title: Paris–Roubaix

text: In 1936 the Belgian, Romain Maes, appeared to win but judges declared Frenchman Georges Speicher the winner and Maes second. Shouting began in the stands and for a moment it looked as though fighting would start, but calm returned and the result was upheld. A Belgian may not have won but there were seven Belgians in the first ten.


title: Paris–Roubaix

text: The result in 1949 took several months and two international conferences to sort out. André Mahé was first but his win was challenged because he took the wrong course. Mahé was in a break of three that reached Roubaix velodrome in the lead, but he was misdirected by officials and entered the track by the wrong gate. Mahé was declared winner but a few minutes later other riders arrived using the correct route and Serse Coppi, brother of famous Fausto, won the sprint for what was assumed to be the minor placings. After a protest and several months, Serse Coppi was named joint winner with Mahé. Mahé said in 2007:


title: Paris–Roubaix

text: "C'est trop bête d'en parler" (It's too stupid to talk about). There was a break. Coppi attacked. His brother Fausto gave him a push to get him away. He wanted his brother to win. I waited a bit and then I attacked and I caught him and the break. Then I went off by myself. I was going to win Paris–Roubaix. At the entrance to the vélodrome, there were crowds everywhere, blocking the way. I looked around for where to go and I was directed round the outside wall of the track, to where the team cars had to park. It wasn't like nowadays, when there's television and everything. Then it was more chaotic and the whole road was blocked. People said I should have known the way into the track. But how do you know a thing like that at the end of Paris–Roubaix, when you've raced all day over roads like that? A gendarme signalled the way to go and that's the way I went.


title: Paris–Roubaix

text: It was a journalist on a motorbike who managed to get up to me. He was shouting 'Not that way! Not that way!' And I turned round in the road and I rode back beneath the outside wall of the grandstand and I saw a gateway that went into the track, a gateway for journalists. And that's the way I went, except that it came out on the other side of the track from the proper entrance. The bunch came in and Serse won the sprint. But then his brother told Serse to go to the judges to object. He told Serse that I hadn't ridden the entire and precise course and that therefore I should be "déclassé". But that was below him. Coppi wanted his brother to have a big victory. He was a great champion, Coppi, but to do what he did, to protest like that to get a victory for his brother, that wasn't dignified for a champion. That was below him. A champion like that should never have stooped that low. I never spoke to him about it. Never did. Why should I?


title: Paris–Roubaix

text: The only other times he rode it were in 1980, when he finished fourth, and in 1982, as the defending champion. When he was criticised, he said: "I don't go into offices and tell people to work harder, yet people ask me to be the strongest on the cobbles." Hinault fell seven times in that race, including 13 km from the finish when a small black dog called Gruson ran out in a bend and ran under his wheel. Hinault had been clear with Roger De Vlaeminck, Hennie Kuiper and Marc Demeyer. The incident made Hinault angry and he raced back to the others and won in Roubaix.


title: Paris–Roubaix

text: He was not the first star to refuse. Jacques Anquetil called it a lottery after puncturing 13 km from the end in 1958 and never took it seriously again.


title: Paris–Roubaix

text: In 2002 only two of the top 20 riders in the UCI table - Jens Voigt and Erik Zabel - were on the start line. The following year only Zabel was there. In 2004 he had stayed at home as well. Philippe Brunel wrote in "L'Équipe":


title: Paris–Roubaix

text: We won't go as far as say that the five-time winner of the Tour [Hinault] - who every year gives the winner his celebration cobble stone on behalf of the organisers — has contributed to the impoverishment ["paupérisation"] of the queen of classics, which would offend him, but his words have contributed to the snub, or the indifference, of those who stay away. The fact isn't new but the phenomenon is getting worse and is concerning. The peloton of stayaways has grown to the point where Paris–Roubaix is now only for a tight group of specialists... especially the Belgians, capable of maintaining high speed on the cobbles.


title: Paris–Roubaix

text: The 1988 race contained a rare spectacle, for the first time in 30 years an early morning breakaway group held on until the finish: 27 kilometres into the race a group of unknown riders broke away and the pack did nothing to chase them down throughout the race. It was on a section of cobblestones outside Roubaix that Thomas Wegmüller (SUI) and Dirk Demol (BEL) broke away from the lead group to try for the victory. Wegmüller was ranked 175th in the world, Demol was a factory worker, ranked just inside the top 500 cyclists. As if the success of the breakaway wasn't enough, Paris–Roubaix was about to deliver a cruel irony.


title: Paris–Roubaix

text: When the two entered Roubaix, Wegmüller ran over a plastic bag that flew out in front of him, which became jammed in his derailleur. Wegmüller was unable to change gears which was crucial for a sprint finish. Demol could have attacked at this moment and won the race with ease, but it was clear he did not want to win in this manner and drafted behind Wegmüller as he got assistance from his team car to remove the bag. They removed most of the bag, but some of it was still jammed and his gears still would not change. Knowing that a bicycle change would be suicidal to his chances, Wegmüller continued on his damaged bike; Demol continued to draft behind him. When it came down to the final sprint, Wegmüller could only watch as Demol easily sprinted ahead to take the victory.


title: Paris–Roubaix

text: Laurent Fignon finished third after a late breakaway from the chasing peloton. Sean Kelly, also part of the chase group containing the top riders in the world, crashed late in the race, after it was already decided the breakaway would not be caught and crossed the finish line with blood covering his face.


title: Paris–Roubaix

text: In 2006 Leif Hoste, Peter Van Petegem and Vladimir Gusev were disqualified for riding through a closed railway level crossing 10 km before the finish and just ahead of an approaching freight train. Fabian Cancellara won and Tom Boonen and Alessandro Ballan were given the remaining places on the podium.


title: Paris–Roubaix

text: In the 2018 race, Michael Goolaerts (Vérandas Willems–Crelan) crashed after suffering a cardiac arrest 100 km into the race. A doctor attempted to resuscitate him on the spot. He was flown to hospital in Lille by helicopter for treatment. At 23:30 local time, his death was announced by his team.


title: Paris–Roubaix

text: The American television channel CBS covered Paris–Roubaix in the 1980s. Theo de Rooij, a Dutchman, had been in a promising position to win the 1985 race but had then crashed, losing his chance of winning. Covered in mud, he offered his thoughts on the race to CBS' John Tesh after the race:


title: Paris–Roubaix

text: "It's a bollocks, this race!” said de Rooij. "You're working like an animal, you don't have time to piss, you wet your pants. You're riding in mud like this, you're slipping ... it’s a pile of shit".


title: Paris–Roubaix

text: Les Amis de Paris–Roubaix – the "friends" of the race – is an enthusiasts' group founded by Jean-Claude Vallaeys in 1983. It is based in France but open to members all over the world. It has its roots in the Paris–Roubaix Cyclo-Touriste of 1972. By 1982 there were 7,242 participants. There and at other events on the course, a petition calling for the cobbles to be saved gathered 10,000 signatures. Jean-Claude Valleys, Jean-François Pescheux and the Vélo-club de Roubaix, which Vallaeys founded in 1966, formed Les Amis de Paris–Roubaix in 1982 at a photo exhibition at the Maison du Nord-Pas de Calais in Paris.


title: Paris–Roubaix

text: Its aim was to find enough stretches of cobbled road to preserve the nature of the race. So many roads had been resurfaced that, as the organiser said, there was a risk that it would become a fast race on smooth roads won by sprinters rather than those who had fought through hell. Alain Bernard, who succeeded Vallaeys, says: "We have succeeded in that. Today, the association looks after the maintenance of these paths of legend, working with local administrations to preserve them. But alongside that, we also do other things to preserve the value of the race, building up an impressive collection of documents, holding exhibitions, honouring former winners, holding tours of the route."


title: Paris–Roubaix

text: Les Amis said in 2007 that it was too late to save the sector of Bersée, which was removed from the race because of its dangerous state in 2007, but restored two years later. The situation is becoming critical, it says, at the Pont Gibus at Wallers, at Mons-en-Pévèle, Pont Thibaut at Ennevelin, the pavé of the Duclos-Lassalle section at Cysoing, and at Camphin-en-Pévèle.


title: Paris–Roubaix

text: The weather is often bad for the riders; it is frequently worse for the "forçats du pavé" (literally 'convicts of the cobbles'), as they call themselves In March 2008, "Les Amis" published an account of their work thus:


title: Paris–Roubaix

text: Saturday 22 March: a cold wind sweeps the plain of Pévèle; alternating showers of hail, melted snow and cold rain. Not a day to venture outside... Nevertheless, at the foot of the Mons-en-Pévèle ridge, silhouettes busy themselves along the soaked cobble roadway. Backs bent against the gusts, they tirelessly scratch at the ground with primitive tools. Who are these dozen souls — A work-gang from a byegone era? Automatons? Treasure-hunters?


title: Paris–Roubaix

text: No, these are members of the "Amis de Paris Roubaix", trying to clean off the mud and crusted earth left on the cobbles by farm work. They are on an important section of Paris–Roubaix and, without their intervention, the greatest of cycling classics, due to be held in only a few days, will not be able to come through... And without these cobbled routes, the Paris–Roubaix would disappear, depriving the whole world of one of sport's most intense and gripping events. This they know, and they'll be back again the two weekends before the race, far from the media and officials who will soon bustle here.


title: Paris–Roubaix

text: The passion that drives them is much stronger than the bad weather. It has nothing to do with the current storms in the cycling world. These discreet servants of the "Queen of the Classics" have only one ambition — to clean the stones so that when the day comes for the champion to be crowned they can hold their cobbled trophy high.


title: Paris–Roubaix

text: Paris–Roubaix is sometimes compared to the other famous cobbled race, the Tour of Flanders in Belgium. Paris–Roubaix is flatter and has more difficult cobbles while the Tour of Flanders contains a series of hills, many on cobbles, like the Koppenberg or Kapelmuur.


title: Paris–Roubaix

text: In addition to Paris–Roubaix and the Tour of Flanders, called the "cobbled classics", other spring races like Omloop Het Nieuwsblad and Gent–Wevelgem feature extensive cobbles.


title: Paris–Roubaix

text: The Paris–Roubaix Cyclo is organised by the Velo Club de Roubaix every other June. This allows amateurs to experience the cobbles, the finishing laps in the vélodrome, and the showers. There is a choice of three levels: 120 km, most of the cobbled sectors; 190 km all the cobbles; or the full 261 km. All finishers receive a small cobblestone on a wooden plinth.


title: Paris–Roubaix

text: From 2021, Paris–Roubaix Femmes - a professional women's race in the UCI Women's World Tour - has been held on Saturday, with the men's race taking place on the Sunday. It follows the same terrain as the men's race, albeit over a shorter distance.


title: Austrian Air Force

text: The Austrian Air Force in its current form was created in May 1955 by the victorious Allied powers, subject to restrictions on its use of guided missiles. The Austrian State Treaty of 1955 committed Austria to permanent neutrality.


title: Austrian Air Force

text: Pilot training started out with a four Yak-11 "Moose" and four Yak-18 "Max" aircraft donated by the Soviet Union, and Austria purchased further light trainer types under the Military Assistance Program. Until 1960 Austria purchased training and support aircraft under the MAP, but no modern fighter aircraft; the role of a fighter was rather inadequately filled by the already outdated, 30 Saab 29 Tunnan bought second-hand from the Swedish Air Force in the early 1960s.


title: Austrian Air Force

text: From 1970, Austria purchased a total of 40 Saab 105 lightweight multi-role aircraft with the intention to deploy them in trainer, reconnaissance, interception and ground attack roles. As it became clear in the 1980s that the sub-sonic aircraft were inadequate for air combat and airspace interdiction, Austria purchased 28 reconditioned Saab 35 Draken fighter aircraft to supersede the Saab 105 as the Austrian Air Force's main interceptor in 1988. The Saab 105 remained in service as a trainer/surveillance aircraft.


title: Austrian Air Force

text: Shortly after, the Draken saw their first major use in airspace interdiction starting 1991 during the Yugoslav Wars, when Yugoslav MiG-21 fighters crossed the Austrian border without permission. In one incident on 28 June a MiG-21 penetrated as far as Graz, causing widespread demands for action. Following repeated border crossings by armed aircraft of the Yugoslav People's Army, changes were suggested to the standing orders for aircraft armament.


title: Austrian Air Force

text: Since 1955, Austria's armed forces had been forbidden to operate any guided missile system, including air-to-air missiles and surface-to-air missiles (SAMs). In the post-Cold War environment, and with gun-armed aircraft a relic of a past age, the Austrian Parliament voted to amend this section of its state treaty and in January 1993 modern AIM-9 Sidewinder missiles were ordered from Sweden to arm its fighter aircraft. A higher performance model of the Sidewinder were purchased directly from the United States; deliveries began in 1995. French Mistral SAMs were purchased to add ground-based protection against air attack. The first Mistrals arrived in 1993 and final deliveries were concluded in 1996.


title: Austrian Air Force

text: The helicopter fleet includes Agusta-Bell (AB) 204s (mainly used for medical evacuation), AB-206s (training and liaison), and AB-212s (used by air-mobile troops and for light transport). 28 French-made Alouette IIIs are available for search-and-rescue tasks, including high mountain operations. The 12 Bell OH-58 Kiowa, a scout helicopter, is mounted with a rapid-firing machine gun, but the air force lacks a true attack helicopter. Most of the helicopters, except the 24 AB-212s, are becoming obsolete. After the 1999 Galtür Avalanche, it became apparent that the Austrian Air Force's helicopter complement were too few in numbers and too limited in design. Therefore, 9 US-built UH-60 Blackhawk helicopters were purchased, to be used for transportation during disasters.


title: Austrian Air Force

text: In 2003 Austria's transport capability was improved when it purchased three C-130 Hercules from the Royal Air Force. These aircraft were needed for the demanding UN peacekeeping missions in which Austria played a role.


title: Austrian Air Force

text: In 2005, the Saab Draken fleet was retired (50 years after the type first flew), to be replaced by the Eurofighter Typhoon. Before the first delivery of Typhoons, 12 F-5 Tiger II were leased from Switzerland as a stopgap measure. The Eurofighter purchase was subject to controversy in Austria, and became a political football for some time, but the 15th and final aircraft was delivered on 24 September 2009. As of 2017, possible corruption affairs surrounding the Eurofighter procurement are still being investigated by the Austrian parliament. In July 2017, as a result of the ongoing controversy, the Austrian Ministry of Defense announced the phasing out of the Typhoon starting 2020, and its replacement by a "militarily more effective and more cost-efficient" air surveillance system.


title: Austrian Air Force

text: Austria's air force is divided into two brigade-level formations: the Air Surveillance Command () in Salzburg tasked with the defense of the Austrian airspace and the Air Support Command () in Hörsching Air Base with helicopters and transport planes.


title: Austrian Air Force

text: All personnel destined to enter service with the Air Force is trained by the Air and Air Defense Personnel School () based at Langenlebarn Air Base. The school is under direct command and control of the Ministry of Defense and controls two flying units:


title: Austrian Air Force

text: After 50 years of service the Austrian Air Force has retired without replacement its Saab 105OE aircraft in January 2021 and has disbanded its Jet Trainer Squadron () at Linz - Hörsching Air Base, which has operated the type. The Squadron's younger pilots will re-qualify for the Eurofighter, the older pilots and the aircraft technicians will re-qualify for the AB 212 helicopters and advanced jet flying training will be outsourced to the Italian Air Force's MB.339 and T.346 jet trainers operating from Lecce - Galatina and Decimomannu.


title: Austrian Air Force

text: Vogler Air Base, north of the town of Hörsching west of Linz, was built as a base for the German Luftwaffe 1938–1940. After the war the USAAF used the base, then named "Camp McCauley – Hörsching" and housing displaced persons, until 1955 when it was returned to the Austrian government.


title: Austrian Air Force

text: Initially used exclusively by the ground forces, the first military aircraft, Yak-18 "Max-A", arrived in 1957. The base was named for First Lieutenant Walter Vogler in 1967.


title: Austrian Air Force

text: The German-built base structures were used jointly by the military and civilian aviation until the 70s when construction of the new civilian area in the northern part of the base was finished.


title: Austrian Air Force

text: FH Vogler is the largest base of the . It houses "3", responsible for overhauls and maintenance of the C-130K Hercules and AB-212.


title: Austrian Air Force

text: Units currently based here are the C-130K Hercules of 4th Air Transport Squadron, Flight Regiment 3; and the AB-212 of 1st and 2nd Helicopter Squadron, Flight Regiment 3.


title: Austrian Air Force

text: Fliegerhorst Brumowski, south of the town of Langenlebarn northwest of Vienna, was built as a base for the Luftwaffe 1938–1940. After the war the base was briefly occupied by Soviet troops before it was taken over by the USAAF, becoming "Air Force Station Tulln – Vienna". In 1946 Pan Am added the base as a destination, and for a short time there were regular flights New York City/Langenlebarn.


title: Austrian Air Force

text: The first Austrian aircraft to arrive were Yak-11 "Moose" and Yak-18 "Max-A" trainers donated by the Soviet Union and Agusta Bell AB47G2 helicopters in late 1955. The base was named for Captain Godwin Brumowski in 1967.


title: Austrian Air Force

text: The base is the headquarters of the (Air Support Wing); it also houses the (Federal School for Aeronautical Engineering) and "1", responsible for overhauls and maintenance of the , S-70A-42 Black Hawk and OH-58B Kiowa.


title: Austrian Air Force

text: Units currently based here are the of 4th Air Squadron, Flight Regiment 1; the S-70A-42 Black Hawk of 1st Helicopter Squadron, Flight Regiment 1, and OH-58B Kiowa of 3rd Helicopter Squadron, Flight Regiment 1.


title: Austrian Air Force

text: Fliegerhorst Hinterstoisser, located north of Zeltweg in a region known as Aichfeld, was built as a base for the Air Force of the 1936–1938. The base was occupied by Soviet troops in the aftermath of the war, but then transferred to the RAF which used the base until 1947.


title: Austrian Air Force

text: In spring 1957 the first aircraft, Piper PA-18/95 Super Cub and Zlin Z-126 Trener, arrived with "1" at the base. Since then it is the main base for the training of new aviators. It was named for Colonel Franz Hinterstoisser in 1967.


title: Austrian Air Force

text: The base houses the 1st Squadron of the (Surveillance Wing). With the retirement of the Saab 35 Draken in 2005 the unit now uses the Eurofighter Typhoon. The first Eurofighter Typhoon arrived in July 2007. The base also houses parts of "2", responsible for overhauls and maintenance of the Saab 105Oe and the Pilatus PC-7 Turbo Trainer, as well as flight school for basic training.


title: Austrian Air Force

text: Units currently based here are the PC-7 Turbo Trainer of flight school; detachments of 2nd Squadron and the Eurofighter Typhoon.


title: Austrian Air Force

text: Fiala-Fernbrugg Air Base (), located north of the town of Aigen im Ennstal on the southern edge of the Totes Gebirge, was built as a base for the Air Force of the 1936–37. At the end of World War II the base became the home base of the only helicopter unit of the Luftwaffe; beginning the tradition of helicopter operations at Aigen im Ennstal.


title: Austrian Air Force

text: Soviet troops occupied the base after the war, but after only a few weeks control switched to US forces. After a few more weeks, the base ended up in British hands. The RAF rebuilt the base and handed it over to Austria in 1947. It was used as a storage depot for the B-Gendarmerie, a paramilitary police force in the western zones.


title: Austrian Air Force

text: After some years of hiatus, the first helicopters, Bell H-13H Sioux arrived in late 1960. The base was named for Captain Benno Fiala von Fernbrugg in 1967.


title: Austrian Air Force

text: The base houses , responsible for overhauls and maintenance on the AS-316B Alouette III. (Alpine landing courses) are conducted at least annually at the base, with officers of foreign air forces as regular attendants.


title: Austrian Air Force

text: Wiener Neustadt Air Base was located northwest of the city and was one of the first airports on the European continent. It opened in 1910 and housed units of the Austro-Hungarian Imperial and Royal Aviation Troops. The base was close to the Wiener Neustädter Flugzeugwerke (WNF) factory in the eastern part of the city and which manufactured Messerschmitt Bf 109s and repaired Junkers bombers and destroyers during World War II. The base was bombed to total destruction during World War II and was rebuilt by the Soviets who operated the base until 1955. The Austrian military took the base over, but didn't use it until 1961. The base houses no units, but Flight Regiment 1's are operating from the base for flight training purposes as well as for training of army parachutists.


title: Austrian Air Force

text: The large Liechtenstein Kaserne on the northern edge of the (Training Area, Gunnery and Bombing Range) is home to a detachment of Flight Regiment 1's OH-58B Kiowas. Known as , it is not only used by military aircraft; helicopters of the Ministry of the Interior also use the base for operations; the task of border surveillance is jointly conducted by the military and civilian authorities. Besides that, Flight Regiment 1's helicopters and are operating from the base or its adjacent meadows and roads on a regular basis.


title: Austrian Air Force

text: The Frundsberg Kaserne in the southern suburbs of Schwaz east of Innsbruck houses a detachment of Flight Regiment 2's AS-316B Alouette IIIs used for SAR and firefighting duties. The helicopters are operating from Schwaz since 1969. The base is earmarked for closure, with the helicopters being redeployed to the nearby Andreas Hofer Kaserne.


title: Austrian Air Force

text: Radar installations sites for mobile air defense systems are also maintained by the Austrian Air Force.


title: Austrian Air Force

text: This radar site is located atop the Kolomannsberg (1,114 m) on the border between Salzburg and Upper Austria north of Thalgau in a region known as the Flachgau. The site is active since January 1968, and provides 24/7 air surveillance since August 1968. Initially a French CSF (now Thales Group) RV376 and a British Marconi (now BAE Systems) S244 height finder were used at the site but were replaced by an Italian Selenia (now Alenia) RAT-31S 3D-radar in 1983. The system was further enhanced with the installation of a Selenia (now Alenia) RAT-31DL 3-D radar in 2003.


title: Austrian Air Force

text: The site is also called (lit. large space radar station) since it has the necessary office and working areas for a complete air traffic/combat control center. It served in this role until 1987 when its tasks were taken over by the EZ/B and is still maintained to provide backup when needed.


title: Austrian Air Force

text: This radar site is located atop the Speikkogel (2,140 m) on the border between Styria and Carinthia in the Koralpe mountains west of Wolfsberg. The site is active since 1986, with its construction and commission severely hampered by the bad weather in the region (partly due to the height above SL) and problems with the radom and the radar itself. A Selenia (now Alenia) RAT-31S 3-D radar is installed, scheduled to receive the RAT-31DL upgrade. The site features a downsized version of the ORS K's control center, but is normally not manned.


title: Austrian Air Force

text: This radar site is located atop the Steinmandl (490 m) north of Ernstbrunn in the Leiser Berge region 40 km north of Vienna. The site is active since 1985; to the immediate west a secondary radar operated by AustroControl is located atop the Buschberg. The ORS uses a Selenia (now Alenia) RAT-31S 3-D radar, but is scheduled to receive the RAT-31DL upgrade. The site was a replacement for the unbuilt one atop the Schneeberg south of Vienna.


title: Austrian Air Force

text: In July 2017, the Austrian Defense Ministry announced that it would replace all their Eurofighter Typhoons by 2020, they stated continued use of its Typhoons over their 30 year life span would cost about 5 billion Euros with the bulk being spent on maintenance. (The cost to operate the aircraft was estimated at $73,000 USD per hour). The Ministry believes that buying a new airframe would save 2 billion Euros over the same period. Austria plans to explore a government-to-government sale or lease agreement, where they could avoid a lengthy and costly tender process with a manufacturer. Possible replacements include the Saab Gripen and the Lockheed Martin F-16. On July 6, 2020 Defence Minister Klaudia Tanner said that the Air Force’s 15 Eurofighters will remain in service as long as the current contract with Airbus runs, as it would be prohibitively expensive to withdraw from the contract. Austria has taken steps to sell the aircraft: "A report in the Austrian newspaper Krone yesterday, September 6, 2020, stated that Defense Minister Klaudia Tanner of the Austrian People’s Party had officially confirmed Indonesia’s interest in buying the 15 Typhoons and that she plans to enter “concrete sales negotiations” with her counterpart in Jakarta". In October 2022, it was reported that Austria was considering purchasing another three Eurofighters.


title: Austrian Air Force

text: As a backup to the fixed sites and to create a better situation image in times of crisis two Selenia (now Alenia) MRCS-403 3-D radars – mobile versions of the RAT-31S – are operated by the since 1979. Several sites (Irrsberg, Hochwechsel) are prepared to host the radar.


title: Austrian Air Force

text: To provide detection of low-flying aircraft the operates six Thomson-CSF (now Thales Group) RAC 3D medium-range 3-D radars – designated TER – mounted on ÖAF 32.403 trucks beginning in 1998. A further 16 of these systems are operated by the Air Defense Bataillons in special target designation configuration to provide early warning and target tracking for the Mistral units.


title: Austrian Air Force

text: The Z/FlAK 85 (/) is the Oerlikon Contraves GDF-001 system. 18 of these guns were purchased in 1965 (under the designation Z/FlAK 65) and used with earlier acquired Oerlikon Contraves FLGer 60 (, fire control radar) Super Fledermaus and new Oerlikon Contraves FLGer 65 Improved Super Fledermaus.


title: Austrian Air Force

text: A second batch of these guns consisting of a further 18 Z/FlAK 65, but this time with FLGer 69, a further improved Super Fledermaus, was purchased in 1973. The FLGer 69 were never issued to the units but returned to the manufacturer in 1973 for eventual replacement with the new FLGer 75 Skyguard beginning in 1976. Improved Skyguards, dubbed FLGer 79 were purchased in 1981.


title: Austrian Air Force

text: The guns itself were upgraded to GDF-005 standard in 1987, designated Z/FlAK 85 since. The FLGer 75/79 have been upgraded to FLGer 98 in the late 90s. A total of 72 guns and 37 Skyguard systems have been acquired over the years.


title: Austrian Air Force

text: The lFAL () Matra (now MBDA) Mistral is not only used with the Air Defense Regiments of the Air Force, but also with the air defense batteries attached to the HQ units of the Army's Brigades.


title: Austrian Air Force

text: 72 of these systems – along with several hundred missiles, the exact number is unknown – have been purchased beginning in 1993. They are used in conjunction with the so-called ZZR (), 16 Thomson-CSF (now Thales Group) RAC 3D medium range 3-D radars in target designation configuration which were purchased in 1998. Nine missile launchers and two radars normally operate in a battery.


title: Statistical area (United States)

text: The United States federal government defines and delineates the nation's metropolitan areas for statistical purposes, using a set of standard statistical area definitions. the U.S. Office of Management and Budget (OMB) defined and delineated 392 metropolitan statistical areas (MSAs) and 547 micropolitan statistical areas (μSAs) in the United States and Puerto Rico. Many of these 939 MSAs and μSAs are, in turn, components of larger combined statistical areas (CSAs) consisting of adjacent MSAs and μSAs that are linked by commuting ties; 551 metropolitan and micropolitan areas are components of the 175 defined CSAs. A collective term for MSAs, μSAs, and CSAs is primary statistical areas (PSAs), though that term is not used by OMB.


title: Statistical area (United States)

text: Metropolitan and micropolitan statistical areas are defined as consisting of one or more adjacent counties or county equivalents with at least one urban core area meeting relevant population thresholds, plus adjacent territory that has a high degree of social and economic integration with the core, as measured by commuting ties. A metropolitan statistical area has at least one urban core with a population of at least 50,000. In a micropolitan statistical area, the largest urban core has a population of at least 10,000 but less than 50,000.


title: Statistical area (United States)

text: The sortable table below shows the number of metropolitan and micropolitan statistical areas in each of the U.S. states, the District of Columbia, and Puerto Rico. For each jurisdiction, it lists:


title: Statistical area (United States)

text: Please note: Because many metropolitan and micropolitan areas overlap jurisdictional boundaries, columns are not additive.


title: Statistical area (United States)

text: The following lists the 569 primary statistical areas (PSAs) of the United States with the following information:


title: Statistical area (United States)

text: The following lists the five primary statistical areas (PSAs) of Puerto Rico with the following information:


title: Florida Gators

text: The Florida Gators are the intercollegiate athletic teams that represent the University of Florida, located in Gainesville. The University of Florida, its athletic program, its alumni and its sports fans are often collectively referred to as the "Gator Nation." The Gators compete in the National Collegiate Athletic Association (NCAA) and the Southeastern Conference (SEC) and are consistently ranked among the top college sports programs in the United States. The University of Florida currently fields teams in nine men's sports and twelve women's sports.


title: Florida Gators

text: All Florida Gators sports teams compete in NCAA Division I, and 20 of the 21 Gators teams compete in the SEC. The sole University of Florida sports team that does not play in the SEC is the women's lacrosse team, which joined the American Athletic Conference beginning in the 2019 lacrosse season because the SEC does not sponsor competition in the sport. The University of Florida was one of the thirteen charter members who joined together to form the new Southeastern Conference in 1932. Previously, the university was a member of the Southern Intercollegiate Athletic Association from 1912 to 1921 and the Southeastern Conference from 1922 until the SEC began play in the fall of 1933.


title: Florida Gators

text: All Florida Gators sports teams have on-campus facilities, and most are located on or near Stadium Road on the north side of campus, including Steve Spurrier-Florida Field at Ben Hill Griffin Stadium for football; the Exactech Arena at the Stephen C. O'Connell Center for basketball, gymnastics, swimming and diving, indoor track and field, and volleyball; and James G. Pressly Stadium for soccer and outdoor track and field. The Katie Seashole Pressly Softball Stadium, the Florida Ballpark, and the Donald R. Dizney Stadium for lacrosse are located on Hull Road on the southwestern side of the campus. The Mark Bostick Golf Course and Scott Linder Stadium for tennis are located on S.W. Second Avenue on the northwestern side of the campus.


title: Florida Gators

text: The Florida Gators athletic program is administered by the University Athletic Association, Inc. (UAA), a private non-profit corporation that reports to the president of the university and its board of trustees. For the 2014–15 school year, the UAA had an operating budget of $103,310,001, projected revenues of $104,064,487, and made a $3.5 million contribution to the university's general fund. Scott Stricklin has been Florida's athletic director since 2016.


title: Florida Gators

text: Beginning in the early 1990s, the Florida Gators has been recognized as one of the premier athletic programs in the Southeastern Conference (SEC) and one of the best in the nation. The SEC has awarded an All-Sports Trophy to the best overall sports program in the conference since 1984, and Florida has won the award 27 times as of 2017. Florida is the only school in the SEC and one of four schools nationally to have won a national championship in the football, men's basketball, and baseball. Every year since 1993, the National Association of Collegiate Directors of Athletics (NACDA) has recognized the Gators athletic program as one of the ten best overall Division I athletic programs in the country in its annual NACDA Directors' Cup standings, including as runners-up on four occasions. The men's athletic program was also the winner of the 2010–11 and 2011–12 Capital One Cup; the women's athletic program won the Capital One Cup in 2013–14, and both programs have placed in the top five in the standings on several other occasions.


title: Florida Gators

text: Among the Gators' recent national championships, the Florida Gators men's basketball team won the 2006 and 2007 NCAA Division I men's basketball tournaments, and the Florida Gators football team won 2007 BCS National Championship Game in football, all in the span of 366 days. Florida is the only school in NCAA Division I history to hold the outright men's basketball and football championships during the same school year. In January 2009, the Gators football team won the 2009 BCS National Championship Game with a 24–14 victory over the Oklahoma Sooners. The Florida Gators baseball team took home its first championship at the 2017 NCAA championship, defeating rival SEC foe the LSU Tigers in two games. The Gators won the NCAA men's indoor track and field championship in three consecutive seasons in 2010, 2011 and 2012, the NCAA women's swimming and diving team national championship in 2010, the NCAA women's tennis championship in 2011, 2012 and 2017, the NCAA men's outdoor track and field championship in 2012, 2013 and 2016, the NCAA women's gymnastics championship in 2013, 2014 and 2015, and the NCAA softball championships in 2014 and 2015. Individual Gator athletes have won 279 individual NCAA championships in boxing, golf, gymnastics, swimming and diving, tennis, and track and field. In 2017, Florida won the baseball national championship for the first time. The Gators swept LSU in the best of three national title series. This earned the university a total of 39 national team championships. Additionally, this put the University of Florida in exclusive company. With the 2017 baseball national championship, Florida became only the fourth school in history to win national championships in football, men's basketball, and baseball. Florida, along with Michigan, Ohio State, and UCLA are the only schools to ever achieve triple crown status. This also put Florida in a category by itself. Florida is the only Southeastern Conference school to accomplish this feat, as well, Florida is the only school in history to have achieved triple crown status in such a short span of time. Florida's first national championships in each of the big three sports (football, men's basketball, and baseball) were won in the span of just 21 years.


title: Florida Gators

text: The University of Florida has been ranked among the nation's top ten NCAA Division I athletic programs every year since 1983–84, an overall ranking that includes both men's and women's sports – the only college sports program ranked in the top ten in the United States for the last thirty-two consecutive years. The National Association of Collegiate Directors of Athletics (NACDA) has recognized the University of Florida as ranking among the top seven Division I programs in its NACDA Directors' Cup standings every year since the NACDA and "USA Today" began awarding the cup in 1993–94. The 2014–15 academic year marked the Gators' thirty-second consecutive year ranked among the nation's top ten best overall collegiate athletic programs, and the twenty-second consecutive year ranked among the top seven Division I programs in the NACDA Directors' Cup standings. Only one other Division I athletic program has matched that feat, and Florida has achieved this record while fielding fewer sports teams than many of the other perennially top-ranked collegiate athletic programs. In the twenty-two years of the NACDA Directors' Cup, the Gators have finished fifth or better in fifteen years, and have never finished lower than seventh; the only other program ranked among the top ten Division I programs every year since 1993–94 is the Stanford Cardinal sports program of Stanford University.


title: Florida Gators

text: Through the end of the 2017–18 school year, the Florida Gators have won 240 Southeastern Conference (SEC) team championships, the most in conference history.


title: Florida Gators

text: The SEC All-Sports Trophy began in 1973 as the Bernie Moore Trophy and tabulated the league's best men's sports program. In 1983, the SEC also began recognizing the best women's sports program in the conference, as well as the best overall SEC sports program. In 1994, the "New York Times" Regional Newspaper Group assumed responsibility for awarding the trophies. In the 39-year history of the awards, Florida has won nineteen Women's Trophies, sixteen Men's Trophies, and twenty-five Overall SEC All-Sports Trophies (including twenty-five of the last twenty-seven).


title: Florida Gators

text: SEC rival Georgia won the overall 2005–06 All-Sports Trophy to snap Florida's record streak at fourteen straight (1990–91 through 2004–05). Florida reclaimed the SEC All-Sports Trophy for the 2006–07 school year, and the Gators again swept the overall, men's, and women's all-sports trophies in every year since. The Gators are the only SEC sports program to earn all three SEC all-sports trophies in a single year, and have swept all three trophies twelve times.


title: Florida Gators

text: In their history of intercollegiate competition, the University of Florida's varsity athletic teams have won 45 national championships (including 40 sponsored by the National Collegiate Athletic Association (NCAA), two by the Association for Intercollegiate Athletics for Women (AIAW), two by the Bowl Championship Series (BCS), and one by the Bowl Alliance). Florida is one of only two Division I schools to hold both major men's championships (football and men's basketball) at the same time (as the 2006 BCS football champions and the 2006 and 2007 NCAA men's basketball champions), and was the first to do so.


title: Florida Gators

text: All of the national championships listed above were sponsored by the NCAA other than football in 1996 (Bowl Alliance), football in 2006 and 2008 (BCS), women's gymnastics in 1982 (AIAW), and women's swimming and diving in 1979 (AIAW).


title: Florida Gators

text: The University of Florida is a founding member of the Southeastern Conference (SEC), one of the nation's premier intercollegiate sports conferences, and 20 of the 21 Gators sports teams compete in the SEC. Since the SEC began play in 1933, Florida's varsity athletic teams have won 255 SEC team championships, more than any other conference member. The women's lacrosse team played its first four seasons in the now-defunct American Lacrosse Conference (ALC), followed by four seasons in the Big East Conference, and joined the American Athletic Conference for the 2019 season (2018–19 school year) and beyond. All of Florida's conference championships are from the SEC unless otherwise stated.


title: Florida Gators

text: For purposes of counting "official" SEC team championships in baseball, men's and women's basketball, soccer, softball, men's and women's tennis, and volleyball, the SEC currently only includes regular season team championships, not tournament championships. The Gators have won an additional 66 SEC tournament titles in these sports which are not included in Florida's total of 255 SEC team championships.


title: Florida Gators

text: Coach Andy Lopez took over the Gators baseball program in 1994, one season after leading the Pepperdine Waves to their first College World Series championship. In 1996, he coached the Gators to a fifty-win season and a College World Series bid. By 2000, the program had seemingly hit a plateau and Lopez was replaced.


title: Florida Gators

text: Pat McMahon became the Gators' head coach in 2001 after coaching the Mississippi State Bulldogs. The Gators' 2005 baseball season was the most successful to date, with the team winning the SEC title, and earning a place in the College World Series for the fifth time in school history. The team advanced to the championship round against the Texas Longhorns, ultimately losing two games to none.


title: Florida Gators

text: Following their 2005 College World Series run, the Gators opened the 2006 season ranked number one in the polls, but struggled to finish 28–28 record (10–20 SEC), and failed to qualify for the NCAA Regionals. After missing the NCAA Regionals again in 2007, McMahon was dismissed.


title: Florida Gators

text: Former Clemson Tigers associate head coach Kevin O'Sullivan became the Gators' new head baseball coach on June 13, 2007. O'Sullivan's Gator teams showed immediate improvement and the Gators finished the 2008 regular season with a 30–24 record (17–13 SEC), and received an invitation to the NCAA Regional in Tallahassee. The 2009 squad finished the regular season with a 38–18 record (19–11 SEC), won the NCAA Regional in Gainesville, and advanced to the Super Regional before losing to the Southern Mississippi Golden Eagles.


title: Florida Gators

text: O'Sullivan's 2010 and 2011 Gators finished their SEC regular season play with identical 22–8 records, won the program's eleventh and twelfth SEC championships, earned a berth in the College World Series in both years, and advanced to the College World Series final in 2011 before getting swept by SEC foe South Carolina in two games.


title: Florida Gators

text: The Gators returned to the College World Series in 2012, 2015, 2016, 2017, and 2018. In June 2017, Florida won their first College World Series championship, sweeping fellow SEC rival LSU in two games in the best-of-three finals.


title: Florida Gators

text: Since 2021, the Gators baseball team plays its home games at Condron Ballpark at Alfred A. McKethan Field.


title: Florida Gators

text: Florida enjoyed limited success in men's basketball before the mid-1980s. Coach Norm Sloan's Gators were invited to the National Invitation Tournament (NIT)—only the second time the team had been invited to a post-season tournament. They returned to the NIT in 1985 and 1986, and made their first appearance in the NCAA basketball tournament in 1987, when guard Vernon Maxwell led the team to the NCAA Sweet Sixteen. The Gators received invitations to the NCAA tournament in 1988 and 1989, but Sloan was forced to resign at the outset of the 1989–90 season as a result of NCAA infractions.


title: Florida Gators

text: Coach Lon Kruger brought renewed success, and his Gators reached the NIT final four in 1991–92. During the 1993–94 season, Andrew DeClercq and Dametri Hill led the Gators to their first NCAA Tournament Final Four following a dramatic overtime victory over the Connecticut Huskies in the NCAA regional semifinal, and a 74-66 win over Boston College in the NCAA Regional Final.


title: Florida Gators

text: Athletic Director Jeremy Foley hired Billy Donovan as Kruger's replacement in 1996. Donovan's recruiting prowess was evident early, as he landed recruiting classes with future NBA players Mike Miller, Udonis Haslem and Matt Bonner. Donovan's Gators advanced to the NCAA tournament final before losing to the Michigan State Spartans in 2000, and demonstrated a consistency previously unknown to the program as they received invitations to the NCAA tournament every year from 1999 through 2007, a team-record eight-year streak.


title: Florida Gators

text: Under Donovan, Florida won its first SEC men's basketball tournament in 2005, when they beat the Kentucky Wildcats in the SEC title game.


title: Florida Gators

text: The 2005–06 team's 17–0 start was the best in team history, surprising many with a young, selfless squad led by four sophomores. The team began the season unranked, but won its second consecutive SEC Tournament championship. On April 3, 2006, the Gators defeated the UCLA Bruins 73–57 in the NCAA tournament championship game to win Florida's first men's national basketball championship. Within days, all five starters announced they would return for another season to try to win back-to-back championships.


title: Florida Gators

text: At the outset of the 2006–07 season, the Gators were ranked No. 1 in both major polls for the first time. The Gators won their second consecutive NCAA national men's basketball championship on April 2, 2007, defeating the Ohio State Buckeyes 84–75. They became the first team since Duke in 1991–92 to win back-to-back tournaments and the first in NCAA history to do so with the same starting line-up. Afterward, Florida's four star juniors announced they would enter the NBA draft.


title: Florida Gators

text: Donovan's Gators returned to championship form in 2010–11, winning the program's fifth regular season SEC championship and fighting their way to the NCAA regional semifinals (the "Elite Eight") before losing 74–71 to the Butler Bulldogs in overtime. In 2011–12, the Gators received a No. 7 seed in the NCAA tournament, exceeded expectations, and advanced to the Elite Eight, headlined by the play of star freshman Bradley Beal and point guard Erving Walker, before falling to Louisville. During the 2013–14 season, the Gators were ranked No. 1 in the nation, won the SEC championship with 36 straight wins during the regular season and tournament games, went on a school-record 30 game winning streak, and advanced to the NCAA Final Four, headlined by the play of four senior veterans: Patric Young, Scottie Wilbekin, Will Yeguete and Casey Prather.


title: Florida Gators

text: The Gators men's basketball team is currently led by coach Todd Golden and plays its home games in the O'Connell Center.


title: Florida Gators

text: Women's basketball was approved as a new women's varsity sport by Florida in March 1972, but began play in 1973 as a club team. In 1975, the Lady Gators debuted as a varsity program under head coach Dr. Paula Welch. The Gators made local headlines in 1976 by winning the "state championship", beating the other three women's college teams located in the state at that time.


title: Florida Gators

text: While traditionally being overshadowed by conference and national basketball powers Tennessee and Georgia, the Lady Gators have made several NCAA tournament appearances and sent players to the WNBA, including DeLisha Milton-Jones. Carol Ross compiled more wins than any other women's basketball coach in Florida's history, and guided the team for twelve seasons from 1991 to 2003, but left to accept the head coaching job at her alma mater, Ole Miss.


title: Florida Gators

text: From 2002 through 2006, the women's basketball team was coached by Carolyn Peck, a former WNBA coach who won a national title with Purdue. Peck was fired midway through the 2006 season (though allowed to finish the season) after enduring the worst losing streak of any Gator sport.


title: Florida Gators

text: Former Gator player and previous Charlotte coach Amanda Butler was named the new women's basketball coach on April 13, 2007. During the 2008–09 season, the Lady Gators received an NCAA tournament bid, and won a first-round game before being defeated by eventual tournament champion Connecticut in the second round. Butler resigned after the 2016–17 season.


title: Florida Gators

text: The Gators women's basketball team is currently coached by Kelly Rae Finley, and plays its home games in the O'Connell Center.


title: Florida Gators

text: The Florida Gators men's cross country team has won three Southeastern Conference championships (SEC), and has competed in eight National Collegiate Athletic Association (NCAA) tournaments. The women's cross country team has also won five SEC Championships (including the most recent), and competed in ten NCAA tournaments. In 2009, the Lady Gators placed seventh at the NCAA cross country championship; in 2010, they won the SEC championship for the second consecutive year.


title: Florida Gators

text: The University of Florida fielded an official varsity football team for the first time in 1906, defeating the Gainesville Athletic Club 6–0 in its first game. Since then, the Gators have played in thirty-seven bowl games, won three national championships (1996, 2006, 2008) and eight Southeastern Conference championships, and produced 138 All-Americans, forty-two National Football League (NFL) first-round draft choices and three Heisman Trophy winners.


title: Florida Gators

text: The Gators' most prominent current football rivals are SEC Eastern Division foes Georgia and Tennessee, annual SEC Western Division opponent LSU, and in-state rival Florida State from the Atlantic Coast Conference (ACC). Florida has historically shared rivalries with Auburn and Miami, too, but those games are no longer played annually and have lessened in intensity.


title: Florida Gators

text: The Gator football team has obtained more victories than any other program in the NCAA's Football Bowl Subdivision (FBS) since 1990, the year Heisman Trophy-winning quarterback Steve Spurrier returned to his alma mater as head coach. The 1996 team, coached by Spurrier and led by another Gator Heisman-winner, Danny Wuerffel, finished with a 12–1 record and won the national championship in the Sugar Bowl, beating rival Florida State 52–20. This win was not without controversy from FSU fans who thought they should not have had to play Florida again. FSU won in the regular season match up 24–21.


title: Florida Gators

text: Urban Meyer became Florida's head football coach in December 2004, and his six teams had great success from 2005 to 2010. The 2006 team won the school's second national championship on January 8, 2007, defeating the number one-ranked Ohio State Buckeyes 41–14. Quarterback Tim Tebow won the Heisman Trophy in 2007, and Florida's Tebow-led 2008 team won the 2009 BCS National Championship Game on January 8, 2009, beating the top-ranked Oklahoma Sooners 24–14, for the Gators' third national championship.


title: Florida Gators

text: The Gators have won the SEC Championship Game seven times in eleven appearances since the SEC instituted the championship game in 1992. The Gators won their first official conference title in 1991, the year before the first SEC conference championship game was played, for a total of eight SEC championships in the last twenty-four seasons.


title: Florida Gators

text: The Gators football team plays its home games in Steve Spurrier-Florida Field at Ben Hill Griffin Stadium, more popularly known as The Swamp, the team's home field since 1930. Billy Napier is the current head coach of the Florida Gators football.


title: Florida Gators

text: The men's golf team has won four NCAA Championships (1968, 1973, 1993, 2001), and has produced two individual NCAA champions, Bob Murphy in 1966 and Nick Gilliam in 2001. The men's golf team has also won fifteen Southeastern Conference (SEC) championships: 1955–56, 1968, 1973–75, 1985, 1989, 1991–94, 1999, 2003, 2011.


title: Florida Gators

text: Numerous former Gator golfers have represented the University of Florida on the PGA Tour, and the program has produced over thirty male athletes who have competed professionally.


title: Florida Gators

text: Buddy Alexander, the long-time head coach for the men's golf team, retired after the 2013–14 season. J. C. Deacon was hired in June 2014, and the 2014–15 season will be his first. The Gators men's golf team plays its home matches at the Mark Bostick Golf Course (formerly known as the "University Golf Course").


title: Florida Gators

text: The women's golf team has won two NCAA team championships (1985, 1986), and has produced one individual NCAA champion, Page Dunlap. The women's golf team has also won eight Southeastern Conference (SEC) championships.


title: Florida Gators

text: Former Lady Gator golfers have regularly represented the University of Florida on the LPGA Tour, and the program has produced over twenty female athletes who have competed in the professional ranks.


title: Florida Gators

text: Emily Glaser is the head coach for the women's team. The Gators women's golf team plays its home matches at the Mark Bostick Golf Course (formerly known as the "University Golf Course").


title: Florida Gators

text: Gymnastics was one of the first women's sports added at the University of Florida and achieved early success by winning the 1982 AIAW national championship. Since the NCAA assumed sponsorship of the national gymnastics championships in 1982, Florida has typically earned invitations to the NCAA national championships (top twelve teams nationally), and advanced to the NCAA "Super Six" eighteen times. Florida won the 2013, 2014 and 2015 NCAA national championships, finished in second-place in 1998 and 2012, and the team has only failed to qualify for the NCAA championships once in the past thirty-three seasons.


title: Florida Gators

text: The Gators have won a total of nine SEC gymnastics championships since 1982. The team's biggest SEC rivals are Alabama and Georgia, both of which are also perennial national contenders. Coach Rhonda Faehn's Gator gymnasts were the SEC champions in both 2012 and 2013, and won the NCAA national team championships in 2013, 2014 and 2015 (the 2014 was as co-champions with Oklahoma).


title: Florida Gators

text: Faehn coached the Gators gymnastics team from 2003 to 2015. Under Faehn, the Gator gymnasts have been nationally competitive and remarkably consistent—finishing in the top seven every year and winning three NCAA national championships.


title: Florida Gators

text: The Gators gymnastics team is now coached by Jenny Rowland. The Gators hold their home meets at the O'Connell Center.


title: Florida Gators

text: In June 2006, the University Athletic Association announced the creation of the new Gators women's lacrosse program, citing the growth of high school lacrosse across the country and the increased availability of Division I competition. Florida became the second Southeastern Conference member university to offer lacrosse as a varsity sport, following Vanderbilt University, and played its inaugural 2010 season in the American Lacrosse Conference (ALC) together with the Commodores. The Gators and Commodores continued to play in the ALC until conference realignment led to the demise of the ALC following the 2014 season, after which both programs became single-sport members of the Big East Conference. The Gators and Commodores played in the Big East through the 2018 season, after which they became charter members of the new women's lacrosse league of the American Athletic Conference (The American). The lacrosse team is the only Gators team that does not compete in the SEC.


title: Florida Gators

text: The Gators lacrosse team has had success over its first nine seasons, including four ALC regular-season championships (2011–2014); two ALC tournament titles (2012, 2014); both regular-season and tournament titles in all of its four Big East seasons (2015–2018); the regular-season and tournament championships in three of its first four seasons as a member of the AAC (2019, 2021, 2022); seven appearances in the NCAA tournament national quarterfinals ("Elite Eight"); and one appearance in the NCAA tournament national semifinals ("Final Four") in 2012. The Gators have been ranked among the top ten women's lacrosse teams in the country each of the last five seasons, and have been ranked as high as No. 1.


title: Florida Gators

text: Amanda O'Leary is the Gators' head coach. Before she was named to jump-start Florida's new program, O'Leary was the head coach at Yale University for 14 seasons, and was honored as a two-time All-American midfielder at Temple University, where she led her team to an NCAA championship in 1988.


title: Florida Gators

text: In only the second season of the Gators lacrosse program, the mostly-sophomores team defeated the defending ALC champion Northwestern Wildcats to win their first regular season conference championship. The Gators completed a perfect 5–0 season three days later by defeating the Vanderbilt Commodores in Nashville.


title: Florida Gators

text: Becky Burleigh was hired to be the first head coach since the women's soccer team began play a varsity sport in 1995. Under Burleigh's leadership, the team quickly became a national contender. In 1998, in the program's fourth season, the Gators won the NCAA national championship by defeating the defending national champion North Carolina Tar Heels 1–0 in the national finals of the NCAA tournament. The women's soccer team has also won thirteen Southeastern Conference regular season championships and ten SEC tournament titles in its eighteen seasons of play. In 2021, Burleigh retired as women's soccer coach.


title: Florida Gators

text: Notable former Gator soccer players include Abby Wambach, who was a member of the U.S. women's national team and scored the game-winning goal in the final game of the 2004 Olympic Games in Athens, Greece; Heather Mitts, who played for the gold medal U.S. national team in the 2008 Olympics in Beijing, China; Melanie Booth, who currently plays for the Canadian women's national soccer team; and Danielle Fotopoulos, who played professionally with the Carolina Courage. Savannah Jordan plays for the Portland Thorns and ranks second in Gators history behind Wambach for goals scored.


title: Florida Gators

text: The Gators women's soccer team is coached by Samantha Bohon and rotates most of its home games between James G. Pressly Stadium and Donald R. Dizney Stadium.


title: Florida Gators

text: The University Athletic Association decided to create the women's varsity softball program in 1995, and the Florida Gators softball team officially started competing in the Southeastern Conference in 1997 under former head coach Larry Ray. Since the beginning of the program, the Florida Gators have had several notable successes, including four SEC championships, eight appearances in the Women's College World Series (2008, 2009, 2010, 2011, 2013, 2014, 2015, & 2017), and back-to-back WCWS national championships (2014 & 2015).


title: Florida Gators

text: The Gators won nine SEC regular season championships (1998, 2008, 2009, 2013, 2016, 2017, 2018, and 2021), as well as five SEC tournament titles (2008, 2009, 2013, 2018, and 2019). In 2009, they played for the NCAA softball championship in the Women's College World Series, losing to the Washington Huskies in the final round. The Gators again advanced to the finals of the 2011 Women's College World Series before falling to the Arizona State Sun Devils. In 2014, the Gators defeated the Alabama Crimson Tide in the first two games of the three-game Women's College World Series to win their first NCAA national championship. In 2015, the Gators repeated as national champions by defeating the Michigan Wolverines in the full three games. The Gators would return to the WCWS finals in 2017 before losing to the Oklahoma Sooners in two games.


title: Florida Gators

text: The current head coach is Tim Walton; the 2015–2016 season will be his tenth as the Gators' coach. He was previously the head coach at Wichita State University and he played baseball for the University of Oklahoma and a minor league team affiliated with the Philadelphia Phillies. Through the end of the 2018 season, Tim Walton has acquired a 722–149 record, while at Florida. Following the conclusion of the 2018 season, Tim Walton was given a 10-year contract extension.


title: Florida Gators

text: The Florida Gators men's swimming and diving teams have won NCAA national championships in 1983 and 1984, and has also won 41 Southeastern Conference team championships – including a stretch of 13 straight from 1956 to 1968. The women's swimming and diving team has won AIAW and NCAA national championships in 1979, 1982 and 2010, and has also won seventeen SEC team championships.


title: Florida Gators

text: The Florida Gators' notable female swimmers include three-time Olympic gold medalist Tracy Caulkins, three-time Olympic gold medalist Nicole Haislett, and four-time Olympic gold medalist Dara Torres, who is also the first American swimmer to compete in five Olympic Games. The Gators' notable male swimmers include Olympic gold medalists Matt Cetlinski, Mike Heath, David Larson, Ryan Lochte, Caeleb Dressel, Anthony Nesty and Martin Zubero.


title: Florida Gators

text: The Gators have had an international flavor since the 1980s, when the Olympic success of Gator swimmers began to attract Canadian, European and Latin American swimmers to train under coach Randy Reese. That tradition continues in the present day under Gregg Troy, the head coach of the Gators men's and women's swimming teams. Troy served as the head coach of the U.S. Olympic men's swim team in 2012. Dale Schultz is the new head coach of the men's and women's diving teams, succeeding long-time coach Donnie Craine in 2014.


title: Florida Gators

text: The Gators swimming and diving teams hold their home meets at the O'Connell Center Natatorium and train in the Carse Swimming Complex.


title: Florida Gators

text: The Florida Gators have one of the strongest and most storied women's tennis programs in NCAA history, and the women's tennis team has won seven NCAA team championships (1992, 1996, 1998, 2003, 2011, 2012, 2017). The team has also produced three individual Gators who have won four NCAA singles championships: Shaun Stafford (1988), Lisa Raymond (1992, 1993), and Jill Craybas (1996). In NCAA championship doubles play, four Gators doubles pairs have won five NCAA doubles championships: Jillian Alexander and Nicole Arendt (1991); Dawn Buth and Stephanie Nickitas (1996, 1997); Whitney Laiho and Jessica Lehnhoff (2001); and Brooke Austin and Kourtney Keegan (2016). The Gator women have also won seven Intercollegiate Tennis Association (ITA) national indoor championships (1988, 1991, 1992, 1996, 1997, 1999, 2017), and twenty-seven SEC team championships. The Gators have also claimed five SEC Tournament titles (1994, 2000, 2005, 2011, and 2016) in Men's Tennis and 20 in Women's Tennis (1982, 1990, 1991, 1992, 1993, 1995, 1996, 1997, 1998, 2000, 2002, 2003, 2004, 2005, 2006, 2010, 2011, 2012, 2013, and 2016).


title: Florida Gators

text: The Gators men's tennis team has a winning tradition and has won 11 Southeastern Conference team championships as of 2021. In 2021, the men's tennis program won its first-ever NCAA team championship. The Gator men have also produced four NCAA singles champions: Mark Merklein (1993), Jeff Morrison (1999), Sam Riffice (2021), and Ben Shelton (2022). Merklein and partner David Blair combined to win an NCAA doubles championship (1994).


title: Florida Gators

text: Bryan Shelton is the head coach of the men's tennis team, and coach Roland Thornqvist leads the women's tennis team. The Florida Gators tennis teams play their home matches at Linder Stadium at the Ring Tennis Complex on the university's campus.


title: Florida Gators

text: The Florida Gators men's track and field team has won six Southeastern Conference indoor championships, and four SEC outdoor championships. After finishing as the runner-up in both the NCAA indoor and outdoor meets in 2009, the men's team went on to win three consecutive NCAA indoor titles (2010-2012) and its first NCAA outdoor title (2012) over the next three years.


title: Florida Gators

text: The women's track and field team won the NCAA indoor championships in 1992 and 2022. The women's team won its first NCAA outdoor title in 2022. In addition, the women's team has won six SEC indoor championships, and four SEC outdoor championships.


title: Florida Gators

text: The head coach for the track and field program is Mike Holloway, and he is responsible for both the men's and women's teams. The assistant coaches are Steve Lemke, Adrain Mann, Nic Petersen, Chris Solinsky, & Mellanee Welty.


title: Florida Gators

text: The Gators men's and women's track and field teams hold their outdoor home meets at Percy Beard Track, which is part of James G. Pressly Stadium.


title: Florida Gators

text: The Gators began competing in women's volleyball in 1984 under coach Marilyn McReavy, but did not become nationally competitive until coach Mary Wise assumed control of Florida's program in 1991. During her 30 seasons as Florida's head coach, Wise has compiled a 979–168 win–loss record (.854), and her Gators teams have won nineteen SEC regular season titles and twelve SEC Tournament titles in her twenty seasons. The Gators have made nineteen trips to the NCAA tournament, including eight NCAA Final Four appearances (1992, 1993, 1996, 1997, 1998, 2002, 2003, 2017). In 2003 and 2017, they advanced to the NCAA national championship final, where they lost to the USC Trojans and Nebraska Cornhuskers, respectively.


title: Florida Gators

text: Florida landed the nation's top 2008 recruiting class, as ranked by Prepvolleyball.com and "Volleyball Magazine", and signed the nation's top recruit and Gatorade National Player of the Year, Kelly Murphy, as well as four other recruits ranked among the top fifty. Murphy garnered First-Team All SEC and a spot on the SEC All-Freshman Team with fellow Gators Colleen Ward and Kristy Jaeckel. Murphy would also gain the SEC Freshman of the Year, AVCA All-South Region Freshman of the Year, the AVCA National Freshman of the Year, and "Volleyball Magazine" Freshman of the Year. She was also honored as an AVCA Third-Team All-American and a "Volleyball Magazine" Second-Team All-American.


title: Florida Gators

text: In the past, the Florida Gators fielded varsity teams in men's boxing and men's wrestling. Gator boxer John Joca, a "Gator Great" member of the University of Florida Athletic Hall of Fame, won the NCAA national boxing championship in the 135-pound weight class in 1940. The Gators boxing team, however, was discontinued in 1943 during World War II, and was never revived after the war. The Gators men's wrestling team was an SEC-sponsored sport from 1970 to 1979; the team won the 1975 SEC championship tournament and placed second during four other seasons. The wrestling team was eliminated as a result of cost-cutting and Title IX compliance issues in 1979. The University Athletic Association, under athletic director Ray Graves and associate director Ruth Alexander, desired to take a proactive role in Title IX compliance by balancing the number of available men's and women's athletic scholarships, and the resulting tight athletic budgets ultimately resulted in the elimination of men's wrestling program.


title: Florida Gators

text: The University of Florida has invested significant capital and effort in the construction, expansion and betterment of its major sports facilities, including the following outdoor stadiums, indoor arenas, and training and practice facilities:


title: Florida Gators

text: The University of Florida began its sports program soon after it was established in Gainesville in 1906, and its teams adopted orange and blue as their official colors soon thereafter. These colors are a probably a combination of the colors of the two primary institutions that merged to form the university, as the East Florida Seminary used orange and black and the Florida Agricultural College used blue and white.


title: Florida Gators

text: As with the school colors, the exact origin of the alligator (almost always shortened to "gator") as the school mascot is not well documented. It likely originated in 1908, when Gainesville merchant Phillip Miller chose the animal to adorn pennants he designed for sale to students. The school did not yet have a mascot, and Miller chose the alligator because it is native to Florida and was not claimed by any other school. The football team subsequently adopted the "Gators" nickname in 1911. There are two theories as how the name become synonymous with the university's athletic programs. One theory is that the 1911 football squad began calling themselves the Gators in honor of team captain Neal "Bo Gator" Storter. Storter himself refuted this explanation, saying that it originated when a sportswriter in South Carolina described Florida's 1911 road upset of Clemson as an "invasion of alligators from Florida" and the players adopted the name. Whatever its origins, the nickname "Gators" (and for a time, "Lady Gators" for women's sports) has been used by the University of Florida's athletic program for over a century.


title: Florida Gators

text: Albert and Alberta are the official costumed mascots of the Florida Gators. Many variations of Albert have been present at sporting events over the decades, and a live alligator was used for many years. The current version of Albert was introduced in the early 1980s, and Alberta joined him in 1984. They are unique among the SEC's mascots as the only male-and-female pair, and are featured together in a life-sized statue outside of UF's Alumni Affairs Building.


title: Florida Gators

text: The University of Florida's marching band is known as "The Pride of the Sunshine", and plays at every home football game, and also performs at various events such as Gator Growl and parades. Florida's fight song is "The Orange and Blue". The Gatorettes are the baton-twirlers, or majorettes, for the marching band. The University of Florida dance team that performs at home basketball games and other sports events is known as the Dazzlers.


title: Florida Gators

text: The "Gator Chomp" is a gesture made by Florida Gators fans and players to show their support of the Florida Gators sports teams. The UF band originated in 1981 and is performed by fully extending one's arms, one over the other, in front of the body with the palms facing each other, and then moved apart and together to symbolize an alligator's mouth. When performed by fans at home football or basketball games, the chomp is often accompanied by Florida's marching band or pep band playing the two-note shark motif from the film "Jaws". Another football fan tradition, both for home and road games, is playing and singing of "We Are the Boys from Old Florida" at the end of every third quarter. Following the death of Gainesville native Tom Petty in October 2017, his song "I Won't Back Down" is also played right after "We Are the Boys" while the entire stadium sings along.


title: Florida Gators

text: "Orange and blue" is a fan cheer that is popular at home sports events, with alternate stadium sections yelling "Orange!", and answering back with their loudest "Blue!" This can go back and forth for several minutes, with both sections competing to be the louder.


title: Florida Gators

text: The football team had a long-time tradition of having George Edmondson Jr. — better known as "Mr. Two Bits"— wandering through the stands with a sign and a whistle to pump up the crowd to the "Two Bits" cheer. Edmondson officially retired in 2008, and was made an honorary alumnus in 2005. His final appearance as Mr. Two Bits was at the last home game of the 2008 season against The Citadel.


title: Florida Gators

text: In June 2020, university president Kent Fuchs announced that the University Athletic Association and the Gator Band would cease the use of the "Gator Bait" cheer at UF sporting events. While the university found no direct link between its usage of the phrase at sporting events and historical depictions of black children being used as alligator bait, it decided to end the chant due to "horrific racist imagery" associated with the phrase. The decision was reached amid a growing demand for social justice in the United States following the murder of George Floyd by Minneapolis police and the ensuing protests.


title: Florida Gators

text: Over 250 notable former Gators athletes and coaches have been inducted into the University of Florida Athletic Hall of Fame. Hall of Fame inductees fall into three categories: "Gator Greats"; "Distinguished Letterwinners"; and "Honorary Letterwinners." Gator Greats are those former athletes who distinguished themselves during their undergraduate sports careers, and include former All-Americans, all-conference selections, winners of major national awards, individual national champions, and those who significantly contributed to national team championships. Distinguished Letterwinners are those former Gators athletes who achieved distinction after graduation, as athletic coaches or administrators, professional athletes, or in public service or other career activities. Honorary Letterwinners are those persons who are not University of Florida alumni and former undergraduate athletes, but have distinguished themselves by their significant contributions to the success of Florida Gators sports teams, including former championship Gators coaches.


title: Florida Gators

text: Gator Greats include Heisman Trophy winners Steve Spurrier, Danny Wuerffel, and Tim Tebow; Pro Football Hall of Fame members Jack Youngblood and Emmitt Smith; Olympic gold medal swimmers Tracy Caulkins, Nicole Haislett and Dara Torres; individual NCAA golf champions Page Dunlap, Nick Gilliam and Bob Murphy; Olympic gold medal soccer players Heather Mitts and Abby Wambach; and individual NCAA tennis champions Jill Craybas, Jeff Morrison and Lisa Raymond.


title: Florida Gators

text: Distinguished Letterwinners include head coaches Doug Dickey, Lindy Infante and Dutch Stanley, as well as U.S. Senator George Smathers.


title: Florida Gators

text: Honorary Letterwinners include former Gators national championship coaches Buster Bishop, Andy Brandi, Randy Reese and Mimi Ryan, former football coach and four-star general James Van Fleet, as well as medical professor Robert Cade, who invented the sports drink Gatorade at UF in the mid-1960s as a rehydration aid to assist Gators athletes.


title: Florida Gators

text: The University of Florida has a reputation and long history of producing athletes who compete in the Olympic Games. Over 160 university alumni, including Florida Gators athletes from over 35 countries, have competed in the Games, winning fifty Olympic gold medals, twenty-nine silver medals and thirty bronze medals (through the end of the 2012 Summer Olympics).


title: Florida Gators

text: The list of notable Gator Olympians and gold medalists includes sprinters Kerron Clement, Dennis Mitchell, and Bernard Williams; marathon runner Frank Shorter; baseball outfielder Brad Wilkerson; basketball forward DeLisha Milton-Jones; soccer forward Abby Wambach; and swimmers Tracy Caulkins, Nicole Haislett and Ryan Lochte.


title: Florida Gators

text: Former Gator Dara Torres is the only American swimmer to compete in five Olympic Games (1984, 1988, 1992, 2000, 2008). At the age of 41, Torres became the oldest swimmer to win an Olympic medal when she won a silver medal in her inclusion in three events in 2008, finishing her career with a total of twelve Olympic medals (including four gold).


title: Baker–Campbell–Hausdorff formula

text: for possibly noncommutative and in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for formula_1 in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in formula_4 and formula_5 and iterated commutators thereof. The first few terms of this series are:


title: Baker–Campbell–Hausdorff formula

text: where "formula_7" indicates terms involving higher commutators of formula_4 and formula_5. If formula_4 and formula_5 are sufficiently small elements of the Lie algebra formula_12 of a Lie group formula_13, the series is convergent. Meanwhile, every element formula_14 sufficiently close to the identity in formula_13 can be expressed as formula_16 for a small formula_4 in formula_12. Thus, we can say that "near the identity" the group multiplication in formula_13—written as formula_20—can be expressed in purely Lie algebraic terms. The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in the Lie group–Lie algebra correspondence.


title: Baker–Campbell–Hausdorff formula

text: If formula_4 and formula_5 are sufficiently small formula_23 matrices, then formula_1 can be computed as the logarithm of formula_25, where the exponentials and the logarithm can be computed as power series. The point of the Baker–Campbell–Hausdorff formula is then the highly nonobvious claim that formula_26 can be expressed as a series in repeated commutators of formula_4 and formula_5.


title: Baker–Campbell–Hausdorff formula

text: Modern expositions of the formula can be found in, among other places, the books of Rossmann and Hall.


title: Baker–Campbell–Hausdorff formula

text: The formula is named after Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff who stated its qualitative form, i.e. that only commutators and commutators of commutators, ad infinitum, are needed to express the solution. An earlier statement of the form was adumbrated by Friedrich Schur in 1890 where a convergent power series is given, with terms recursively defined. This qualitative form is what is used in the most important applications, such as the relatively accessible proofs of the Lie correspondence and in quantum field theory. Following Schur, it was noted in print by Campbell (1897); elaborated by Henri Poincaré (1899) and Baker (1902); and systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906). The first actual explicit formula, with all numerical coefficients, is due to Eugene Dynkin (1947). The history of the formula is described in detail in the article of Achilles and Bonfiglioli and in the book of Bonfiglioli and Fulci.


title: Baker–Campbell–Hausdorff formula

text: For many purposes, it is only necessary to know that an expansion for formula_1 in terms of iterated commutators of formula_4 and formula_5 exists; the exact coefficients are often irrelevant. (See, for example, the discussion of the relationship between Lie group and Lie algebra homomorphisms in Section 5.2 of Hall's book, where the precise coefficients play no role in the argument.) A remarkably direct existence proof was given by Martin Eichler, see also the "Existence results" section below.


title: Baker–Campbell–Hausdorff formula

text: In other cases, one may need detailed information about formula_1 and it is therefore desirable to compute formula_1 as explicitly as possible. Numerous formulas exist; we will describe two of the main ones (Dynkin's formula and the integral formula of Poincaré) in this section.


title: Baker–Campbell–Hausdorff formula

text: where the sum is performed over all nonnegative values of formula_37 and formula_38, and the following notation has been used:


title: Baker–Campbell–Hausdorff formula

text: The series is not convergent in general; it is convergent (and the stated formula is valid) for all sufficiently small formula_4 and formula_5.


title: Baker–Campbell–Hausdorff formula

text: The first few terms are well-known, with all higher-order terms involving and commutator nestings thereof (thus in the Lie algebra):


title: Baker–Campbell–Hausdorff formula

text: The above lists all summands of order 6 or lower (i.e. those containing 6 or fewer 's and 's). The (anti-)/symmetry in alternating orders of the expansion, follows from . A complete elementary proof of this formula can be found in the article on the derivative of the exponential map.


title: Baker–Campbell–Hausdorff formula

text: There are numerous other expressions for formula_1, many of which are used in the physics literature. A popular integral formula is


title: Baker–Campbell–Hausdorff formula

text: For a matrix Lie group formula_48 the Lie algebra is the tangent space of the identity "I", and the commutator is simply ; the exponential map is the standard exponential map of matrices,


title: Baker–Campbell–Hausdorff formula

text: The formulas for the various formula_56's is "not" the Baker–Campbell–Hausdorff formula. Rather, the Baker–Campbell–Hausdorff formula is one of various expressions for formula_56's "in terms of repeated commutators of formula_4 and formula_5". The point is that it is far from obvious that it is possible to express each formula_56 in terms of commutators. (The reader is invited, for example, to verify by direct computation that formula_61 is expressible as a linear combination of the two nontrivial third-order commutators of formula_4 and formula_5, namely formula_64 and formula_65.) The general result that each formula_56 is expressible as a combination of commutators was shown in an elegant, recursive way by Eichler.


title: Baker–Campbell–Hausdorff formula

text: That is to say, since each formula_56 with formula_69 is expressible as a linear combination of commutators, the trace of each such terms is zero.


title: Baker–Campbell–Hausdorff formula

text: Suppose formula_4 and formula_5 are the following matrices in the Lie algebra formula_72 (the space of formula_73 matrices with trace zero):


title: Baker–Campbell–Hausdorff formula

text: It is then not hard to show that there does not exist a matrix formula_1 in formula_77 with formula_20. (Similar examples may be found in the article of Wei.)


title: Baker–Campbell–Hausdorff formula

text: This simple example illustrates that the various versions of the Baker–Campbell–Hausdorff formula, which give expressions for in terms of iterated Lie-brackets of and , describe "formal" power series whose convergence is not guaranteed. Thus, if one wants to be an actual element of the Lie algebra containing and (as opposed to a formal power series), one has to assume that and are small. Thus, the conclusion that the product operation on a Lie group is determined by the Lie algebra is only a local statement. Indeed, the result cannot be global, because globally one can have nonisomorphic Lie groups with isomorphic Lie algebras.


title: Baker–Campbell–Hausdorff formula

text: Concretely, if working with a matrix Lie algebra and formula_79 is a given submultiplicative matrix norm, convergence is guaranteed if


title: Baker–Campbell–Hausdorff formula

text: If formula_4 and formula_5 commute, that is formula_83, the Baker–Campbell–Hausdorff formula reduces to formula_84.


title: Baker–Campbell–Hausdorff formula

text: Another case assumes that formula_85 commutes with both formula_4 and formula_5, as for the nilpotent Heisenberg group. Then the formula reduces to its "first three terms".


title: Baker–Campbell–Hausdorff formula

text: This is the degenerate case used routinely in quantum mechanics, as illustrated below. In this case, there are no smallness restrictions on formula_4 and formula_5. This result is behind the "exponentiated commutation relations" that enter into the Stone–von Neumann theorem. A simple proof of this identity is given below.


title: Baker–Campbell–Hausdorff formula

text: Another useful form of the general formula emphasizes expansion in terms of "Y" and uses the adjoint mapping notation formula_94:


title: Baker–Campbell–Hausdorff formula

text: which is evident from the integral formula above. (The coefficients of the nested commutators with a single formula_5 are normalized Bernoulli numbers.)


title: Baker–Campbell–Hausdorff formula

text: Now assume that the commutator is a multiple of formula_5, so that formula_98. Then all iterated commutators will be multiples of formula_5, and no quadratic or higher terms in formula_5 appear. Thus, the formula_101 term above vanishes and we obtain:


title: Baker–Campbell–Hausdorff formula

text: "Theorem": If formula_98, where formula_103 is a complex number with formula_104 for all integers formula_105, then we have formula_106


title: Baker–Campbell–Hausdorff formula

text: Again, in this case there are no smallness restriction on formula_4 and formula_5. The restriction on formula_103 guarantees that the expression on the right side makes sense. (When formula_110 we may interpret formula_111.) We also obtain a simple "braiding identity":


title: Baker–Campbell–Hausdorff formula

text: If formula_4 and formula_5 are matrices, one can compute formula_26 using the power series for the exponential and logarithm, with convergence of the series if formula_4 and formula_5 are sufficiently small. It is natural to collect together all terms where the total degree in formula_4 and formula_5 equals a fixed number formula_121, giving an expression formula_122. (See the section "Matrix Lie group illustration" above for formulas for the first several formula_122's.) A remarkably direct and concise, recursive proof that each formula_122 is expressible in terms of repeated commutators of formula_4 and formula_5 was given by Martin Eichler.


title: Baker–Campbell–Hausdorff formula

text: Alternatively, we can give an existence argument as follows. The Baker–Campbell–Hausdorff formula implies that if and are in some Lie algebra formula_127 defined over any field of characteristic 0 like formula_128 or formula_129, then


title: Baker–Campbell–Hausdorff formula

text: can formally be written as an infinite sum of elements of formula_12. [This infinite series may or may not converge, so it need not define an actual element in formula_12.] For many applications, the mere assurance of the existence of this formal expression is sufficient, and an explicit expression for this infinite sum is not needed. This is for instance the case in the Lorentzian construction of a Lie group representation from a Lie algebra representation. Existence can be seen as follows.


title: Baker–Campbell–Hausdorff formula

text: We consider the ring formula_133 of all non-commuting formal power series with real coefficients in the non-commuting variables and . There is a ring homomorphism from to the tensor product of with over ,


title: Baker–Campbell–Hausdorff formula

text: The elements "X" and "Y" are primitive, so formula_141 and formula_142 are grouplike; so their product formula_143 is also grouplike; so its logarithm formula_144 is primitive; and hence can be written as an infinite sum of elements of the Lie algebra generated by and .


title: Baker–Campbell–Hausdorff formula

text: The universal enveloping algebra of the free Lie algebra generated by and is isomorphic to the algebra of all non-commuting polynomials in and . In common with all universal enveloping algebras, it has a natural structure of a Hopf algebra, with a coproduct . The ring used above is just a completion of this Hopf algebra.


title: Baker–Campbell–Hausdorff formula

text: This is a particularly useful formula which is commonly used to conduct [[Unitary transformation (quantum mechanics)|unitary transforms in quantum mechanics]]. By defining the iterated commutator,


title: Baker–Campbell–Hausdorff formula

text: This formula can be proved by evaluation of the derivative with respect to of , solution of the resulting differential equation and evaluation at ,


title: Baker–Campbell–Hausdorff formula

text: Taking formula_154 gives one of the special cases of the Baker–Campbell–Hausdorff formula described above:


title: Baker–Campbell–Hausdorff formula

text: This variation is commonly used to write coordinates and vielbeins as pullbacks of the metric on a Lie group.


title: Baker–Campbell–Hausdorff formula

text: For example, writing formula_158 for some functions formula_159 and a basis formula_160 for the Lie algebra, one readily computes that


title: Baker–Campbell–Hausdorff formula

text: The usefulness of this expression comes from the fact that the matrix is a vielbein. Thus, given some map formula_166 from some manifold to some manifold , the metric tensor on the manifold can be written as the pullback of the metric tensor formula_167 on the Lie group ,


title: Baker–Campbell–Hausdorff formula

text: The metric tensor formula_167 on the Lie group is the Cartan metric, the Killing form. For a (pseudo-)Riemannian manifold, the metric is a (pseudo-)Riemannian metric.


title: Baker–Campbell–Hausdorff formula

text: A special case of the Baker–Campbell–Hausdorff formula is useful in quantum mechanics and especially quantum optics, where and are Hilbert space operators, generating the Heisenberg Lie algebra. Specifically, the position and momentum operators in quantum mechanics, usually denoted formula_4 and formula_171, satisfy the canonical commutation relation:


title: Baker–Campbell–Hausdorff formula

text: where formula_173 is the identity operator. It follows that formula_4 and formula_171 commute with their commutator. Thus, if we "formally" applied a special case of the Baker–Campbell–Hausdorff formula (even though formula_4 and formula_171 are unbounded operators and not matrices), we would conclude that


title: Baker–Campbell–Hausdorff formula

text: This "exponentiated commutation relation" does indeed hold, and forms the basis of the Stone–von Neumann theorem.


title: Baker–Campbell–Hausdorff formula

text: A related application is the annihilation and creation operators, and . Their commutator is "central", that is, it commutes with both and . As indicated above, the expansion then collapses to the semi-trivial degenerate form:


title: Baker–Campbell–Hausdorff formula

text: This example illustrates the resolution of the displacement operator, , into exponentials of annihilation and creation operators and scalars.


title: Baker–Campbell–Hausdorff formula

text: This degenerate Baker–Campbell–Hausdorff formula then displays the product of two displacement operators as another displacement operator (up to a phase factor), with the resultant displacement equal to the sum of the two displacements,


title: Baker–Campbell–Hausdorff formula

text: since the Heisenberg group they provide a representation of is nilpotent. The degenerate Baker–Campbell–Hausdorff formula is frequently used in quantum field theory as well.


title: Donald Neilson

text: Donald Neilson (born Donald Nappey; 1 August 1936 – 18 December 2011), alias the “Black Panther,” was a British armed robber, kidnapper, and multiple murderer. From 1971, he committed a series of robberies of sub-post offices; in 1974, Neilson killed three men during these robberies. In 1975, he kidnapped Lesley Whittle, an heiress from Highley, Shropshire, who died in his captivity. He was arrested later that year, convicted of four counts of murder, and sentenced to life imprisonment in July 1976. Neilson remained in prison until his death in 2011.


title: Donald Neilson

text: Neilson, born Donald Nappey, was aged ten in January 1947, when his 33-year-old mother died from breast cancer. He was said to have had an unhappy childhood, and was caught shopbreaking in 1948. But due to his age and circumstances, Neilson was given a police caution or stern warning. In April 1955, an 18-year-old Neilson married 20-year-old Irene Tate. In 1958, his wife persuaded him to leave the army, following a period as a national serviceman in Kenya, Aden, and Cyprus as part of the King's Own Yorkshire Light Infantry.


title: Donald Neilson

text: Their daughter, Kathryn, was born in 1960. Four years after his daughter's birth, Nappey changed the family name to “Neilson,” so that the little girl would not suffer the bullying and abuse he had endured at school and in the army because of his surname’s similarity to the word nappy.


title: Donald Neilson

text: According to David Bell and Harry Hawkes, Nappey bought a taxi business from a man named Neilson, and decided to use that name instead of the former. An alternative theory, proposed by a lodger, Lena Fearnley, who stayed with the Neilson family in the early 1960s, is that Neilson took the name from an ice-cream van (carrying the brand name “Lord Neilson”), from which he and Irene often bought ice-cream for their daughter. Fearnley told the BBC in an interview that he told her, “I like that name.”


title: Donald Neilson

text: Neilson committed over 400 house burglaries without detection during his early days of crime. Before he became notorious as the “Black Panther,” he was sought under a variety of nicknames, such as “The Phantom” and “Handy Andy.” To confuse the police, he adopted a different modus operandi every few weeks. For example: he would steal a radio from each house, and abandon it nearby; when that pattern of behaviour was established, he would drop it, and do something else.


title: Donald Neilson

text: Proceeds from simple housebreakings were low, however, and after stealing guns and ammunition from a house in Cheshire, he upped his criminal activity, turning to robbing small post offices. Neilson committed eighteen such crimes between 1971 and 1974. His crimes became progressively more violent as he sought to protect himself from occupants prepared to resist and defend their property.


title: Donald Neilson

text: In February 1972, he gained entry during the night to a sub-post office located in a private home in Rochdale Road, Heywood, Lancashire. Leslie Richardson, the postmaster, and his wife woke to find a hooded man in their bedroom. Richardson leapt out of bed to tackle the intruder while his wife phoned the police. During the struggle, Neilson showed Richardson his sawn-off shotgun and warned, in a fake West Indian accent, “This is loaded.” Richardson saw that the gun was pointing up at the ceiling, and realized that there was no danger of anyone being shot.


title: Donald Neilson

text: Richardson rejoined, “We'll find out if it's loaded!” and pulled the trigger himself, blasting two holes in the ceiling. The fight continued, and Richardson pulled off Neilson's black hood. Neilson stamped on Richardson's feet, breaking several toes, and kneed him in the groin. As Richardson collapsed on the floor, Neilson made his escape empty-handed. Richardson gave police a description of the masked intruder, which turned out to be inaccurate in many respects. Several other photofits of Neilson were similarly unhelpful to the police, but one, made by sub-postmistress Margaret Grayland, was extremely accurate.


title: Donald Neilson

text: Neilson committed his first three murders in 1974. During post office robberies, he shot dead two sub-postmasters and the husband of a sub-postmistress, as well as brutally battering sub-postmistress Margaret Grayland. He killed Donald Skepper in Harrogate in North Yorkshire in February 1974, Derek Astin of Baxenden in Lancashire in September 1974, and Sidney Grayland in Langley, West Midlands during November 1974.


title: Donald Neilson

text: The Baxenden murder resulted in Neilson being dubbed the “Black Panther,” as, during an interview with a local television reporter, Astin's wife, Marion, described her husband's killer as being “so quick, he was like a panther.” Alluding to the killer's dark clothing, the enterprising reporter ended his piece by asking, “Where is this Black Panther?” and the nickname stuck. Neilson was linked to the post office shootings after he shot security guard Gerald Smith six times while checking a ransom trail. Forensic examination showed the bullets were fired from the same .22 LR pistol used to murder both Astin and Grayland.


title: Donald Neilson

text: Lesley Whittle (1957–14 January 1975) was a 17-year-old girl when she became Neilson's youngest victim. Whittle was the daughter of noted coach transport business owner George Whittle and his mistress. Whittle had left his entire fortune to his mistress and their children, Ronald and Lesley. After reading about a family dispute over George Whittle's will, Neilson planned for three years to gain some of this fortune.


title: Donald Neilson

text: On 14 January 1975, Neilson entered the Whittle family home in Highley, Shropshire, and kidnapped Lesley from her bedroom. Neilson calculated that the family would not materially miss £50,000 of their fortune, and so made a subsequent demand for that sum in a note left at the family home. A series of police bungles and other circumstances resulted in Whittle’s brother, Ronald, being unable to deliver the ransom money at the designated time and place demanded by the kidnapper.