text:
Talk:Sandpit

catogorise
--> yer exactly - how do you catogorise sandpitts? --Infinitive definition 12:50, 4 April 2007 (UTC)

Litterbox
--> I think that in the first paragraph, the line "Homeowners with pets should be concerned that the sandpit does not become a litterbox." should be eliminated. According to the official policy of Wikipedia, What Wikipedia is not, "Wikipedia articles should not include instructions, advice (legal, medical, or otherwise) or suggestions"

Therefore, as of July 26, 2007, I am deleting "Homeowners with pets should be concerned that the sandpit does not become a litterbox." from the Sandpit article.

--LivingforHim 19:40, 26 July 2007 (UTC)

 * I can agree with following such rules, but in the interest of not just deleting valuable points, perhaps such so-called "advice" can be rephrased in a more informative way, e.g. something like: "Homeowners and property managers often take precautions against neighborhood pets, or their own pets, from using the sandbox as a litterbox, as such usage can create a sanitation hazard for the children who play in the sandbox." Try to cite any such claims if you can (e.g. http://schoolnet.gov.mt/healthandsafety/Newsletters/06-01.pdf, http://www.thriftyfun.com/tf710924.tip.html , http://cats.about.com/od/faqsonlitterboxmatters/f/sandbox.htm , etc.) - Blueguy 65.0.221.140 (talk) 17:11, 25 September 2009 (UTC)

Protecting
OK you need to protect this page if you are going to have sandbox redirect here. —The preceding unsigned comment was added by Special:Contributions/ (talk)
 * Hrmm, I was unaware of that existing re-direct. I'll take a look at what can be done about that, thanks for the head's up! Ariel ♥ Gold 09:53, 21 August 2007 (UTC)

Probably need to protect this page.
"Sandpit" directs here. UK users will mistake sandboxes for sandpits. —Preceding unsigned comment added by 163.166.135.44 (talk) 09:25, 11 August 2008 (UTC)

"Sandpit" or "Sandbox"?
So, which term is most used? As you might know, the title of a Wikipedia article should use the most commonly used name. Judging from a Google comparison, the term "sandbox" seems more often used in computing, while "sandpit" seems more commonly used to refer to the real thing. It looks like the current situation is the correct one, but I could be wrong of course. Cheers, Face 10:31, 11 August 2008 (UTC)

 * I've never heard the term "sandpit" before, so while it may be used, it's certainly not as popular as "sandbox", especially in the USA. (The only time I've heard "pit" used to refer to a sandbox is in the case of a "long jump pit", which has a very different purpose than a playground sandbox.)  It seems like naming this article "sandpit" instead of "sandbox" is to artificially inflate the usage of "sandpit" just so you can make a distinction between the original and more recent metaphorical uses.  Consider the Google search build sandbox vs. build sandpit, (using "build" so as to eliminate some of the metaphorical results):  Not only does the first search have many more results, but the the results take you to well known sites which have chosen to use the word "sandbox" over the less common "sandpit".  The first 9 results I got (bobvila.com, lowes.com, wikihow.com, thisoldhouse.com, about.com, doityourself.com, ehow.com, diynetwork.com, wisegeek.com) are all sites I've visited before, unlike the random assortment of sites I got with [build sandpit]. - Blueguy 65.0.221.140 (talk) 16:36, 25 September 2009 (UTC)

 * Sandpit is British, Sandbox is American. Wikipedia does not preference English variations on neutral subjects such as this. The most common term is not relevant for this kind of thing. A Brit probably created the article, so the article will remain in standard English. "Thems the rules" as you might say. McLerristarr (Mclay1) (talk) 03:41, 2 August 2010 (UTC)
 * Also, calling the article "Sandpit" helps differentiate between it and WP:Sandbox. McLerristarr (Mclay1) (talk) 03:45, 2 August 2010 (UTC)
 * Also - "Sandpit" is the thing that children play in, while "sandbox" is also used in computing terms to mean "test area". Therefore the distinction is useful here. EuroSong talk 21:44, 18 November 2016 (UTC)

In Canada, a sandbox is a man made container usually filled with sand or water (which is sometimes called a water station), most kindergarden classrooms have them. A sandpit is a hole dug into the ground and filled with sand, very common in playscapes (natural playgrounds). — Preceding unsigned comment added by 99.229.112.135 (talk) 13:53, 2 June 2011 (UTC)

Please put a lock on this page
I got 3 warnings for editing this page mistaken it for SandBox, maybe it needs a lock? In UK, Sandpits and Sandboxes are synonyms--86.135.86.189 (talk) 16:38, 18 December 2008 (UTC)
 * Sorry you got warnings for an honest mistake. We already have a notice at the top of the article, and a bit "Attention New Users" notice at the top of the page you see when you edit it, but I can see how it's possible to miss those. I'm hesitant to request that this article be semi-protected, and I don't think an admin would be likely to grant the request anyway, as that would hinder future users who wanted to edit it constructively. I will, however, add an additional note to the editing box itself.
 * Don't worry about it too much, by the way. No real harm was done. The article was quickly reverted, and while you received warnings, you weren't blocked so you can keep editing so long as it's done constructively. I hope you decide to stick around! Visit WP:WELCOME for some helpful advice and information for new users. --Icarus (Hi!) 19:23, 18 December 2008 (UTC)

History
I came looking for information on when & where sandpits originated, the article could do with some expansion in that area. Ryan Paddy (talk) 22:10, 4 March 2009 (UTC)

 * I quite agree. That's what I came to the article hoping to find out, but there's nothing in this article that isn't pretty much common knowledge. - Blueguy 65.0.221.140 (talk) 16:41, 25 September 2009 (UTC)

I tried to find some reliable sources on the historical aspect of specifically sandboxes, but a quick search mostly found information on history of playgrounds in general, and not much outside of North America. Here's a source that shows they were common as early as the 1930s. The Playground article has some general historical information as well. Now, if I were to guess, I'd say the sandbox idea originated from parents watching their children play in sand at the beach, which led me to search for sandplay in general where I had better luck. Google Scholar turned up a promising article, here's a short snippet, "The use of sand in children’s programs has a long history, starting with a series of sand gardens developed in Berlin in the 1880s. These play areas, consisting of piles of sand encased in wooden borders, inspired the construction of America’s first supervised playgrounds: ten sand gardens in poor areas of Boston in 1887". I'll leave it to someone else to add content to the article if they wish. On a side note, while doing this search I came across another interesting Wikipedia article: Cold War playground equipment! -- œ ™ 03:43, 16 November 2015 (UTC)

External links modified (January 2018)
Hello fellow Wikipedians,

I have just modified one external link on Sandpit. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20110610200102/http://www.breswaoutdoorfurniture.co.nz/kumeushow.htm to http://www.breswaoutdoorfurniture.co.nz/kumeushow.htm

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Diff. English variant tags
The template above tells us to use UK English, but, when looking at the text editor in the main article, it tells us to use. Which one is it? 2601:4A:C300:412:1C33:1F1:9584:59C8 (talk) 18:36, 12 July 2020 (UTC)

add { { Not a sandbox } } tag perhaps?
Does this belong on the article? I can't add it myself (edit protection), so I am hoping somebody adds it or finds that it does not belong.

73.45.143.221 (talk) 23:39, 9 October 2021 (UTC)

title: Talk:Sandpit


text:
Taiwan Sign Language

Taiwan Sign Language (TSL; ) is the sign language most commonly used by the deaf and hard of hearing in Taiwan.

History
The beginnings of Taiwan Sign Language date from 1895.

The origins of TSL developed from Japanese Sign Language during Japanese rule. TSL is considered part of the Japanese Sign Language family.

TSL has some mutual intelligibility with both Japanese Sign Language and Korean Sign Language; it has about a 60% lexical similarity with JSL. There are two main dialects of TSL centered on two of the three major sign language schools in Taiwan: one in Taipei, the other in Tainan City. There is a variant based in Taichung, but this sign language is essentially the same as the Tainan school.

After the ROC took over Taiwan, Taiwan absorbed an influx of Chinese Sign Language users from China who influenced TSL through teaching methods and loanwords.

Serious linguistic research into TSL began in the 1970s and is continuing at present. The first International Symposium on Taiwan Sign Language Linguistics was held on March 1–2, 2003, at National Chung Cheng University in Minxiong, Chiayi, Taiwan.

Functional markers
TSL, like other sign languages, incorporates nonmanual markers with lexical, syntactic, discourse, and affective functions. These include brow raising and furrowing, frowning, head shaking and nodding, and leaning and shifting the torso.

In popular culture
The 2020 psychological-thriller The Silent Forest uses a large amount of the Taipei variant of TSL in the dialogue.

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

title: Taiwan Sign Language


text:
User:Jaimenote

 Name : Jaime Ray   Website : Jaimenote   Location : San Francisco Bay Area   Current Advocacy : None   Contact : [mailto:jaimenote@gmail.com Jaime Ray]

title: User:Jaimenote


text:
Grammy Award for Best Comedy Album

The Grammy Award for Best Comedy Album is presented by the National Academy of Recording Arts and Sciences of the United States to "honor artistic achievement in comedy." The award was awarded yearly from 1959 to 1993 and then from 2004 to present day.

History
There have been several minor changes to the name of the award over this time:

 * From 1959 to 1967 it was Best Comedy Performance
 * From 1968 to 1991 it was known as Best Comedy Recording
 * From 1992 to 1993 and from 2004 to the present day it was awarded as Best Comedy Album

In 1960 and 1961 two separate awards were presented for the best spoken and for the best musical comedy performance.

In 1994, after four consecutive years of wins by classical music comedy albums, the award was restricted to spoken word comedy albums and moved into the "spoken" field. From then through 2003, it was awarded as the Grammy Award for Best Spoken Comedy Album.

In 2004 the award was reinstated within the comedy field as the Grammy Award for Best Comedy Album, once again allowing musical comedy works to be considered.

Bill Cosby holds the record for most consecutive wins, with six earned between 1965 and 1970. Peter Schickele (of P.D.Q. Bach fame) is the runner-up, with four wins between 1990 and 1993.

Recipients

The winner is the first-named artist and work for each year, or, for 1960 and 1961, category. For 2024, no winner has yet been chosen as of the edit date.

Artists with multiple wins

 * 7 wins
 * Bill Cosby

 * 5 wins
 * George Carlin
 * Dave Chappelle
 * Richard Pryor

 * 4 wins
 * Peter Schickele
 * Robin Williams

 * 3 wins
 * Chris Rock
 * Louis C.K.
 * "Weird Al" Yankovic

 * 2 wins
 * Lewis Black
 * Steve Martin

Artists with multiple nominations

 * 16 nominations
 * George Carlin

 * 13 nominations
 * Bill Cosby

 * 10 nominations
 * Richard Pryor
 * "Weird Al" Yankovic

 * 8 nominations
 * Jonathan Winters

 * 7 nominations
 * Jim Gaffigan
 * Patton Oswalt

 * 6 nominations
 * Lewis Black
 * Cheech & Chong
 * Stan Freberg
 * Kathy Griffin
 * Garrison Keillor
 * Robin Williams

 * 5 nominations
 * Dave Chappelle
 * Louis C.K.
 * Steve Martin
 * Chris Rock
 * Peter Schickele
 * Flip Wilson

 * 4 nominations
 * Mel Brooks
 * Margaret Cho
 * Jeff Foxworthy
 * Al Franken
 * Homer and Jethro
 * Carl Reiner

 * 3 nominations
 * Erma Bombeck
 * Lenny Bruce
 * Rodney Dangerfield
 * Bob Elliott
 * The Firesign Theatre
 * Flight of the Conchords
 * Ray Goulding
 * George Lopez
 * Elaine May
 * Dennis Miller
 * Monty Python
 * National Lampoon
 * Mike Nichols
 * Adam Sandler
 * Jerry Seinfeld
 * Allan Sherman
 * Sarah Silverman
 * Lily Tomlin

 * 2 nominations
 * Shelley Berman
 * Archie Campbell
 * David Cross
 * Ellen DeGeneres
 * Jimmy Fallon
 * Craig Ferguson
 * David Frye
 * Whoopi Goldberg
 * Kevin Hart
 * Sam Kinison
 * Robert Klein
 * Lisa Lampanelli
 * Larry the Cable Guy
 * Jackie Mason
 * Eddie Murphy
 * Bob Newhart
 * Trevor Noah
 * Tig Notaro
 * Don Rickles
 * Mort Sahl
 * Harry Shearer
 * Smothers Brothers
 * Ray Stevens
 * Judy Tenuta
 * Ron White
 * Steven Wright

title: Grammy Award for Best Comedy Album


text:
Talk:IDEN

Merger proposal
Formal request has been received to merge: WiDEN into iDEN; dated January 2016; discuss here. GenQuest "Talk to Me" 13:36, 1 February 2016 (UTC)
 * Rationale: After the merger I propose a cleanup and proper integration of the existing content.

 * I took action and proceeded with the merge as no opposition raised since the opening of the merger proposal. Nightwalker-87 (talk) 12:01, 28 February 2016 (UTC)

title: Talk:IDEN


text:
SVNS

The SVNS, known as the HSBC SVNS for sponsorship reasons, is an annual series of international rugby sevens tournaments run by World Rugby featuring national sevens teams. Organised for the first time in the 1999–2000 season as the IRB World Sevens Series, the competition was formed to promote an elite-level of international rugby sevens and develop the game into a viable commercial product. The competition has been sponsored by banking group HSBC since 2014.

The season's circuit consists of eight tournaments held in five continents, generally beginning in November or December and ending in May or June. All tournaments feature the same 12 teams.

Teams compete for the World Rugby Series title by accumulating points based on their finishing position in each tournament. The bottom four teams play a repechange tournament against the top four teams of the World Rugby Sevens Challenger Series.

New Zealand had originally dominated the Series, winning each of the first six seasons from 1999–2000 to 2004–05, but since then, Fiji, South Africa, Samoa and Australia have each won season titles. England, Argentina and the United States have placed in the top three for several seasons but have not won the series title.

The International Olympic Committee's decision in 2009 to add rugby sevens to the Summer Olympics beginning in 2016 has added a boost to rugby sevens and to the World Sevens Series; this boost has led to increased exposure and revenues, leading several of the core teams to field fully professional squads.

International sevens
The first international rugby sevens tournament was held in 1973 in Scotland, which was celebrating a century of the Scottish Rugby Union. Seven international teams took part, with England defeating Ireland 22–18 in the final to take the trophy. The Hong Kong Sevens annual tournament began in 1976. Over the next two decades the number of international sevens competitions increased. The most notable was the Rugby World Cup Sevens with Scotland hosting the inaugural event in 1993, along with rugby joining the Commonwealth Games program in 1998.

World Series early years
The first season of the World Sevens Series was the 1999–2000 season. At the Series launch, the chairman of the International Rugby Board, Vernon Pugh, described the IRB's vision of the role of this new competition: "this competition has set in place another important element in the IRB’s drive to establish rugby as a truly global sport, one with widespread visibility and steadily improving standards of athletic excellence." New Zealand and Fiji dominated the first series, meeting in the final in eight of the ten season tournaments, and New Zealand narrowly won, overtaking Fiji by winning the last tournament of the series.

New Zealand won the first six seasons in a row from 1999–2000 to 2004–05, led by players such as Karl Te Nana and Amasio Valence. The number of stops in the series varied over the seasons, but experienced a contraction from 11 tournaments in 2001–02 to 7 tournaments in 2002–03 due to the global recession. In the 2005–06 season Fiji clinched the season trophy on the last tournament of the season finishing ahead of England. New Zealand regained the trophy in 2006–07 season in the last tournament of the season.

South Africa was the next team to win the series after taking home the 2008–09 title. In the 2009–10 season, Samoa who finished seventh the previous year shocked the world – led by 2010 top try-scorer and World Rugby Sevens Player of the Year Mikaele Pesamino – by winning four of the last five tournaments to overtake New Zealand and win the series.

Olympic era and professionalism
The number of core teams expanded from 12 to 15 for the 2011–12 series. Qualification for these places was played out at the 2012 Hong Kong Sevens. Canada (returning to core status for the first time since 2008), Spain and Portugal joined the 12 core teams for the next season. The Japan event also made a return for the first time since 2001 (lasting until 2015). New Zealand continued their dominance by finishing on top.

Argentina was originally planned to begin hosting a tenth event with Mar Del Plata the venue in the 2012–13 season, giving the tour an event on each continent, but when Argentina joined the Rugby Championship those plans were shelved. With the same schedule, New Zealand again were the winners over South Africa. They took it again in 2013–14 with Spain the first team to be relegated after finishing last during that season with Japan replacing them.

Heading into the 2014–15 season, the top four teams qualifying to the 2016 Summer Olympics, with Fiji, South Africa, New Zealand and Great Britain all qualifying through. The 2014–15 season and 2015–16 season were won by Fiji – the first time a team other than New Zealand won back-to-back season titles – led by 2015 and 2016 season Dream Team nominee Osea Kolinisau The two seasons also yielded teams winning their first tournaments – the United States won the 2015 London Sevens to finish the season in sixth overall; Kenya won the 2016 Singapore Sevens, and Scotland won the 2016 London Sevens. Prior to the 2015–16 season World Rugby did a comprehensive review of all nine tournament hosts and adjusted the schedule, dropping two sites (Japan and Scotland), and adding three sites (France, Singapore and Canada) to the calendar.

In the 2016–17 series, a dominant and consistent display by South Africa saw them reach the finals of the 2016–17 series rounds on eight occasions, winning five of these. As a result, South Africa were series champions with victory in the penultimate round in Paris. The season was a qualifier for the 2018 Rugby World Cup Sevens with the top four teams that had not already qualified, coming from this season. The teams that made it through to the World Cup via this method were Canada, Argentina, Scotland and Samoa.

Tournament hosts
The World Series will consist of 8 scheduled tournament stops from the 2023–24 season, which generally fall in the same order and timeframes. From 2020 to 2022, however, several of these events had to be cancelled due to impacts of the COVID-19 pandemic.

Key: Event added for the 2023–24 season

Core teams
A group of core teams, currently 12 in number, is announced for each season based on performances in the previous season. Each core team has a guaranteed place in all of that season's events. The core teams have been selected through a designated promotion/relegation process since the 2012–13 season. A new system from the 2023–24 season, will see 12 core teams, with up to 4 being relegated each year.

Key: * indicates a tied placing

 * Notes

Invited teams
Non-core teams are also invited to compete in every season of the World Rugby Sevens Series under previous formats. With 15 core teams, there was generally only one invited team at each 16-team tournament. Before 2012–13, when there were only 12 core teams, four places at each tournament were usually available to invited teams.

Key: * indicates a tied placing

Promotion and relegation
In 2019, World Rugby announced a plan to create a second-tier competition that would allow the best thirteen sevens teams, with the addition of three invited teams, from their region to compete in a similar style format to the Sevens Series for the potential of gaining promotion to the World Rugby Sevens Series and becoming a core team. This breaks from the usual format of promotion and relegation in the sevens series.

From 2013–14 series to 2018–19 the promotion/relegation was as follows:
 * One team is relegated and one team is promoted each year.
 * The core team that finishes bottom of the table at the end of the season series is relegated.
 * The team that wins the 12-team qualifying tournament at the Hong Kong Sevens is promoted.

From 2020 to 2023 the style of promotion/relegation was as such:

 * One team is relegated and one team is promoted each year.
 * The core team that finishes bottom of the table at the end of the season series is relegated to the Challenger Series.
 * Eight teams will compete for promotion in the Hong Kong Sevens event after qualifying through the Challenger Series.

From 2024 onwards the style of promotion/relegation will be as such:

 * Teams ranked 9–12 will play in the promotion/relegation part of the grand final where they will be joined by the top 4 teams from the Challenger Series
 * Top 4 teams from this event are promoted to the SVNS Series
 * The bottom 4 teams are relegated to the Challenger Series

 * Notes

Other qualifying
The World Series results are sometimes used as a qualifier for other tournaments. For example, the top four teams of the 2014–15 series automatically qualified for the 2016 Summer Olympics. Similarly, certain teams from the 2016–17 series qualified for the 2018 Rugby World Cup Sevens.

Top-6 placings by season
Summary of the top six placegetters for each series:

 * Notes

Top-6 placings by team
Tally of top six placings in the series for each team, updated after the most recently completed 2022-23 season (obtained by summing the placings of each team as recorded in the above table of results by season).

SVNS League Results
Summary of the top six placegetters for the regular season (since 2024):

Events won
List of legs won by each team since 1999.

Updated on 5 May 2024.

Format
Rugby sevens is a fast-paced version of rugby union with seven players each side on a full-sized rugby field. Games are much shorter, lasting seven minutes each half. The game is quicker and faster-scoring than 15-a-side rugby, which explains part of its appeal. It also gives players the space for superb feats of individual skill. Sevens is traditionally played in a two-day tournament format. Currently, in a normal event, 16 teams are entered.

World Rugby operates satellite tournaments in each continent alongside the Sevens World Series which serve as qualifiers for Series events; in 2012–13 they also determined the entrants in the World Series Pre-Qualifier, and since 2013–14 determine the entrants in the Core Team Qualifier.

In each tournament, the teams are divided into pools of four teams, who play a round-robin within the pool. Points are awarded in each pool on a different schedule from most rugby tournaments—3 for a win, 2 for a draw, 1 for a loss, 0 for a no-show. In case teams are tied after pool play, the tiebreakers are:

 * 1) Head-to-head result between the tied teams.
 * 2) Difference in points scored and allowed during pool play.
 * 3) Difference in tries scored and allowed during pool play.
 * 4) Points scored during pool play.
 * 5) Coin toss.

As of the 2009–10 series, four trophies are awarded in each tournament. In descending order of prestige, they are the Cup, whose winner is the overall tournament champion, Plate, Bowl and Shield. Each trophy is awarded at the end of a knockout tournament.

In a normal event, the top two teams in each pool advance to the Cup competition. The four quarterfinal losers drop into the bracket for the Plate. The Bowl is contested by the third and fourth-place finishers in each pool, while the Shield is contested by the losing quarterfinalists of the Bowl.

A third-place match is now conducted between the losing Cup semifinalists in all tournaments; this was introduced for the 2011–12 series.

In 2012–13, the season-ending London Sevens expanded to 20 teams, with 12 competing for series points and eight involved in the Core Team Qualifier. With the promotion place now determined at the Hong Kong Sevens, the London Sevens returned to the traditional 16-team format in 2013–14.

The 2024 Rugby Perth Sevens features nine men’s and six women’s rounds over six months. The calendar includes stopovers in many of the usual destinations, from London to Langford, plus three new cities (Malaga, Seville and Toulouse) to replace traditional hosts Australia and New Zealand.

Hong Kong 7s
The Hong Kong Sevens (an anomaly as a three-day event) is the most famous sevens tournament. The Hong Kong Sevens had 24 teams through the 2011–12 series, but has featured 28 teams since 2012–13, with 15 core teams and the winner of the HSBC Asian Sevens Series competing for series points. At the 2013 event, the remaining 12 teams were those in the World Series Pre-Qualifier; from 2014 forward, the remaining 12 teams are those in the Core Team Qualifier. In Hong Kong, the Shield was awarded for the first time in 2010.

Originally, the six pool winners of the Hong Kong Sevens, plus the two highest-finishing second-place teams, advanced to the Cup.

In 2010 and 2011, a different system was used:
 * The losing quarterfinalists in the Cup competition contested the Plate competition.
 * The four remaining second-place teams and the four best third-place teams, which contested the Plate in previous years, competed for the Bowl.
 * The remaining eight teams in the competition, which contested the Bowl in previous years, competed for the Shield.

In the transitional year of 2012, the Hong Kong Sevens was split into two separate competitions. The 12 core teams competed for the Cup, Plate and Bowl under a format similar to that of a regular event. The 12 invited teams all competed for the Shield, with the top three sides in that competition also earning core status for 2012–13.

From 2013 on, the Hong Kong Sevens was played under the same 16-team format used in the rest of the series, with typically 15 core teams plus an invited team (for Hong Kong, usually the winner of the HSBC Asian Sevens Series) competing in the main draw of the tournament. In line with changes which began at the start of the 2016–17 World Rugby Sevens Series, the duration of the Cup final was reduced from 20 minutes to 14 minutes in 2017. In that season, the number of trophies was also reduced to two; the main Cup contested by the top eight teams from the pool stage, and a Challenge Trophy contested by the bottom eight teams from the pool stage.

Points schedule
The season championship is determined by points earned in each tournament. World Rugby introduced a new scoring system for the 2011–12 series, in which all teams participating in a tournament are guaranteed points. Initially, World Rugby announced the new points schedule only for the standard 16-team events; the allocations for the Hong Kong Sevens were announced later. A new scoring system was introduced in 2019–20 requiring teams to play for 7th, 11th and 15th places, previously teams had tied for 7th–8th 11th–12th and 15th–16th places.

The points schedule used at each standard event until 2023 is summarised below.

 * {|class = "wikitable" style="text-align:center;"

! Place ! Status ! Points !colspan=3| !colspan=3| !colspan=3|
 * Cup winner, gold medalist
 * 22
 * Cup runner-up, silver medalist
 * 19
 * 3rd-place winner, bronze medalist
 * 17
 * 4
 * 3rd-place loser
 * 15
 * 3rd-place winner, bronze medalist
 * 17
 * 4
 * 3rd-place loser
 * 15
 * 3rd-place loser
 * 15
 * 5
 * 5th-place winner
 * 13
 * 6
 * 5th-place loser
 * 12
 * 7
 * 7th-place winner
 * 11
 * 8
 * 7th-place loser
 * 10
 * 8
 * 7th-place loser
 * 10
 * 10
 * 9
 * 9th-place winner
 * 8
 * 10
 * 9th-place loser
 * 7
 * 11
 * 11th-place winner
 * 6
 * 12
 * 11th-place loser
 * 5
 * 12
 * 11th-place loser
 * 5
 * 5
 * 13
 * 13th-place winner
 * 4
 * 14
 * 13th-place loser
 * 3
 * 15
 * 15th-place winner
 * 2
 * 16
 * 15th-place loser
 * 1
 * }
 * 16
 * 15th-place loser
 * 1
 * }

From 2023–24 the following points schedule is used for each event:
 * {|class = "wikitable" style="text-align:center;"

! Place ! Status ! Points !colspan=3| !colspan=3|
 * Cup winner, gold medalist
 * 20
 * Cup runner-up, silver medalist
 * 18
 * 3rd-place winner, bronze medalist
 * 16
 * 4
 * 3rd-place loser
 * 14
 * 3rd-place winner, bronze medalist
 * 16
 * 4
 * 3rd-place loser
 * 14
 * 3rd-place loser
 * 14
 * 5
 * 5th-place winner
 * 12
 * 6
 * 5th-place loser
 * 10
 * 7
 * 7th-place winner
 * 8
 * 8
 * 7th-place loser
 * 6
 * 8
 * 7th-place loser
 * 6
 * 6
 * 9
 * 9th-place winner
 * 4
 * 10
 * 9th-place loser
 * 3
 * 11
 * 11th-place winner
 * 2
 * 12
 * 11th-place loser
 * 1
 * }
 * 12
 * 11th-place loser
 * 1
 * }

Tie-breaking: If two or more teams are level on overall series points, the following tie-breakers are used:
 * 1) Overall difference in points scored and allowed during the season.
 * 2) Total try count during the season.
 * 3) If neither of the above produces a winner, the teams are considered tied.

TV and media
The tour received 1,147 hours of air time in 2005–06; 530 of which was live, and was broadcast to 136 countries. By 2008–09, the hours of air time had increased to over 3,300, with 35 broadcasters airing the series in 139 countries and 15 languages. Broadcast time increased further in 2009–10, with 3,561 hours of air time (1,143 hours live) carried by 34 broadcasters in 141 countries and 16 languages. In 2010–11, 3,657 hours of coverage were aired (1,161 hours live), with the same number of broadcasters as the previous season but six new countries added. For that season, Sevens World Series programming was available in 332 million homes worldwide, with a potential audience of 760 million.

Sponsorship
The International Rugby Board reached a 5-year deal with HSBC in October 2010 that granted them status as the first-ever title sponsor of the Sevens World Series. Through the agreement, HSBC acquired title naming rights to all tournaments in the World Series, beginning with the Dubai Sevens on 3 December 2010. HSBC has since sub-licensed the naming rights to individual tournaments, while retaining its name sponsorship of the overall series. A renewed, 4-year deal was announced before the 2015–16 Series, this deal was also expanded to include the World Rugby Women's Sevens Series.

Player contracts and salaries
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Function (mathematics)

In mathematics, a function from a set $X$ to a set $Y$ assigns to each element of $X$ exactly one element of $Y$. The set $X$ is called the domain of the function and the set $Y$ is called the codomain of the function.

Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

A function is often denoted by a letter such as $f$, $g$ or $h$. The value of a function $f$ at an element $x$ of its domain (that is the element of the codomain that is associated to $x$) is denoted by $f(x)$; for example, the value of $f$ at $x = 4$ is denoted by $f(4)$. Commonly, a specific function is defined by means of an expression depending on $x$, such as $$f(x)=x^2+1;$$ in this case, some computation, called function evaluation, may be needed for deducing the value of the function at a particular value; for example, if $$f(x)=x^2+1,$$ then $$f(4)=4^2+1=17.$$

Given its domain and its codomain, a function is uniquely represented by the set of all pairs $(x, f (x))$, called the graph of the function, a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.

Functions are widely used in science, engineering, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.

Definition

A function $f$ from a set $X$ to a set $Y$ is an assignment of one element of $Y$ to each element of $X$. The set $X$ is called the domain of the function and the set $Y$ is called the codomain of the function.

If the element $y$ in $Y$ is assigned to $x$ in $X$ by the function $f$, one says that $f$ maps $x$ to $y$, and this is commonly written $$y=f(x).$$ In this notation, $x$ is the argument or variable of the function. A specific element $x$ of $X$ is a value of the variable, and the corresponding element of $Y$ is the value of the function at $x$, or the image of $x$ under the function.

A function $f$, its domain $X$, and its codomain $Y$ are often specified by the notation $$f: X\to Y.$$ One may write $$x\mapsto y$$ instead of $$y=f(x)$$, where the symbol $$\mapsto$$ (read 'maps to') is used to specify where a particular element $x$ in the domain is mapped to by $f$. This allows the definition of a function without naming. For example, the square function is the function $$x\mapsto x^2.$$

The domain and codomain are not always explicitly given when a function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if $$f:\R\to\R$$ is a real function, the determination of the domain of the function $$x\mapsto 1/f(x)$$ requires to know the zeros of $f.$ This is one of the reasons for which, in mathematical analysis, "a function from $X$ to $Y$ " may refer to a function having a proper subset of $X$ as a domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable whose domain is a proper subset of the real numbers, typically a subset that contains a non-empty open interval. Such a function is then called a partial function.

The range or image of a function is the set of the images of all elements in the domain.

A function $f$ on a set $S$ means a function from the domain $S$, without specifying a codomain. However, some authors use it as shorthand for saying that the function is $f : S → S$.

Formal definition

The above definition of a function is essentially that of the founders of calculus, Leibniz, Newton and Euler. However, it cannot be formalized, since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of set theory. This set-theoretic definition is based on the fact that a function establishes a relation between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, a binary relation between two sets $X$ and $Y$ is a subset of the set of all ordered pairs $$(x, y)$$ such that $$x\in X$$ and $$y\in Y.$$ The set of all these pairs is called the Cartesian product of $X$ and $Y$ and denoted $$X\times Y.$$ Thus, the above definition may be formalized as follows.

A function with domain $X$ and codomain $Y$ is a binary relation $R$ between $X$ and $Y$ that satisfies the two following conditions:
 * For every $$x$$ in $$X$$ there exists $$y$$ in $$Y$$ such that $$(x,y)\in R.$$
 * $$(x,y)\in R$$ and $$(x,z)\in R$$ imply $$y=z.$$

Still more formally, without referring explicitly to the concept of a relation, but using more notation (including set-builder notation):

A function is formed by three sets, the domain $$X,$$ the codomain $$Y,$$ and the graph $$R$$ that satisfy the three following conditions.
 * $$R\subset \{(x,y) \mid x\in X, y\in Y\}$$
 * $$\forall x\in X, \exists y\in Y, \left(x, y\right) \in R \qquad$$
 * $$(x,y)\in R \land (x,z)\in R \implies y=z\qquad$$

Partial functions
Partial functions are defined similarly to ordinary functions, with the "total" condition removed. That is, a partial function from $X$ to $Y$ is a binary relation $R$ between $X$ and $Y$ such that, for every $$x\in X,$$ there is at most one $y$ in $Y$ such that $$(x,y) \in R.$$

Using functional notation, this means that, given $$x\in X,$$ either $$f(x)$$ is in $Y$, or it is undefined.

The set of the elements of $X$ such that $$f(x)$$ is defined and belongs to $Y$ is called the domain of definition of the function. A partial function from $X$ to $Y$ is thus a ordinary function that has as its domain a subset of $X$ called the domain of definition of the function. If the domain of definition equals $X$, one often says that the partial function is a total function.

In several areas of mathematics the term "function" refers to partial functions rather than to ordinary functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain.

In calculus, a real-valued function of a real variable or real function is a partial function from the set $$\R$$ of the real numbers to itself. Given a real function $$f:x\mapsto f(x)$$ its multiplicative inverse $$x\mapsto 1/f(x)$$ is also a real function. The determination of the domain of definition of a multiplicative inverse of a (partial) function amounts to compute the zeros of the function, the values where the function is defined but not its multiplicative inverse.

Similarly, a function of a complex variable is generally a partial function with a domain of definition included in the set $$\Complex$$ of the complex numbers. The difficulty of determining the domain of definition of a complex function is illustrated by the multiplicative inverse of the Riemann zeta function: the determination of the domain of definition of the function $$z\mapsto 1/\zeta(z)$$ is more or less equivalent to the proof or disproof of one of the major open problems in mathematics, the Riemann hypothesis.

In computability theory, a general recursive function is a partial function from the integers to the integers whose values can be computed by an algorithm (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether $0$ belongs to its domain of definition (see Halting problem).

Multivariate functions

A multivariate function, multivariable function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed.

Formally, a function of $n$ variables is a function whose domain is a set of $n$-tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. Commonly, an $n$-tuple is denoted enclosed between parentheses, such as in $$(1,2,\ldots, n).$$ When using functional notation, one usually omits the parentheses surrounding tuples, writing  $$f(x_1,\ldots,x_n)$$ instead of $$f((x_1,\ldots,x_n)).$$

Given $n$ sets $$X_1,\ldots, X_n,$$ the set of all $n$-tuples $$(x_1,\ldots,x_n)$$ such that $$x_1\in X_1, \ldots, x_n\in X_n$$ is called the Cartesian product of $$X_1,\ldots, X_n,$$ and denoted $$X_1\times\cdots\times X_n.$$

Therefore, a multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain.
 * $$f: U\to Y,$$

where the domain $n$ has the form
 * $$U\subseteq X_1\times\cdots\times X_n.$$

If all the $$X_i$$ are equal to the set $$\R$$ of the real numbers or to the set $$\C$$ of the complex numbers, one talks respectively of a function of several real variables or of a function of several complex variables.

Notation
There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below.

Functional notation
The functional notation requires that a name is given to the function, which, in the case of a unspecified function is often the letter $U$. Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in
 * $$f(x), \quad \sin(3),\quad \text{or}\quad f(x^2+1).$$

The argument between the parentheses may be a variable, often $f$, that represents an arbitrary element of the domain of the function, a specific element of the domain ($n = 0$ in the above example), or an expression that can be evaluated to an element of the domain ($$x^2+1$$ in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let $$f(x)=\sin(x^2+1)$$".

When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write $3$ instead of $sin x$.

Functional notation was first used by Leonhard Euler in 1734. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case, a roman type is customarily used instead, such as "$sin(x)$" for the sine function, in contrast to italic font for single-letter symbols.

The functional notation is often used colloqually for referring to a function and simultaneously naming its argument, such as in "let $$f(x)$$ be a function". This is an abuse of notation that is useful for a simpler formulation.

Arrow notation
Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. For example, $$x\mapsto x+1$$ is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of $$\R$$ is implied.

The domain and codomain can also be explicitly stated, for example:
 * $$\begin{align}

\operatorname{sqr}\colon \Z &\to \Z\\ x &\mapsto x^2.\end{align}$$

This defines a function $sin$ from the integers to the integers that returns the square of its input.

As a common application of the arrow notation, suppose $$f: X\times X\to Y;\;(x,t) \mapsto f(x,t)$$ is a function in two variables, and we want to refer to a partially applied function $$X\to Y$$ produced by fixing the second argument to the value $sqr$ without introducing a new function name. The map in question could be denoted $$x\mapsto f(x,t_0)$$ using the arrow notation. The expression $$x\mapsto f(x,t_0)$$ (read: "the map taking $x$ to $x$ of $f$ comma $x$ nought") represents this new function with just one argument, whereas the expression $t_{0}$ refers to the value of the function $t$ at the point $f(x_{0}, t_{0})$.

Index notation
Index notation may be used instead of functional notation. That is, instead of writing $(x_{0}, t_{0})$, one writes $$f_x.$$

This is typically the case for functions whose domain is the set of the natural numbers. Such a function is called a sequence, and, in this case the element $$f_n$$ is called the $f$th element of the sequence.

The index notation can also be used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map $$x\mapsto f(x,t)$$ (see above) would be denoted $$f_t$$ using index notation, if we define the collection of maps $$f_t$$ by the formula $$f_t(x)=f(x,t)$$ for all $$x,t\in X$$.

Dot notation
In the notation $$x\mapsto f(x),$$ the symbol $n$ does not represent any value; it is simply a placeholder, meaning that, if $x$ is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, $x$ may be replaced by any symbol, often an interpunct "$f (x)$". This may be useful for distinguishing the function $⋅$ from its value $f (⋅)$ at $x$.

For example, $$ a(\cdot)^2$$ may stand for the function $$ x\mapsto ax^2$$, and $ \int_a^{\, (\cdot)} f(u)\,du$ may stand for a function defined by an integral with variable upper bound: $ x\mapsto \int_a^x f(u)\,du$.

Specialized notations
There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis, linear forms and the vectors they act upon are denoted using a dual pair to show the underlying duality. This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application. In category theory and homological algebra, networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.

Functions of more than one variable
In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function $x$ can be defined as mapping any pair of real numbers $$(x, y)$$ to the sum of their squares, $$x^2 + y^2$$. Such a function is commonly written as $$f(x, y)=x^2 + y^2$$ and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as $$f(w,x, y)$$, $$f(w,x, y, z)$$.

Other terms
A function may also be called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds). In particular map may be used in place of homomorphism for the sake of succinctness (e.g., linear map or map from $f$ to $G$ instead of group homomorphism from $H$ to $G$). Some authors reserve the word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function.

Some authors, such as Serge Lang, use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term mapping for more general functions.

In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.

Whichever definition of map is used, related terms like domain, codomain, injective, continuous have the same meaning as for a function.

Specifying a function
Given a function $$f$$, by definition, to each element $$x$$ of the domain of the function $$f$$, there is a unique element associated to it, the value $$f(x)$$ of $$f$$ at $$x$$. There are several ways to specify or describe how $$x$$ is related to $$f(x)$$, both explicitly and implicitly. Sometimes, a theorem or an axiom asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function $$f$$.

By listing function values
On a finite set a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. For example, if $$A = \{ 1, 2, 3 \}$$, then one can define a function $$f: A \to \mathbb{R}$$ by $$f(1) = 2, f(2) = 3, f(3) = 4.$$

By a formula
Functions are often defined by an expression that describes a combination of arithmetic operations and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain. For example, in the above example, $$f$$ can be defined by the formula $$f(n) = n+1$$, for $$n\in\{1,2,3\}$$.

When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the zeros of auxiliary functions. Similarly, if square roots occur in the definition of a function from $$\mathbb{R}$$ to $$\mathbb{R},$$ the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.

For example, $$f(x)=\sqrt{1+x^2}$$ defines a function $$f: \mathbb{R} \to \mathbb{R}$$ whose domain is $$\mathbb{R},$$ because $$1+x^2$$ is always positive if $H$ is a real number. On the other hand, $$f(x)=\sqrt{1-x^2}$$ defines a function from the reals to the reals whose domain is reduced to the interval $x$. (In old texts, such a domain was called the domain of definition of the function.)

Functions can be classified by the nature of formulas that define them:
 * A quadratic function is a function that may be written $$f(x) = ax^2+bx+c,$$ where $f (x)$ are constants.
 * More generally, a polynomial function is a function that can be defined by a formula involving only additions, subtractions, multiplications, and exponentiation to nonnegative integer powers. For example, $$f(x) = x^3-3x-1$$ and $$f(x) = (x-1)(x^3+1) +2x^2 -1$$ are polynomial functions of $$x$$.
 * A rational function is the same, with divisions also allowed, such as $$f(x) = \frac{x-1}{x+1},$$ and $$f(x) = \frac 1{x+1}+\frac 3x-\frac 2{x-1}.$$
 * An algebraic function is the same, with $[−1, 1]$th roots and roots of polynomials also allowed.
 * An elementary function is the same, with logarithms and exponential functions allowed.

Inverse and implicit functions
A function $$f : X\to Y,$$ with domain $n$ and codomain $X$, is bijective, if for every $Y$ in $y$, there is one and only one element $Y$ in $x$ such that $a, b, c$. In this case, the inverse function of $X$ is the function $$f^{-1} : Y \to X$$ that maps $$y\in Y$$ to the element $$x\in X$$ such that $y = f(x)$. For example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the exponential function, that maps the real numbers onto the positive numbers.

If a function $$f: X\to Y$$ is not bijective, it may occur that one can select subsets $$E\subseteq X$$ and $$F\subseteq Y$$ such that the restriction of $f$ to $f$ is a bijection from $E$ to $E$, and has thus an inverse. The inverse trigonometric functions are defined this way. For example, the cosine function induces, by restriction, a bijection from the interval $F$ onto the interval $[0, π]$, and its inverse function, called arccosine, maps $[−1, 1]$ onto $[−1, 1]$. The other inverse trigonometric functions are defined similarly.

More generally, given a binary relation $[0, π]$ between two sets $R$ and $X$, let $Y$ be a subset of $E$ such that, for every $$x\in E,$$ there is some $$y\in Y$$ such that $y = f(x)$. If one has a criterion allowing selecting such a $X$ for every $$x\in E,$$ this defines a function $$f: E\to Y,$$ called an implicit function, because it is implicitly defined by the relation $y$.

For example, the equation of the unit circle $$x^2+y^2=1$$ defines a relation on real numbers. If $x R y$ there are two possible values of $R$, one positive and one negative. For $−1 < x < 1$, these two values become both equal to 0. Otherwise, there is no possible value of $y$. This means that the equation defines two implicit functions with domain $y$ and respective codomains $[−1, 1]$ and $[0, +∞)$.

In this example, the equation can be solved in $(−∞, 0]$, giving $$y=\pm \sqrt{1-x^2},$$ but, in more complicated examples, this is impossible. For example, the relation $$y^5+y+x=0$$ defines $y$ as an implicit function of $y$, called the Bring radical, which has $$\mathbb R$$ as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and $x$th roots.

The implicit function theorem provides mild differentiability conditions for existence and uniqueness of an implicit function in the neighborhood of a point.

Using differential calculus
Many functions can be defined as the antiderivative of another function. This is the case of the natural logarithm, which is the antiderivative of $x = ± 1$ that is 0 for $1/x$. Another common example is the error function.

More generally, many functions, including most special functions, can be defined as solutions of differential equations. The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for $x = 1$.

Power series can be used to define functions on the domain in which they converge. For example, the exponential function is given by $e^x = \sum_{n=0}^{\infty} {x^n \over n!}$. However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its Taylor series in some interval, this power series allows immediately enlarging the domain to a subset of the complex numbers, the disc of convergence of the series. Then analytic continuation allows enlarging further the domain for including almost the whole complex plane. This process is the method that is generally used for defining the logarithm, the exponential and the trigonometric functions of a complex number.

By recurrence
Functions whose domain are the nonnegative integers, known as sequences, are sometimes defined by recurrence relations.

The factorial function on the nonnegative integers ($$n\mapsto n!$$) is a basic example, as it can be defined by the recurrence relation
 * $$n!=n(n-1)!\quad\text{for}\quad n>0,$$

and the initial condition
 * $$0!=1.$$

Representing a function
A graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.

Graphs and plots

Given a function $$f : X\to Y,$$ its graph is, formally, the set
 * $$G=\{(x,f(x))\mid x\in X\}.$$

In the frequent case where $n$ and $X$ are subsets of the real numbers (or may be identified with such subsets, e.g. intervals), an element $$(x,y)\in G$$ may be identified with a point having coordinates $x = 0$ in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a plot that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the graph of the function. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the square function
 * $$x\mapsto x^2,$$

consisting of all points with coordinates $$(x, x^2)$$ for $$x\in \R,$$ yields, when depicted in Cartesian coordinates, the well known parabola. If the same quadratic function $$x\mapsto x^2,$$ with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates $$(r,\theta) =(x,x^2),$$ the plot obtained is Fermat's spiral.

Tables
A function can be represented as a table of values. If the domain of a function is finite, then the function can be completely specified in this way. For example, the multiplication function $$f:\{1,\ldots,5\}^2 \to \mathbb{R}$$ defined as $$f(x,y)=xy$$ can be represented by the familiar multiplication table

On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed, interpolation can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:

Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.

Bar chart
A bar chart can represent a function whose domain is a finite set, the natural numbers, or the integers. In this case, an element $Y$ of the domain is represented by an interval of the $x$-axis, and the corresponding value of the function, $x, y$, is represented by a rectangle whose base is the interval corresponding to $y$ and whose height is $sin x$ (possibly negative, in which case the bar extends below the $x$-axis).

General properties
This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.

Standard functions
There are a number of standard functions that occur frequently:
 * For every set $x$, there is a unique function, called the empty function, or empty map, from the empty set to $x$. The graph of an empty function is the empty set. The existence of empty functions is needed both for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements. Under the usual set-theoretic definition of a function as an ordered triplet (or equivalent ones), there is exactly one empty function for each set, thus the empty function $$\varnothing \to X$$ is not equal to $$\varnothing \to Y$$ if and only if $$X\ne Y$$, although their graphs are both the empty set.
 * For every set $x$ and every singleton set $f(x)$, there is a unique function from $x$ to $f(x)$, which maps every element of $X$ to $X$. This is a surjection (see below) unless $X$ is the empty set.
 * Given a function $$f: X\to Y,$$ the canonical surjection of $X$ onto its image $$f(X)=\{f(x)\mid x\in X\}$$ is the function from $X$ to $∅ × X$ that maps $X$ to $\{s\}$.
 * For every subset $s$ of a set $X$, the inclusion map of $f$ into $X$ is the injective (see below) function that maps every element of $x$ to itself.
 * The identity function on a set $A$, often denoted by $\{s\}$, is the inclusion of $X$ into itself.

Function composition
Given two functions $$f: X\to Y$$ and $$g: Y\to Z$$ such that the domain of $A$ is the codomain of $X$, their composition is the function $$g \circ f: X \rightarrow Z$$ defined by
 * $$(g \circ f)(x) = g(f(x)).$$

That is, the value of $$g \circ f$$ is obtained by first applying $f(X)$ to $f(x)$ to obtain $id_{X}$ and then applying $f$ to the result $A$ to obtain $x$. In the notation the function that is applied first is always written on the right.

The composition $$g\circ f$$ is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both $$g \circ f$$ and $$f \circ g$$ satisfy these conditions, the composition is not necessarily commutative, that is, the functions $$g \circ f$$ and $$ f \circ g$$ need not be equal, but may deliver different values for the same argument. For example, let $y = f(x)$ and $g$, then $$g(f(x))=x^2+1$$ and $$ f(g(x)) = (x+1)^2$$ agree just for $$x=0.$$

The function composition is associative in the sense that, if one of $$(h\circ g)\circ f$$ and $$h\circ (g\circ f)$$ is defined, then the other is also defined, and they are equal, that is, $$(h\circ g)\circ f = h\circ (g\circ f).$$ Therefore, it is usual to just write $$h\circ g\circ f.$$

The identity functions $$\operatorname{id}_X$$ and $$\operatorname{id}_Y$$ are respectively a right identity and a left identity for functions from $X$ to $X$. That is, if $g$ is a function with domain $f$, and codomain $y$, one has $$f\circ \operatorname{id}_X = \operatorname{id}_Y \circ f = f.$$

Image and preimage
Let $$f: X\to Y.$$ The image under $X$ of an element $Y$ of the domain $f$ is $g(y) = g(f(x))$. If $f(x) = x^{2}$ is any subset of $g(x) = x + 1$, then the image of $X$ under $Y$, denoted $(g ∘ f )(c) = #$, is the subset of the codomain $f(x)$ consisting of all images of elements of $f$, that is,
 * $$f(A)=\{f(x)\mid x\in A\}.$$

The image of $A$ is the image of the whole domain, that is, $X$. It is also called the range of $x$, although the term range may also refer to the codomain.

On the other hand, the inverse image or preimage under $X$ of an element $A$ of the codomain $f$ is the set of all elements of the domain $f(A)$ whose images under $A$ equal $f$. In symbols, the preimage of $f$ is denoted by $$f^{-1}(y)$$ and is given by the equation
 * $$f^{-1}(y) = \{x \in X \mid f(x) = y\}.$$

Likewise, the preimage of a subset $Y$ of the codomain $f$ is the set of the preimages of the elements of $f(X)$, that is, it is the subset of the domain $X$ consisting of all elements of $B$ whose images belong to $Y$. It is denoted by $$f^{-1}(B)$$ and is given by the equation
 * $$f^{-1}(B) = \{x \in X \mid f(x) \in B\}.$$

For example, the preimage of $$\{4, 9\}$$ under the square function is the set $$\{-3,-2,2,3\}$$.

By definition of a function, the image of an element $B$ of the domain is always a single element of the codomain. However, the preimage $$f^{-1}(y)$$ of an element $y$ of the codomain may be empty or contain any number of elements. For example, if $Y$ is the function from the integers to themselves that maps every integer to 0, then $$f^{-1}(0) = \mathbb{Z}$$.

If $$f : X\to Y$$ is a function, $X$ and $X$ are subsets of $B$, and $x$ and $A$ are subsets of $B$, then one has the following properties:
 * $$A\subseteq B \Longrightarrow f(A)\subseteq f(B)$$
 * $$C\subseteq D \Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)$$
 * $$A \subseteq f^{-1}(f(A))$$
 * $$C \supseteq f(f^{-1}(C))$$
 * $$f(f^{-1}(f(A)))=f(A)$$
 * $$f^{-1}(f(f^{-1}(C)))=f^{-1}(C)$$

The preimage by $f$ of an element $y$ of the codomain is sometimes called, in some contexts, the fiber of $X$ under $y$.

If a function $y$ has an inverse (see below), this inverse is denoted $$f^{-1}.$$ In this case $$f^{-1}(C)$$ may denote either the image by $$f^{-1}$$ or the preimage by $f$ of $f$. This is not a problem, as these sets are equal. The notation $$f(A)$$ and $$f^{-1}(C)$$ may be ambiguous in the case of sets that contain some subsets as elements, such as $$\{x, \{x\}\}.$$ In this case, some care may be needed, for example, by using square brackets $$f[A], f^{-1}[C]$$ for images and preimages of subsets and ordinary parentheses for images and preimages of elements.

Injective, surjective and bijective functions
Let $$f : X\to Y$$ be a function.

The function $y$ is injective (or one-to-one, or is an injection) if $C$ for every two different elements $D$ and $f$ of $f$. Equivalently, $f$ is injective if and only if, for every $$y\in Y,$$ the preimage $$f^{-1}(y)$$ contains at most one element. An empty function is always injective. If $C$ is not the empty set, then $f$ is injective if and only if there exists a function $$g: Y\to X$$ such that $$g\circ f=\operatorname{id}_X,$$ that is, if $b$ has a left inverse. Proof: If $X$ is injective, for defining $f$, one chooses an element $$x_0$$ in $X$ (which exists as $f$ is supposed to be nonempty), and one defines $f$ by $$g(y)=x$$ if $$y=f(x)$$ and $$g(y)=x_0$$ if $$y\not\in f(X).$$ Conversely, if $$g\circ f=\operatorname{id}_X,$$ and $$y=f(x),$$ then $$x=g(y),$$ and thus $$f^{-1}(y)=\{x\}.$$

The function $f$ is surjective (or onto, or is a surjection) if its range $$f(X)$$ equals its codomain $$Y$$, that is, if, for each element $$y$$ of the codomain, there exists some element $$x$$ of the domain such that $$f(x) = y$$ (in other words, the preimage $$f^{-1}(y)$$ of every $$y\in Y$$ is nonempty). If, as usual in modern mathematics, the axiom of choice is assumed, then $g$ is surjective if and only if there exists a function $$g: Y\to X$$ such that $$f\circ g=\operatorname{id}_Y,$$ that is, if $X$ has a right inverse. The axiom of choice is needed, because, if $X$ is surjective, one defines $g$ by $$g(y)=x,$$ where $$x$$ is an arbitrarily chosen element of $$f^{-1}(y).$$

The function $f$ is bijective (or is a bijection or a one-to-one correspondence) if it is both injective and surjective. That is, $f$ is bijective if, for every $$y\in Y,$$ the preimage $$f^{-1}(y)$$ contains exactly one element. The function $f$ is bijective if and only if it admits an inverse function, that is, a function $$g : Y\to X$$ such that $$g\circ f=\operatorname{id}_X$$ and $$f\circ g=\operatorname{id}_Y.$$ (Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward).

Every function $$f: X\to Y$$ may be factorized as the composition $$i\circ s$$ of a surjection followed by an injection, where $f$ is the canonical surjection of $g$ onto $Y$ and $f$ is the canonical injection of $y$ into $f$. This is the canonical factorization of $f$.

"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement "$f(a) ≠ f(b)$ maps $a$ onto $f(X)$" differs from "$f(X)$ maps $f$ into $X$", in that the former implies that $Y$ is surjective, while the latter makes no assertion about the nature of $f$. In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage of being more symmetrical.

Restriction and extension
If $$f : X \to Y$$ is a function and $X$ is a subset of $B$, then the restriction of $$f$$ to S, denoted $$f|_S$$, is the function from $f$ to $f$ defined by
 * $$f|_S(x) = f(x)$$

for all $S$ in $X$. Restrictions can be used to define partial inverse functions: if there is a subset $S$ of the domain of a function $$f$$ such that $$f|_S$$ is injective, then the canonical surjection of $$f|_S$$ onto its image $$f|_S(S) = f(S)$$ is a bijection, and thus has an inverse function from $$f(S)$$ to $Y$. One application is the definition of inverse trigonometric functions. For example, the cosine function is injective when restricted to the interval $s$. The image of this restriction is the interval $X$, and thus the restriction has an inverse function from $i$ to $Y$, which is called arccosine and is denoted $x$.

Function restriction may also be used for "gluing" functions together. Let $ X=\bigcup_{i\in I}U_i$ be the decomposition of $f$ as a union of subsets, and suppose that a function $$f_i : U_i \to Y$$ is defined on each $$U_i$$ such that for each pair $$i, j$$ of indices, the restrictions of $$f_i$$ and $$f_j$$ to $$U_i \cap U_j$$ are equal. Then this defines a unique function $$f : X \to Y$$ such that $$f|_{U_i} = f_i$$ for all $[0, π]$. This is the way that functions on manifolds are defined.

An extension of a function $[−1, 1]$ is a function $[−1, 1]$ such that $[0, π]$ is a restriction of $X$. A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.

Here is another classical example of a function extension that is encountered when studying homographies of the real line. A homography is a function $$h(x)=\frac{ax+b}{cx+d}$$ such that $S$. Its domain is the set of all real numbers different from $$-d/c,$$ and its image is the set of all real numbers different from $$a/c.$$ If one extends the real line to the projectively extended real line by including $S$, one may extend $i$ to a bijection from the extended real line to itself by setting $$h(\infty)=a/c$$ and $$h(-d/c)=\infty$$.

In calculus
The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus. At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth. But the definition was soon extended to functions of several variables and to functions of a complex variable. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.

Functions are now used throughout all areas of mathematics. In introductory calculus, when the word function is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as real analysis and complex analysis.

Real function
A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval. In this section, these functions are simply called functions.

The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous, differentiable, and even analytic. This regularity insures that these functions can be visualized by their graphs. In this section, all functions are differentiable in some interval.

Functions enjoy pointwise operations, that is, if $f$ and $g$ are functions, their sum, difference and product are functions defined by
 * $$\begin{align}

(f+g)(x)&=f(x)+g(x)\\ (f-g)(x)&=f(x)-g(x)\\ (f\cdot g)(x)&=f(x)\cdot g(x)\\ \end{align}.$$ The domains of the resulting functions are the intersection of the domains of $f$ and $g$. The quotient of two functions is defined similarly by
 * $$\frac fg(x)=\frac{f(x)}{g(x)},$$

but the domain of the resulting function is obtained by removing the zeros of $h$ from the intersection of the domains of $f$ and $g$.

The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function $$x\mapsto \frac 1x,$$ whose graph is a hyperbola, and whose domain is the whole real line except for 0.

The derivative of a real differentiable function is a real function. An antiderivative of a continuous real function is a real function that has the original function as a derivative. For example, the function $x\mapsto\frac 1x$ is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for $S$, is a differentiable function called the natural logarithm.

A real function $f$ is monotonic in an interval if the sign of $$\frac{f(x)-f(y)}{x-y}$$ does not depend of the choice of $g$ and $g$ in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function $f$ is monotonic in an interval $g$, it has an inverse function, which is a real function with domain $arccos$ and image $f$. This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the exponential function.

Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. For example, the sine and the cosine functions are the solutions of the linear differential equation
 * $$y''+y=0$$

such that
 * $$\sin 0=0, \quad \cos 0=1, \quad\frac{\partial \sin x}{\partial x}(0)=1, \quad\frac{\partial \cos x}{\partial x}(0)=0.$$

Vector-valued function
When the elements of the codomain of a function are vectors, the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its velocity vector is a vector-valued function.

Some vector-valued functions are defined on a subset of $$\mathbb{R}^n$$ or other spaces that share geometric or topological properties of $$\mathbb{R}^n$$, such as manifolds. These vector-valued functions are given the name vector fields.

Function space
In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support (that is, they are zero outside some compact set) form a function space that is at the basis of the theory of distributions.

Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.

Multi-valued functions
Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a neighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point $$x_0,$$ there are several possible starting values for the function.

For example, in defining the square root as the inverse function of the square function, for any positive real number $$x_0,$$ there are two choices for the value of the square root, one of which is positive and denoted $$\sqrt {x_0},$$ and another which is negative and denoted $$-\sqrt {x_0}.$$ These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single smooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positive $x$, one value for 0 and no value for negative $y$.

In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the implicit function that maps $f$ to a root $I$ of $$x^3-3x-y =0$$ (see the figure on the right). For $ad − bc ≠ 0$ one may choose either $$0, \sqrt 3,\text{ or } -\sqrt 3$$ for $I$. By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval $x$ and the image is $x$; for the second one, the domain is $y$ and the image is $x$; for the last one, the domain is $x$ and the image is $[−2, 2]$. As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single multi-valued function of $[−1, 1]$ that has three values for $∞$, and only one value for $x = 1$ and $f(I)$.

Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions. The domain to which a complex function may be extended by analytic continuation generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets $[−2, ∞)$ for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets $y = 0$. There are generally two ways of solving the problem. One may define a function that is not continuous along some curve, called a branch cut. Such a function is called the principal value of the function. The other way is to consider that one has a multi-valued function, which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the monodromy.

In the foundations of mathematics
The definition of a function that is given in this article requires the concept of set, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.

For example, the singleton set may be considered as a function $$x\mapsto \{x\}.$$ Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definition for these weakly specified functions.

These generalized functions may be critical in the development of a formalization of the foundations of mathematics. For example, Von Neumann–Bernays–Gödel set theory, is an extension of the set theory in which the collection of all sets is a class. This theory includes the replacement axiom, which may be stated as: If $[1, ∞)$ is a set and $(−∞, 2]$ is a function, then $−2 < y < 2$ is a set.

In alternative formulations of the foundations of mathematics using type theory rather than set theory, functions are taken as primitive notions rather than defined from other kinds of object. They are the inhabitants of function types, and may be constructed using expressions in the lambda calculus.

In computer science
In computer programming, a function is, in general, a piece of a computer program, which implements the abstract concept of function. That is, it is a program unit that produces an output for each input. However, in many programming languages every subroutine is called a function, even when there is no output, and when the functionality consists simply of modifying some data in the computer memory.

Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions. For example,  is a function that takes three functions as arguments, and, depending on the result of the first function (true or false), returns the result of either the second or the third function. An important advantage of functional programming is that it makes easier program proofs, as being based on a well founded theory, the lambda calculus (see below).

Except for computer-language terminology, "function" has the usual mathematical meaning in computer science. In this area, a property of major interest is the computability of a function. For giving a precise meaning to this concept, and to the related concept of algorithm, several models of computation have been introduced, the old ones being general recursive functions, lambda calculus and Turing machine. The fundamental theorem of computability theory is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The Church–Turing thesis is the claim that every philosophically acceptable definition of a computable function defines also the same functions.

General recursive functions are partial functions from integers to integers that can be defined from via the operators Although defined only for functions from integers to integers, they can model any computable function as a consequence of the following properties:
 * constant functions,
 * successor, and
 * projection functions
 * composition,
 * primitive recursion, and
 * minimization.
 * a computation is the manipulation of finite sequences of symbols (digits of numbers, formulas, ...),
 * every sequence of symbols may be coded as a sequence of bits,
 * a bit sequence can be interpreted as the binary representation of an integer.

Lambda calculus is a theory that defines computable functions without using set theory, and is the theoretical background of functional programming. It consists of terms that are either variables, function definitions (𝜆-terms), or applications of functions to terms. Terms are manipulated through some rules, (the $y ≤ −2$-equivalence, the $(−∞, −1]$-reduction, and the $y$-conversion), which are the axioms of the theory and may be interpreted as rules of computation.

In its original form, lambda calculus does not include the concepts of domain and codomain of a function. Roughly speaking, they have been introduced in the theory under the name of type in typed lambda calculus. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus.

Subpages

 * List of types of functions
 * List of functions
 * Function fitting
 * Implicit function

Generalizations

 * Higher-order function
 * Homomorphism
 * Morphism
 * Microfunction
 * Distribution
 * Functor

Related topics

 * Associative array
 * Closed-form expression
 * Elementary function
 * Functional
 * Functional decomposition
 * Functional predicate
 * Functional programming
 * Parametric equation
 * Set function
 * Simple function

title: Function (mathematics)


text:
Artichoke (disambiguation)

The globe artichoke (Cynara cardunculus var. scolymus) is a species of thistle cultivated as a food. Artichoke may also refer to:

Plants

 * Artichoke cactus, a species of cactus
 * Jerusalem artichoke, a species of sunflower with an edible tuber
 * Stachys affinis, also known as Chinese artichoke or Artichoke betony, an herbaceous perennial plant of the family Lamiaceae
 * Artichoke Garlic

Places

 * Artichoke, Minnesota, an unincorporated community in the United States
 * Artichoke Lake, a lake in Minnesota
 * Artichoke River, a river of Minnesota, United States
 * Artichoke Township, Minnesota, a civil township in the United States

Other uses

 * Artichoke (band), an indie pop band in Los Angeles
 * Artichoke (company), a creative company specialising in arts events
 * Artichoke Creek (disambiguation)
 * PH Artichoke, a designer light fixture manufactured by Louis Poulsen
 * Project ARTICHOKE, a CIA operation
 * Artichoke gall, a growth caused by a wasp on oak trees

title: Artichoke (disambiguation)


text:
Talk:Free variables and bound variables

style
The transition from the list of examples to the proposed precise definition, which I suspect may be too narrow, is abrupt. The article needs polishing, by someone familiar with logic and with the lambda calculus. Michael Hardy 22:51 Feb 23, 2003 (UTC)

The see also link for Closure appears to be the wrong kind of closure. It should probably be Closure (computer science). Glenn Willen (Talk)   16:18, 2 Sep 2004 (UTC)

 * I wonder if the explanation free to be varied as was recently inserted in the article is not misleading. My guess is that the etimologically bound variable precedes free variable. CSTAR 05:47, 13 Dec 2004 (UTC)

 * OK, I'm thinking about your first point. As to your second point, the words free and bound in common language go together so intimately that it's difficult to imagine one of the two having been introduced earlier than the other.  Michael Hardy 23:47, 13 Dec 2004 (UTC)

 * What I had in mind was the following. (As I know you know) bound is used in a related sense: A variable binding is similar to an assignment (in both mathematical languages and in programming languages). Alternatively, we can say a variable is bound to a value. This sense of bound is certainly different than being within the scope of a binder although I am not exactly sure how to express the difference. CSTAR 00:19, 14 Dec 2004 (UTC)

PIVOT variable = BOUND variable DUMMY variable is (special case of) bound variable that can have only values 0 or 1.

 * The above is utter nonsense and is already covered in the article. Statisticians use the term "dummy variable" to mean something that can only be either 0 or 1.  That is a completely different concept from that of dummy variable discussed in this article.  And the article already says that.  The "dummy variables" used by statisticians that only take the values 0 or 1 are NOT bound variables at all. Michael Hardy 22:31, 29 Apr 2005 (UTC)

 * I searched for "dummy variable" hopeing to get to the "indicator variable" page. There is a fundamental difference between the mathematical dummy variable and statistical dummy variable.  Nevertheless, I think those who are searching for the statistical version (as I was) may be confused by what they read here.  Unless they read through the article, they will not see the link to "indicator variable".  In my opinion, rather than having a single line buried in the text stating that "unfortunately the term dummy variable is used by many statisticians to mean an indicator variable or some variant thereof", I have added a "for the statisitcal dummy variable..." link at the top.  I think this should remove much of the confusion (at least the initial confusion I had). Mustard 20:41, 20 January 2006 (UTC)

 * We need to get rid of the redirect and have a proper article on statistical dummy variables. The link to indicator variables only makes things worse JQ 23:36, 23 April 2006 (UTC)

 * Fixed. David-Sarah Hopwood (talk) 14:20, 27 July 2008 (UTC)

Recursive function link
Michael Hardy reverted the link I edited this morning. There are several places you could link the words recursive function to: I'm not planning to edit the page again; decide for yourself which of the above links is the meaning you want. Note that recursive function is a disambiguation page, so you shouldn't link there.CMummert 02:13, 17 July 2006 (UTC)
 * computable function
 * Recursion
 * Recursion (computer science)

 * Within this general area of mathematics, it usually means a computable function. Michael Hardy 17:53, 17 July 2006 (UTC)

 * That usage doesn't make sense in the sentence
 * Similarly, an identifier bound to a recursive function is also technically a free variable within its own body but is treated specially.
 * CMummert 17:55, 17 July 2006 (UTC)
 * I'm not going to vouch for all changes made throughout WP by CMummert from computable to recursive (since I haven't looked at them all) but I think in the instance being discussed here, his change is correct. Recursion in this instance means (a least) fixed point of some equation function. --CSTAR 19:04, 17 July 2006 (UTC)

There is no explaination of what a bound variable is in software development.
Can someone please define what a bound variable is in computer programming. —Preceding unsigned comment added by 71.237.89.49 (talk) 21:47, 2 March 2008 (UTC)

 * I would like to point out more importantly that there is no explanation of what a bound variable is ANYWHERE. There is only an explanation of what it is not.  In the first section a free variable is clearly defined, but then it jumps directly to an example of a bound variable, but not a definition. Emry (talk) 07:27, 9 October 2008 (UTC)

 * Whether a variable is bound or not depends always on the context. --77.1.51.135 (talk) 11:24, 7 February 2011 (UTC)

Free variables in programming
Free variables only apply to lexical scoping, there's no distinction between bound and and free variables in dynamic scope. 85.221.147.47 (talk) 12:01, 10 August 2009 (UTC)

That's what the article claims, and it also states that this is the reason why closures only exist in lexical scoping, but PHP 5.3 is getting closures and supposedly PHP isn't lexically scoped. So how does that work?--Subversive Sound (talk) 01:14, 13 March 2010 (UTC)

Contradiction
"   *Ashleyi met heri. (interpretation #1: her = Ashley)    Ashleyi met herj. (interpretation #2: her = a female that is not Ashley)

The first interpretation is impossible, but the second interpretation is grammatical (and in this case, is the only interpretation)."

I think that the author might mean this would seem to be a contradiction in objective ordinary experience, not a grammatical error as the context of the article suggests. "Ashley met her(self)" is a valid statement that may be written for a number of reasons, probably to highlight a contradiction, discontinuity, unusual condition or perception. It might be said in realty, as the effective summary of a situation, from a certain point of view, without it being a grammatical error. For instance, Ashley may have never seen herself in a mirror, or met her twin who was previously unknown for the first time. If one fails to perceive an extensive mirror as part of a wall, and sees it as a passage way, one might meet oneself. Various medical or mental or memory conditions might lead to a case where one effectively "meets" oneself. It may be used in rhetoric. It may be used in pure fiction for dramatic effect, i.e. in romatic fiction to bring about an emotional response or as part of speculative fiction to bring a contradictory set of conditions to a focus. Since the article seems to be making a grammatical argument and uses the words "impossible" and "only interpretation", it would seem to have overreached. This point may seem picky, but, a collection of such logical overreaches and similar rigidity erects barriers to automated natural language interpretation, interpreting perception, or interpreting fiction. — Preceding unsigned comment added by 209.151.137.168 (talk) 15:35, 21 August 2013 (UTC)

 * No, you have misunderstood. The point has nothing to do with ‘objective ordinary experience’. The point is about grammar.

 * Basically, if you are reporting the event that Ashley has met herself, then you can only say Ashley met herself. You can't say Ashley met her to report that event. I think that you recognize this because you added (self) to the sentence. And, if you are reporting the event that Ashley has met another female that is Ashley, then you can only say Ashley met her. You can't say Ashley met herself. So, the point was that herself (a reflexive) is always a bound variable while her can be either a free variable or an obligatorily unbound variable depending on where the variable occurs within the syntax of a sentence.

 * Now whether or not people can meet themselves via a mirror or psychosis or whatever is a completely different issue. Maybe we can change the verb met to hit since it's obvious that people can hit themselves as well as other people. Therefore, we make it clear that we are really talking about grammar and variable assignment in natural language. – ishwar   (speak)  02:21, 8 September 2013 (UTC)

alpha conversion
A brief description of alpha conversion would be nice. Currently, a google of alpha conversion points one to either the haskell wiki, or to the Lambda calculus article, here. But of course, alpha conversion is a far more general concept than that, and its awkward to tell my readers that they have to first grok lambda calculus before they can understand this more basic concept. 67.198.37.16 (talk) 23:15, 20 July 2017 (UTC)

title: Talk:Free variables and bound variables