x^*_1 \wedge y^*_1 \wedge \dotsb \wedge x^*_n \wedge y^*_n.$$

Authors variously define ωn or (−1)n/2ωn as the standard volume form. An occasional factor of n! may also appear, depending on whether the definition of the alternating product contains a factor of n! or not. The volume form defines an orientation on the symplectic vector space (V, ω).

Symplectic map
Suppose that (V, ω) and (W, ρ) are symplectic vector spaces. Then a linear map f : V → W is called a symplectic map if the pullback preserves the symplectic form, i.e. fρ = ω, where the pullback form is defined by (fρ)(u, v) = ρ(f(u), f(v)). Symplectic maps are volume- and orientation-preserving.

Symplectic group
If V = W, then a symplectic map is called a linear symplectic transformation of V. In particular, in this case one has that ω(f(u), f(v)) = ω(u, v), and so the linear transformation f preserves the symplectic form. The set of all symplectic transformations forms a group and in particular a Lie group, called the symplectic group and denoted by Sp(V) or sometimes Sp(V, ω). In matrix form symplectic transformations are given by symplectic matrices.

Subspaces
Let W be a linear subspace of V. Define the symplectic complement of W to be the subspace
 * $$W^\perp = \{v \in V \mid \omega(v,w) = 0 \mbox{ for all } w \in W\}.$$

The symplectic complement satisfies:
 * $$\begin{align}

\left(W^\perp\right)^\perp &= W \\ \dim W + \dim W^\perp &= \dim V. \end{align}$$

However, unlike orthogonal complements, W⊥ ∩ W need not be 0. We distinguish four cases:
 * W is symplectic if W⊥ ∩ W = {0}. This is true if and only if ω restricts to a nondegenerate form on W. A symplectic subspace with the restricted form is a symplectic vector space in its own right.
 * W is isotropic if W ⊆ W⊥. This is true if and only if ω restricts to 0 on W. Any one-dimensional subspace is isotropic.
 * W is coisotropic if W⊥ ⊆ W. W is coisotropic if and only if ω descends to a nondegenerate form on the quotient space W/W⊥. Equivalently W is coisotropic if and only if W⊥ is isotropic. Any codimension-one subspace is coisotropic.
 * W is Lagrangian if W = W⊥. A subspace is Lagrangian if and only if it is both isotropic and coisotropic. In a finite-dimensional vector space, a Lagrangian subspace is an isotropic one whose dimension is half that of V. Every isotropic subspace can be extended to a Lagrangian one.

Referring to the canonical vector space R2n above,
 * the subspace spanned by {x1, y1} is symplectic
 * the subspace spanned by {x1, x2} is isotropic
 * the subspace spanned by {x1, x2, ..., xn, y1} is coisotropic
 * the subspace spanned by {x1, x2, ..., xn} is Lagrangian.

Heisenberg group
A Heisenberg group can be defined for any symplectic vector space, and this is the typical way that Heisenberg groups arise.

A vector space can be thought of as a commutative Lie group (under addition), or equivalently as a commutative Lie algebra, meaning with trivial Lie bracket. The Heisenberg group is a central extension of such a commutative Lie group/algebra: the symplectic form defines the commutation, analogously to the canonical commutation relations (CCR), and a Darboux basis corresponds to canonical coordinates – in physics terms, to momentum operators and position operators.

Indeed, by the Stone–von Neumann theorem, every representation satisfying the CCR (every representation of the Heisenberg group) is of this form, or more properly unitarily conjugate to the standard one.

Further, the group algebra of (the dual to) a vector space is the symmetric algebra, and the group algebra of the Heisenberg group (of the dual) is the Weyl algebra: one can think of the central extension as corresponding to quantization or deformation.

Formally, the symmetric algebra of a vector space V over a field F is the group algebra of the dual, Sym(V) := F[V∗], and the Weyl algebra is the group algebra of the (dual) Heisenberg group W(V) = F[H(V∗)]. Since passing to group algebras is a contravariant functor, the central extension map H(V) → V becomes an inclusion Sym(V) → W(V).

title: Symplectic vector space


text:
Canton of Solothurn

The canton of Solothurn or canton of Soleure (Kanton Solothurn; Chantun Soloturn; Canton de Soleure; Canton Soletta) is a canton of Switzerland. It is located in the northwest of Switzerland. The capital is Solothurn.

History
The foundation of the village of Salodurum took place in the time of the Roman emperor Tiberius. The territory of the canton comprises land acquired by the former town, mainly in the Middle Ages. For that reason the shape of the canton is irregular and includes two exclaves along the French border, separated from the rest of the canton by Basel-Land, which form separate districts of the canton. In 1481, the canton became a member of the military alliance of the former Swiss confederation. At the end of the Reformation, Solothurn maintained its Catholic religion. Between 1798 and 1803 the canton was part of the Helvetic Republic. In 1803 Solothurn was one of the 19 Swiss cantons that were reconstituted by Napoleon (Mediation). In 1830, the population rebelled against the aristocratic regime and the canton became definitely liberal-democratic. Even though the population was strictly Roman Catholic, Solothurn did not join the Catholic separatist movement (Sonderbund) in 1845–7. Similarly, the federal constitutions of 1848 and 1874 were approved. The current constitution of the canton dates from 1987.

Geography
The canton is located in the north-west of Switzerland. To the west and south lie the cantons of Jura and Bern, to the east is Aargau. To the north the canton is bounded by the canton of Basel-Landschaft. Parts of two of the districts are exclaves and are located along the border of France (Grand Est). The lands are drained by the Aare river and its tributaries. The landscape is mostly flat, but it includes the foothills of the Jura massif. Part of this, the massif of the Weissenstein, overlooks Solothurn and the Mittelland from the north and has views of the Bernese Alps. The flat lands are a plain created by the Aare river. The total area of the canton is 791 km².

Districts
From 2005, Solothurn's ten districts are merged pairwise into five electoral districts, termed Amtei. From 2005, the districts have only a statistical significance.
 * Bucheggberg, Amtei Wasseramt-Bucheggberg
 * Dorneck, Amtei Dorneck-Thierstein (unofficially Schwarzbubenland)
 * Gäu, Amtei Thal-Gäu
 * Gösgen, Amtei Olten-Gösgen (unofficially Niederamt)
 * Lebern, Amtei Solothurn-Lebern
 * Olten, Amtei Olten-Gösgen
 * Solothurn, Amtei Solothurn-Lebern
 * Thal, Amtei Thal-Gäu
 * Thierstein, Amtei Dorneck-Thierstein
 * Wasseramt, Amtei Wasseramt-Bucheggberg

Municipalities
There are 125 municipalities in the canton.

Demographics
The population is mostly German-speaking. In 2000 about 44% of the population was Roman Catholic, with most of the remainder being Protestant (31%). The population of the canton (as of ) is. , the population included 46,898 foreigners, or about 18.7% of the total population.

Historical population
The historical population is given in the following table:

Economy
Up to the 19th century agriculture was the main economic activity in the canton. Agriculture is still of importance, but manufacturing and the service industry are now more significant. The industries of the canton are specialized in watches, jewellery, textiles, paper, cement and auto parts. Until recently the manufacturing of shoes was an important economic activity, but global competition thought that the Swiss canton was not competitive enough.

The canton is home to the Gösgen Nuclear Power Plant near Däniken which started operation in 1979.

Federal election results

 * FDP before 2009, FDP.The Liberals after 2009
 * "*" indicates that the party was not on the ballot in this canton.

Transport
The canton has good connections with other parts of Switzerland, both by rail and by road. There is a railway junction at Olten with direct trains to Geneva, Zürich, Basel and the Ticino via Lucerne.

title: Canton of Solothurn


text:
Genesis 3

Genesis 3 may refer to:

 * Chapter 3 of the biblical Book of Genesis describing the fall of man
 * A model of the Sega Genesis game console
 * The 2013 edition of the esports tournament GENESIS

title: Genesis 3


text:
Daresbury

Daresbury is a village and civil parish in the Borough of Halton, Cheshire, England. At the 2011 census, it had a population of 246.

History
The name means "Deor's fortification", derived from an Old English personal name and the word burh (a fortified place).

The population was recorded over time at 134 in the census of 1801, 157 in 1851, 153 in 1901, 235 in 1951 and reaching 216 by 2001.

Opened in 1850 on the Birkenhead Railway, the area was served by Daresbury railway station in the nearby village of Moore. The station closed to passengers in 1952 and to goods in 1965, although the line through the station site remains in use.

Daresbury was the birthplace of Alice's Adventures in Wonderland author Lewis Carroll, in All Saints' Vicarage. The village has a Lewis Carroll Visitor Centre and in the parish church of All Saints, a Lewis Carroll window, which includes an image of the Cheshire Cat.

Landmarks
Built in 1759, Daresbury Hall is a former Georgian country house which has been used as a private dwelling, military hospital and residential home. A designated Grade II* listed building since 1952, it is on the Heritage at Risk Register due to being in very poor condition. A major fire in June 2016 destroyed much of the building, which has since been secured with scaffold and mothballed in a roofless condition.

Governance
Daresbury was a township and chapelry in Runcorn parish of Bucklow Hundred, which became a civil parish in 1866. The village was administered as part of Runcorn Rural Sanitary District from 1875, then from 1894 Runcorn Rural District. Local government reorganisation on 1 April 1974 transferred the area to the Borough of Halton.

Daresbury is also an electoral ward. However, the boundary of the ward is different from (and larger than) the parish boundary, and includes the parishes of Moore, Sandymoor and Preston Brook. The total ward population in 2011 was 5,401. Daresbury is represented nationally by the Weaver Vale parliamentary constituency.

Economy
Near the village is the Science and Technology Facilities Council's Sci-Tech Daresbury science and innovation campus, which includes the Daresbury Laboratory and the Cockcroft Institute. The Synchrotron Radiation Source (SRS) facility at Daresbury Laboratory was in operation between 1981 and 2008.

Daresbury Park is to the south west of the village, at junction 11 of the M56 motorway. The 225 acre site provides office and warehouse space for several businesses and includes the Daresbury Park Hotel.

Transport
The village of Daresbury lies to the east of the realigned A56 road. The B5356 Daresbury Lane heads eastwards from the village towards Hatton and Stretton, while the A558 Daresbury Expressway heads westwards towards Runcorn and crosses the Bridgewater Canal.

Community and culture
In 2006, the annual Creamfields dance festival was held in Daresbury after relocating from the disused Liverpool airport site it had occupied for the six previous years. This saw 40,000 revellers partying from 3pm-6am to a line-up that included live performances from the Prodigy and Zutons, as well as DJ sets from the likes of Sasha, Paul Oakenfold, 2 Many DJ's, Green Velvet and DJ Shadow. As of 2019, the festival has been an annual event at the site.

title: Daresbury


text:
Classification yard

A classification yard (American English, as well as the Canadian National Railway), marshalling yard (British, Hong Kong, Indian, and Australian English, as well as the Canadian Pacific Railway) or shunting yard (Central Europe) is a railway yard found at some freight train stations, used to separate railway cars onto one of several tracks. First the cars are taken to a track, sometimes called a lead or a drill. From there the cars are sent through a series of switches called a ladder onto the classification tracks. Larger yards tend to put the lead on an artificially built hill called a hump to use the force of gravity to propel the cars through the ladder.

Freight trains that consist of isolated cars must be made into trains and divided according to their destinations. Thus the cars must be shunted several times along their route in contrast to a unit train, which carries, for example, automobiles from the plant to a port, or coal from a mine to the power plant. This shunting is done partly at the starting and final destinations and partly (for long-distance-hauling) in classification yards.

Flat
Flat yards are constructed on flat ground, or on a gentle slope, not enough to allow a free-fall operation without locomotives. Freight cars are pushed by a locomotive and coast to their required location.

Gravity
Gravity yards were invented in the 19th century, saving shunting engines and instead letting the cars roll by gravity was seen as a major benefit, whereas the larger amount of manual work required to stop the rolling cars in the classification tracks was judged to be not that important. Gravity yards were a historical step in the development of classification yards and were later judged as inferior to hump yards, because it became clear that shunting engines were needed anyway (at least in inclement weather like strong winds or icy temperatures when the oil in the bearings became thick), and because manual labour was getting relatively more and more expensive. Thus, only a few gravity yards were ever built, sometimes requiring massive earthwork (one example is the first German gravity yard at Dresden). The historic technique of a gravity yard is today partly presented in Chemnitz-Hilbersdorf gravity yard (museum).

Most gravity yards were built in Germany (especially in the kingdom of Saxony) and in Great Britain (so Edgehill, 1873), a few also in some other European countries, for example Łazy yard near Zawiercie on the Warsaw–Vienna Railway (in Poland). In the US, there were very few old gravity yards; one of the few gravity yards in operation today is CSX's Readville Yard south of Boston, Massachusetts.

Hump

Hump yards are the largest and most effective classification yards, with the largest shunting capacity, often several thousand cars a day. They work similarly to gravity yards, but the falling gradient is limited to a small part of the yard, namely the hump. It is the heart of the yard—a lead track on a small hill over which an engine pushes the cars. Single cars, or a block of coupled cars, are uncoupled at or just before the crest of the hump, and roll by gravity onto their destination tracks in the area of the yard where the cars are sorted, called the classification bowl. The first hump in Germany (Leipzig) was built in 1858 and in France (Saint-Etienne) in 1863.

The speed of the cars rolling down from the hump into the classification bowl must be regulated according to whether they are full or empty, heavy or light freight, varying number of axles, whether there are few or many cars on the classification tracks, and varying weather conditions, including temperature, wind speed and direction. In regards to speed regulation, there are two types of hump yards—with or without mechanisation by retarders. In the old non-retarder yards braking was usually done in Europe by railroaders who laid skates onto the tracks. The skate or wheel chock was manually (or, in rare cases, mechanically) placed on one or both of the rails so that the treadles or rims of the wheel or wheels caused frictional retardation and resulted in the halting of the railway car. In the United States this braking was done by riders on the cars. In the modern retarder yards this work is done by mechanized "rail brakes" called retarders, which brake the cars by gripping the wheels. They are operated either pneumatically or hydraulically. Pneumatic systems are prevalent in the United States, France, Belgium, Russia and China, while hydraulic systems are used in Germany, Italy and the Netherlands.

Classification bowls in Europe typically consist of 20 to 40 tracks, divided into several fans or balloons of tracks, usually with eight classification tracks following a retarder in each one, often 32 tracks altogether. In the United States, many classification bowls have more than 40 tracks, which are often divided into six to ten classification tracks in each balloon loop.

Bailey Yard in North Platte, Nebraska, United States, the world's largest classification yard, is a hump yard. Other large American hump yards include Argentine Yard in Kansas City, Kansas, Robert Young Yard in Elkhart, Indiana, Clearing Yard in Chicago, Illinois, Englewood Yard in Houston, Texas, and Waycross Rice Yard in Waycross, Georgia. Notably, in Europe, Russia and China, all major classification yards are hump yards. Europe's largest hump yard is that of Maschen near Hamburg, Germany; it is only slightly smaller than Bailey Yard. The second largest is in the port of Antwerp, Belgium. Most hump yards are single yards with one classification bowl, but some, mostly very large, hump yards have two of them, one for each direction, thus are double yards, such as the Maschen, Antwerp, Clearing, and Bailey yards.

Almost all gravity yards have been retrofitted with humps and are worked as hump yards. Examples include Chemnitz Hilbersdorf (today Saxon Railway Museum), Dresden Friedrichstadt and Nürnberg (Nuremberg) Rbf (Rbf: Rangierbahnhof, "classification yard"), in Germany.

title: Classification yard


text:
G2 (mathematics)

In mathematics, G2 is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras $$\mathfrak{g}_2,$$ as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.

The compact form of G2 can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation (a spin representation).

History
The Lie algebra $$\mathfrak{g}_2$$, being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras. On May 23, 1887, Wilhelm Killing wrote a letter to Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, which we now call $$\mathfrak{g}_2$$.

In 1893, Élie Cartan published a note describing an open set in $$\mathbb{C}^5$$ equipped with a 2-dimensional distribution—that is, a smoothly varying field of 2-dimensional subspaces of the tangent space—for which the Lie algebra $$\mathfrak{g}_2$$ appears as the infinitesimal symmetries. In the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2-dimensional distribution is closely related to a ball rolling on another ball. The space of configurations of the rolling ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting.

In 1900, Engel discovered that a generic antisymmetric trilinear form (or 3-form) on a 7-dimensional complex vector space is preserved by a group isomorphic to the complex form of G2.

In 1908 Cartan mentioned that the automorphism group of the octonions is a 14-dimensional simple Lie group. In 1914 he stated that this is the compact real form of G2.

In older books and papers, G2 is sometimes denoted by E2.

Real forms
There are 3 simple real Lie algebras associated with this root system:

 * The underlying real Lie algebra of the complex Lie algebra G2 has dimension 28. It has complex conjugation as an outer automorphism and is simply connected. The maximal compact subgroup of its associated group is the compact form of G2.
 * The Lie algebra of the compact form is 14-dimensional. The associated Lie group has no outer automorphisms, no center, and is simply connected and compact.
 * The Lie algebra of the non-compact (split) form has dimension 14. The associated simple Lie group has fundamental group of order 2 and its outer automorphism group is the trivial group. Its maximal compact subgroup is SU(2) × SU(2)/(−1,−1). It has a non-algebraic double cover that is simply connected.

Dynkin diagram and Cartan matrix
The Dynkin diagram for G2 is given by.

Its Cartan matrix is:

\left [\begin{array}{rr} 2 & -3 \\ -1 & 2 \end{array}\right] $$

Roots of G2
A set of simple roots for can be read directly from the Cartan matrix above. These are (2,−3) and (−1, 2), however the integer lattice spanned by those is not the one pictured above (from obvious reason: the hexagonal lattice on the plane cannot be generated by integer vectors). The diagram above is obtained from a different pair roots: $$\alpha = \left( 1, 0 \right)$$ and $\beta = \sqrt{3}\left(\cos{\frac{5\pi}{6}},\sin{\frac{5\pi}{6}}\right) = \frac{1}{2}\left(-3,\sqrt{3} \right)$.

The remaining (positive) roots are $A = \alpha + \beta,\, B = 3\alpha + \beta,\, \alpha + A = 2\alpha + \beta \,\,{\rm and }\,\, A + B = 3\alpha + 2\beta$.

Although they do span a 2-dimensional space, as drawn, it is much more symmetric to consider them as vectors in a 2-dimensional subspace of a three-dimensional space. In this identification α corresponds to e₁−e₂, β to −e₁ + 2e₂−e₃, A to e₂−e₃ and so on. In euclidean coordinates these vectors look as follows:

The corresponding set of simple roots is:
 * e₁−e₂ = (1,−1,0), and −e₁+2e₂−e₃ = (−1,2,−1)

Note: α and A together form root system identical to A₂, while the system formed by β and B is isomorphic to A₂.

Weyl/Coxeter group
Its Weyl/Coxeter group $$G = W(G_2)$$ is the dihedral group $$D_6$$ of order 12. It has minimal faithful degree $$\mu(G) = 5$$.

Special holonomy
G2 is one of the possible special groups that can appear as the holonomy group of a Riemannian metric. The manifolds of G2 holonomy are also called G2-manifolds.

Polynomial invariant
G2 is the automorphism group of the following two polynomials in 7 non-commutative variables.

 * $$C_1 = t^2+u^2+v^2+w^2+x^2+y^2+z^2$$
 * $$C_2 = tuv + wtx + ywu + zyt + vzw + xvy  + uxz  $$ (± permutations)

which comes from the octonion algebra. The variables must be non-commutative otherwise the second polynomial would be identically zero.

Generators
Adding a representation of the 14 generators with coefficients A, ..., N gives the matrix:
 * $$A\lambda_1+\cdots+N\lambda_{14}=

\begin{bmatrix} 0 & C &-B & E &-D &-G &F-M \\ -C & 0 & A & F &-G+N&D-K&-E-L \\ B &-A & 0 &-N & M & L & -K \\ -E &-F & N & 0 &-A+H&-B+I&C-J\\ D &G-N &-M &A-H& 0 & J &I \\ G &K-D& -L&B-I&-J & 0 & -H \\ -F+M&E+L& K &-C+J& -I & H & 0 \end{bmatrix}$$

It is exactly the Lie algebra of the group
 * $$G_2=\{g\in \mathrm{SO}(7):g^*\varphi=\varphi, \varphi = \omega^{123} + \omega^{145} + \omega^{167} + \omega^{246} - \omega^{257} - \omega^{347} - \omega^{356}\}$$

There are 480 different representations of $$G_2$$ corresponding to the 480 representations of octonions. The calibrated form, $$\varphi$$ has 30 different forms and each has 16 different signed variations. Each of the signed variations generate signed differences of $$G_2$$ and each is an automorphism of all 16 corresponding octonions. Hence there are really only 30 different representations of $$G_2$$. These can all be constructed with Clifford algebra using an invertible form $$3e_{1234567}\pm\varphi$$ for octonions. For other signed variations of $$\varphi$$, this form has remainders that classify 6 other non-associative algebras that show partial $$G_2$$ symmetry. An analogous calibration in $$\mathrm{Spin}(15)$$ leads to sedenions and at least 11 other related algebras.

Representations

The characters of finite-dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are :

 * 1, 7, 14, 27, 64, 77 (twice), 182, 189, 273, 286, 378, 448, 714, 729, 748, 896, 924, 1254, 1547, 1728, 1729, 2079 (twice), 2261, 2926, 3003, 3289, 3542, 4096, 4914, 4928 (twice), 5005, 5103, 6630, 7293, 7371, 7722, 8372, 9177, 9660, 10206, 10556, 11571, 11648, 12096, 13090....

The 14-dimensional representation is the adjoint representation, and the 7-dimensional one is action of G2 on the imaginary octonions.

There are two non-isomorphic irreducible representations of dimensions 77, 2079, 4928, 30107, etc. The fundamental representations are those with dimensions 14 and 7 (corresponding to the two nodes in the Dynkin diagram in the order such that the triple arrow points from the first to the second).

described the (infinite-dimensional) unitary irreducible representations of the split real form of G2.

The embeddings of the maximal subgroups of G2 up to dimension 77 are shown to the right.

Finite groups
The group G2(q) is the points of the algebraic group G2 over the finite field Fq. These finite groups were first introduced by Leonard Eugene Dickson in for odd q and  for even q. The order of G2(q) is q6(q6 − 1)(q2 − 1). When q ≠ 2, the group is simple, and when q = 2, it has a simple subgroup of index 2 isomorphic to 2A2(32), and is the automorphism group of a maximal order of the octonions. The Janko group J1 was first constructed as a subgroup of G2(11). introduced twisted Ree groups 2G2(q) of order q3(q3 + 1)(q − 1) for q = 32n+1, an odd power of 3.

title: G2 (mathematics)


text:
Gundam Wing: Endless Waltz

Gundam Wing: Endless Waltz, known in Japan as New Mobile Report Gundam Wing: Endless Waltz (新機動戦記ガンダムW: ENDLESS WALTZ), is the sequel to Mobile Suit Gundam Wing, both of which are set in the "After Colony" timeline, an alternate universe to that of the original Mobile Suit Gundam series. Aside from being a continuation to the Gundam Wing TV series, it also reveals details regarding the pasts of the five Gundam pilots and the true objective behind "Operation Meteor."

Endless Waltz originally premiered in Japan as a three-part OVA in 1997. It was later released as a theatrical compilation film in 1998, including additional scenes and an altered musical score.

Synopsis
It is the year After Colony 196, and the battles between Earth and the space colonies have ended. Treize Khushrenada is dead and OZ has come to an end. This gives birth to the Earth Sphere Unified Nation (ESUN) and the Preventers. Seeing they won't be needed anymore, the Gundam pilots (except Chang Wufei) send their suits into the sun. However, this peace would not last, for a rebellion occurs on the newly completed colony, L3 X-18999. Led by seven-year-old Mariemaia Khushrenada, Treize's illegitimate daughter, the rebellion kidnaps Relena Darlian, now the Vice Foreign Minister of the ESUN, during a diplomatic mission to X-18999. As the Gundam pilots investigate further, they discover that Mariemaia is merely a puppet controlled by her grandfather Dekim Barton, a former advisor to martyred colony leader Heero Yuy, who is using X-18999 to go through with the original Operation Meteor, as a contingency plan in case the ESUN doesn't comply. The Gundam pilots must prevent Dekim from seizing power over the ESUN. The Gundams are retrieved from their course to sun to Earth's orbit thanks to Quatre and the Maganacs. The pilots use their Gundams one last time to fight against Dekim's forces, not killing anyone. In the end, Dekim is killed by one of his own soldiers, Earth and its colonies are at peace once again and all mobile suits (including the Gundams) are forever destroyed and never seen again.

Characters

 * Heero Yuy
 * Pilot of the XXXG-00W0 Wing Gundam Zero, who was code-named after the assassinated pacifist of the same name. Though he brought an end to the war between Earth and the colonies, Heero must now prevent the Mariemaia Army from disrupting the peace.
 * Pilot of the XXXG-00W0 Wing Gundam Zero, who was code-named after the assassinated pacifist of the same name. Though he brought an end to the war between Earth and the colonies, Heero must now prevent the Mariemaia Army from disrupting the peace.

 * Relena Darlian
 * An ally of the Gundam pilots and the strongest political advocate for peace between Earth and the colonies, who is kidnapped by the Mariemaia Army.
 * An ally of the Gundam pilots and the strongest political advocate for peace between Earth and the colonies, who is kidnapped by the Mariemaia Army.

 * Duo Maxwell
 * Pilot of the XXXG-01D2 Gundam Deathscythe Hell, who assists his fellow Gundam pilots in maintaining the peace they fought so hard to attain.
 * Pilot of the XXXG-01D2 Gundam Deathscythe Hell, who assists his fellow Gundam pilots in maintaining the peace they fought so hard to attain.

 * Trowa Barton
 * Pilot of the XXXG-01H2 Gundam Heavyarms Kai, who infiltrates the Mariemaia Army in order to prevent its revised version of Operation Meteor.
 * Pilot of the XXXG-01H2 Gundam Heavyarms Kai, who infiltrates the Mariemaia Army in order to prevent its revised version of Operation Meteor.

 * Quatre Raberba Winner
 * }
 * Pilot of the XXXG-01SR2 Gundam Sandrock Kai, who comes up with the idea of sending the Gundams into the Sun, but is forced to retrieve them after the Mariemaia Army declares war on the Earth.

 * Chang Wufei
 * Pilot of the XXXG-01S2 Altron Gundam, who joins forces with the Mariemaia Army and becomes an enemy toward the other Gundam pilots.
 * Pilot of the XXXG-01S2 Altron Gundam, who joins forces with the Mariemaia Army and becomes an enemy toward the other Gundam pilots.

 * Zechs Merquise
 * A former enemy of the Gundam pilots, now their ally and a member of the Preventers (code name: "Wind"), as well as the pilot of the new OZ-00MS2B Tallgeese III mobile suit.
 * A former enemy of the Gundam pilots, now their ally and a member of the Preventers (code name: "Wind"), as well as the pilot of the new OZ-00MS2B Tallgeese III mobile suit.

 * Lucrezia Noin
 * An ally of the Gundam pilots and a member of the Preventers (code name: "Fire").
 * An ally of the Gundam pilots and a member of the Preventers (code name: "Fire").

 * Sally Po
 * Another ally of the Gundam pilots and a member of the Preventers (code name: "Water").
 * Another ally of the Gundam pilots and a member of the Preventers (code name: "Water").

 * Lady Une
 * Another former enemy of the Gundam pilots, now their ally and the head of the Preventers (code name: "Gold").
 * Another former enemy of the Gundam pilots, now their ally and the head of the Preventers (code name: "Gold").

 * Mariemaia Khushrenada
 * The young daughter of the deceased Treize Khushrenada, who is manipulated by her grandfather Dekim Barton into leading a rebellion against the Earth with the intentions of conquering it.
 * The young daughter of the deceased Treize Khushrenada, who is manipulated by her grandfather Dekim Barton into leading a rebellion against the Earth with the intentions of conquering it.

 * Narrator

Mecha redesigns
One of the most notable features of Endless Waltz was the massive redesigns all five of the Gundams from the end of the Gundam Wing TV series received, courtesy of the artist Hajime Katoki. The Gundams in Endless Waltz have more stylized appearances, reflected in the even more demonic design of the Gundam Deathscythe Hell, the more dragon-like design of the Altron Gundam, and the new "angel-winged" design of the Wing Zero Gundam. Despite the dramatically different designs of the Gundams, the story in Endless Waltz retcons them as if the original designs from the TV series never existed. However, this changed after the release of the Glory of Losers manga series.

Theme songs

 * OVA Ending Theme: "White Reflection" by Two-Mix
 * Movie Ending Theme: "Last Impression" by Two-Mix

Distribution
In North America, Endless Waltz premiered on Canada's YTV on September 11, 2000 and on Cartoon Network in the U.S. on November 10, 2000. It was later released to VHS, UMD, and DVD by Bandai Entertainment, with the DVD edition containing both the OVA and compilation film versions on one disc. Due to the closure of Bandai Entertainment, the OVA and film went out-of-print. On October 11, 2014 at their 2014 New York Comic Con panel, Sunrise announced they will be releasing all of the Gundam franchise, including Endless Waltz in North America though distribution from Right Stuf Inc., beginning in Spring 2015. Right Stuf had re-released Endless Waltz on Blu-ray and DVD in December 2017.

Reception
The initial airing of the OVA on November 10, 2000, was Cartoon Network's second highest-rated program ever at the time, only being topped by the Funimation's in-house dub of Dragon Ball Z. Helen McCarthy in 500 Essential Anime Movies commented that "the giant robot fights are as good as ever" and that "the art direction and design is excellent".

title: Gundam Wing: Endless Waltz