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Miller Huggins

Miller James Huggins (March 27, 1878 – September 25, 1929) was an American professional baseball player and manager. Huggins played second base for the Cincinnati Reds (1904–1909) and St. Louis Cardinals (1910–1916). He managed the Cardinals (1913–1917) and New York Yankees (1918–1929), including the Murderers' Row teams of the 1920s that won six American League (AL) pennants and three World Series championships.

Huggins was born in Cincinnati. He received a degree in law from the University of Cincinnati, where he was also captain on the baseball team. Rather than serve as a lawyer, Huggins chose to pursue a professional baseball career. He played semi-professional and minor league baseball from 1898 through 1903, at which time he signed with the Reds.

As a player, Huggins was adept at getting on base. He was also an excellent fielding second baseman, earning the nicknames "Rabbit", "Little Everywhere", and "Mighty Mite" for his defensive prowess and was later considered an intelligent manager who understood the fundamentals of the game. Despite fielding successful teams for the Yankees in the 1920s, he continued to make personnel changes in order to maintain his teams' superiority in the AL. He was elected to the National Baseball Hall of Fame by the Veterans Committee in 1964.

Early life
Miller James Huggins was born on March 27, 1878, in Cincinnati, Ohio, where his father, an Englishman, worked as a grocer. His mother was a native of Cincinnati. He had two brothers and one sister.

Huggins attended Woodward High School and Walnut Hills High School, both in Cincinnati.

College career
Huggins attended the University of Cincinnati, where he studied law and played college baseball for the Cincinnati Bearcats baseball team. A shortstop, he was named team captain of the Bearcats in 1900. Seeing him consumed with baseball, his law professors summoned him to justify why they should keep him in the law program.

Huggins' father, a devout Methodist, objected to his son playing baseball on Sundays. But Huggins played semi-professional baseball in 1898 for the Cincinnati Shamrocks, a team organized by Julius Fleischmann, where he played under the pseudonym "Proctor" due to his father's opposition and his amateur status. In 1900, he played for Fleischmann's semiprofessional team based in the Catskill Mountains, the Mountain Tourists, leading the team with a .400 batting average.

After receiving his law degree from Cincinnati, Huggins realized that he could make even more money playing baseball, and as such William Howard Taft, one of Huggins' law professors, advised him to play baseball. He was admitted to the bar, but never practiced law.

Minor leagues
Huggins began his playing career in minor league baseball with the Mansfield Haymakers of the Class B Interstate League in 1899. He continued his minor league apprenticeship with the St. Paul Saints of the American Association from 1901 through 1903. After starting his career as an exclusively right-handed hitter, he began to bat from the left side in 1902 in response to his offensive struggles in the 1901 season while also moving to second base during his time at St. Paul.

Huggins handled 19 fielding chances, 11 putouts and nine assists, without committing an error in a game with the Saints in 1902; the previous Major League Baseball (MLB) record being 18, set by Fred Dunlap in 1882. In 1903, he pulled off the first delayed steal in recorded baseball history. Fleischmann, part-owner of the Cincinnati Reds of the National League (NL), kept an eye on Huggins while he played for St. Paul.

Cincinnati Reds (1904–1909)
The Reds duly purchased his contract from the Saints before the 1904 season. He made his MLB debut on April 15, 1904, and proved very adept at getting on base. He batted .264 with the Reds that season and improved in the 1906 season, finishing with a .292 batting average and 41 stolen bases, while spending considerable time developing his upper-body strength.

Although Huggins hoped to be selected as Ned Hanlon's successor as Cincinnati's manager after the 1907 season, the Reds instead went with John Ganzel. In 1908, he played with the Reds in the Cuban-American Major League Clubs Series. Hampered by a broken ankle and torn ligaments in his shoulder, he slumped to .209 in 1909.

St. Louis Cardinals (1910–1917)
Before the 1910 season, the Reds traded Huggins, along with Frank Corridon and Rebel Oakes, to the St. Louis Cardinals in return for Fred Beebe and Alan Storke. He set an MLB record on June 1, 1910, with six plate appearances but no at bats, with four walks and two sacrifice flies. In the same year, he batted .265 for the Cardinals and led the NL in walks.

On July 13, 1911, he tied the NL record for successful fielding chances in a game with 16. At the end of the season, he finished sixth in the voting for the Chalmers Award for Most Valuable Player. In 1912, he hit over .300 for the first time in his career. Huggins became player-manager for the Cardinals after the 1912 season, succeeding Roger Bresnahan. Team owner Helene Hathaway Britton preferred Huggins' "gentlemanly" manner over Bresnahan's rougher personality.

With the acquisition of speed in a trade with the Pittsburgh Pirates, including Dots Miller, Art Butler, Cozy Dolan and Chief Wilson, the Cardinals contended for the NL pennant in 1914. Finishing in third place, it was the Cardinals' best finish since 1876, but they fell back to sixth in 1915 and last in 1916. When Britton sold the team after that season, she offered Huggins a chance to buy a part of the team. While he was attempting to raise money from the Fleischmann family, Britton sold the team to a group headed by Samuel Breadon, who hired Branch Rickey to run the team's day-to-day operations in the front office.

Huggins had coached the young Rogers Hornsby, helping him to correct his batting stance, and Hornsby duly succeeded him as the team's starting second baseman in 1917 as Huggins ended his playing career. He managed the team during 1917, the last year of his contract, but was not retained.

New York Yankees (1918–1929)
With the New York Yankees of the American League (AL) not performing well, Yankees owners Jacob Ruppert and Tillinghast L'Hommedieu Huston sought to replace "Wild" Bill Donovan as manager. Ban Johnson, AL president, suggested Huggins to Ruppert as a replacement for Donovan. Huston, who had been in Europe at the time that Ruppert had made the appointment, disliked Huggins and wanted to hire Wilbert Robinson, his drinking buddy. Ruppert himself had been put off by Huggins' wool cap and practice of smoking pipes in public, which he felt was the mark of the working class. However, Ruppert interviewed Huggins upon Johnson's recommendation, and agreed that Huggins knew much about baseball. Ruppert offered the job to Huggins, who initially did not want to take the position, as the Yankees were in no better a position than the Cardinals. J. G. Taylor Spink of The Sporting News eventually convinced Huggins to accept the offer, and he signed a two-year contract. The hiring of Huggins drove a wedge between the two co-owners that culminated in Huston selling his shares of the team to Ruppert in 1922.

Having taken charge Huggins did not shy away from making personnel changes. Upon taking over the Yankees, he stressed fundamental baseball, drilling the Yankees in the art of the bunt. Huggins also made his first player transaction, acquiring Del Pratt and Eddie Plank from the St. Louis Browns trading Nick Cullop, Joe Gedeon, Fritz Maisel, Les Nunamaker, Urban Shocker and $15,000 ($0 in current dollar terms), a move that led to criticism in the press.

In Huggins' first season with the Yankees, the team finished fourth in the AL. After that season, he obtained Ernie Shore, Dutch Leonard and Duffy Lewis from the Boston Red Sox, for Ray Caldwell, Frank Gilhooley, Slim Love, and Roxy Walters. The following year he traded Pratt, Muddy Ruel, Hank Thormahlen and Sammy Vick to the Red Sox for Waite Hoyt, Harry Harper, Mike McNally and Wally Schang. Huggins signed a one-year contract to remain with the Yankees for a reported $12,000.

Huston continually took the side of his players in any argument they had against Huggins, criticizing Huggins in the press when the Yankees lost the pennant in 1920. Meanwhile, Ruppert was at best a lukewarm advocate of Huggins. Babe Ruth resisted Huggins' discipline; he did not respect Huggins due to his small stature, soft-spoken nature, and inability to fight, and Huggins was unable to enforce punishments on Ruth, despite being well educated. The Yankees finished third in the AL in 1919 and 1920. Huggins signed a one-year contract to remain with the team in 1921.

Coming into the 1921 season, Huggins was still experiencing criticism in the press. Hugh Fullerton wrote that "in the past Huggins has not shone as a leader of men". By that season, Huggins developed Aaron Ward, Wally Pipp, and Bob Shawkey. The Yankees won their first AL pennant in 1921, reaching the World Series for the first time in franchise history, silencing his critics in the press. However, they lost the 1921 World Series to the New York Giants. In response, Huggins sought to add more pitching talent.

Before the 1922 season, Huggins acquired Johnny Mitchell from the Vernon Tigers of the Pacific Coast League and traded fan favorite Roger Peckinpaugh along with Rip Collins, Bill Piercy, and Jack Quinn to the Red Sox for Everett Scott, Bullet Joe Bush, and Sad Sam Jones. By this time, Ruppert hired Ed Barrow as the team's business manager, and he aided Huggins in player transactions. The Yankees repeated as AL champions that season, but lost the 1922 World Series, again to the Giants. With the newly gained confidence of his owners, Huggins was retained as manager.

The Yankees won their first World Series in 1923, preventing the Giants from repeating as champions for the third consecutive season. They were unable to retain their title the following season however, finishing second in the AL to the Washington Senators.

Huggins had come to regret his trade of Urban Shocker to the Browns. In St. Louis, the spitballer Shocker had come into his own as a starter, racking up four straight 20-win seasons (1920–23) and leading the American League with 27 wins in 1921 and strikeouts the following year, when he won 24 games. Shocker was reacquired for Bullet Joe Bush, Milt Gaston and Joe Giard in December 1924.

However, through 42 games of the 1925 season, the Yankees struggled, falling to seventh place in the eight-team AL, $13 1/2$ games out of first place. Huggins made wholesale changes to the Yankees' lineup, as he replaced Ward at second base with Howard Shanks, catchers Steve O'Neill and Wally Schang with Benny Bengough, and, most notably, Pipp with Lou Gehrig at first base, beginning Gehrig's record consecutive games played streak. Among the team's regulars, only Babe Ruth, Joe Dugan, and Bob Meusel remained in the lineup. However, the team continued to struggle; amid rumors that he might replace Huggins, Ruppert stated that "Miller Huggins will be manager as long as he cares to be". The Yankees fell to seventh place in the AL that season.

With Ruppert's full support, Huggins' duties with the Yankees included keeping Ruth in line. Unafraid of his star player, Huggins and Ruth often clashed. Huggins suspended Ruth indefinitely on August 29, 1925, for "misconduct off the playing field", while also fining him $5,000 ($0 in current dollar terms), and as it was an away game in St. Louis, Ruth was ordered to pay his own way back to New York. The actual reason was because he arrived late for batting practice after yet another night out the night before, though it was a culmination of his pranks and insistence of recruiting less able teammates to go clubbing with him, causing them to nurse hangovers. Ruth responded by claiming Ruppert would rescind the fine and suspension, and that he would never play for Huggins again, believing that Ruppert would side with him over Huggins. However, Ruppert insisted that the fine would stand and that Ruth would be suspended for as long as Huggins desired. After apologizing to Huggins and Ruppert, Ruth was reinstated on September 5. Ruth did not challenge Huggins' authority again.

Huggins restructured the team for the 1926 season, giving starting jobs to Mark Koenig and Tony Lazzeri. That season, Huggins won his fourth pennant with the Yankees in 1926, marking the first time that a team won a pennant after finishing seventh the year prior. However, the Yankees lost the 1926 World Series to the Cardinals in seven games.

Winning the pennant once again in the 1927 season, Huggins matched Cap Anson's mark of five pennants in seven seasons. That year, the Yankees benefited from the development of George Pipgras and Wilcy Moore, and set an American League record with 110 regular season victories, winning the AL by 19 games. The Yankees swept the Pittsburgh Pirates in the 1927 World Series. This team became known as Murderers' Row, and is considered one of the greatest teams in baseball history.

Huggins remained confident in his team's ability to repeat as AL champions in 1928 season. Huggins supplemented his team by acquiring Bill Dickey from the minor leagues. He acquired Stan Coveleski, who was attempting to return to his peak years, but released him in August when the former star continued to struggle. The Yankees reached the 1928 World Series, winning their sixth pennant in eight years, and defeated the Cardinals 4 games to 0. This was the first time a team swept their opponents in consecutive World Series'.

Huggins continued to tinker with his roster during the offseason. He traded Dugan, Mike Gazella, Rosy Ryan and Pat Collins, and acquired Lyn Lary from the Pacific Coast League. He tried Lary at third base and Leo Durocher at shortstop, while he attempted to acquire George Uhle and Ed Morris, but was unsuccessful in both cases. The Yankees fell behind the Philadelphia Athletics in the standings during the 1929 season and as it became clear that the Yankees would not win the AL pennant in 1929, Huggins began consulting with coaches Art Fletcher and Bob Shawkey about the future of the team, including how to replace Bob Meusel in left field. However, by August 1929, Huggins began losing weight and complained of feeling ill.

Death
Huggins fell ill on September 20, 1929, and checked into Saint Vincent's Catholic Medical Center for erysipelas. His condition was complicated by the development of influenza with high fever. The Yankees' club physician, in consultation with other doctors, decided to administer blood transfusions. But despite their best efforts, Huggins died at the age of 51 on September 25, 1929, of pyaemia. The American League canceled its games for September 27, the day of his funeral, and his viewing at Yankee Stadium drew thousands of tearful fans. A moment of silence was held for Huggins before the start of Game 4 of the 1929 World Series (at Philadelphia's Shibe Park, after which the A's overcame an 8–0 Cubs advantage with 10 runs in the last of the seventh for a spectacular 10–8 come-from-behind victory and a 3–1 Series advantage). He was interred in Spring Grove Cemetery in his native Cincinnati.

The Yankees found it difficult to replace Huggins. Art Fletcher managed the team for its final 11 games of the 1929 season, but he did not want to manage the team full-time. After the season, Ruppert offered the job in turn to Fletcher, Donie Bush and Eddie Collins, all of whom declined. Eventually, "Bob the Gob" Shawkey agreed to serve as the Yankees manager for the 1930 season, leading the team to a third-place finish.

Playing and managing profile
Huggins was listed at 5 ft and 140 lbs. His small stature inspired the nicknames "Mighty Mite" and the "Mite Manager". He was also known as "Rabbit" and "Little Everywhere" for his ability to cover ground in the infield. An excellent leadoff batter and defensive second baseman, he ended his playing career with a .265 batting average and .956 fielding percentage. He led the league in walks four times and regularly posted an on-base percentage near .400. He scored 100 or more runs three times and regularly stole 30 or more bases for 324 lifetime steals. On the downside, in 1914 he set the single season caught-stealing record in the National League, when he was thrown out 36 times (as opposed to 32 successful steals).

Huggins finished his managerial career with a 1413–1134 record. His 1413 wins as a manager ranks 23rd all-time (as of the start of the 2012 season). He learned his managerial strategy by observing Hanlon and Bresnahan. His managerial style at first emphasized speed, base-stealing, hit- and-run plays and "slap" (i.e., contact) hitting, but the acquisition of superstar slugger Ruth dictated a change of emphasis towards power and not giving away outs. For the slugging Yankees of the 1920s, he recruited power hitters and consistent (as opposed to brilliant) pitchers.

Legacy
In 1915, umpire and sportswriter Billy Evans, writing about the scarcity of competent second basemen in baseball, listed Huggins, Collins, Pratt, Johnny Evers, and Nap Lajoie as the best in the game. He later wrote that Huggins was "one of the greatest managers I have ever met". Bill James ranked Huggins as the 37th best second baseman of all time in 2001 in his The New Bill James Historical Baseball Abstract.

The Yankees dedicated a monument to Huggins on May 30, 1932, placing it in front of the flagpole in center field at Yankee Stadium. Huggins was the first Yankees legend granted this honor. Later, posthumous monuments were added there for Lou Gehrig in 1941, and Babe Ruth in 1949. These always remained within the field of play, so long as Yankees Stadium maintained its original commodious outfield dimensions (that reached 461 feet in right center).

Many years later, with the outfield shrunk to barely over 400 feet in a sweeping stadium remodel, the memorials were relocated to a "Monument Park" created behind the centerfield fence, and dedicated in 1976. Since that time, additional monuments have been placed for Mickey Mantle (1996), Joe DiMaggio (1999), and George Steinbrenner (2010), again all posthumous.

The plaque on Huggins’ monument describes him as "A splendid character who made priceless contributions to baseball." The Yankees also named a field at Al Lang Stadium, their spring training home in Florida, after Huggins.

Huggins was included on the ballot for the National Baseball Hall of Fame in 1937, 1938, 1939, 1942, 1945, 1946, 1948, and 1950, failing to receive the number of votes required for election on those occasions. Named in the Honor Rolls of Baseball in 1946, the Veterans Committee elected Huggins to the Hall of Fame in February 1964, and he was posthumously inducted that summer.

Personal life
Huggins was a private man who kept to himself. He lived in Cincinnati during the winters while playing for the Reds and Cardinals, but began to make St. Petersburg, Florida, his winter home while managing the Yankees. Huggins did not marry, and lived with his sister while in Cincinnati.

Huggins invested in real estate holdings in Florida, although he sold them in 1926 (three years before the stock market crash, fortunately for him) as they took too much of his time away from baseball. He enjoyed playing golf and billiards in his spare time.

He was portrayed by Ernie Adams in The Pride of the Yankees, Fred Lightner in The Babe Ruth Story, Bruce Weitz in Babe Ruth, and Joe Ragno in The Babe.

title: Miller Huggins


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Durex

Durex is a brand of condoms and personal lubricants owned by the British company Reckitt and currently led by Ben Wilson. It was initially developed in London under the purview of the London Rubber Company and British Latex Products Ltd, where it was manufactured between 1932 and 1994. The London Rubber Company was formed in 1915, and the Durex brand name ("Durability, reliability, and excellence") was launched in 1929, although London Rubber did not begin manufacturing own-brand condoms until 1932, in collaboration with a rubber technology student from Poland named Lucian Lundau. The first book on The London Rubber Company and the history of Durex condoms, written by Jessica Borge, was published in September 2020 by McGill-Queen's University Press.

The London Rubber Company later merged with SSL International, and has since 2010 been owned by Reckitt Benckiser. It is one of the best-selling condom brands around the world, with 30% of the global market. In 2006, Durex condoms were the second-best-selling brand of condoms in the United States, with Trojan condoms being the first. It is supplied by, among others, the company Karex.

In 2007 the last factory making Durex condoms in the UK stopped manufacturing and production has since moved to China, India and Thailand. The modern range includes a wide variety of latex condom, including the Sheik and Ramses brands in North America, and the Avanti condom. Durex also provides a range of lubricants and sex toys.

Although Durex was not an official sponsor of the Olympic Games, Durex provided 150,000 free condoms to more than 10,000 athletes that competed in the 2012 Summer Olympics in London.

China
Although condoms are not a widely used form of contraception in China, with only around 10% of sexually active people using them, among Chinese condom users Durex has a dominant market share of 45% as of 2015. Due to a relatively conservative culture surrounding sex, Durex's marketing in China is often indirect and subtle, often involving lighthearted humour. Marketing is done almost exclusively on social media. Durex has operated an account on the microblogging site Weibo in 2011, which had 2.65 million followers and more than 20,000 posts as of 2018. Durex targets Chinese users on Sina Weibo aged from 20 to 40, and primarily men under 35 years old. In 2011, Durex began cross-industry advertising tie-ins with the antivirus software company Trend Micro, using the common themes of "antivirus" and "security".

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

title: Durex


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Mary Arden, Lady Arden of Heswall

Mary Howarth Arden, Baroness Mance,, PC (born 23 January 1947), known professionally as Lady Arden of Heswall, is a former Justice of the Supreme Court of the United Kingdom. Before that, she was a judge of the Court of Appeal of England and Wales.

Early life and education
Mary Howarth Arden was born in Liverpool, the daughter of Lieutenant-Colonel Eric Cuthbert Arden, of Heswall, Cheshire, a solicitor who had served with the Royal Garrison Artillery, and Mary Margaret (née Smith). Her grandfather was a partner in Gamon Arden and Co., a Liverpool firm of solicitors. Her father and brother, Roger, joined the family firm which merged with Hill Dickinson in 2007. She was brought up in south Liverpool and educated at Huyton College. She read law at Girton College, Cambridge, where she gained a starred first and an LLM, and an LLM degree at Harvard Law School in 1970 as a Kennedy Scholar.

Career
She was called to the bar at Gray's Inn in 1971, and joined Lincoln's Inn in 1973. She practised at Erskine Chambers from 1971 to 1993, mainly in company law. She became a QC in 1986, and served as Attorney-General of the Duchy of Lancaster. She is an Honorary Fellow of Girton College, Cambridge, Royal Holloway, University of London, and of Liverpool John Moores University.

She was appointed to the High Court on 30 April 1993, becoming the first female High Court judge to be assigned to the Chancery Division. As is customary, she was appointed a Dame Commander of the Order of the British Empire (DBE) at the same time. Arden was chairman of the Law Commission from 1996 to 1999. On 2 October 2000, she was also appointed to the Court of Appeal.

Other posts held by Arden include her membership of the Steering Group of the Company Law Review, and a current post as Head of International Judicial Relations for England and Wales. She is a member of the advisory board, Centre of Commercial Law; of the Council of the Statute Law Society; of the Board of the Institute of Advanced Legal Studies, as well as being President of the Trinity Hall Law Society, and of the Association of Women Barristers. She is a member of the Permanent Court of Arbitration in the Hague.

Arden's appointment to the Supreme Court of the United Kingdom, replacing her husband Lord Mance on the court, was announced in June 2018, to take effect on 1 October 2018. On entering her office, she took the judicial courtesy title of Lady Arden of Heswall. She was the only woman out of the eleven judges of the court between the retirement of Lady Black of Derwent in January 2021 and the appointment of Lady Black's successor, Lady Rose of Colmworth in April 2021.

By the time she retired in 2022, she was the eldest of the justices (Lord Lloyd-Jones was the second-eldest at 69). Following the retirement (and death) of Lord Kerr of Tonaghmore in 2020 she was the only remaining member to have held judicial office before 31 March 1995 and was thus allowed to retire at 75 instead of 70.

Personal life
She married Jonathan Mance (now Lord Mance, former Deputy President of the Supreme Court of the United Kingdom) in the Lady Chapel at Liverpool Cathedral in 1973; they have three children together:
 * Abigail Mance, a speech therapist
 * Jessica Mance, a barrister
 * Henry Mance, chief features writer at the Financial Times.

Arden's husband, Lord Mance, joined the Queen's Bench Division in October 1993, making them the first married couple to sit on the High Court bench. The couple also became the first married couple both to sit on the Court of Appeal.

By virtue of her marriage, Lady Arden of Heswall is entitled to be styled The Lady Mance, but is instead known by her own judicial title.

Honours

 * She was appointed as a Queen's Counsel (QC) in 1986.
 * She was appointed Dame Commander of the Order of the British Empire in the Civil Division on 30 April 1993 upon being made a High Court Judge. This gave her the right to be referred to as Dame Mary Arden.
 * She was elected an Honorary Fellow of Girton College, Cambridge in 1995.
 * She was appointed to the Court of Appeal on 2 October 2000. At this time She was appointed to the Privy Council of the United Kingdom. This gave her the honorific style "The Right Honourable" for life.
 * She is an Honorary Fellow of Royal Holloway, University of London and Liverpool John Moores University.
 * In 2002 she was awarded the honorary degree of Doctor of Laws (LLD) from the University of Nottingham.

Judgments
Arden's judgments include:

 * Hutchinson Personal Communications Ltd v. Hook Advertising Ltd - English contract law case, ruling that clients cannot take creative work pitched to them speculatively without a prior agreement.
 * Price Meats v Barclays Bank
 * Bankway Properties Ltd v Pensfold-Dunsford
 * Pennington v Waine - English trusts law concerning the requirements for a trust to be properly constituted, and the operation of constructive trusts.
 * Item Software (UK) Ltd v Fassihi and others
 * Citibank v MBIA
 * Collier v P & MJ Wright (Holdings) Ltd - English contract law concerning the doctrine of consideration and promissory estoppel in relation to "alteration promises".
 * McCarthy v Secretary of State for the Home Department
 * C (A Child) v XYZ CC
 * R. (on the application of S) v Secretary of State for the Home Office
 * Cooper v Attorney General
 * Relfo Ltd v Varsani - English unjust enrichment law concerning to what extent enrichment of the defendant must be at the expense of the claimant.

title: Mary Arden, Lady Arden of Heswall


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User:Elpenmaster

Elpenmaster Contributions
I have added to Yucca Valley High School, Morongo Valley, California, and Hitman: Contracts.

About Elpenmaster
I have a webpage Random Consequences. I really like rain and snow.

title: User:Elpenmaster


text: Category:Kant scholars

title: Category:Kant scholars


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Cocoloring

In graph theory, a cocoloring of a graph G is an assignment of colors to the vertices such that each color class forms an independent set in G or in the complement of G. The cochromatic number z(G) of G is the fewest colors needed in any cocolorings of G. The graphs with cochromatic number 2 are exactly the bipartite graphs, complements of bipartite graphs, and split graphs.

As the requirement that each color class be a clique or independent is weaker than the requirement for coloring (in which each color class must be an independent set) and stronger than for subcoloring (in which each color class must be a disjoint union of cliques), it follows that the cochromatic number of G is less than or equal to the chromatic number of G, and that it is greater than or equal to the subchromatic number of G.

Cocoloring was named and first studied by. characterizes critical 3-cochromatic graphs, while describe algorithms for approximating the cochromatic number of a graph. defines a class of perfect cochromatic graphs, analogous to the definition of perfect graphs via graph coloring, and provides a forbidden subgraph characterization of these graphs.

title: Cocoloring


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Talk:Erasure (duo)

Robot Freaking Unicorn
\m/ —Preceding unsigned comment added by 99.7.82.30 (talk) 03:38, 15 March 2010 (UTC)

Real Release Dates
Just a nit here - Solisbury Hill & Make Me Smile (Come Up & See Me) were released in 2003, not in the "1980s" or "1990s" as the lead in sentence states.

Pet Shop Boys Comparison
Reason why we should compare Erasure and Pet Shop Boys - Both lead singers are homosexuals and the other member plays the synth. There were other Pop duos in the 80's.

Gay Icon Project
In my effort to merge the now-deleted list from the article Gay icon to the Gay icons category, I have added this page to the category. I engaged in this effort as a "human script", adding everyone from the list to the category, bypassing the fact-checking stage. That is what I am relying on you to do. Please check the article Gay icon and make a judgment as to whether this person or group fits the category. By distributing this task from the regular editors of one article to the regular editors of several articles, I believe that the task of fact-checking this information can be expedited. Thank you very much. Philwelch 20:31, 24 Mar 2005 (UTC)
 * I'd say that they can be classed as gay icons, since they are well known among the gay community and have songs with lyrics which LGBT people can relate to. Beno1000 22:46, 21 May 2006 (UTC)

Ridiculous
"As pop craftsmen, their work bears comparison with that of Irving Berlin, Lerner and Loewe, Rodgers and Hart, Gershwin, and other songwriting luminaries."

Are there some cites of critics to support this grandiose claim? Or is it just one WP editor who thinks so?

"Their productivity and record sales have declined in the last decade, Erasure remains an inevitable presence on the British singles charts, scoring at least one Top 30 hit with the release of every new album. Few bands have demonstrated the durability of Erasure."

Hitting the top 30 doesn't sound like such great shakes to me. And Erasure's only been around since 1986? (1985 actually-Ed) There are plenty of bands with a longer commercial lifetime than that.(Like who?-Ed) (And Erasure have had as many top ten's and top twenty's as Pet Shop Boys, last year they made number 3 with Breathe. They are still a successful pop act-Ed)

Back on goes the NPOV tag. Wasted Time R 02:38, 21 July 2005 (UTC)

I wrote somewhere that they'll be the only synth pop band remembered. I exaggerated. Excuse me, Erasure and all other synth pop bands. Gregorik (talk) 19:00, 3 May 2008 (UTC)

Suggestion
What about locking this page because of continuous vandalism? see history —Preceding unsigned comment added by 78.98.208.232 (talk) 20:55, 13 May 2010 (UTC)

I suggest to add the adjective "electronic" ... Few electronic bands have demonstrated the durability of Erasure..." as this is the truth, so that the article will be entirely neutral.

"...through 1995, Erasure were the greatest pop phenomenon in Britain since ABBA." WHAT????? Shouldn't a truly great pop phenom have at least one #1 song?70.188.137.18 17:12, 11 August 2006 (UTC)K. Rudasill

NPOV tag withdrawal
I've just replaced the inapropriate comparisons with more pertinent ones and added the adjective "electronic" when needed. Hence, I suggest to put out the NPOV tag.

 * I did some further toning down and general simplification, and removed the NPOV tag.   Wasted Time R 20:17, 25 July 2005 (UTC)

"(additionally, Bell was openly gay from the beginning, and he reflected this in his lyrics at a time when society as a whole was more ignorant around queer issues than it is today - making Erasure's success all the more notable)" This quote, while very true, seems only loosely relevant to the section it's in. Also it creates quite the run-on sentence. I'm not sure what to do with it, though. --Loltard 08:10, 6 December 2006 (UTC)

updating charts and tables
Xinger: If you are going to add chart statistics and do major work to the albums/singles formatting then do the work on a subpage of your own, then transfer it over to the main article once all the research is done and the formatting is complete. Moving two songs ("Chains Of Love" and "A Little Respect") out of the main singles table and into their own on the bottom of the section destroys the chronology of the article and makes everything look sloppy. If you want to include Modern Rock Chart peaks then find out the stats on ALL their Modern Rock entries, not just a few. Keep the article as one found in an encylopaedia... doing it bit by bit on the main article page leaves it in disarray. -- eo 00:44, 31 August 2005 (UTC)

Music Videos?
Where can I view Erasure music videos? Specifically I'm looking for "Make Me Smile". -- AS Artimour 23:25, 5 October 2006 (UTC)
 * YouTube maybe? -- eo 23:33, 5 October 2006 (UTC)

"Make Me Smile" video is featured on the double-DVD "Hits! The Videos". Look at the tracklisting of this DVD with the link below : http://www.erasureinfo.com/releases/videos/dvd_hits.html

Handwritten Q&A with Vince & Andy
I already posted one photo on the Erasure main page (and have others) - but there are a few things I'm not going to upload to wikipedia, namely these two handwritten questionnaires I did with the band 20 (ack!) years ago:

 * http://andwhatsnext.com/qa-with-vince-clarke-of-erasure/
 * http://andwhatsnext.com/qa-with-andy-bell-of-erasure/

So if anyone wants to get anything from those to report here, or to link to them otherwise, please do. (I'm not doing it myself because of COI concerns.)

Andwhatsnext 03:17, 7 March 2007 (UTC)

Moving page??
I'm disputing this for now. I think moving a page with this many links to it without a discussion first should not have occurred. As there is (I assume) a yet-to-be-created page called Erasure (logic) and no other articles using the title of "Erasure" - perhaps it would be best to simply put a disambiguation/other uses link at the top of each article? The bands is certainly what is most associated with the word "Erasure". - eo 14:50, 18 March 2007 (UTC)
 * P.S. I've noticed that the Talk Page was not moved along with the main article? Clicking on the Talk Page for Erasure (the disambiguation page) leads to Erasure (the band) Talk Page.  Something wasn't done correctly. - eo 14:52, 18 March 2007 (UTC)

Arguements for creating the disambiguation page: There is an extensive set of logic pages on wikipedia, with the exception of erasure. The uses of erasure for logic and for a band name are disjoint.

Arguments against creating the disambiguation page: There are around 50 links to the Erasure page that intend the band. There are more pop fans than logicians in the world.

Very sorry, I didn't know the conventions for such edits. Was just trying to do the right thing. I read the docs on disambiguation pages and didn't see advisories about discussing with maintainers.

What's the right thing now? —The preceding unsigned comment was added by EricGPrud (talk • contribs) 15:21, 18 March 2007 (UTC).
 * No need for apologies - if others feel that (band) works better then that is fine. Actually a lot more than 50 articles link to the band (50 is just the default view setting when clicking on "What links here") so in a case such as this I'd like to see some discussion first.  I've asked an admin about moving the band article back to "Erasure" - not sure if that will happen right away, but in the meantime some input from other editors will definitely help. - eo 15:31, 18 March 2007 (UTC)

Is EBX5 - EBX8 a rumor or true?
I saw a list for EBX5 - EBX8. Is this true? It seems logical, but I don't want to get excited unless there is a reliable source.
 * EBX 5:

 * Who Needs Love Like That (remix/Hamburg)
 * Always
 * Run to the Sun
 * I Love Saturday
 * Stay With Me
 * EBX6:

 * Fingers & Thumbs (Cold Summer's Day)
 * Rock Me Gently
 * In My Arms
 * Don't Say Your Love Is Killing Me
 * Rain
 * EBX7:

 * Freedom
 * Moon & the Sky
 * Solsbury Hill
 * Make Me Smile (Come Up and See Me)
 * Oh L'amour (remix/August)
 * EBX8:

 * Breathe
 * Don't Say You Love Me
 * Here I Go Impossible Again
 * All This Time Still Falling Out of Love
 * Boy
 * Thanks - Antmusic 18:37, 3 April 2007 (UTC)
 * Where'd you see it? It's a long-running rumor.  Unless you see it on Mute or Erasure's websites I wouldn't expect anything. - eo 18:59, 3 April 2007 (UTC)

Popularity in Brazil
Erasure made a HUGE success in Brazil in the late 80's and early 90's, when songs like "A Little Respect", which was included in the soundtrack of a Brazilian soap opera called "O Sexo dos Anjos" ("The Sex of the Angels"), "Stop!" (mainly the remix version by Mark Saunders), "Oh L'Amour" (live version), "Star", which was included in the soundtrack of a Brazilian soap opera called "Rainha da Sucata" ("Queen of the Strap"), "Love To Hate You", and mainly "Blue Savannah" conquered many Brazilian fans for the band. And they still have a strong and supporting fan base here in my country, including there is a Brazilian website dedicated to this fantastic duo, erasure.com.br. So they surely should tour here again in the near future!

(Please, Andy, forget the ridiculously pitiable machist behavior of those guys - a bunch of brainless headbangers - during your last Brazilian tour on 1997 in the Close-Up Planet Festival...) - Erasurian 15:26, 15 December 2010 (UTC)

Well, they came back to Brazil this year and they made nothing less than 5 very well received shows here! Therefore, you can easily conclude that they continue being one of the most popular pop acts in my country. Erasurian (talk) 13:29, 15 August 2011 (UTC)

"is" vs. "are"
A lot of edits have been made recently saying that correct syntax would be to say something like "Erasure are an English band." But I'm pretty sure this is incorrect, it should be "Erasure is an English band." Even if a band contains two people, it is still a single band, so singular words should be used. Also the word "erasure" is a singular noun.

I see that some Wikipedia articles about bands with plural names like The Rolling Stones, The Who, and The Beatles use "are", and some bands with singular names like Tool, Dave Matthews Band, and Green Day are singular. I also see the opposite. Is there a Wiki standard for this?

Personally I think that "The Rolling Stones is an English band" makes more sense "The Rolling Stones are an English band" because "band" is still singular. Fnordware (talk) 02:40, 19 July 2011 (UTC)
 * In British English we tend to use plural (see Depeche Mode, Duran Duran, Suede (band), Pulp (band), Oasis (band), Blur (band), etc.) --Garik 11 (talk) 04:57, 19 July 2011 (UTC)

Citations for Don't Say You Love Me 'unique' downloads
Added a citation for evidence showing that each custom mix variant made and purchased of DSYLM was "limited to a single download". I've referenced an article by freelance journalist Sean McManus [here]. He's published, but not sure if he would be regarded a 'reliable' source, or if that matters in this instance. If it's a problem, info on the unique mp3s also exists on the Erasure biography page on the [Guardian website here] and briefly on the official Erasure website's 21 March 2005 news [Create Your Own Mix Of The New Single!] Opwerty (talk) 18:40, 2 August 2011 (UTC)

Spelling mistakes
There are several spelling mistakes and grammar errors in the section titled Releases 1995-2007

In spite of a return to three-minute pop songs, the 1997 album Cowboy did not restore the success of their 1986–1994 era. Cowboy enjoyed a short-lived success, peaking at number ten in the UK but lasting only two weeks in the UK Top 40. In the U.S. billboard charts it was one of their most successful records. The first single "In my arms" reached number 13 in the U.K. and entered the Top 2 in the U.S. Dance charts.The second single "Don´t say your love os killing me" made the Top 20 in the uk. The third single "Rain" was also only released in Germany.

Billboard should be a cap B and In my arms should be re-written as In My Arms, same goes for Don't Say Your Love Is Killing Me, as well as the spelling of 'Is' as 'of' ???? Did a child write this?

Also Don't Say Your Love Is Killing Me, did not make the UK Top 20, it only reached #23, therefore making the Top 30.

Oh l´amour was released. Original a commercial flop - It should be Originally.

Surely this should read - Oh L´Amour was released. Originally a commercial flop

The third single was a double a-side, features new versions of Here I go impossible again and All this time still falling out of love It maded the U.K. Top 20.

Here I Go Impossible Again/ All This Time Still Falling Out of Love did not make the UK Top 20, it peaked at #25. and Maded? That should be spelt simply as 'made'

If the page is to be kept locked so people can't correct daft mistakes like this themselves then please at least check it over first.

Upcoming releases
"To celebrate their first 30 years in the music industry, Erasure will be releasing an updated version of "Sometimes" as a single and a brand new compilation album entitled Always: The Very Best of Erasure. Release dates are 23 October and 30 October 2015 respectively."

Don't we usually have to treat upcoming stuff differently than this, lest Wikipedia becomes a news/rumor site of some sort? — Preceding unsigned comment added by 195.249.185.2 (talk) 00:14, 15 November 2015 (UTC)

Removed Depeche Mode discography
I have taken the liberty of commenting out the Depeche Mode discography at the bottom of this page. I see no real reason as to why the main Erasure page should have this discography as part of it. The fact that one half of Erasure was a member of the band for just one album doesn't seem justification enough to me and a link to the Depeche Mode page is all that is needed (which is in the document already). If the Depeche Mode discography is to be there, then why not a Yazoo one? And one for Andy Bell's solo career? And Vince's other projects? Opwerty (talk) 12:58, 16 February 2016 (UTC)
 * Agree! Thanks. - eo (talk) 14:01, 16 February 2016 (UTC)

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Replace new wave with hi-NRG
@Hiddenstranger It is anachronistic and ahistorical in the extreme to describe Erasure, an entirely synth-based act, as 'new wave' when they formed half a decade after the new wave had crested. In the UK, new wave denoted a more commercialised or 'artistic', though still guitar-based, variation on punk, entirely separate from synth-pop. Defining acts included Elvis Costello and The Jam, as well as American groups like Talking Heads and Blondie, but the new wave was more or less finished after 1980. For more on this, see what I wrote here and here, as well as on the talk page for synth-pop. I would also ask why it is that every source describing these synthpop groups as ‘new wave’ seems to be from the 21st-century, rather than contemporaneous with the groups’ peak success in the 1980s. Erasure, far from being a group of the new wave period, formed in 1985 and were an entirely synth-based pop and dance group. Their dance music was mostly in the electronic disco style known as hi-NRG, as is noted by several sources, and the group is listed on the Wiki page "list of hi-NRG artists and songs". The other day, I edited this page (while logged out, as it happened) to trim the genres listed down to 4, as Binksternet had previously done when Msftwin95 added sources for hi-NRG and dance-pop. Binksternet took the same approach toward the page for Aztec Camera, so to remain on his good side I suggest we choose four genres to represent the band, even if more are actually allowed. New wave has no place on this page, so it ought to go. If you could provide a third opinion again, @Ceoil, I'd be grateful. Many thanks. Janglyguitars (talk) 19:26, 3 August 2023 (UTC)

Requested move 21 September 2023

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: Moved. Apparent consensus to move below as a WP:NOPRIMARY. estar8806 (talk) ★ 00:59, 20 October 2023 (UTC)

– Is this the primary topic? Reading the 2007 discussion above, it seems this title was defaulted to when there were few other topics with the name. But there are now, not least of which is the poetic form. |Erasure_code|Erasure_(artform)|LGBT_erasure|Bisexual_erasure Pageviews by far favor the musical group, but I question whether it's more significant than just the concept of erasure (as erasing, social invisibility, etc.). (And as a side note, erasure poetry's pageviews aren't helped by its present confusing title; i.e., google doesn't give it a knowledge panel) Hameltion (talk | contribs) 00:19, 21 September 2023 (UTC) — Relisting. Natg 19 (talk) 01:46, 12 October 2023 (UTC)
 * Erasure → Erasure (duo)
 * Erasure (disambiguation) → Erasure
 * Support. There is no primary topic here. Rreagan007 (talk) 03:32, 21 September 2023 (UTC)
 * https://wikinav.toolforge.org/?language=en&title=Erasure shows an actually significant interest in the current assumed primary topic with 32k views, and the hatnote clicks don't even reach top 20, with just 85 clicks. In turn, https://wikinav.toolforge.org/?language=en&title=Erasure_%28disambiguation%29 shows pretty much negligible traffic. Now, this could be because the navigation is all messed up, but with this kind of a ratio it's doubtful. I think the implicit usage argument is so strong that we need a more coherent significance argument to overrule it. --Joy (talk) 11:11, 21 September 2023 (UTC)
 * Oppose. Based on page views, the duo are the primary topic. —Jonny Nixon (talk) 06:48, 24 September 2023 (UTC)
 * Support per nom.--Ortizesp (talk) 05:00, 25 September 2023 (UTC)
 * Support per nom. No primary In ictu oculi (talk) 07:44, 26 September 2023 (UTC)
 * Support per nom. -- Necrothesp (talk) 13:37, 26 September 2023 (UTC)
 * Oppose – looking at the evidence given, I'm not convinced that the pop duo aren't the primary topic (see 's comment). Sceptre (talk) 19:42, 29 September 2023 (UTC)
 * Well, now that you mention it @Sceptre, the evidence given is pretty scarce, so maybe we shouldn't jump to conclusions at this point already. For example, let's start with looking at what does https://www.google.com/search?tbm=bks&q=erasure&pws=0 show. For me, it doesn't seem to show anything about the pop duo and I went as far as page 7. This is a bit strange, I would expect that the music topic has at least some prominent book coverage, which would in turn be quickly found by the search engine. An argument could be made that despite WP:NOTDICT, the term is inherently so ambiguous that it doesn't make sense to short-circuit navigation to a single topic because it could WP:ASTONISH the average reader. We could also look for parallels with Deletion, Removal, Expungement, Excision, Crossing out, Striking out, ... --Joy (talk) 09:52, 30 September 2023 (UTC)
 * Your Google search was focused on "books" topic !!
 * When one does a general Google search, it is obvious that the pop duo is the primary topic !
 * https://www.google.com/search?q=erasure
 * AIRAZUR (talk) 08:10, 7 October 2023 (UTC)
 * @AIRAZUR yes, of course it focused on the books topic, that is a resource we commonly use to examine academic coverage of a topic which can be most relevant to the encyclopedia, because it gives us a glimpse into what is the potential of having reliably sourced articles. The results of a general search can likewise be relevant, but because of the nature of general search engines, which are obviously not encylopedias, there's often a distinction on topics that are popular, novel, etc. --Joy (talk) 09:08, 7 October 2023 (UTC)
 * Oppose Google Trends, page views statistics both show obviously that the duo Erasure is the primary topic for the word. AIRAZUR (talk) 07:11, 6 October 2023 (UTC)
 * Oppose. Both |Erasure|Erasure_(album)|Erasure_(novel)|LGBT_erasure|Erasure_(heraldry)|Erasure_(artform)|Data_erasure|Type_erasure recent pageviews and Google indicate that the duo is what most readers want and expect under this title. Station1 (talk) 02:35, 12 October 2023 (UTC)
 * Support if anything, the most common use is data erasure (such as erasing written text, files, harddisks), of which the computer form has an article -- 65.92.244.127 (talk) 09:31, 12 October 2023 (UTC)
 * Support per nom—blindlynx 23:10, 12 October 2023 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

post-move
Cf. Talk:Erasure. --Joy (talk) 16:08, 11 January 2024 (UTC)

title: Talk:Erasure (duo)


text:
Robert Mallet

Robert Mallet MRIA MRGSI (3 June 1810 – 5 November 1881) was an Irish geophysicist, civil engineer, and inventor who distinguished himself by research concerning earthquakes (and is sometimes known as the father of seismology). His son, Frederick Richard Mallet, was a geologist who worked in India.

Early life
Mallet was born in Dublin, on 3 June 1810, the son of factory owner John Mallet. He was educated at Trinity College, Dublin, entering it at the age of 16 and graduating in science and mathematics in 1830 at the age of 20.

Career
After his graduation, he joined his father's iron foundry business and helped build the business into one of the most important engineering works in Ireland, supplying ironwork for railway companies, the Fastnet Rock lighthouse, and a swing bridge over the River Shannon at Athlone. He also helped manufacture the characteristic iron railings that surround Trinity College and which bear his family name at the base.

Mallet was elected to the Royal Irish Academy in 1832 at the early age of 22. He also enrolled in the British Association for the Advancement of Science in 1835 which helped finance much of his research of seismology.

In 1838 he became a life member of the Royal Geological Society of Ireland, and acted as its President from 1846–48. From 1848–1849 he managed the construction of the Fastnet Rock lighthouse, southwest of Cape Clear.

Seismological work
On 9 February 1846 he presented to the Royal Irish Academy his paper, "On the Dynamics Of Earthquakes", which is considered to be one of the origins of modern seismology. He is also credited with inventing the word "seismology" and some related words, e.g. the isoseismal map, which he used for his research. He also invented the term epicentre.

From 1852 to 1858, he was prepared (with his son, John William Mallet) his work, The Earthquake Catalogue of the British Association (1858), and performed blasting experiments to determine the speed of seismic propagation in sand and solid rock.

On 16 December 1857, the area around Padula, Italy, was devastated by the Great Neapolitan earthquake which caused 11,000 deaths. At the time it was the third largest known earthquake in world history and has been estimated to have been of magnitude 7.0 on the Moment magnitude scale. Mallet, with letters of endorsement from Charles Lyell and Charles Darwin, petitioned the Royal Society of London and received a grant of £150 to go to Padula and record the devastation personally. The resulting report was presented to the Royal Society as the Report on the Great Neapolitan Earthquake of 1857. It was a major scientific work and made great use of the then new process of photography to record the devastation caused by the earthquake. In 1862, he published the "Great Neapolitan Earthquake of 1857: The First Principles of Observational Seismology" in two volumes; he brought forward evidence to show that the depth below the Earth's surface, from where the impulse of the Neapolitan earthquake originated, was about 8–9 geographical miles.

One of Mallet's papers was Volcanic Energy: an Attempt to develop its True Origin and Cosmical Relations, in which he sought to show that volcanic heat may be attributed to the effects of crushing, contortion and other disturbances in the crust of the earth; the disturbances resulting in the formation of lines of fracture, more or less vertical, down which water would find its way, and if the temperature generated be sufficient volcanic eruptions of steam or lava would follow.

Other work

During the Crimean War he designed a 42 LT mortar of 36 in calibre capable of throwing a 2400 lb shell a distance of 1.5 mi. The huge mortar was built in sections to allow transport, but was too late to be used in action. An example has been preserved at the Royal Artillery base in Woolwich and one is displayed before the Royal Armouries Fort Nelson near Portsmouth.

Mallet was elected Fellow of the Royal Society in 1854, and in 1861 relocated to London, where he became a consulting engineer and edited The Practical Mechanic's Journal. He was awarded the Telford Medal by the Institution of Civil Engineers in 1859, followed by the Cunningham Medal of the Royal Irish Academy for his research into the theory of earthquakes in 1862 and the Wollaston medal of the Geological Society of London in 1877, the Geological Society's greatest award.

Blind for the last seven years of his life, he died at Stockwell, London, on 5 November 1881 and is buried at West Norwood Cemetery.

title: Robert Mallet


text: User talk:67.71.63.47

title: User talk:67.71.63.47


text: User talk:165.121.198.160

title: User talk:165.121.198.160


text: Category:Saturn

title: Category:Saturn


text:
True-range multilateration

True-range multilateration (also termed range-range multilateration and spherical multilateration) is a method to determine the location of a movable vehicle or stationary point in space using multiple ranges (distances) between the vehicle/point and multiple spatially-separated known locations (often termed "stations"). Energy waves may be involved in determining range, but are not required.

True-range multilateration is both a mathematical topic and an applied technique used in several fields. A practical application involving a fixed location occurs in surveying. Applications involving vehicle location are termed navigation when on-board persons/equipment are informed of its location, and are termed surveillance when off-vehicle entities are informed of the vehicle's location.

Two slant ranges from two known locations can be used to locate a third point in a two-dimensional Cartesian space (plane), which is a frequently applied technique (e.g., in surveying). Similarly, two spherical ranges can be used to locate a point on a sphere, which is a fundamental concept of the ancient discipline of celestial navigation — termed the altitude intercept problem. Moreover, if more than the minimum number of ranges are available, it is good practice to utilize those as well. This article addresses the general issue of position determination using multiple ranges. In two-dimensional geometry, it is known that if a point lies on two circles, then the circle centers and the two radii provide sufficient information to narrow the possible locations down to two – one of which is the desired solution and the other is an ambiguous solution. Additional information often narrow the possibilities down to a unique location. In three-dimensional geometry, when it is known that a point lies on the surfaces of three spheres, then the centers of the three spheres along with their radii also provide sufficient information to narrow the possible locations down to no more than two (unless the centers lie on a straight line).

True-range multilateration can be contrasted to the more frequently encountered pseudo-range multilateration, which employs range differences to locate a (typically, movable) point. Pseudo range multilateration is almost always implemented by measuring times-of-arrival (TOAs) of energy waves. True-range multilateration can also be contrasted to triangulation, which involves the measurement of angles.

Terminology
There is no accepted or widely-used general term for what is termed true-range multilateration here. That name is selected because it: (a) is an accurate description and partially familiar terminology (multilateration is often used in this context); (b) avoids specifying the number of ranges involved (as does, e.g., range-range; (c) avoids implying an application (as do, e.g., DME/DME navigation or trilateration) and (d) and avoids confusion with the more common pseudo-range multilateration.

Obtaining ranges
For similar ranges and measurement errors, a navigation and surveillance system based on true-range multilateration provide service to a significantly larger 2-D area or 3-D volume than systems based on pseudo-range multilateration. However, it is often more difficult or costly to measure true=ranges than it is to measure pseudo ranges. For distances up to a few miles and fixed locations, true-range can be measured manually. This has been done in surveying for several thousand years –  e.g., using ropes and chains.

For longer distances and/or moving vehicles, a radio/radar system is generally needed. This technology was first developed circa 1940 in conjunction with radar. Since then, three methods have been employed:
 * Two-way range measurement, one party active –  This is the method used by traditional radars (sometimes termed primary radars) to determine the range of a non-cooperative target, and now used by laser rangefinders. Its major limitations are that: (a) the target does not identify itself, and in a multiple target situation, mis-assignment of a return can occur; (b) the return signal is attenuated (relative to the transmitted signal) by the fourth power of the vehicle-station range (thus, for distances of tens of miles or more, stations generally require high-power transmitters and/or large/sensitive antennas); and (c) many systems utilize line-of-sight propagation, which limits their ranges to less than 20 miles when both parties are at similar heights above sea level.
 * Two-way range measurement, both parties active –  This method was reportedly first used for navigation by the Y-Gerät aircraft guidance system fielded in 1941 by the Luftwaffe. It is now used globally in air traffic control – e.g., secondary radar surveillance and DME/DME navigation. It requires that both parties have both transmitters and receivers, and may require that interference issues be addressed.
 * One-way range measurement –  The time of flight (TOF) of electromagnetic energy between multiple stations and the vehicle is measured based on transmission by one party and reception by the other. This is the most recently developed method, and was enabled by the development of atomic clocks; it requires that the vehicle (user) and stations having synchronized clocks. It has been successfully demonstrated (experimentally) with Loran-C and GPS.

Solution methods
True-range multilateration algorithms may be partitioned based on Any pseudo-range multilateration algorithm can be specialized for use with true-range multilateration.
 * problem space dimension (generally, two or three),
 * problem space geometry (generally, Cartesian or spherical) and
 * presence of redundant measurements (more than the problem space dimension).

Two Cartesian dimensions, two measured slant ranges (trilateration)

An analytic solution has likely been known for over 1,000 years, and is given in several texts. Moreover, one can easily adapt algorithms for a three dimensional Cartesian space.

The simplest algorithm employs analytic geometry and a station-based coordinate frame. Thus, consider the circle centers (or stations) C1 and C2 in Fig. 1 which have known coordinates (e.g., have already been surveyed) and thus whose separation $$U$$ is known. The figure 'page' contains C1 and C2. If a third 'point of interest' P (e.g., a vehicle or another point to be surveyed) is at unknown point $$(x,y)$$, then Pythagoras's theorem yields

 * $$ \begin{align}

r_1^2 & = x^2 + y^2 \\[4pt] r_2^2 & = (U-x)^2 + y^2 \end{align} $$

Thus,

Note that $$y$$ has two values (i.e., solution is ambiguous); this is usually not a problem.

While there are many enhancements, Equation $$ is the most fundamental true-range multilateration relationship. Aircraft DME/DME navigation and the trilateration method of surveying are examples of its application. During World War II Oboe and during the Korean War SHORAN used the same principle to guide aircraft based on measured ranges to two ground stations. SHORAN was later used for off-shore oil exploration and for aerial surveying. The Australian Aerodist aerial survey system utilized 2-D Cartesian true-range multilateration. This 2-D scenario is sufficiently important that the term trilateration is often applied to all applications involving a known baseline and two range measurements.

The baseline containing the centers of the circles is a line of symmetry. The correct and ambiguous solutions are perpendicular to and equally distant from (on opposite sides of) the baseline. Usually, the ambiguous solution is easily identified. For example, if P is a vehicle, any motion toward or away from the baseline will be opposite that of the ambiguous solution; thus, a crude measurement of vehicle heading is sufficient. A second example: surveyors are well aware of which side of the baseline that P lies. A third example: in applications where P is an aircraft and C1 and C2 are on the ground, the ambiguous solution is usually below ground.

If needed, the interior angles of triangle C1-C2-P can be found using the trigonometric law of cosines. Also, if needed, the coordinates of P can be expressed in a second, better-known coordinate system—e.g., the Universal Transverse Mercator (UTM) system—provided the coordinates of C1 and C2 are known in that second system. Both are often done in surveying when the trilateration method is employed. Once the coordinates of P are established, lines C1-P and C2-P can be used as new baselines, and additional points surveyed. Thus, large areas or distances can be surveyed based on multiple, smaller triangles—termed a traverse.

An implied assumption for the above equation to be true is that $$r_1$$ and $$r_2$$ relate to the same position of P. When P is a vehicle, then typically $$r_1$$ and $$r_2$$ must be measured within a synchronization tolerance that depends on the vehicle speed and the allowable vehicle position error. Alternatively, vehicle motion between range measurements may be accounted for, often by dead reckoning.

A trigonometric solution is also possible (side-side-side case). Also, a solution employing graphics is possible. A graphical solution is sometimes employed during real-time navigation, as an overlay on a map.

Three Cartesian dimensions, three measured slant ranges

There are multiple algorithms that solve the 3-D Cartesian true-range multilateration problem directly (i.e., in closed-form) – e.g., Fang. Moreover, one can adopt closed-form algorithms developed for pseudo range multilateration. Bancroft's algorithm (adapted) employs vectors, which is an advantage in some situations.

The simplest algorithm corresponds to the sphere centers in Fig. 2. The figure 'page' is the plane containing C1, C2 and C3. If P is a 'point of interest' (e.g., vehicle) at $$(x,y,z)$$, then Pythagoras's theorem yields the slant ranges between P and the sphere centers:

 * $$ \begin{align}

r_1^2 & = x^2 + y^2 + z^2 \\[4pt] r_2^2 & = (x-U)^2 + y^2 + z^2 \\[4pt] r_3^2 & = (x-V_x)^2 + (y-V_y)^2 + z^2 \end{align} $$

Thus, the coordinates of P are:

The plane containing the sphere centers is a plane of symmetry. The correct and ambiguous solutions are perpendicular to it and equally distant from it, on opposite sides.

Many applications of 3-D true-range multilateration involve short ranges—e.g., precision manufacturing. Integrating range measurement from three or more radars (e.g., FAA's ERAM) is a 3-D aircraft surveillance application. 3-D true-range multilateration has been used on an experimental basis with GPS satellites for aircraft navigation. The requirement that an aircraft be equipped with an atomic clock precludes its general use. However, GPS receiver clock aiding is an area of active research, including aiding over a network. Thus, conclusions may change. 3-D true-range multilateration was evaluated by the International Civil Aviation Organization as an aircraft landing system, but another technique was found to be more efficient. Accurately measuring the altitude of aircraft during approach and landing requires many ground stations along the flight path.

Two spherical dimensions, two or more measured spherical ranges
This is a classic celestial (or astronomical) navigation problem, termed the altitude intercept problem (Fig. 3). It's the spherical geometry equivalent of the trilateration method of surveying (although the distances involved are generally much larger). A solution at sea (not necessarily involving the Sun and Moon) was made possible by the marine chronometer (introduced in 1761) and the discovery of the 'line of position' (LOP) in 1837. The solution method now most taught at universities (e.g., U.S. Naval Academy) employs spherical trigonometry to solve an oblique spherical triangle based on sextant measurements of the 'altitude' of two heavenly bodies. This problem can also be addressed using vector analysis. Historically, graphical techniques – e.g., the intercept method – were employed. These can accommodate more than two measured 'altitudes'. Owing to the difficulty of making measurements at sea, 3 to 5 'altitudes' are often recommended.

As the earth is better modeled as an ellipsoid of revolution than a sphere, iterative techniques may be used in modern implementations. In high-altitude aircraft and missiles, a celestial navigation subsystem is often integrated with an inertial navigation subsystem to perform automated navigation—e.g., U.S. Air Force SR-71 Blackbird and B-2 Spirit.

While intended as a 'spherical' pseudo range multilateration system, Loran-C has also been used as a 'spherical' true-range multilateration system by well-equipped users (e.g., Canadian Hydrographic Service). This enabled the coverage area of a Loran-C station triad to be extended significantly (e.g., doubled or tripled) and the minimum number of available transmitters to be reduced from three to two. In modern aviation, slant ranges rather than spherical ranges are more often measured; however, when aircraft altitude is known, slant ranges are readily converted to spherical ranges.

Redundant range measurements
When there are more range measurements available than there are problem dimensions, either from the same C1 and C2 (or C1, C2 and C3) stations, or from additional stations, at least these benefits accrue:
 * 'Bad' measurements can be identified and rejected
 * Ambiguous solutions can be identified automatically (i.e., without human involvement) -- requires an additional station
 * Errors in 'good' measurements can be averaged, reducing their effect.

The iterative Gauss–Newton algorithm for solving non-linear least squares (NLLS) problems is generally preferred when there are more 'good' measurements than the minimum necessary. An important advantage of the Gauss–Newton method over many closed-form algorithms is that it treats range errors linearly, which is often their nature, thereby reducing the effect of range errors by averaging. The Gauss–Newton method may also be used with the minimum number of measured ranges. Since it is iterative, the Gauss–Newton method requires an initial solution estimate.

In 3-D Cartesian space, a fourth sphere eliminates the ambiguous solution that occurs with three ranges, provided its center is not co-planar with the first three. In 2-D Cartesian or spherical space, a third circle eliminates the ambiguous solution that occurs with two ranges, provided its center is not co-linear with the first two.

One-time application versus repetitive application
This article largely describes 'one-time' application of the true-range multilateration technique, which is the most basic use of the technique. With reference to Fig. 1, the characteristic of 'one-time' situations is that point P and at least one of C1 and C2 change from one application of the true-range multilateration technique to the next. This is appropriate for surveying, celestial navigation using manual sightings, and some aircraft DME/DME navigation.

However, in other situations, the true-range multilateration technique is applied repetitively (essentially continuously). In those situations, C1 and C2 (and perhaps Cn, n = 3,4,...) remain constant and P is the same vehicle. Example applications (and selected intervals between measurements) are: multiple radar aircraft surveillance (5 and 12 seconds, depending upon radar coverage range), aerial surveying, Loran-C navigation with a high-accuracy user clock (roughly 0.1 seconds), and some aircraft DME/DME navigation (roughly 0.1 seconds). Generally, implementations for repetitive use: (a) employ a 'tracker' algorithm (in addition to the multilateration solution algorithm), which enables measurements collected at different times to be compared and averaged in some manner; and (b) utilize an iterative solution algorithm, as they (b1) admit varying numbers of measurements (including redundant measurements) and (b2) inherently have an initial guess each time the solution algorithm is invoked.

Hybrid multilateration systems
Hybrid multilateration systems – those that are neither true-range nor pseudo range systems – are also possible. For example, in Fig. 1, if the circle centers are shifted to the left so that C1 is at $$x_1^\prime = - \tfrac{1}{2} U, y_1^\prime = 0$$ and C2 is at $$x_2^\prime = \tfrac{1}{2} U, y_2^\prime = 0$$ then the point of interest P is at

 * $$ \begin{align}

x^\prime & = \frac { (r_1^\prime + r_2^\prime)(r_1^\prime - r_2^\prime) } {2 U}  \\[4pt] y^\prime & = \pm \frac { \sqrt{ (r_1^\prime + r_2^\prime)^2 - U^2 } \sqrt{ U^2 - (r_1^\prime - r_2^\prime)^2 } } {2 U} \end{align} $$

This form of the solution explicitly depends on the sum and difference of $$r_1^\prime$$ and $$r_2^\prime$$ and does not require 'chaining' from the $$x^\prime$$-solution to the $$y^\prime$$-solution. It could be implemented as a true-range multilateration system by measuring $$r_1^\prime$$ and $$r_2^\prime$$.

However, it could also be implemented as a hybrid multilateration system by measuring $$r_1^\prime + r_2^\prime$$ and $$r_1^\prime - r_2^\prime$$ using different equipment – e.g., for surveillance by a multistatic radar with one transmitter and two receivers (rather than two monostatic radars). While eliminating one transmitter is a benefit, there is a countervailing 'cost': the synchronization tolerance for the two stations becomes dependent on the propagation speed (typically, the speed of light) rather that the speed of point P, in order to accurately measure both $$r_1^\prime \pm r_2^\prime$$.

While not implemented operationally, hybrid multilateration systems have been investigated for aircraft surveillance near airports and as a GPS navigation backup system for aviation.

Preliminary and final computations

The position accuracy of a true-range multilateration system—e.g., accuracy of the $$(x,y)$$ coordinates of point P in Fig. 1 -- depends upon two factors: (1) the range measurement accuracy, and (2) the geometric relationship of P to the system's stations C1 and C2. This can be understood from Fig. 4. The two stations are shown as dots, and BLU denotes baseline units. (The measurement pattern is symmetric about both the baseline and the perpendicular bisector of the baseline, and is truncated in the figure.) As is commonly done, individual range measurement errors are taken to be independent of range, statistically independent and identically distributed. This reasonable assumption separates the effects of user-station geometry and range measurement errors on the error in the calculated $$(x,y)$$ coordinates of P. Here, the measurement geometry is simply the angle at which two circles cross—or equivalently, the angle between lines P-C1 and P-C2. When point P- is not on a circle, the error in its position is approximately proportional to the area bounded by the nearest two blue and nearest two magenta circles.

Without redundant measurements, a true-range multilateration system can be no more accurate than the range measurements, but can be significantly less accurate if the measurement geometry is not chosen properly. Accordingly, some applications place restrictions on the location of point P. For a 2-D Cartesian (trilateration) situation, these restrictions take one of two equivalent forms:
 * The allowable interior angle at P between lines P-C1 and P-C2: The ideal is a right angle, which occurs at distances from the baseline of one-half or less of the baseline length; maximum allowable deviations from the ideal 90 degrees may be specified.
 * The horizontal dilution of precision (HDOP), which multiplies the range error in determining the position error: For two dimensions, the ideal (minimum) HDOP is the square root of 2 ($$\sqrt{2} \approx 1.414$$), which occurs when the angle between P-C1 and P-C2 is 90 degrees; a maximum allowable HDOP value may be specified. (Here, equal HDOPs are simply the locus of points in Fig. 4 having the same crossing angle.)

Planning a true-range multilateration navigation or surveillance system often involves a dilution of precision (DOP) analysis to inform decisions on the number and location of the stations and the system's service area (two dimensions) or service volume (three dimensions). Fig. 5 shows horizontal DOPs (HDOPs) for a 2-D, two-station true-range multilateration system. HDOP is infinite along the baseline and its extensions, as only one of the two dimensions is actually measured. A user of such a system should be roughly broadside of the baseline and within an application-dependent range band. For example, for DME/DME navigation fixes by aircraft, the maximum HDOP permitted by the U.S. FAA is twice the minimum possible value, or 2.828, which limits the maximum usage range (which occurs along the baseline bisector) to 1.866 times the baseline length. (The plane containing two DME ground stations and an aircraft in not strictly horizontal, but usually is nearly so.) Similarly, surveyors select point P in Fig. 1 so that C1-C2-P roughly form an equilateral triangle (where HDOP = 1.633).

Errors in trilateration surveys are discussed in several documents. Generally, emphasis is placed on the effects of range measurement errors, rather than on the effects of algorithm numerical errors.

Applications

 * Land surveying using the trilateration method
 * Aerial surveying
 * Maritime archeology surveying
 * DME/DME RNAV aircraft navigation
 * Multiple radar integration (e.g., FAA ERAM)
 * Celestial navigation using the altitude intercept method
 * Intercept method—Graphical solution to the altitude intercept problem
 * Calibrating laser interferometers
 * SHORAN, Oboe, Gee-H—Aircraft guidance systems developed for 'blind' bombing
 * JTIDS (Joint Tactical Information Distribution System) -- U.S./NATO system that (among other capabilities) locates participants in a network using inter-participant ranges
 * USAF SR-71 Blackbird aircraft—Employs astro-inertial navigation
 * USAF B-2 Spirit aircraft—Employs astro-inertial navigation
 * Experimental Loran-C technique

Advantages and disadvantages for vehicle navigation and surveillance
Navigation and surveillance systems typically involve vehicles and require that a government entity or other organization deploy multiple stations that employ a form of radio technology (i.e., utilize electromagnetic waves). The advantages and disadvantages of employing true-range multilateration for such a system are shown in the following table.

True-range multilateration is often contrasted with (pseudo range) multilateration, as both require a form of user ranges to multiple stations. Complexity and cost of user equipage is likely the most important factor in limiting use of true-range multilateration for vehicle navigation and surveillance. Some uses are not the original purpose for system deployment – e.g., DME/DME aircraft navigation.

title: True-range multilateration


text:
Bernard Morin

Bernard Morin (3 March 1931 in Shanghai, China – 12 March 2018) was a French mathematician, specifically a topologist.

Early life and education
Morin lost his sight at the age of six due to glaucoma, but his blindness did not prevent him from having a successful career in mathematics. He received his Ph.D. in 1972 from the Centre National de la Recherche Scientifique.

Career
Morin was a member of the group that first exhibited an eversion of the sphere, i.e., a homotopy which starts with a sphere and ends with the same sphere but turned inside-out. He also discovered the Morin surface, which is a half-way model for the sphere eversion, and used it to prove a lower bound on the number of steps needed to turn a sphere inside out.

Morin discovered the first parametrization of Boy's surface (earlier used as a half-way model), in 1978. His graduate student François Apéry, in 1986, discovered another parametrization of Boy's surface, which conforms to the general method for parametrizing non-orientable surfaces.

Morin worked at the Institute for Advanced Study in Princeton, New Jersey. Most of his career, though, he spent at the University of Strasbourg.

 *  Morin's surface.

title: Bernard Morin


text:
Andrei Medvedev (tennis)

Andrei Medvedev (Андрій Медведєв; born 31 August 1974) is a Ukrainian former professional tennis player. Medvedev reached the final of the 1999 French Open, the French Open semifinals in 1993, and won four Masters titles during his career, achieving a career-high ranking of world No. 4 in May 1994.

Career
In 1991, Medvedev won the junior singles title at the French Open.

As a 17 year old the following year, Medvedev won his first two ATP Tour titles in Genoa and Stuttgart (where he beat then-world No. 2 Stefan Edberg, and finished the season ranked within the world's top 25.

His most successful tournament was the Hamburg Masters (formerly the German Open), which he won three times (1994, 1995 and 1997). He reached a career-high singles ranking of world No. 4.

In the late 1990s, Medvedev's form and results began to flounder until he unexpectedly reached the final of the 1999 French Open where — ranked 100 — he defeated Dinu Pescariu, Pete Sampras, Byron Black, Arnaud Di Pasquale, Gustavo Kuerten and Fernando Meligeni en route. Medvedev dominated the first two sets of the final against Andre Agassi before Agassi mounted a come-from-behind victory, which allowed him to complete a career Grand Slam. Afterwards, Medvedev did not score further notable results, and retired from the tour in 2001.

One main rival of Medvedev was Sergi Bruguera. While their head-to-head record ended deadlocked at 5–5, Bruguera was able to win their two most important matches — the semifinals and quarterfinals of the 1993 and 1994 French Opens, respectively, with Bruguera winning both matches in straight sets.

In the French Open tournament, Medvedev lost six times to the eventual champion (1992–95, 1997 and 1999).

Personal life
His sister, Natalia Medvedeva, formerly a top 25 player on the WTA Tour, partnered with Andrei to represent Ukraine at the seventh Hopman Cup in 1995, finishing runners-up to Germany's Boris Becker and Anke Huber (Medvedev's girlfriend back then) in the final.

Main achievements

 * 1991 Won junior French Open, beating Thomas Enqvist in the final
 * 1992 Won the title in Stuttgart (Outdoor) with the strongest draw in the history of the event
 * 1993 Semifinalist at the French Open and Masters in Frankfurt
 * 1994 Won the titles in Monte Carlo and Hamburg (Super 9 events)
 * 1995 Won the title in Hamburg
 * 1997 Won the title in Hamburg title for the third time in four years
 * 1999 Reached the final of the French Open

title: Andrei Medvedev (tennis)


text:
Operation Lüttich

Operation Lüttich (7–13 August 1944) was the codename of the Nazi German counter-attack during the Battle of Normandy, which occurred near U.S. positions near Mortain, in northwestern France. Lüttich is the German name for the city of Liège, Belgium. In British and American histories of the Second World War, the German Operation Lüttich is known as the Mortain counter-attack, which Hitler ordered to regain territory gained by the First United States Army during Operation Cobra by reaching the coast of the Avranches region, which is at the base of the Cotentin peninsula, in order to isolate the units of the Third United States Army that had advanced into Brittany.

The Germans' main force was the XLVII Panzer Corps, with two Heer and one-and-a-half Waffen-SS Panzer Divisions. Despite initial success against the defending U.S. VII Corps, the Germans were soon halted, and the Allies inflicted severe losses on the attacking troops, eventually destroying most of the German tanks involved in the attack. Although fighting continued around Mortain for six days, the American forces had regained the initiative within a day of the opening of the German attack.

As the German commanders on the spot had warned Hitler in vain, there was little chance of the attack succeeding, and the concentration of their armoured reserves at the western end of the front in Normandy soon led to disaster, as they were outflanked to their south and the front to their east collapsed, resulting in many of the German troops in Normandy being trapped in the Falaise Pocket.

Background
On 25 July 1944, following six weeks of attritional warfare along a stalemated front, American forces under Lieutenant General Omar Bradley mounted an attack code-named Operation Cobra, which broke through the German defenses near Saint-Lô. Almost the entire western half of the German front in Normandy collapsed, and on 1 August, American forces captured Avranches. With the capture of this town at the base of the Cotentin peninsula, and an intact bridge at Pontaubault nearby, the American forces had "turned the corner"; the German front could no longer be anchored against the sea at its western end and American forces could advance west and south into Brittany. The U.S. Third Army, commanded by Lieutenant General George S. Patton, was activated the same day. Despite German air attacks against the bridge at Pontaubault, Patton pushed no less than seven divisions across it during the next three days, and units of his army began advancing almost unopposed towards the Brittany ports.

Beginning on 30 July, the British Second Army, under Lieutenant-General Miles C. Dempsey, mounted a supporting attack (code-named Operation Bluecoat) on the eastern flank of the American armies. Much of the German armored reserves being rushed west to halt the American breakthrough were diverted to face this new threat. Meanwhile, the U.S. continued its attacks to widen the corridor around Avranches. Although the Germans held the vital road junction of Vire, U.S. VII Corps, under Major General J. Lawton Collins, captured Mortain, 19 mi east of Avranches, on 3 August.

The next day, although Major General Troy H. Middleton's U.S. VIII Corps continued to advance west through Brittany toward the ports of Brest and Lorient, Bradley ordered Patton to drive eastward with the main body of the U.S. Third Army, around the open German flank and into the German rear areas. Major General Wade H. Haislip's U.S. XV Corps advanced no less than 75 mi during the next three days, and by 7 August they were approaching Le Mans, formerly the location of the headquarters of the German 7th Army, and still an important logistic center.

German command and decisions
Generalfeldmarshall Günther von Kluge was the German supreme commander in the West. After Generalfeldmarschall Erwin Rommel was injured by Allied aircraft on 17 July, Kluge also took over direct command of Army Group B, the formation conducting the battle in Normandy. He had warned Hitler on 22 July that the collapse of the front was imminent, but Hitler continued to order him to stand fast.

On 2 August, Hitler, fearing the loss of U-boat pens on the French coast, sent a directive to Kluge ordering "an immediate counter-attack between Mortain and Avranches". General Walter Warlimont—the Deputy Chief of Staff at OKW, the German armed forces headquarters—was also sent to Kluge's headquarters to ensure these orders were complied with. Kluge suggested that there was no chance of success, and the German forces in Normandy should retire to the Seine River, pivoting on the intact defences south of Caen, but on 4 August, Hitler categorically ordered the attack to be launched. He demanded that eight of the nine Panzer Divisions in Normandy be used in the attack, and that the Luftwaffe commit its entire reserve, including 1,000 fighter aircraft. According to Hitler, three qualifications had to be met for the attack to proceed. "Von Kluge must believe in it. He must be able to detach enough armour from the main front in Normandy to create an effective striking force, and he must achieve surprise".

Although ordered to wait "until every tank, gun and plane was assembled", Kluge and SS General Paul Hausser (commanding the German 7th Army, which held the western part of the front) decided to attack as soon as possible, before the overall situation deteriorated further. The main striking force assigned was the XLVII Panzer Corps, commanded by General Hans Freiherr von Funck. Instead of eight Panzer Divisions, only four—one of them incomplete—could be relieved from their defensive tasks and assembled in time; the 2nd Panzer Division, 116th Panzer Division, the 2nd SS Panzer Division and part of the 1st SS Panzer Division, with a total of about 300 tanks. The Panzer Corps was supported by two Infantry Divisions and five Kampfgruppen, formed from the remnants of the Panzer Lehr Division and four equally battered infantry divisions.

Kluge ordered the attack to be mounted on the night of 6/7 August. To avoid alerting American forces to the attack, there would be no preparatory artillery bombardments. The intention was to hit the U.S. 30th Infantry Division, commanded by Major-General Leland Hobbs, east of Mortain, then cut through American defenses to reach the coast. Had surprise been achieved, the attack might well have succeeded, but Allied decoders at Ultra had intercepted and decrypted the orders for Operation Lüttich by August 4. As a result, Bradley was able to obtain air support from both the US 9th Air Force and the RAF.

German attacks

At 22:00 on 6 August, von Funck reported that his troops were still not concentrated, and the commander of the 116th Panzer Division "had made a mess of things". In fact, this officer (Gerhard von Schwerin) had been so pessimistic about the operation that he had not even ordered his tank units to take part. This delay disjointed the German attack, but on the German left flank, the SS Panzer troops attacked the positions of the American 30th Infantry Division east of Mortain shortly after midnight. The Germans achieved temporary surprise, as the Ultra documents had arrived at U.S. First Army Headquarters too late to alert the troops to the immediate assault. They briefly captured Mortain but were unable to breach the lines of the 30th Division, as the 2nd Battalion of the 120th Infantry Regiment commanded Hill 314, the dominant feature around Mortain. Although cut off, they were supplied by parachute drops. Of the 700 men who defended the position until 12 August, over 300 were killed or wounded.

To the north, the 2nd Panzer Division attacked several hours later, aiming south-west toward Avranches. It managed to penetrate several miles into the American lines, before being stopped by the 35th Infantry Division and a combat command of the 3rd Armored Division only 2 mi short of Avranches. The German High Command ordered the attacks to be renewed before the afternoon, so that Avranches could be taken.

Allied air strikes—the offensive stalls
By noon of 7 August, the early morning fog had dispersed, and large numbers of Allied aircraft appeared over the battlefield. With the advance knowledge of the attack provided by Ultra, the U.S. 9th Air Force had been reinforced by the RAF Second Tactical Air Force. Despite assurances by the Luftwaffe that German forces would have adequate air support, the Allied aircraft quickly achieved complete control of the airspace over Mortain. The Luftwaffe reported that its fighters were engaged by Allied aircraft from the moment they took off, and were unable even to reach the battlefield. In the open ground east of Mortain, the German Panzers became exposed targets, especially for rocket-firing Hawker Typhoon fighter bombers of the RAF. The RAF Typhoon, which was selected by General Eisenhower to attack the armoured divisions, carried eight 60 lb rockets which had the destructive power of a 6-inch naval gun shell and even with a 40-50 yard error they were extremely destructive especially when fired in salvo. Typhoons flew 294 sorties that afternoon, firing 2,088 rockets and dropping 80 short tons (73 t) of bombs. They engaged the German formations while the US 9th Air Force prevented German fighters from intervening. General Speidel, Chief of Staff to Field-Marshal Von Kluge, wrote after the war "The armoured operation was completely wrecked exclusively by the Allied Air Forces supported by a highly trained ground wireless organization". Author and Typhoon pilot, Desmond Scott states that the battle of Mortain "proved conclusively that major ground offensives can be defeated by the use of tactical air power alone". Dwight D. Eisenhower, the Supreme Allied Commander, said of the Typhoons; "The chief credit in smashing the enemy's spearhead, however, must go to the rocket-firing Typhoon aircraft of the Second Tactical Air Force ..The result of the strafing was that the enemy attack was effectively brought to a halt, and a threat was turned into a great victory."

Allied fighter bomber attacks in the Battle of Mortain produced claims of the pilots of over two hundred German tanks destroyed. This was disputed by subsequent Army studies of the battlefield, which showed that most of the destroyed German armor was knocked out by gunfire from ground forces and that the main effect of the Allied air attacks had been to destroy the unarmored vehicles and troops in the German offensive, as well as forcing the German armor to disperse to cover (thus disrupting its operations). The rockets and bombs of the Allied fighter-bombers were too poor to destroy more than a few of the German tanks, but fear of being incinerated inside a tank if a rocket or bomb struck it caused some of the inexperienced German crews to abandon their tanks intact under air attack. In one British study, it was found that the average Typhoon pilot firing a barrage of all eight rockets had only a four percent chance of striking a target the size of a tank.

American counter-moves

Through 7 August, American troops had continued to press south near Vire, on the right flank of the German attack. The 116th Panzer Division—which was supposed to advance in this sector—was actually driven back. In the afternoon, the 1st SS and 116th Panzer Divisions made renewed attacks, but the flanks of the Mortain positions had been sealed off, allowing the American VII Corps to contain the German advance.

Meanwhile, Bradley had sent two armoured combat commands against the German southern (left) flank. On 8 August, one of these (from the U.S. 2nd Armored Division) was attacking the rear of the two SS Panzer Divisions. Although fighting would continue around Mortain for several more days, there was no further prospect of any German success. The Germans issued orders to go on to the defensive along the entire front, but poorly communicated orders resulted in this being impossible to achieve, with some German forces retreating, and others preparing to hold their ground.

As the U.S. First Army counter-attacked German units near Mortain, units of Patton's 3rd Army were advancing unchecked through the open country to the south of the German armies, and had taken Le Mans on 8 August. The same day, the First Canadian Army attacked the weakened German positions south of Caen in Operation Totalize and threatened to break through to Falaise, although this attack stalled after two days. In desperation, Hitler ordered the attacks against Mortain to be renewed with greater intensity, demanding that the 9th Panzer Division, almost the only formation opposing Patton's advance east from Le Mans, be transferred to Mortain to take part in the attack. General Heinrich Eberbach—commander of Panzergruppe West—was ordered to form a new headquarters, named "Panzer Group Eberbach", to command the renewed offensive. Kluge—who feared he was about to be implicated by the Gestapo in the 20 July Plot—acquiesced in this apparently suicidal order. Eberbach's proposed counter-attack was soon overtaken by events, and was never mounted.

Aftermath
By 13 August, the offensive had fully halted, with German forces being driven out of Mortain. The Germans had lost 120 tanks and assault guns to Allied counter-attacks and air strikes, more than two-thirds of their committed total. As Hitler ordered German forces in Normandy to hold their positions, the U.S. VII and XV Corps were swinging east and north toward Argentan. The German attack west left the 7th Army and Panzergruppe West in danger of being encircled by Allied forces. As American forces advanced on Argentan, British and Canadian forces advanced on Falaise, threatening to cut off both armies in the newly formed Falaise Pocket.

Although American casualties in Operation Lüttich were significantly lighter than in previous operations, certain sectors of the front, notably the positions held by the 30th Division around Mortain, took severe casualties. By the end of 7 August alone, nearly 1,000 men of the 30th Division had been killed. Estimates for American casualties from 6–13 August vary from 2,000-3,000 fatalities, with an unknown number of wounded.

On 14 August, Canadian forces launched Operation Tractable, in conjunction with American movements northwards to Chambois. On 19 August, a brigade of the Polish 1st Armoured Division linked up with forces of the 90th U.S. Infantry Division, sealing off some 50,000 German troops in the pocket. By 21 August, German attempts to reopen the gap had been thwarted, and all German troops trapped in the pocket surrendered to Allied forces, effectively putting an end to the German 7th Army.

title: Operation Lüttich