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World Cyber Games

The World Cyber Games (WCG) is an international esports competition with multi-game titles in which hundreds of esports athletes from around the world participate in a variety of competitions also known as Esports Olympics. WCG events attempt to emulate a traditional sporting tournament, such as the Olympic Games; events included an official opening ceremony, and players from various countries competing for gold, silver, and bronze medals. WCG are held every year in cities around the world. The WCG 2020 competition received nearly 650 million views worldwide.

General
World Cyber Games is one of the largest global esports tournaments, with divisions in various countries. The World Cyber Games, created by International Cyber Marketing CEO Yooseop Oh and backed financially by Samsung, was considered the e-sports Olympics; events included an official opening ceremony, and players from various countries competing for gold, silver and bronze medals. The organization itself had an official mascot, and used an Olympic Games inspired logo. Organizations from each participating country conducted preliminary events at a regional level, before conducting national finals to determine the players best suited to represent them in the main World Cyber Games tournament event. All events had areas for spectators, but the tournament could also be viewed over internet video streams.

Besides providing a platform for tournament gaming, the World Cyber Games was used as a marketing tool; sponsors, such as Samsung, using the space around the venue to set up product demonstrations and stalls. In addition, advertisers saw the event as a good means to reach young male audiences, who may not be exposed to traditional advertising streams via television.

History
In 2000, the World Cyber Games was formed, and an event was held titled "The World Cyber Game Challenge", which began with an opening ceremony on 7 October. The event was sponsored by the Republic of Korea's Ministry of Culture and Tourism, Ministry of Information and Communications, and Samsung. It brought together teams from 17 countries to compete against each other in PC games including Quake III Arena, FIFA 2000, Age of Empires II, and StarCraft: Brood War. The tournament ended on 15 October 2000. The competition initially had 174 competitors from 17 different countries with a total prize purse of $20,000.

In 2001, the World Cyber Games held their first main event, hosted in Seoul, Korea, with a prize pool of $300,000 USD. National preliminaries were held between March and September, with the main tournament running between 5 December to 9 December. The World Cyber Games quoted an attendance of 389,000 competitors in the preliminaries, with 430 players advancing to the final tournament; teams from 24 countries in total were involved in the tournament.

In 2002, the World Cyber Games held a larger event in Daejeon, Korea with a prize pool of US$1,300,000; 450,000 competitors took part in the preliminary events, with 450 ultimately making it through to the final tournament. The 2003 tournament, which took place in Seoul again, saw an even bigger prize pool of US$2,000,000, and was the first World Cyber Games tournament to feature a console based competition, with the game Halo: Combat Evolved on the Xbox.

In 2004, the World Cyber Games held a tournament in San Francisco, California, United States, the first tournament outside of its home country. At this stage, the prize pool was at US$2,500,000; with 642 players competing in the grand final. The tournament has since been hosted in various countries around the world; including Singapore in 2005 and Monza, Italy in 2006 - at this time Microsoft became a major sponsor to the event, who would provide software and hardware for all the events through to 2008. In addition, all games played at the tournament would be based exclusively on Windows PC's or the Xbox console.

In 2006, the prize purse had risen to $462,000, and the event had grown to 9 different competitions and 700 qualified participants from 70 different countries.

In 2007, the event was hosted in Seattle, Washington, United States, with a total prize pool of US$4,000,000. In 2008, the tournament was hosted in Cologne, Germany; it was the first World Cyber Games tournament to incorporate a mobile-game based tournament, with Asphalt 4: Elite Racing, In 2009, the tournament was held in Chengdu, China, and featured a special promotion of the game Dungeon & Fighter. The tournament was also coincided to run alongside the World Cyber Games debut reality television show, WCG Ultimate Gamer. Season 2 of WCG Ultimate Gamer was aired between August and October 2010.

In 2014 February, the CEO Brad Lee announced the closing of WCG. Several partners described difficulty working with the CEO and the organization.

In March 2017, the former Samsung owned WCG Trademark was transferred to Korean Publisher Smilegate. Plans to develop the WCG "into the world's top digital entertainment festival in the future". World Cyber Games 2018 should have been hosted in Bangkok on April 26–29, 2018, but was cancelled subsequently. World Cyber Games 2019 was hosted in Xi'an, China on July 18–21, 2019. The WCG 2020 competition received nearly 650 million views worldwide.

title: World Cyber Games


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Oblique

Oblique may refer to:
 * an alternative name for the character usually called a slash (punctuation) ( / )
 * Oblique angle, in geometry
 * Oblique triangle, in geometry
 * Oblique lattice, in geometry
 * Oblique leaf base, a characteristic shape of the base of a leaf
 * Oblique angle, a synonym for Dutch angle, a cinematographic technique
 * Oblique (album), by jazz vibraphonist Bobby Hutcherson
 * Oblique (film), a 2008 Norwegian film
 * Oblique (Vasarely), a 1966 collage, by Victor Vasarely
 * Oblique banded rattail, a fish also known as a rough-head whiptail
 * Oblique case, in linguistics
 * Oblique argument, in linguistics
 * Oblique correction, in particle physics
 * Oblique motion, in music
 * Oblique order, a military formation
 * Oblique projection, in geometry and drawing, including cavalier and cabinet projection
 * Oblique reflection, in Euclidean geometry
 * Oblique shock, in gas dynamics
 * Oblique type, a form of type that slants slightly to the right in typography.
 * Oblique wing, in aircraft design
 * Oblique icebreaker, a special type of icegoing ship

Anatomy

 * Oblique arytenoid muscle, in the neck
 * Oblique cord, near the elbow point
 * Oblique fissure, separating the inferior and superior lobes of the lungs
 * Oblique muscle (disambiguation), any of several in the human body

 * Abdominal muscles
 * Abdominal external oblique muscle
 * Abdominal internal oblique muscle
 * Oblique strain, an injury of either of these muscles, common in baseball

 * Eye muscles
 * Inferior oblique muscle
 * Superior oblique muscle

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

title: Oblique


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WCG

WCG may refer to:

 * World Cyber Games, an international e-sports event
 * World Combat Games, a sports event
 * Wide Color Gamut
 * World Community Grid, for scientific research computing
 * Worldwide Church of God, renamed Grace Communion International in 2009
 * WCG (firm), an advertising agency
 * WCG (college), a group of UK colleges
 * WCG (Wide DC electric goods), a classification of locomotives of India

title: WCG


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9th (Scottish) Division

The 9th (Scottish) Division, was an infantry division of the British Army during the First World War, one of the Kitchener's Army divisions raised from volunteers by Lord Kitchener to serve on the Western Front during the First World War.

After the 1st South African Infantry Brigade Group joined in early 1916, the division was known colloquially as the Jock and Springboks.

Background
A 9th Division had been formed for service during the Second Boer War, and was commanded by Henry Edward Colvile. In 1902, a 9th Division was as formed and was commanded by Edward Pemberton Leach, but it was broken-up at some point prior to the start of the war.

First World War
In the Battle of Loos, notable for being the first battle in which British forces used poison gas, the 9th (Scottish) Division assaulted the Hohenzollern Redoubt, the 5th Camerons suffered horrific casualties, and Corporal James Dalgleish Pollock gained a Victoria Cross for his actions.

The 9th (Scottish) Division took part in major fighting during the Somme offensive. Notably it relieved the 30th Division at Montauban and later attacked German positions at Bernafay Wood, where it succeeded in capturing vital objectives and forcing a German withdrawal. In the Somme offensive, the 9th (Scottish) Division liberated the village of Longueval; the village now has a statue of a Scottish piper at its crossroads that commemorates this fact (see Caterpillar Valley Cemetery) and also other pipers who served in the First World War.

Order of battle
The division comprised the following brigades:

 * 26th Brigade
 * 8th (Service) Battalion, Black Watch (Royal Highlanders)
 * 7th (Service) Battalion, Seaforth Highlanders (Ross-shire Buffs, Duke of Albany's)
 * 8th (Service) Battalion, Gordon Highlanders (transferred to 15th (Scottish) Division 7 May 1916)
 * 5th (Service) Battalion, Queen's Own Cameron Highlanders
 * 1/5th Battalion, Loyal North Lancashire Regiment, Territorial Force (TF) (joined December 1915 left January 1916)
 * 10th (Service) Battalion, Argyll and Sutherland Highlanders (transferred from 27th Brigade 6 May 1916, transferred to 32nd Division 17 February 1918)
 * 26th Machine Gun Company (formed 29 January 1916 left to move into 9th Battalion Machine Gun Corps (M.G.C.) 1 March 1918)
 * 26th Trench Mortar Battery (joined 15 June 1916)

 * 27th Brigade
 * 11th (Service) Battalion, Royal Scots (Lothian Regiment)
 * 12th (Service) Battalion, Royal Scots (Lothian Regiment)
 * 6th (Service) Battalion, Royal Scots Fusiliers (transferred to 15th (Scottish) Division 7 May 1916)
 * 10th (Service) Battalion, Argyll and Sutherland Highlanders (transferred to 26th Brigade 7 May 1916)
 * 6th (Service) Battalion, King's Own Scottish Borderers (transferred from 28th Brigade 6 May 1916)
 * 9th (Service) Battalion, Cameronians (Scottish Rifles) (transferred from 28th Brigade 6 May 1916, transferred to the 14th (Light) Division in February 1918)
 * 27th Machine Gun Company (formed 23 December 1915, left to move into 9th Battalion M.G. C. 1 March 1918)
 * 27th Trench Mortar Battery (joined July 1916)

(The 28th Brigade was broken up in May 1916 and replaced with the 1st South African Brigade. It reformed in September 1918)
 * 28th Brigade
 * 6th (Service) Battalion, King's Own Scottish Borderers (transferred to 27th Brigade 6 May 1916)
 * 9th (Service) Battalion, Cameronians (Scottish Rifles) (transferred to 27th Brigade 6 May 1916, returned September 1918)
 * 10th (Service) Battalion, Highland Light Infantry (transferred to South African Brigade 6 May 1916)
 * 11th (Service) Battalion, Highland Light Infantry (transferred to South African Brigade 6 May 1916)
 * 2nd Battalion, Royal Scots Fusiliers (transferred from South African Brigade 13 September 1918)
 * 1st Battalion, Royal Newfoundland Regiment (joined September 1918)
 * 28th Machine Gun Company (formed 3 January 1916 transferred to South African Brigade 6 May 1916)
 * 28th Trench Mortar Battery(joined 11 September 1918)

(Replaced the 28th Brigade in May 1916 and was replaced in turn by a reformed 28th Brigade on 13 September 1918)
 * 1st South African Brigade
 * 1st Regiment, South African Infantry (merged with 2nd and 4th Battalions on 24 April 1918 as the South African (Composite) Regiment, regained separate identity on 1 September 1918)
 * 2nd Regiment, South African Infantry (merged with 1st Battalion, reformed 1 September 1918)
 * 3rd Regiment, South African Infantry (disbanded 18 February 1918)
 * 4th Regiment, South African Infantry (merged with 1st Battalion, reformed 1 September 1918)
 * 10th (Service) Battalion, Highland Light Infantry (transferred from 28th Brigade 6 May 1916, merged with 11th Battalion on 14 May 1916 forming 10/11th Battalion and transferred to 15th (Scottish) Division)
 * 11th (Service) Battalion, Highland Light Infantry (transferred from 28th Brigade 6 May 1916 merged with 10th Battalion)
 * 3/4th Battalion, Royal West Kent Regiment, TF (joined 6 June 1917, left 15 June 1917)
 * 3/10th Battalion, Middlesex Regiment, TF (joined 6 June 1917, left 23 July 1917)
 * 2nd Battalion, Royal Scots Fusiliers (joined 26 April 1918, transferred to 28th Brigade 13 September 1918)
 * 9th (Service) Battalion, Cameronians (Scottish Rifles) (joined 21 April 1918, transferred to 28th Brigade 12 September 1918)
 * 28th Machine Gun Company (transferred from 28th Brigade 6 May 1916 left to move into 9th Battalion M.G.C. 1 March 1918)
 * South African Trench Mortar Battery (joined 13 June 1916)

Divisional Troops
 * 6th (Service) Battalion, Bedfordshire Regiment (left March 1915)
 * 6th (Service) Battalion, Leicestershire Regiment (left April 1915)
 * 9th (Service) Battalion, Seaforth Highlanders (joined December 1914, became Divisional Pioneer Battalion early 1915)
 * 10th Motor Machine Gun Battery (joined 30 April 1915, left 11 June 1916)
 * 197th Company, M.G.C. (joined 19 December 1916, left to move into 9th Battalion M.G.C. 1 March 1918)
 * 9th Battalion, M.G.C. (formed 1 March 1918 absorbing the brigade MG companies)
 * 11th Motor Machine Gun Battery (joined 7 October 1918, left 7 November 1918)
 * Divisional Mounted Troops
 * B Squadron, 1/1st Glasgow Yeomanry (joined 15 May 1915, left 10 May 1916)
 * 9th Divisional Cyclist Company, Army Cyclist Corps (formed 1 December 1914, left 26 June 1916)
 * 9th Divisional Train Army Service Corps
 * 104th, 105th, 106th and 107th Companies
 * 21st Mobile Veterinary Section Army Veterinary Corps
 * 212th Divisional Employment Company (formed by 23 June 1917)

Royal Artillery
 * L Brigade, Royal Field Artillery (R.F.A.)
 * LI Brigade, R.F.A.
 * LII Brigade, R.F.A. (left 8 January 1917)
 * LIII (Howitzer) Brigade, R.F.A. (broken up 11 September 1916)
 * 9th Divisional Ammunition Column R.F.A.
 * 9th Heavy Battery, Royal Garrison Artillery (left 16 May 1915)
 * V.9 Heavy Trench Mortar Battery R.F.A. (joined May 1916, left February 1918)
 * X.9, Y.9 and Z.9 Medium Mortar Batteries R.F.A. (joined April 1916; on 13 February 1918, Z broken up and distributed among X and Y batteries)

Royal Engineers
 * 63rd Field Company
 * 64th Field Company
 * 90th Field Company (joined January 1915)
 * 9th Divisional Signals Company

Royal Army Medical Corps
 * 27th Field Ambulance
 * 28th Field Ambulance
 * 29th Field Ambulance (left May 1916)
 * South African Field Ambulance (joined May 1916, left 13 September 1918)
 * 2/1st (East Lancashire) Field Ambulance (joined 26 September 1918)
 * 20th Sanitary Section (left 29 March 1917)

General Officers Commanding
Commanders have included:
 * 27 August – 11 October 1914 Major-General Colin Mackenzie
 * 11 October – 26 October 1914 Brigadier-General Spencer Scrase-Dickens (acting)
 * 26 October – 31 December 1914 Lieutenant-General Sir Charles Fergusson
 * 31 December 1914 – 21 January 1915 Brigadier-General Spencer Scrase-Dickens (acting)
 * 21 January – 8 September 1915 Major-General Herman Landon
 * 9 September – 27 September 1915 Major-General George Thesiger
 * 27 September 1915 Major-General Edward Bulfin (temporary)
 * 28 September 1915 – 2 December 1916 Major-General William Furse
 * 2 December 1916 – 4 March 1918 Major-General Henry Lukin
 * 4 March – 13 March 1918 Brigadier-General Henry Hugh Tudor (acting)
 * 13 March – 16 March 1918 Major-General Cyril Blacklock
 * 16 March – 24 March 1918 Brigadier-General Henry Hugh Tudor (acting)
 * 24 March – 28 March 1918 Major-General Cyril Blacklock
 * 28 March 1918 – 1919 Major-General Henry Hugh Tudor

title: 9th (Scottish) Division


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Talk:President of the United States in Congress assembled

Former content was merged into the Talk page at Talk:President of the Continental Congress. Lou I 21:23, 20 Jan 2004 (UTC)


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Initialization

Initialization may refer to:

 * Booting, a process that starts computer operating systems
 * Initialism, an abbreviation formed using the initial letters of words or word parts
 * In computing, formatting a storage medium like a hard disk or memory. Also, making sure a device is available to the operating system.
 * Initialization (programming)

title: Initialization


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Adamski

Adam Paul Tinley (born 4 December 1967 ), known professionally as Adamski, as well as Sonny Eriksson, is an English DJ, musician, singer and record producer, prominent at the time of acid house for his tracks "N-R-G" and "Killer", a collaboration with Seal, which was a No. 1 song in the UK in 1990.

Career
Tinley was born in Lymington, Hampshire, England. As a youngster, influenced by punk rock and John Peel, he formed his first band The Stupid Babies when he was 11 and living in New Forest in England. He persuaded his 5-year-old brother Dominic to sing while he strummed a small guitar, and sent a demo tape to the indie label Fast Product, run by The Human League's manager Bob Last. "Everyone thought that was a really precocious and strange thing for an 11 year-old to do," Adamski recalls "but I just thought that's what everybody did". The kiddie-punk tracks were released on a sampler. When alternative BBC Radio 1 DJ John Peel started playing their song "Babysitters" the band caused quite a stir, receiving positive write-ups in music magazines like Smash Hits and Melody Maker. He performed with his brother Mark Tinley, and Johnny Slut of the band Specimen, as Diskord Datkord. They released their only single in 1988, an electroid cover of "Identity" by punk band X-Ray Spex. It was single of the week in NME.

In March 1989, Adamski was booked for his first solo gig at Le Petit Prince Restaurant in Kentish Town, run by his manager Phil Smith. Lenny D, promoter of nightclub Heaven, happened to be walking past, and was convinced by Smith to book Adamski for an all-dayer at Heaven. He quickly catapulted into the upper echelons of the nascent rave scene. Within a few weeks, Adamski was playing to 8,000 people at Sunrise Festival at Santa Pod Raceway and, after a bidding war, signed to MCA Records, producing the first rave record on MCA called Liveandirect.

He had success with this first release, which was a collection of tracks recorded live at various raves. It contained a short, live version of his first single "N-R-G", as well as "I Dream of You", which appeared on a free 4-track 7" vinyl single given away with the music paper Record Mirror in 1989.  The cover of the single "N-R-G" featured a mocked up Lucozade bottle with the word "Lucozade" replaced with "N-R-G".

Adamski toured many clubs with his portable keyboard set up, playing long sets, with an MC, Daddy Chester, and later with Seal. In front of his keyboard was a UK car number plate with the word ADAMSKI on it. Early versions of future singles "Killer" and "Future Love Paradise" were played on some of the Seal dates.

The album, Adamski's Thing, was issued in late 1998 on Trevor Horn's ZTT Records label, recorded at Adrian Sherwood’s On-U Sound studios. The style followed the trend started with his 1992 album Naughty, with guitars, strings, raw vocals and introspective lyrics, but maintaining a rhythmic dance sensibility. Adamski's Thing spawned two singles, "Intravenous Venus" and "One of the People" (a record featuring dance vocalist Gerideau, that got to No. 56 in the UK Singles Chart).

Throughout the late 1990s, Tinley shifted focus to his DJ career. He soon adopted a new moniker, Adam Sky, touring Europe, and making the odd UK appearance such as playing at the electroclash night Nag Nag Nag in London in 2002 – run by his old friend Jonny Slut. As a producer, his songs at that time also included a collaboration with musician Danny Williams. In 2007, he released a remake of The Pop Group's 1979 single "We Are All Prostitutes" with Mark Stewart, which appeared on a number of compilations. His single "ApeX" was released on Kitsuné Music in 2006. In 2009, he began to release more material on Shir Khan's record label.

In 2009, he found his way back to the UK to start his Futurewaltz project, working in the 3/4 time signature. Tinley’s latest alter ego is cyberbilly Sonny Eriksson - a self-styled Teddy Boy fusing rockabilly and psychobilly styles with the hypnotic, strange and electronic sounds of today and beyond.

In 2020, Adamski released Free to Kill Again, featuring 10 new interpretations of "Killer" with guest features by Boy George, Nina Hagen, Adrian Sherwood, Mykki Blanco, Hannah Hu and others. In 2022, Adamski released "Black Butterfly", featuring Robert Owens. The track is a tribute to Mina Smallman's daughters Nicole and Bibaa, with all proceeds donated to the charity Million Women Rise. Remixes by done by Captain Mustache, Leeroy Thornhil (The Prodigy), Mr. C (Superfreq), and Shadow Child. In October 2022, Adamski released "Black Star Acid" on Boys Noize Records.

Albums

 * Liveandirect (1989) –   UK No. 47
 * Doctor Adamski's Musical Pharmacy (1990) – UK No. 8
 * Naughty (1992)
 * Adamski vs The Sentinels (1993)
 * Adamski's Thing (1998)
 * Killer – The Best of Adamski (1999)
 * Mutant Pop (1999)
 * This is 3-Step EP (2014)
 * Revolt (2015)
 * The Sound of Sonny Eriksson (2017)
 * The Spirit of Sonny Eriksson (EP) (2018)
 *  Re:nrgise (2019)
 * Free to Kill Again (2020)

Remixes

 * Miss Kittin and The Hacker – "Stock Exchange" (2003)
 * Mignon – "Demons of Love" (2004)
 * ESKA – "She's in the Flowers" (2015)

title: Adamski


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Internet suite

An Internet suite is an Internet-related software suite. Internet suites usually include a web browser, e-mail client (often with a news client and address book), download manager, HTML editor, and an IRC client.

The diversity of Internet suite offerings was greatest in the mid-1990s, when proprietary web browser vendors felt it more profitable to sell entire retail suites of applications on compact disc. However, by the end of the first browser war, the Internet suite market dwindled to one or two competitors every few years. In reaction, alternative routes of profit or funding were sought. Opera Software, for example, moved away from offering the Opera Internet suite with embedded advertisements to a completely ad-free product. Mozilla (spun out of Netscape) began to separate the Mozilla Application Suite into separate projects: Firefox, Thunderbird, and Sunbird, but Internet suite development continued with Seamonkey, which is still active under the Mozilla umbrella. A further cause of depression in the diversity of Internet suites was the rise of Ajax-based web applications (webmail, for example) which replicated most functions of their desktop client equivalents while offering cross-platform portability through web browsers, portability not offered as easily by desktop clients inside Internet suites and outside web browsers.

Examples

 * Arachne
 * Cyberdog
 * Mozilla Application Suite and SeaMonkey internet suite
 * Gnuzilla
 * K Desktop Environment
 * MSN Explorer
 * Netscape Communicator
 * Netscape
 * Opera (version 12.18 and earlier)
 * Vivaldi

title: Internet suite


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Professional Adventure Writer

Professional Adventure Writer or PAW (sometimes called PAWS for Professional Adventure Writing System) is a program that allows the user to write textual adventure games with graphic illustrations. It was written by Tim Gilberts, Graeme Yeandle and Phil Wade, based on Yeandle's earlier system called The Quill.

PAW was published by Gilsoft in 1987 and quickly gained a loyal following. PAW improved over The Quill in several ways. In particular, its textual input parser was more sophisticated, meaning inputs were no longer confined to the two-word telegraphic verb noun (e.g. "GO WEST; TAKE LAMP") style. PAW also supported NPCs, different character sets, and full use of the memory of the 128K ZX Spectrum. However, unlike The Quill, the PAW no longer supported other computer systems like the BBC Micro or the Commodore 64. Over 400 games were written using PAW.

To ensure that as much text as possible can be used, PAWS compresses the descriptions by replacing the most common letters combinations by tokens, using characters greater than 127 (the ones that, in the Spectrum, are used for storing the BASIC tokens).

In 2001, WinPAW was written by Douglas Harter. It could read adventures written in PAW, but ran under MS-Windows and had a few extensions to the original. The adventures made in WinPAW could only be played using the MS Windows runtime. In 2009, InPAWS was released in its first version. It allows to extract PAW adventures, edit them or create from scratch and write back a database for PAW for either Amstrad CPC or ZX Spectrum. Thus, it also allows PAW adventures to be ported between the systems.

Graeme Yeandle also released an updated version of the CP/M version of PAW for MS-DOS and called it PC Adventure Writer.

title: Professional Adventure Writer


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10th (Irish) Division

The 10th (Irish) Division, was one of the first of Kitchener's New Army K1 Army Group divisions (formed from Kitchener's 'first hundred thousand' new volunteers), authorized on 21 August 1914, after the outbreak of the Great War. It included battalions from the various provinces of Ireland. It was led by Irish General Bryan Mahon and fought at Gallipoli, Salonika and Palestine. It was the first of the Irish Divisions to take to the field and was the most travelled of the Irish formations. The division served as a formation of the United Kingdom's British Army during World War I.

History

Formed in Ireland on 21 August 1914, the 10th Division was sent to Gallipoli where, as part of General Sir Frederick Stopford's IX Corps, at Suvla Bay on 7 August it participated in the Landing at Suvla Bay and the August offensive. Some battalions of the division were landed at Anzac and fought at Chunuk Bair.

In September 1915, when the Suvla front became a stalemate, the division was moved to Salonika where it remained for two years and fought the Battle of Kosturino.

The division moved to Egypt in September 1917 where it joined General Chetwode's XX Corps. It fought in the Third Battle of Gaza which succeeded in breaking the resistance of the Turkish defenders in southern Palestine.

Heavy losses on the Western Front following Operation Michael, the great German spring offensive in 1918, resulted in the transfer of ten of the division's battalions from Palestine to France, their place being taken by Indian Army units. This left only one British battalion per brigade. The remainder of the division remained in Palestine until the end of the war with Turkey on 31 October 1918.

On 12 November 1918 the Division concentrated at Sarafand, ready for moving back to Egypt. By 1 December it had returned to Cairo.

Order of battle
The division comprised the following brigades:

 * 29th Brigade:
 * 5th Battalion, Royal Irish Regiment (left June 1915 to become the divisional pioneer battalion)
 * 6th (Service) Battalion, Royal Irish Rifles (disbanded May 1918)
 * 5th (Service) Battalion, Connaught Rangers (left April 1918)
 * 6th (Service) Battalion, Prince of Wales's Leinster Regiment (Royal Canadians) (left April 1918)
 * 10th (Service) Battalion, Hampshire Regiment (joined March left October 1915)
 * 1st Battalion, Prince of Wales's Leinster Regiment (Royal Canadians) (joined November 1916)
 * 29th Machine Gun Company (formed 10 May 1916 left to move into 10th Battalion Machine Gun Corps (M.G.C.) 7 May 1918)
 * 29th Trench Mortar Battery (joined 2 October 1916 as No 7 Stokes Mortar Battery transferred to Divisional TMB 17 October 1917)

The brigade was reorganised with Indian Army units from April to June 1918

 * 1st Battalion, Prince of Wales's Leinster Regiment (Royal Canadians)
 * 1st Battalion, 54th Sikhs (joined 27 April 1918)
 * 1st Battalion, 101st Grenadiers (joined 30 April 1918)
 * 2nd Battalion, 151st Sikh Infantry (joined 10 June 1918)

 * 30th Brigade :
 * 6th (Service) Battalion, Royal Munster Fusiliers (left 30 April 1918)
 * 7th (Service) Battalion, Royal Munster Fusiliers (absorbed by the 6th Battalion 3 November 1916)
 * 6th (Service) Battalion, Royal Dublin Fusiliers (left 27 May 1918)
 * 7th (Service) Battalion, Royal Dublin Fusiliers (left 30 April 1918)
 * 1st Battalion, Royal Irish Regiment (joined 3 November 1916)
 * 30th Machine Gun Company (formed 10 May 1916, left to move into 10th Battalion M.G.C. 7 May 1918)
 * 30th Trench Mortar Battery (joined 28 September 1916 as No 8 Stokes Mortar Battery, transferred to Divisional TMB 17 October 1917)

The brigade was reorganised with Indian Army units from April to June 1918

 * 1st Battalion, Royal Irish Regiment
 * 38th Dogras (joined 29 April 1918)
 * 1st Battalion, Kashmir Rifles (joined 30 April 1918)
 * 46th Punjabis (joined 25 May 1918)

 * 31st Brigade :
 * 5th (Service) Battalion, Royal Inniskilling Fusiliers (left 28 May 1918)
 * 6th (Service) Battalion, Royal Inniskilling Fusiliers (left 2 May 1918)
 * 5th (Service) Battalion, Princess Victoria's (Royal Irish Fusiliers) (left 30 April 1918)
 * 6th (Service) Battalion, Princess Victoria's (Royal Irish Fusiliers) (absorbed by the 5th Battalion 2 November 1916, )
 * 2nd Battalion, Princess Victoria's (Royal Irish Fusiliers) (joined 2 November 1916)
 * 31st Machine Gun Company (formed 11 May 1916, left to move into 10th Battalion M.G.C. 7 May 1918)
 * 31st Trench Mortar Battery (joined 17 October 1916, transferred to Divisional TMB 17 October 1917)

The brigade was reorganised with Indian Army units from April to June 1918

 * 6th (Service) Battalion, Princess Victoria's (Royal Irish Fusiliers)
 * 74th Punjabis (joined 29 April 1918)
 * 2nd Battalion, 101st Grenadiers (joined 1 May 1918)
 * 38th (Service) Battalion, Royal Fusiliers (attached 11 June – 17 July 1918)
 * 2nd Battalion, 42nd Deoli Regiment (joined 18 July 1918); Pioneers :

Divisional Troops
 * 5th (Service) Battalion, Royal Irish Regiment (joined as Divisional Pioneer Battalion June 1915, left April 1918)
 * 2nd Battalion, 155th Pioneers (Indian pioneers)(from July 1918)
 * Divisional Trench Mortar Battery (formed 17 October 1917, broken up 9 June 1918)
 * Divisional Mounted Troops
 * 10th Divisional Cyclist Company, Army Cyclist Corps (left 7 December 1916)
 * 10th Divisional Train Army Service Corps
 * 108th, 109th, 110th and 111th Companies (left October 1915)
 * 471st, 472nd, 473rd and 474th Companies (joined October 1915 from 52nd Division)
 * 25th Mobile Veterinary Section Army Veterinary Corps
 * 212th Divisional Employment Company (formed by 23 June 1917)

Royal Artillery
 * LIV Brigade, Royal Field Artillery (R.F.A.) (left 29 August 1917)
 * LV Brigade, R.F.A. (left January 1916)
 * LVI Brigade, R.F.A. (left January 1916)
 * LVII (Howitzer) Brigade, R.F.A. (left 28 August 1917)
 * 10th Divisional Ammunition Column R.F.A. (the original column did not go overseas with the Division. The 29th Divisional Ammunition Column joined in Egypt in October 1915. Suffered losses when transport “Marquette” torpedoed off Salonika on 23 October. Numbers were made up by men, horses and equipment from 42nd Division Ammunition Column. Formally renumbered 10th DAC on 4 March 1916)
 * LXVII Brigade, R.F.A. (joined October 1915)
 * LXVIII Brigade, R.F.A. (joined October 1915)
 * 10th Heavy Battery Royal Garrison Artillery (R.G.A.) (joined March 1915, left by 10 August 1915)
 * 15th Heavy Battery R.G.A. (joined 10 August 1915, left by 19 December 1915)
 * IV Highland (Mountain) Brigade, R.G.A. (joined 13 August 1915)
 * 2nd Mountain Battery R.G.A. (joined 30 December 1915, left 27 February 1916)
 * CXXXII (Howitzer) Brigade, R.F.A. (joined 26 April 1916, broken up 25 January 1917)
 * Hong Kong & Singapore Mountain Battery R.G.A. (joined 1 September 1918, left 26 October 1918)

Royal Engineers
 * 65th Field Company (left 14 July 1918)
 * 66th Field Company
 * 85th Field Company (joined January 1915)
 * 10th Divisional Signals Company
 * 18/3 Sappers & Miners (joined by 17 July 1918)

Royal Army Medical Corps
 * 30th, 31st and 32nd Field Ambulances (left 20 May 1918)
 * 154th, 165th and 166th Camel Field Ambulances (joined 20 May 1918)
 * 21st Sanitary Section (left 31 July 1915, rejoined October 1915, left again 22 October 1917)
 * 18th Sanitary Section (joined 22 October 1917)

Battles and engagements
Gallipoli Campaign
 * The landing at Suvla.
 * Battle of Sari Bair.
 * Capture of Chocolate Hill.
 * Hill 60.

Salonika front
 * Battle of Kosturino.
 * Retreat from Serbia.
 * Capture of the Karajokois.
 * Capture of Yenikoi.

Sinai and Palestine Campaign
 * Third Battle of Gaza.
 * Capture of the Sheria Position.
 * Capture of Jerusalem.
 * Defence of Jerusalem.
 * Tell 'Asure.
 * Battle of Nablus.

General Officers Commanding
Commanders included:
 * 24 August 1914 – 16 August 1915: Lieutenant-General Sir Bryan Mahon
 * 16 August – 19 August 1915: Brigadier-General F. F. Hill (acting)
 * 19 August – 23 August 1915: Major-General William Peyton
 * 23 August – 18 November 1915: Lieutenant-General Sir Bryan Mahon
 * 18 November – 20 December 1915: Brigadier-General L. L. Nicol (acting)
 * 20 December 1915 – 12 July 1916: Major-General John Longley
 * 12 July – 24 September 1916: Brigadier-General L. L. Nicol (acting)
 * 24 September 1916 – 11 June 1918: Major-General John Longley
 * 11 June – 18 August 1918: Brigadier-General E. M. Morris (acting)
 * 18 August 1918 – June 1919: Major-General John Longley
 * June 1919 – 1921: Major-General Sir George Gorringe

Great War Memorials

 * Irish National War Memorial Gardens Dublin.
 * Island of Ireland Peace Park Messines, Belgium.
 * Menin Gate Memorial Ypres, Belgium.
 * Ulster Tower Memorial Thiepval, France.
 * Dojran Celtic Cross, Doiran Lake, Macedonia.

title: 10th (Irish) Division


text:
Étale cohomology

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

History
Étale cohomology was introduced by, using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality part of the conjectures in 1960 using p-adic methods), and the remaining conjecture, the analogue of the Riemann hypothesis was proved by Pierre Deligne (1974) using ℓ-adic cohomology.

Further contact with classical theory was found in the shape of the Grothendieck version of the Brauer group; this was applied in short order to diophantine geometry, by Yuri Manin. The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such as Poincaré duality and the Lefschetz fixed-point theorem in this context.

Grothendieck originally developed étale cohomology in an extremely general setting, working with concepts such as Grothendieck toposes and Grothendieck universes. With hindsight, much of this machinery proved unnecessary for most practical applications of the étale theory, and gave a simplified exposition of étale cohomology theory. Grothendieck's use of these universes (whose existence cannot be proved in Zermelo–Fraenkel set theory) led to some speculation that étale cohomology and its applications (such as the proof of Fermat's Last Theorem) require axioms beyond ZFC. However, in practice étale cohomology is used mainly in the case of constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with care the necessary objects can be constructed without using any uncountable sets, and this can be done in ZFC, and even in much weaker theories.

Étale cohomology quickly found other applications, for example Deligne and George Lusztig used it to construct representations of finite groups of Lie type; see Deligne–Lusztig theory.

Motivation
For complex algebraic varieties, invariants from algebraic topology such as the fundamental group and cohomology groups are very useful, and one would like to have analogues of these for varieties over other fields, such as finite fields. (One reason for this is that Weil suggested that the Weil conjectures could be proved using such a cohomology theory.) In the case of cohomology of coherent sheaves, Serre showed that one could get a satisfactory theory just by using the Zariski topology of the algebraic variety, and in the case of complex varieties this gives the same cohomology groups (for coherent sheaves) as the much finer complex topology. However, for constant sheaves such as the sheaf of integers this does not work: the cohomology groups defined using the Zariski topology are badly behaved. For example, Weil envisioned a cohomology theory for varieties over finite fields with similar power as the usual singular cohomology of topological spaces, but in fact, any constant sheaf on an irreducible variety has trivial cohomology (all higher cohomology groups vanish).

The reason that the Zariski topology does not work well is that it is too coarse: it has too few open sets. There seems to be no good way to fix this by using a finer topology on a general algebraic variety. Grothendieck's key insight was to realize that there is no reason why the more general open sets should be subsets of the algebraic variety: the definition of a sheaf works perfectly well for any category, not just the category of open subsets of a space. He defined étale cohomology by replacing the category of open subsets of a space by the category of étale mappings to a space: roughly speaking, these can be thought of as open subsets of finite unbranched covers of the space. These turn out (after a lot of work) to give just enough extra open sets that one can get reasonable cohomology groups for some constant coefficients, in particular for coefficients Z/nZ when n is coprime to the characteristic of the field one is working over.

Some basic intuitions of the theory are these:

 * The étale requirement is the condition that would allow one to apply the implicit function theorem if it were true in algebraic geometry (but it isn't — implicit algebraic functions are called algebroid in older literature).
 * There are certain basic cases, of dimension 0 and 1, and for an abelian variety, where the answers with constant sheaves of coefficients can be predicted (via Galois cohomology and Tate modules).

Definitions
For any scheme X the category Et(X) is the category of all étale morphisms from a scheme to X. It is an analogue of the category of open subsets of a topological space, and its objects can be thought of informally as "étale open subsets" of X. The intersection of two open sets of a topological space corresponds to the pullback of two étale maps to X. There is a rather minor set-theoretical problem here, since Et(X) is a "large" category: its objects do not form a set.

A presheaf on a topological space X is a contravariant functor from the category of open subsets to sets. By analogy we define an étale presheaf on a scheme X to be a contravariant functor from Et(X) to sets.

A presheaf F on a topological space is called a sheaf if it satisfies the sheaf condition: whenever an open subset is covered by open subsets Ui, and we are given elements of F(Ui) for all i whose restrictions to Ui ∩ Uj agree for all i, j, then they are images of a unique element of F(U). By analogy, an étale presheaf is called a sheaf if it satisfies the same condition (with intersections of open sets replaced by pullbacks of étale morphisms, and where a set of étale maps to U is said to cover U if the topological space underlying U is the union of their images). More generally, one can define a sheaf for any Grothendieck topology on a category in a similar way.

The category of sheaves of abelian groups over a scheme has enough injective objects, so one can define right derived functors of left exact functors. The étale cohomology groups Hi(F) of the sheaf F of abelian groups are defined as the right derived functors of the functor of sections,

 * $$F \to \Gamma(F)$$

(where the space of sections Γ(F) of F is F(X)). The sections of a sheaf can be thought of as Hom(Z, F) where Z is the sheaf that returns the integers as an abelian group. The idea of derived functor here is that the functor of sections doesn't respect exact sequences as it is not right exact; according to general principles of homological algebra there will be a sequence of functors H 0, H 1, ... that represent the 'compensations' that must be made in order to restore some measure of exactness (long exact sequences arising from short ones). The H 0 functor coincides with the section functor Γ.

More generally, a morphism of schemes f : X → Y induces a map f∗ from étale sheaves over X to étale sheaves over Y, and its right derived functors are denoted by Rqf∗, for q a non-negative integer. In the special case when Y is the spectrum of an algebraically closed field (a point), Rqf∗(F ) is the same as Hq(F ).

Suppose that X is a Noetherian scheme. An abelian étale sheaf F over X is called finite locally constant if it is represented by an étale cover of X. It is called constructible if X can be covered by a finite family of subschemes on each of which the restriction of F is finite locally constant. It is called torsion if F(U) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.

ℓ-adic cohomology groups
In applications to algebraic geometry over a finite field Fq with characteristic p, the main objective was to find a replacement for the singular cohomology groups with integer (or rational) coefficients, which are not available in the same way as for geometry of an algebraic variety over the complex number field. Étale cohomology works fine for coefficients Z/nZ for n co-prime to p, but gives unsatisfactory results for non-torsion coefficients. To get cohomology groups without torsion from étale cohomology one has to take an inverse limit of étale cohomology groups with certain torsion coefficients; this is called ℓ-adic cohomology, where ℓ stands for any prime number different from p. One considers, for schemes V, the cohomology groups

 * $$H^i(V, \mathbf{Z}/\ell^k\mathbf{Z})$$

and defines the ℓ-adic cohomology group

 * $$H^i(V,\mathbf{Z}_\ell) = \varprojlim H^i(V, \mathbf{Z}/\ell^k\mathbf{Z})$$

as their inverse limit. Here Zℓ denotes the ℓ-adic integers, but the definition is by means of the system of 'constant' sheaves with the finite coefficients Z/ℓkZ. (There is a notorious trap here: cohomology does not commute with taking inverse limits, and the ℓ-adic cohomology group, defined as an inverse limit, is not the cohomology with coefficients in the étale sheaf Zℓ; the latter cohomology group exists but gives the "wrong" cohomology groups.)

More generally, if F is an inverse system of étale sheaves Fi, then the cohomology of F is defined to be the inverse limit of the cohomology of the sheaves Fi

 * $$H^q(X, F) = \varprojlim H^q(X, F_i),$$

and though there is a natural map

 * $$H^q(X,\varprojlim F_i) \to \varprojlim H^q(X, F_i),$$

this is not usually an isomorphism. An ℓ-adic sheaf is a special sort of inverse system of étale sheaves Fi, where i runs through positive integers, and Fi is a module over Z/ℓi Z and the map from Fi+1 to Fi is just reduction mod Z/ℓi Z.

When V is a non-singular algebraic curve of genus g, H1 is a free Zℓ-module of rank 2g, dual to the Tate module of the Jacobian variety of V. Since the first Betti number of a Riemann surface of genus g is 2g, this is isomorphic to the usual singular cohomology with Zℓ coefficients for complex algebraic curves. It also shows one reason why the condition ℓ ≠ p is required: when ℓ = p the rank of the Tate module is at most g.

Torsion subgroups can occur, and were applied by Michael Artin and David Mumford to geometric questions. To remove any torsion subgroup from the ℓ-adic cohomology groups and get cohomology groups that are vector spaces over fields of characteristic 0 one defines

 * $$H^i(V,\mathbf{Q}_\ell)=H^i(V,\mathbf{Z}_\ell)\otimes\mathbf{Q}_\ell.$$

This notation is misleading: the symbol Qℓ on the left represents neither an étale sheaf nor an ℓ-adic sheaf. The etale cohomology with coefficients in the constant etale sheaf Qℓ does also exist but is quite different from $$H^i(V,\mathbf{Z}_\ell)\otimes\mathbf{Q}_\ell$$. Confusing these two groups is a common mistake.

Properties
In general the ℓ-adic cohomology groups of a variety tend to have similar properties to the singular cohomology groups of complex varieties, except that they are modules over the ℓ-adic integers (or numbers) rather than the integers (or rationals). They satisfy a form of Poincaré duality on non-singular projective varieties, and the ℓ-adic cohomology groups of a "reduction mod p" of a complex variety tend to have the same rank as the singular cohomology groups. A Künneth formula also holds.

For example, the first cohomology group of a complex elliptic curve is a free module of rank 2 over the integers, while the first ℓ-adic cohomology group of an elliptic curve over a finite field is a free module of rank 2 over the ℓ-adic integers, provided ℓ is not the characteristic of the field concerned, and is dual to its Tate module.

There is one way in which ℓ-adic cohomology groups are better than singular cohomology groups: they tend to be acted on by Galois groups. For example, if a complex variety is defined over the rational numbers, its ℓ-adic cohomology groups are acted on by the absolute Galois group of the rational numbers: they afford Galois representations.

Elements of the Galois group of the rationals, other than the identity and complex conjugation, do not usually act continuously on a complex variety defined over the rationals, so do not act on the singular cohomology groups. This phenomenon of Galois representations is related to the fact that the fundamental group of a topological space acts on the singular cohomology groups, because Grothendieck showed that the Galois group can be regarded as a sort of fundamental group. (See also Grothendieck's Galois theory.)

Calculation of étale cohomology groups for algebraic curves
The main initial step in calculating étale cohomology groups of a variety is to calculate them for complete connected smooth algebraic curves X over algebraically closed fields k. The étale cohomology groups of arbitrary varieties can then be controlled using analogues of the usual machinery of algebraic topology, such as the spectral sequence of a fibration. For curves the calculation takes several steps, as follows. Let Gm denote the sheaf of non-vanishing functions.

Calculation of H1(X, Gm)
The exact sequence of étale sheaves

 * $$1\to \mathbf{G}_m\to  j_*\mathbf{G}_{m, K}\to \bigoplus_{x\in |X|}i_{x*}\mathbf{Z}\to  1$$

gives a long exact sequence of cohomology groups

 * $$\begin{align}

0 &\to H^0(\mathbf{G}_m)\to  H^0(j_*\mathbf{G}_{m, K})\to \bigoplus\nolimits_{x\in |X|}H^0(i_{x*}\mathbf{Z})  \to \\ &\to H^1(\mathbf{G}_m)\to  H^1(j_*\mathbf{G}_{m, K})\to  \bigoplus\nolimits_{x\in |X|}H^1(i_{x*}\mathbf{Z})  \to \\ &\to \cdots \end{align}$$

Here j is the injection of the generic point, ix is the injection of a closed point x, Gm,K is the sheaf Gm on Spec K (the generic point of X), and Zx is a copy of Z for each closed point of X. The groups H i(ix* Z) vanish if i > 0 (because ix* Z is a skyscraper sheaf) and for i = 0 they are Z so their sum is just the divisor group of X. Moreover, the first cohomology group H 1(X, j∗Gm,K) is isomorphic to the Galois cohomology group H 1(K, K*) which vanishes by Hilbert's theorem 90. Therefore, the long exact sequence of étale cohomology groups gives an exact sequence

 * $$K\to \operatorname{Div}(X)\to H^1(\mathbf{G}_m)\to 1$$

where Div(X) is the group of divisors of X and K is its function field. In particular H 1(X, Gm) is the Picard group Pic(X) (and the first cohomology groups of Gm are the same for the étale and Zariski topologies). This step works for varieties X of any dimension (with points replaced by codimension 1 subvarieties), not just curves.

Calculation of Hi(X, Gm)
The same long exact sequence above shows that if i ≥ 2 then the cohomology group H i(X, Gm) is isomorphic to H i(X, j*Gm,K), which is isomorphic to the Galois cohomology group H i(K, K*). Tsen's theorem implies that the Brauer group of a function field K in one variable over an algebraically closed field vanishes. This in turn implies that all the Galois cohomology groups H i(K, K*) vanish for i ≥ 1, so all the cohomology groups H i(X, Gm) vanish if i ≥ 2.

Calculation of Hi(X, μn)
If μn is the sheaf of n-th roots of unity and n and the characteristic of the field k are coprime integers, then:

 * $$H^i (X, \mu_n) = \begin{cases} \mu_n(k) & i =0 \\ \operatorname{Pic}_n(X) & i = 1 \\ \mathbf{Z}/n\mathbf{Z} & i =2 \\ 0 & i \geqslant 3 \end{cases}$$

where Picn(X) is group of n-torsion points of Pic(X). This follows from the previous results using the long exact sequence

 * $$\begin{align}

0 &\to H^0(X, \mu_n)\to H^0(X, \mathbf{G}_m)\to H^0(X, \mathbf{G}_m)\to \\ &\to H^1(X, \mu_n)\to  H^1(X, \mathbf{G}_m)\to  H^1(X, \mathbf{G}_m)\to  \\ &\to H^2(X, \mu_n)\to  H^2(X, \mathbf{G}_m)\to  H^2(X, \mathbf{G}_m) \to \\ &\to \cdots \end{align}$$

of the Kummer exact sequence of étale sheaves

 * $$1 \to \mu_n \to  \mathbf{G}_m \xrightarrow{(\cdot)^n} \mathbf{G}_m \to  1.$$

and inserting the known values

 * $$ H^i (X, \mathbf{G}_m) = \begin{cases}  k^* & i = 0 \\ \operatorname{Pic}(X) & i =1 \\ 0 &i \geqslant 2 \end{cases}$$

In particular we get an exact sequence

 * $$1\to H^1(X, \mu_n)\to \operatorname{Pic}(X)\xrightarrow{\times n} \operatorname{Pic}(X)\to  H^2(X, \mu_n)\to  1.$$

If n is divisible by p this argument breaks down because p-th roots of unity behave strangely over fields of characteristic p. In the Zariski topology the Kummer sequence is not exact on the right, as a non-vanishing function does not usually have an n-th root locally for the Zariski topology, so this is one place where the use of the étale topology rather than the Zariski topology is essential.

Calculation of H i(X, Z/nZ)
By fixing a primitive n-th root of unity we can identify the group Z/nZ with the group μn of n-th roots of unity. The étale group H i(X, Z/nZ) is then a free module over the ring Z/nZ and its rank is given by:

 * $$\operatorname{rank}(H^i(X, \mathbf{Z}/n\mathbf{Z})) = \begin{cases} 1 & i =0 \\ 2g & i=1 \\1 & i = 2\\0 & i \geqslant 3 \end{cases}$$

where g is the genus of the curve X. This follows from the previous result, using the fact that the Picard group of a curve is the points of its Jacobian variety, an abelian variety of dimension g, and if n is coprime to the characteristic then the points of order dividing n in an abelian variety of dimension g over an algebraically closed field form a group isomorphic to (Z/nZ)2g. These values for the étale group H i(X, Z/nZ) are the same as the corresponding singular cohomology groups when X is a complex curve.

Calculation of Hi(X, Z/pZ)
It is possible to calculate étale cohomology groups with constant coefficients of order divisible by the characteristic in a similar way, using the Artin–Schreier sequence

 * $$0\to \mathbf{Z}/p\mathbf{Z}\to  K\ \xrightarrow{{}\atop x\mapsto x^p-x}\ K\to  0$$

instead of the Kummer sequence. (For coefficients in Z/pnZ there is a similar sequence involving Witt vectors.) The resulting cohomology groups usually have ranks less than that of the corresponding groups in characteristic 0.

Examples of étale cohomology groups

 * If X is the spectrum of a field K with absolute Galois group G, then étale sheaves over X correspond to continuous sets (or abelian groups) acted on by the (profinite) group G, and étale cohomology of the sheaf is the same as the group cohomology of G, i.e. the Galois cohomology of K.
 * If X is a complex variety, then étale cohomology with finite coefficients is isomorphic to singular cohomology with finite coefficients. (This does not hold for integer coefficients.) More generally the cohomology with coefficients in any constructible sheaf is the same.
 * If F is a coherent sheaf (or Gm) then the étale cohomology of F is the same as Serre's coherent sheaf cohomology calculated with the Zariski topology (and if X is a complex variety this is the same as the sheaf cohomology calculated with the usual complex topology).
 * For abelian varieties and curves there is an elementary description of ℓ-adic cohomology. For abelian varieties the first ℓ-adic cohomology group is the dual of the Tate module, and the higher cohomology groups are given by its exterior powers. For curves the first cohomology group is the first cohomology group of its Jacobian. This explains why Weil was able to give a more elementary proof of the Weil conjectures in these two cases: in general one expects to find an elementary proof whenever there is an elementary description of the ℓ-adic cohomology.

Poincaré duality and cohomology with compact support
The étale cohomology groups with compact support of a variety X are defined to be

 * $$H_c^q(X, F) = H^q(Y, j_!F)$$

where j is an open immersion of X into a proper variety Y and j! is the extension by 0 of the étale sheaf F to Y. This is independent of the immersion j. If X has dimension at most n and F is a torsion sheaf then these cohomology groups $$H_c^q(X, F)$$ with compact support vanish if q > 2n, and if in addition X is affine of finite type over a separably closed field the cohomology groups $$H^q(X, F)$$ vanish for q > n (for the last statement, see SGA 4, XIV, Cor.3.2).

More generally if f is a separated morphism of finite type from X to S (with X and S Noetherian) then the higher direct images with compact support Rqf! are defined by

 * $$R^qf_!(F)=R^qg_*(j_!F)$$

for any torsion sheaf F. Here j is any open immersion of X into a scheme Y with a proper morphism g to  S (with f = gj), and as before the definition does not depend on the choice of j and Y. Cohomology with compact support is the special case of this with S a point. If f is a separated morphism of finite type then Rqf! takes constructible sheaves on X to constructible sheaves on S. If in addition the fibers of f have dimension at most n then Rqf! vanishes on torsion sheaves for q > 2n. If X is a complex variety then Rqf! is the same as the usual higher direct image with compact support (for the complex topology) for torsion sheaves.

If X is a smooth algebraic variety of dimension N and n is coprime to the characteristic then there is a trace map

 * $$\operatorname{Tr}: H_c^{2N}(X, \mu_n^N) \rightarrow \mathbf{Z}/n\mathbf{Z}$$

and the bilinear form Tr(a ∪ b) with values in Z/nZ identifies each of the groups

 * $$H^i_c(X,\mu_n^N)$$

and

 * $$H^{2N-i}(X,\mathbf{Z}/n\mathbf{Z})$$

with the dual of the other. This is the analogue of Poincaré duality for étale cohomology.

An application to curves
This is how the theory could be applied to the local zeta-function of an algebraic curve.

Theorem. Let $X$ be a curve of genus $g$ defined over $F_{p}$, the finite field with $p$ elements. Then for $n ≥ 1$

 * $$\#X \left (\mathbf F_{p^n} \right ) = p^n + 1 -\sum_{i=1}^{2g} \alpha_i^n,$$

where $α_{i}$ are certain algebraic numbers satisfying $|α_{i}| = √p$.

This agrees with $P^{1}(F_{p^{n}})$ being a curve of genus $0$ with $p^{n} + 1$ points. It also shows that the number of points on any curve is rather close (within $2gp^{n / 2}$) to that of the projective line; in particular, it generalizes Hasse's theorem on elliptic curves.

Idea of proof
According to the Lefschetz fixed-point theorem, the number of fixed points of any morphism $f : X → X$ is equal to the sum

 * $$\sum_{i=0}^{2 \dim(X)} (-1)^i \operatorname{Tr} \left (f|_{H^i(X)} \right ).$$

This formula is valid for ordinary topological varieties and ordinary topology, but it is wrong for most algebraic topologies. However, this formula does hold for étale cohomology (though this is not so simple to prove).

The points of $X$ that are defined over $F_{p^{n}}$ are those fixed by $F^{n}$, where $F$ is the Frobenius automorphism in characteristic $p$.

The étale cohomology Betti numbers of $X$ in dimensions 0, 1, 2 are 1, 2g, and 1 respectively.

According to all of these,

 * $$\#X \left (\mathbf F_{p^n} \right ) = \operatorname{Tr} \left (F^n|_{H^0(X)} \right )- \operatorname{Tr} \left (F^n|_{H^1(X)} \right ) + \operatorname{Tr} \left (F^n|_{H^2(X)} \right ).$$

This gives the general form of the theorem.

The assertion on the absolute values of the $α_{i}$ is the 1-dimensional Riemann Hypothesis of the Weil Conjectures.

The whole idea fits into the framework of motives: formally [X] = [point] + [line] + [1-part], and [1-part] has something like $√p$ points.

title: Étale cohomology


text:
Talk:Euler equations (fluid dynamics)

Old discussions and comments
I would assume that the first equation expresses the conservation of mass and is the equation of continuity, the second one is the conservation of momentum and is the true Euler equation. The third is the conservation of enthalpy, and is the enthalpy or energy equation. None of them incorporates outer force field effects, wich would be nice tough. --- should I put stuff about Rankine-Hugoniot conditions here or under shock waves?

Mo ena ene test lor eulers la. Ki pou fou????

Is it possible for someone to define all the variables in this article? Rtdrury 10:59, 19 December 2005 (UTC)

-

As an answer to : "in particular, it is not intuitively clear why this equation is correct and :$$ \left(\partial/{\partial t}+{\mathbf u}\cdot\nabla\right)(\rho{\mathbf u})+\nabla p=0 $$ is incorrect" : the reason is simple enough, Newton's second law states Force=mass*(derivative of u) not Force=derivative of (mass*u), where u is the velocity, therefore it is clear that the formula given above cannot be correct if the mass density is not constant (like in the flow of a compressible medium). This is a common misconception of Newton law, which for example leads to incorrect results for the eqution of motion of rockets. Gringo.ch 10:44, 3 October 2006 (UTC)

Incorrect! If you read Principia, Newton's law states that F = dp/dt, where p is the momentum. This must be the correct form because it can be shown that it is equivalent to both Lagrangian and Hamiltonian dynamics, for generalized coordinate q and its canonical momentum p. --GringoNegro

Yes, you are right, sorry I wasn't clear enough. My point was that, when you have systems where the mass is not constant in time (say, a droplet falling while losing mass to evaporation) then you have to be a bit careful on how to apply Newton's 2nd law. In that example, blindly writing F=dp/dt => mg=d/dt(mv)=dm/dt v+ m dv/dt would yield the acceleration of the droplet as a=dv/dt=g-v/m*dm/dt which is not the correct answer a=g, whereas just writing F=m(t)*dv/dt gives you directly the right solution (isotropic evaporation assumed here, no air friction). To get the right answer with the former approach one should account also for the evaporated material. Gringo.ch 17:01, 28 February 2007 (UTC)

It doesn't give the right answer because, as you say, there are other quantities that must be accounted for as forces on the left hand side--that's one of the main difficulties of Newtonian mechanics: expressing all the forces acting on a body. That's also why Lagrangian dynamics is so appealing. As we both agree that F = dp/dt (as long as F is the correct F) is the correct equation, we still haven't answered why "it is not intuitively clear why this equation is correct and :$$ \left(\partial/{\partial t}+{\mathbf u}\cdot\nabla\right)(\rho{\mathbf u})+\nabla p=0 $$ is incorrect" --GringoNegro

What about this: we are following a mass element dm in the flow, which has a volume dV(t) which depends on time. The density is then r=dm/dV, which changes in time, but dm is constant. Then we have that F=dp/dt. The force per volume is -grad p, hence F = -grad p *dV(t)=d/dt(dm*v(t))=dm*dv/dt, which can be written as -grad p= r* dv/dt. Transformation from the streaming system to the laboratory system (d/dt goes into d/dt + v grad ) delivers the Euler equation. So the confusion seems to arise in the step going from the force F to the force per volume, right? Gringo.ch 11:27, 1 March 2007 (UTC)

While it could be argued that the non-conservation form of the Euler equations is less intuitive, it remains the standard form used in analytic solution of the Euler equations. If there are no objections, I will do some rewording to place the conservation and non-conservation forms of the equations on a more equal footing. On another note, could someone explain "Although the Euler equations formally reduce to potential flow in the limit of vanishing Mach number, this is not helpful in practice, essentially because the approximation of incompressibility is almost invariably very close."? It seems to me that if the approximation of incompressibility is very close, an assumption of incompressibility would be a very good one to make. Cheers. Chrisjohnson 22:24, 16 March 2007 (UTC)
 * Should the note about the other equation not being true dropped? Also, the appended comment of both being "correct"? (what does that exactly mean? I can see both are correct only if the fluid is incompressible) --Daniel (talk) 12:51, 27 April 2008 (UTC)

I would like to object to the remark about the use of the ideal gas law as $$ p=\rho(\gamma-1)e $$ which is only true for caloric perfect gases and relatively cold mixtures of caloric perfect gases (i.e. gas mixtures around 300K). In fact, $$ p=\rho(\gamma-1)e $$ is only true if $$ e = c_v T $$, which is only relatively true for caloric perfect gases, and $$ nR=C_p-C_v $$ which is not true for mixtures of ideal gases that are allowed to react with each other. While a mixture of ideal gases is not ideal it is still possible to use the ideal gas law in it's original formulation if no other assumptions are made.

There are thus two problems with the rewriting of the ideal gas law as $$ p=\rho(\gamma-1)e $$:

1) $$c_v $$ is a function of temperature and is defined as $$ \left(\partial e / \partial T \right)_p$$. If $$c_v $$ is assumed to be constant the equation $$ e = c_v T$$ is valid, but only then.

2) For a mixture of ideal gases the number of moles $$ n $$ changes with temperaure and pressure. Since $$C_p$$ and $$C_v$$ are defined as differentials of the internal energy $$ nR=C_p-C_v $$ will not always be true. It is, however, still possible to use the ideal gas law in it's original formulation for such a gas.

While it's a good approximation for many gases it is unnecessary to refer to a version of the ideal gas law that makes assumptions that are not always true. The best choice would be to use $$ pV = nRT $$ since this equation is always valid, even for non-ideal reacting gas-mixtures where n changes with temperature and pressure. (I know that the ideal gas law isn't correct in itself but if one wants to use the ideal gas law, one should use it correctly or have to make the assumptions, and check their validity, themself.)

Per Öberg - Nov 15 2007 (With minor changes Nov 21) —Preceding unsigned comment added by 130.236.50.246 (talk) 09:23, 15 November 2007 (UTC)

Stiff equation of state
The article says "The well-known Bernoulli's equation can be derived by integrating Euler's equation along a streamline, under the assumption of constant density and a sufficiently stiff equation of state."

Would it be correct to say that an assumption of sufficiently stiff equation of state is equivalent to saying "sufficiently incompressible"?

If so, I would suggest that we say

 * Integration of Euler's equation for incompressible flow, (specifically an assumption of constant density and a sufficiently stiff equation of state), yields the well-known Bernoulli's equation.

--Hroðulf (or Hrothulf) (Talk) 14:51, 23 March 2010 (UTC)

 * "Sufficiently stiff equation of state" is not equivalent to "sufficiently incompressible". Neither depends on the other. Constant density, however, already implies "suffuciently incompressible", indeed it means "not compressible at all". So I would agree with substituting "incompressible flow" for "constant density", but the content of the rest of the sentence should not be changed. Cs32en   Talk to me  18:45, 23 March 2010 (UTC)

 * Thanks. You corrected a couple of my misunderstandings.

 * Could you explain the precise meaning of "stiffness of the equation of state"? My only recollection of the term stiff equations is from the field of numerical analysis, and the top hits for stiff equations of state in Google and Google Books are advanced pieces in astrophysics and nuclear physics (interesting but unusual fluids.)

 * Landau and Lifshitz (Vol 6 Ch 1) integrate Euler's momentum equation to get forms of Bernoulli's equation (twice, once for constant density as does Wikipedia, and once for the general case.) If I understand the calculation correctly, (and there is every chance I don't), they do so by examining steady flow, and by constraining thermal conductivity and viscosity to be unimportant.  Is the formulation in the article a way of saying the same thing, or another way of getting to the adiabatic and isentropic constraints that L&L use?

 * --Hroðulf (or Hrothulf) (Talk) 16:39, 24 March 2010 (UTC)

the gravity term (!)
I think there should a gravity term rho * g be included in the momentum equation since dm * Dv/Dt = dF = -dP*dA + dm*g ... rho * Dv/Dt = -grad(P) + rho * g. —Preceding unsigned comment added by 193.190.253.144 (talk) 22:33, 5 October 2010 (UTC)
 * Body forces, of which gravity is an important one, are not included in the presented form of the Euler equations. -- Crowsnest (talk) 11:10, 6 October 2010 (UTC)

Removed comment
I removed these comments from the conservation and component section:

"in particular, it is not intuitively clear why this equation is correct and $\left(\partial/{\partial t}+{\mathbf u}\cdot\nabla\right)(\rho{\mathbf u})+\nabla p=0$ is incorrect) -- Both forms are 'correct'"

"FR - This note represents the author's opinion of what he/she believes is more intuitive. As opposed the this author, I would say that it is the non-conservation form that is more directly linked to the classic representation of Newton's second law F = m du/dt when a fixed mass is followed in time. For Fluids the non-conservation expression is just: mass/volume du/dt = F/Volume, where du/dt is the derivative of the velocity of the fluid particles as they pass through a certain point in the flow field (= substantial derivative, typically written as Du/Dt to remind the reader that an Eulerian frame of reference is used). The non-conservation form maps directly to the interpretation of Newton's 2nd law for fixed mass particles. What is more, the conservation form can be easily reworked from the non-conservation form by means of the continuity equation. That is using additional information (conservation of mass) thereby obscuring Newton's 2nd law. Of course, the conservation form can also be directly derived from principles of momentum changes, representing an alternative interpretation of Newton's 2nd law. This is then just a limiting case of the Reynolds Transport Theorem for which the control volume size is shrunk to an infinitesimal one. But, whether this shows the 'direct connection' to Newton's 2nd law can be argued as I have done. Therefore I would suggest to delete the opinions from this page. Best Regards, FR"

PAR (talk) 06:27, 22 February 2012 (UTC)

What about external forces?
What about external forces, such as gravity? Are they not supposed to be included in these equations? Or maybe this fluid isn't affected by gravity??

At least, the section about the Bernoulli equation for unsteady potential flow states that the equation is obtained by integrating the momentum equations of the Euler equations, and that equation does contain gravity. —Kri (talk) 16:29, 9 March 2012 (UTC)

 * External forces can off course be added to the momentum equations, and might also appear in the energy equation if they perform work. -- Crowsnest (talk) 14:01, 10 March 2012 (UTC)

 * Well, you say that they can off course be added, but they are not present in the equations in their current form in the article, which has to be the most important thing. Why are they not there? They are obviously supposed to be there. Do you agree with me? —Kri (talk) 10:55, 12 March 2012 (UTC)

 * Sure I agree with you. There are however not many textbooks of renowned fluid-dynamics scientists including them in the equations. Emphasis is often on how to present/derive them in a conservative form. And further, in aerodynamics (where they are mostly used) gravity is often of minor importance. So a reliable source including the external (body) forces, like gravity, has to be found. -- Crowsnest (talk) 18:04, 12 March 2012 (UTC)

 * What do you mean by that a reliable source including the external (body) forces has to be found? —Kri (talk) 23:15, 15 March 2012 (UTC)

 * Ah, never mind. I realized that you where talking about a literary source and not a source that should be included in the equations. —Kri (talk) 23:17, 15 March 2012 (UTC)

Merger of Streamline curvature theorem
There is hardly anything to find on "streamline curvature theorem": 4 papers in Japanese scientific journals. For a fundamental theorem in fluid dynamics this seems very unlikely, one would expect tons of search results. The same is the case for the term "Euler's normal equation".

What can be found is several references to the Euler equations in "streamline coordinates" or "path coordinates". These have a component normal to the streamline, the subject of the article proposed to merge here, and along the streamline one recovers Bernoulli's equation.

See:
 * This book is the reference in the article, and talks about "Euler's equation in streamline coordinates".
 * This book is the reference in the article, and talks about "Euler's equation in streamline coordinates".
 * This book is the reference in the article, and talks about "Euler's equation in streamline coordinates".

See also this Google search for the combination "Euler", "equations", "streamline" & "coordinates".

So my suggestion would be to merge Streamline curvature theorem into a section in this article, e.g. "Steady flow in streamline coordinates". -- Crowsnest (talk) 14:15, 11 May 2012 (UTC)

A section ”Steady flow in streamline coordinates" is added according to the proposal. If no problems in this section, I will merge Streamline curvature theorem into this section. くま兄やん (talk) 11:28, 17 May 2012 (UTC)
 * That looks really very good, くま兄やん, to my opinion. I changed Streamline curvature theorem into a redirect to this article in order to finish the merger. -- Crowsnest (talk) 12:54, 17 May 2012 (UTC)
 * Thank you, Crowsnest, for merging. くま兄やん (talk) 13:09, 17 May 2012 (UTC)

Minus signs
Energy equation in convective form is given as

$${D e \over Dt} = \frac p \rho \nabla \cdot \mathbf u$$

Should not it be

$${D e \over Dt} = -\frac p \rho \nabla \cdot \mathbf u$$ ?

See, e.g., http://www.mathematik.uni-dortmund.de/~kuzmin/afc2.pdf, beginning of section 2, which has $$\nabla(\rho H \mathbf v)$$, with $$\rho H = \rho E + p$$.

Also the section of "Enthalpy conservation" in the article seems to have lost a minus in $${D e \over Dt} = {D h \over Dt} - \frac 1 \rho \left({D p \over Dt} + \frac p \rho {D \rho \over Dt} \right)$$

It should be $${D e \over Dt} = {D h \over Dt} - \frac 1 \rho \left({D p \over Dt} - \frac p \rho {D \rho \over Dt} \right)$$ 192.198.151.43 (talk) 13:56, 29 July 2015 (UTC)

The above sign errors were corrected on 30 July 2015. There is another sign error in the 3rd equation of Sec. 4.2 on Enthalpy Conservation. It was:

$${D e \over Dt}= {D h \over Dt} + \frac 1 \rho \left(p \nabla \cdot \mathbf u - {D p \over Dt} \right)$$

and I am now changing it to:

$${D e \over Dt}= {D h \over Dt} - \frac 1 \rho \left( p \nabla \cdot \mathbf u + {D p \over Dt} \right)$$

Plasma g (talk) 16:42, 17 August 2015 (UTC)

Error?
(I give up using wikipedia math expressions:)

Section: "Incompressible Euler equations with constant and uniform density"

Contain the equation: du/dt + u. grad u = - grad w + g

the u. grad u term is not very clear. Clearer to write (u.grad) u. see p.1-2

This is made worse by the equation: du_j/dt+Sum_i u_j d(u_i + w e_i)/dr_i == 0 where to my eyes the second term is plainly wrong.

should be: du_j/dt +sum_i u_i d u_j/dr_i + dw/dr_j == 0 see p.1  — Preceding unsigned comment added by 194.47.168.94 (talk) 09:01, 10 May 2017 (UTC)

Deduction of the form of sound speed valid for ideal gases
Firstly, in my opinion, the second equation in the deduction of the sound speed for ideal gases in chapter Euler equations (fluid dynamics) is wrong. There should be a minus sign instead of a plus sign. Can somebody confirm that?

Second, division of the second equation by ρ−γ dρ does not yield the third equation, because ρ−γ-1 divided by ρ−γ yields ρ−1 instead of ρ.

Belgariath (talk) 12:08, 12 November 2019 (UTC)

singularity with a computer-aided demonstration
A preprint, posted online in October 2022, states that, under certain boundary conditions, the Euler equations fall in a mathematical singularity. It was the first time in which a singularity was discovered through a computer-aided demonstration. (source: Wired). 95.237.45.179 (talk) 20:59, 29 December 2022 (UTC)

Spontaneous stochasticity
This article notes that the eqns are ill-posed, without also specifying eqn of state. Additional refs for this are:
 * C. De Lellis and L. Szekelyhidi, On admissibility criteria for weak solutions of the Euler equations, Archive for Rational Mechanics and analysis 195, 225 (2010).
 * C. De Lellis and H. Kwon, On non-uniqueness of Holder continuous globally dissipative Euler flows, arXiv:2006.06482 (2020)
 * S. Daneri, E. Runa, and L. Szekelyhidi, Non-uniqueness for the Euler equations up to Onsager’s critical exponent, Annals of PDE 7, 1 (2021)

At high Reynolds numbers, this apparently leads to "Eulerian spontaneous stochasticity" during turbulent fluid motion:
 * A. A. Mailybaev, Spontaneously stochastic solutions in one-dimensional inviscid systems, Nonlinearity 29, 2238 (2016).
 * S. Thalabard, J. Bec, and A. A. Mailybaev, From the butterfly effect to spontaneous stochasticity in singular shear flows, Communications Physics 3, 1 (2020).

I'd like to read more about how/why the eqns are ill-posed without the eqn of state, and a description of spontaneous stochasticity. 67.198.37.16 (talk) 21:11, 20 March 2024 (UTC)

Error in Second Euler Equation For Compressible Fluid
The equation is supposed to represent conservation of momentum for the general case where rho can be a function of space and time. In that case, the equation should be D/Dt (rho * u) = -grad p + g. NotSqwerty (talk) 06:21, 5 May 2024 (UTC)

title: Talk:Euler equations (fluid dynamics)


text:
Conway Cabal

The Conway Cabal was a group of senior Continental Army officers in late 1777 and early 1778 who aimed to have George Washington replaced as commander-in-chief of the Army during the American Revolutionary War. It was named after Brigadier General Thomas Conway, whose letters criticizing Washington were forwarded to the Second Continental Congress. When these suggestions (which were often little more than criticisms and expressions of discontent with either Washington or the general course of the war) were made public, supporters of Washington mobilized to assist him politically. Conway ended up resigning from the army, and General Horatio Gates, a leading candidate to replace Washington, issued an apology for his role in events.

No formal requests were ever made asking for Washington's removal as commander in chief. There was no sign of any formal conspiracy amongst the various malcontents, although Washington was concerned that there might be one. It was the only major political threat to Washington's command during the war.

Background

In the fall of 1777, British forces captured Philadelphia, the seat of the Second Continental Congress, which was forced to relocate to York, Pennsylvania. A series of military setbacks (principally the losses at Brandywine in September and Germantown in October) caused many in the Continental Army and Congress to question George Washington's leadership of the war effort. In contrast, the northern army of General Horatio Gates had won a signal victory over John Burgoyne's forces, compelling Burgoyne to surrender his entire army after the Battles of Saratoga. Gates controversially claimed credit for the victory. Some historians feel that this was more due to the actions of Benedict Arnold, who, in the first battle on September 19, effectively and almost independently defended his positions against repeated British assaults. It was also alleged that Gates had failed to provide Arnold with adequate reinforcements which would have turned the battle into an outright American victory, although there is not universal agreement on this matter. Gates was politically well connected to Congress. Some congressmen such as Richard Henry Lee, John Adams, and Samuel Adams wanted tighter Congressional control of the war effort and supported Gates. Although John Adams did not specifically call for Washington to be replaced, he worried that Washington was being made into a military idol, and was fearful of the effects of this upon republicanism.

Military custom dictated that, after Saratoga, Gates would have sent his official report to Washington, who was his immediate superior. However, Gates sidestepped Washington and sent his report directly to Congress. Washington sent his staff officer, Colonel Alexander Hamilton, to meet Gates and tell him on Washington's behalf to send three of his brigades to Washington's troops outside Philadelphia. The logic was that Washington required more troops to fight British commander William Howe's forces, which had just taken the capital, whereas Gates had no major British force to contend with. Gates desired to hold onto the brigades and suggested that another British force might attack. He then agreed to send only one 600-man brigade, which Hamilton discovered was the weakest of the three requested. Finally, Hamilton exacted a promise from Gates to send two brigades. At the same time, Gates wrote to Washington, rebuking him for sending Hamilton.

"Conspiracy" is perhaps too strong a term to use in describing varied actions taken by disaffected officers and Congressional delegates unhappy with the course of the war. Most of those involved only shared the view that Washington was a less-than-perfect commander in chief, and very few of their activities were coordinated. General Gates was used as a stalking horse to replace Washington, and had himself also engaged in some lobbying for the command, but he was not responsible for the strong response within the Congress. Opposition to Washington's command in Pennsylvania was anchored by Thomas Mifflin, a former Congressional delegate and also a former quartermaster of the Continental Army who had previously worked closely with Washington. His view of Washington as a rank amateur was supported by Lee, Benjamin Rush, and probably others. A number of French officers who had been commissioned into the Continental Army were also critical of Washington. These notably included Johann de Kalb, Louis Lebègue Duportail, and Thomas Conway.

Conway's letter
Thomas Conway was an Irishman who was educated in France and had served in its military. Recruited by American diplomat Silas Deane, he arrived at Washington's headquarters in Morristown, New Jersey in the spring of 1777. With Washington's support, Congress made him a brigadier general in the Continental Army, and he served with some distinction under Washington during the Philadelphia campaign. His time in combat included distinguished service at Brandywine and Germantown. In October 1777, Conway began lobbying Congress for a promotion to major general, including in his writings criticisms of Washington. Washington in turn had grown to distrust Conway, finding his personal conduct arrogant and unbearable. Conway had publicly admitted that his desire for promotion was rooted in the fact that if he became a major general in the Continental Army, he could become a brigadier general once he returned to the French service. Washington opposed Conway's promotion, as he felt there were many American-born officers senior in rank to Conway and more deserving of promotion who would be upset by such a move. He identified Conway as someone "without conspicuous merit" and that his promotion would "give a fatal blow to the existence of the army." He continued adding, "It will be impossible for me to be of any further service if such insuperable difficulties are thrown in my way." This was seen as an implicit threat to resign. As part of Conway's efforts at self-promotion, he wrote a letter to Gates in which he was reported to have said, "Heaven has been determined to save your Country; or a weak General and bad Counsellors would have ruind [sic] it."

General William Alexander (Lord Stirling) learned of this quotation in a drunken revelation by Gates' adjutant, James Wilkinson. Alexander forwarded the quote in a letter to Washington, who received it on November 8, 1777. This led Washington to consider the possibility that his subordinates were scheming behind his back to replace him. He was already aware that Gates in particular was politically well-connected and popular due to his success at Saratoga.

Washington wrote Conway a brief letter: "Sir, a letter which I received last Night contained the following paragraph. In a letter from Genl Conway to Genl Gates he says", and then quotes the above passage. Conway immediately acknowledged having written to Gates, but denied having written the quoted passage. Conway also again criticized Washington in this letter, writing that "although your advice in council is commonly sound and proper, you have often been influenc'd by men who Were not equal to you in point of experience, Knowledge, or judgement." Washington never saw the actual letter Conway sent to Gates. Henry Laurens did however, and sent Washington an excerpt: "What pity there is but one General Gates! but the more I see of this Army the less I think it fit for general Action under its actual Chiefs [...] I speak [to] you sincerely & wish I could serve under you."

General Mifflin, a man Washington also distrusted, and to whom Conway had also expressed concerns over the army's leadership, informed Gates of what had transpired. Gates wrote a letter to Washington complaining that "[t]hese letters have been stealingly copied" by persons unknown to him, and that he was sending the letter to Laurens, not Washington. This attitude did nothing to placate Washington, and his relationship with Gates deteriorated.

Board of War
Washington had suggested the creation of a Board of War to supply the army and requested that he be able to appoint an experienced foreign officer to serve as Inspector General. Mifflin and Lee turned the suggestion into a program which would make the Board of War into the top military authority, with the Inspector General supervising Washington and reporting directly to the Board of War. The two managed to get this plan through Congress. From Conway's letter Washington assumed that he and Mifflin were close. In mid-November, Congress decided to appoint Mifflin and Gates to the Board of War. Conway's letter of resignation was referred to the board, which rejected it.

Conclusion
Conway received his coveted promotion of Inspector General of the Army on December 13. Eventually, Washington revealed Wilkinson as his source for the letter. Gates was forced to apologize and lie about his correspondence with Conway. General Conway submitted his resignation to Congress in April 1778, which accepted it. Washington, still unsatisfied, urged his supporters to challenge Conway and his allies to duels. Wilkinson turned on Gates and challenged him, but, according to Wilkinson, Gates wept and apologized for any offense caused and the duel was called off for a time. A second duel was fought between them in August 1778, but neither party was injured. On July 4, Conway engaged Brigadier General John Cadwalader in a duel. Cadwalader shot him in the mouth and the bullet exited through his head. "I have stopped the damned rascal's lying tongue at any rate," he said. As Conway suffered from his wound, he wrote an apology to Washington, which was never answered. After recovering, he returned to France.

There is no evidence that any of the participants ever formally attempted to oust Washington. It was not properly a "cabal." Rather, it was a loose network of military leaders and congressmen who lacked confidence in Washington and, in at least some cases, wanted him replaced with General Gates. After the Conway Cabal, the nation largely rallied behind Washington, who was seen more as a figure of national unity. No noticeable attempt was made to replace him for the remainder of the war.

title: Conway Cabal