title: Touhou Project

text: Each album (except "Dolls in Pseudo Paradise," "Unknown Flower, Mesmerizing Journey," and "Rainbow-Colored Septentrion") includes a booklet written by ZUN documenting the activities of the , a self-described "club of necromancers" in Kyoto, Japan. In the loose collection of stories, club members Renko Usami and Maribel Hearn research and discuss various topics related to Gensokyo and the paranormal; "Dolls in Pseudo Paradise" includes an unrelated story about the fate of eight thieves spirited away to Gensokyo.


title: Touhou Project

text: In 2006 and 2007, ZUN released , a five-volume collection of PC-98 soundtracks that includes several unused themes. The albums respectively cover "Lotus Land Story", "Mystic Square", "Story of Eastern Wonderland", "Phantasmagoria of Dim.Dream", and "Highly Responsive to Prayers". Each track in the collection was enhanced with the addition of a sixth FM synthesis channel, which was originally reserved for sound effects in the games:


title: Touhou Project

text: Original soundtracks for the six official fighting games have also been released. The albums primarily consist of arrangements by of Twilight Frontier, with several arrangements and/or new compositions by ZUN:


title: Touhou Project

text: "ZUN's Music Collection", "Akyu's Untouched Score", and the fighting game soundtracks have been released for digital download on Google Play and the iTunes Store (Japan, US). Several music CDs have accompanied copies of official print works; these are listed below.


title: Touhou Project

text: In the shooting games, the player's bullet power increases on a linear scale as the player collects power-ups dropped by enemies, and eventually maxes out. The player can also collect "point" icons to earn extra lives; the amount needed grows exponentially as the player's score rises. The player can enter "focus mode" by holding the shift key by default, which slows down the player's movement, makes their hitbox visible (from "Perfect Cherry Blossom" onwards), and generally focuses the player's attack to make it more powerful. The graze counter, missing from "Mountain of Faith" and "Story of Eastern Wonderland", tracks how many bullets entered the character sprite but avoided the hitbox, rewarding the player with a score bonus for taking risks.


title: Touhou Project

text: The player can use a "bomb" or "spell card", similar to the "bomb" in many other shooting games. Although the player has a limited number at any given time, losing a life replenishes the current number of bombs up to a certain amount. With some exceptions, use of a bomb will make the user temporarily invulnerable, clear many of the bullets, and cause heavy damage to any enemies on screen. The overall effect the bomb has varies by character and by game. The player can use one during a short period after being hit by a bullet (called a "deathbomb") to avoid loss of a life. The amount of time the player has available to deathbomb is usually around 0.3 seconds (8 frames). Bosses have attack phases, which are also referred to as spell cards, but with bosses the term applies to a prolonged pattern of movements and shots that last until the player depletes the boss' health by a certain amount or the time runs out.


title: Touhou Project

text: Each individual game of the main series from "Perfect Cherry Blossom" to "Wily Beast and Weakest Creature" has additional items that affect some aspect of gameplay, such as scoring or gaining extra lives/bombs. For example, "Perfect Cherry Blossom" has "cherry points", which are used mostly in scoring, but can grant temporary invulnerability (known as the "supernatural border"); "Imperishable Night" has "time points", which are essential for advancing to later stages, and also determine if the player gets to challenge a boss's 'final spell' on normal or higher difficulties; "Mountain of Faith" has "faith" points, which boost the score the player receives upon gathering point items and bonuses for clearing spell cards without dying or using a spell card.


title: Touhou Project

text: Each main "Touhou" game has four difficulty levels ("easy", "normal", "hard", and "lunatic") with each one being harder than the previous. Regardless of difficulty choice, there are six stages in each game that become progressively difficult. The only exceptions to this are "Story of Eastern Wonderland" and "Embodiment of Scarlet Devil" on easy difficulty, both of which only have five stages.


title: Touhou Project

text: In addition to the four main difficulties, there is an extra stage, unlocked after completing the game without using any continues. The extra stage is more difficult than "normal" and less difficult than "hard", and contains an especially long boss fight (usually with ten spell cards). Bosses in extra stages are usually immune to bombs. The only games that allow players to reach the extra stage by completing the easy difficulty are "Story of Eastern Wonderland", "Mystic Square", "Perfect Cherry Blossom", "Imperishable Night", "Fairy Wars", "Double Dealing Character", "Legacy of Lunatic Kingdom", "Hidden Star in Four Seasons", "Wily Beast and Weakest Creature", and "Unconnected Marketeers". In "Perfect Cherry Blossom", there is a phantasm extra stage where the difficulty is increased even further. The phantasm stage is accessed by beating the extra stage and capturing at least 60 spells with any of the characters and weapon types combined.


title: Touhou Project

text: The plots of the "Touhou Project" revolve around the strange phenomena that occur in the fictional realm of , which ZUN designed with a human village in some remote mountain recesses in Japan. Originally, it was simply called "a remote separated land of a human village in an eastern country." Long before "Touhou Project"'s story begins, many non-humans like "yōkai" lived with some humans in the area. After a few humans disappeared into Gensokyo, many humans became afraid of approaching this area, while others settled there to exterminate "yōkai". However, as time went on, humans developed civilization and multiplied in number, and thus "yōkai" worried about how the balance between humans and "yōkai" would be affected. 500 years before "Embodiment of Scarlet Devil" (EoSD), the "yōkai" sage Yukari Yakumo developed the "boundary of phantasm and substance," which was favored by the yōkai and protected the balance. This was called the "Yōkai Expansion Project" and made Gensokyo a phantasmal world that automatically called out to the weakened "yōkai" of the outside world. Other things that disappear from the outside world, like extinct animals, lost tools, and architecture, appear in Gensokyo. Since Gensokyo was a plot of land in Japan that is separated by a barrier, it is Japan that is immediately outside of this barrier.


title: Touhou Project

text: As a result of the seal, Gensokyo became inaccessible from the outside world, and similarly, those in Gensokyo were unable to leave. Gensokyo's existence could not be confirmed from the outside world, nor could the outside world be confirmed within Gensokyo. As a result, the isolated community developed its own civilization, separate from the outside world. Although separated by a barrier, it is a bordering world to its outside, as opposed to being in a parallel universe. There are no seas in Gensokyo, since it is landlocked. In Gensokyo, there are few humans, and various kinds of "yōkai". Some species include magicians, beasts, therianthropes, vampires, "bōrei", tengu, mermaids, kappa, and "yōkai" (a kind of miscellaneous group). There are others species that could be "yōkai" depending on definition, like fairies, spirits, "yūrei", "onryō", poltergeists, hermits, oni, and gods which are all portrayed in human female form.


title: Touhou Project

text: In present Gensokyo, presented in all "Touhou Project" games since EoSD, magical and spiritual qualities prevail compared to the outside world where unscientific phenomena were dismissed as "superstition" around the time of the Meiji era. The only known gateway from the outside world into Gensokyo is the Hakurei Shrine on the border of Gensokyo. The spell card rules were also established to keep up the relationship between humans and "yōkai" in a mock style, which was necessary for the preservation of the balance of Gensokyo. The "Great Hakurei Barrier," managed by past Hakurei "miko", was constructed several decades before EoSD, which is described as a "barrier of common sense," and is thus a strong logical barrier that not even "yōkai" can pass through. The "yōkai" opposed its construction at first before understanding its usefulness.


title: Touhou Project

text: In Gensokyo, events called "incidents" occur once in a while. An incident is an event that affects all of Gensokyo and is of unknown cause at the time it occurs. "Touhou Project" mainly focuses on incidents that affects the entirety of Gensokyo in its stories, but there are also works like "Mountain of Faith" that are centered on lesser-scale events.


title: Touhou Project

text: Frequently, incidents are due to a "yōkai" whim or curiosity, and usually Reimu Hakurei would go to investigate it and find and chastise the perpetrator. While Reimu is usually the one to resolve incidents, there are cases where Marisa Kirisame and other characters would resolve them. When a major incident occurs, the spirits and fairies are affected by the incident and experience an increase in power for the duration incident.


title: Touhou Project

text: With its focus on bishōjo characters, the "Touhou" series possesses a large cast compared to other shooting games. While they are not developed nearly to the standards of a story-based game, many players love them. One example is Hong Meiling, affectionately called Chūgoku (China), the stage 3 boss of "Embodiment of Scarlet Devil", who won a popularity contest in Japan out of all "Touhou" characters. Among the vast array of characters, only six named characters are male. One (Genjii) is a turtle, one is a cat (Sokrates), two (Youki Konpaku and Myouren Hijiri) are only mentioned in passing, one (Rinnosuke Morichika) who is a merchant is only featured in the serialized novel, and the last (Unzan) is a cloud-like entity. There is also one before all these (Shingyoku), but this character has three forms of different genders, so this character may not exactly be male.


title: Touhou Project

text: Though each game features a collection of different characters, the main protagonist of the series is always Reimu Hakurei, joined by Marisa Kirisame after the second game. Exceptions to this include "Shoot the Bullet" and "Double Spoiler" (Aya Shameimaru is playable on both, and Hatate Himekaidou can become playable in "Double Spoiler"), "Fairy Wars" (which has Cirno as the sole playable character), "Impossible Spell Card" and "Gold Rush" (where only Seija Kijin is playable), and "Violet Detector" (where only Sumireko Usami is playable).


title: Touhou Project

text: The "Touhou Project" is a one-man project by Jun'ya Ōta (usually under the pseudonym ZUN), who does all the graphics, music, and programming alone for the bullet hell games, with the exceptions of the portrait art in "Fairy Wars," which was done by Makoto Hirasaka. The fighting games, "Immaterial and Missing Power", "Scarlet Weather Rhapsody", "Touhou Hisōtensoku", "Hopeless Masquerade", "Urban Legend in Limbo", and "Antinomy of Common Flowers", were dual efforts with Twilight Frontier, in which ZUN wrote the music and story, and Twilight Frontier created the art and gameplay.


title: Touhou Project

text: ZUN's first interest in developing video games came during his high school years. While most shoot 'em up games utilise a military or science fiction theme, ZUN wanted a game with a miko main character, and a Shinto aesthetic. ZUN was part of his school's orchestra club, and originally wanted to create music for video games. He went to college, hoping to compose music for fighting games, since they were popular at the time due to "Street Fighter II". As he did not know anybody else who was making games that he could put his music in, he made his own games for this purpose, which led to the first "Touhou" game, "Highly Responsive to Prayers", being released in 1996. The first game was originally intended as a practice in programming. "Touhou" only became a shooting game series from the second game onwards, because the popularity of shooting games had revived due to "RayForce" and ZUN had long been a fan of such games. ZUN remarked how the general theme and direction of "Touhou" only started coming together in the sixth game, "Embodiment of Scarlet Devil".


title: Touhou Project

text: ZUN develops his games with Visual Studio, Adobe Photoshop, and Cubase, according to his interview in "Bohemian Archive in Japanese Red".


title: Touhou Project

text: Many derivative works based on "Touhou" have been created since the release of "Embodiment of Scarlet Devil", including dōjinshi, dōjin music (many of the songs had been made remixing the original games' music), dōjin anime, and dōjin games (even though the "Touhou" games themselves are dōjin games). The vast scope of "Touhou Project" derivatives prompted commentary, noting that "Touhou Project" became an unmissable aspect of Japanese consumer generated media. These dōjin activities are mostly responsible for adding original attributes to characters that ZUN may not have intended. ZUN, for the most part, had acknowledged, appreciated, and even encouraged these derivative works by imposing very few restrictions on the use of his works. The major restrictions are on unauthorized commercial distribution (as opposed to dōjin) and the spoiling of endings; proper attribution to Team Shanghai Alice is a requirement. ZUN stated himself that he did not want the "Touhou Project" game series to be officially commercialized.


title: Touhou Project

text: The first publication of "Touhou" derivative doujinshi occurred during December 2003, following the release of "Perfect Cherry Blossom"; seven circles sold "Touhou" derivative works at Comiket 65 in December 2003. At Comiket 74 in August 2008, a total of 885 circles had "Touhou" derivative works on display or for sale, out of a total of 35,000 circles participating at Comiket. At Comiket 77 (December 2009), 2,372 circles were dedicated to "Touhou", breaking the previous record held by "The Prince of Tennis" at Comiket 66 (August 2004), which had 2,130 circles. At Comiket 85 (December 2013), "Touhou" was still in the lead, with 2,272 participating circles selling "Touhou" derivatives, compared to "Kuroko's Basketball", the second most popular franchise at that year's Comiket, which had 1,462 circles. Throughout the mid 2010s, "Touhou" popularity at Comiket began to diminish. From 2015-2017, "Kantai Collection" was consistently the most popular series at Comiket. By 2019, "Touhou" was the fourth most popular series at Comiket, with less than half the circles at Comiket than the "Fate" series.


title: Touhou Project

text: The dōjin games based on "Touhou" include adaptations of other game series' mechanics with "Touhou" characters, such as "" (a parody of the "Castlevania" games with the title being a play on the Japanese game of the series, "Akumajō Dracula"), "Age of Ethanols" (based on the "Age of Empires" series), and "Touhou Puppet Dance Performance" (based on the "Pokémon" games).


title: Touhou Project

text: At Tokyo Game Show 2014, ZUN announced a collaborative project with PlayStation bringing unofficial, fan-made "Touhou Project" dōjin games to the PlayStation 4 (PS4) and PlayStation Vita platforms. Yuyuko Saigyouji and Reimu Hakurei also made cameo character appearances in Square Enix's game "Lord of Vermilion Re:2".


title: Touhou Project

text: The music of "Touhou" has been a particular source of interest, and many arrangement CDs are sold at Comiket and other Japanese conventions. Among the most popular derivatives are the series of Flash music videos created by the dōjin music circle IOSYS, which are often shared on otaku internet forums and Nico Nico Douga. One such song, "Marisa Stole the Precious Thing", was parodied in derivative anime music videos. "Marisa Stole the Precious Thing", alongside "Cirno's Perfect Math Class", "Night of Nights", and "Bad Apple" were included in the dōjin game "Touhou Spell Bubble." Toby Fox, the creator of "Undertale", cited "Touhou" as one of his musical influences.


title: Touhou Project

text: Several fanmade anime have been made for "Touhou". Albeit created by amateur studios, the projects have sometimes included appearance by professional voice actors, such as Rie Tanaka. Dōjin anime of note include by the circle Manpuku Jinja in 2011, by the dōjin circle Maikaze, and "Hifuu Club Activity Record ~ The Sealed Esoteric History" () by the Chinese dōjin circle Kyoto Fantasy Troupe. A derivative anime from a commercial anime studio came in the form of by Ufotable to celebrate the 10th anniversary of Japanese goods chain Animate as a promotional video for the store combining "Touhou" with Animate's mascot, Meito Anizawa.


title: Touhou Project

text: The "Touhou Project" was nominated for the 11th annual Media Arts Awards held by Japan's Agency for Cultural Affairs, under the Entertainment category, where "Touhou Project" eventually lost to Nintendo's "Wii Sports" for the Grand Prize award. The series was inducted into the Guinness World Records in October 2010 as the "most prolific fan-made shooter series".


title: Touhou Project

text: A major Internet meme based on "Touhou" is , which centers around the disembodied, deformed heads of "Touhou" characters, often referred to as "yukkuris". This meme originated from a disfigured attempt to draw the main characters Reimu and Marisa with Shift JIS art. Yukkuris became so popular that the phrase "Yukkuri shite itte ne!!!" won bronze for 2008's "Net Slang of the Year" in Japan. Yukkuris also appear in Internet advertisements, the anime "Natsu no Arashi!" and "Pani Poni Dash!", and most notably, the front page of 2channel, one of the most visited Japanese websites.


title: Touhou Project

text: The is the largest of the many dōjin conventions hosting only "Touhou" content. Although the coordinator of this convention has nothing to do with Team Shanghai Alice officially, the name "Hakurei Shrine Reitaisai" was given by ZUN himself. It first started in 2004 as a way for Team Shanghai Alice to publicly distribute the trial version for their upcoming games, "Imperishable Night" and "Immaterial and Missing Power". In addition, the 2004 Reitaisai featured a total of 114 participating circles. Since then, many "Touhou" derivative works are gathered and sold there. Commencing every year in April or May, the convention has been hosted in Ōta, Tokyo, in 2004; Naka-ku, Yokohama, in 2005; Sunshine City, Tokyo, in 2006 and 2007; and the Tokyo Big Sight from 2008 onwards. In 2010 Reitaisai SP, an additional Reitaisai to be held every autumn due to increasing popularity, was started, but it was cancelled after 2011, and later brought back and rebranded as Autumn Reitaisai from 2014 onwards. The 2011 Reitaisai was originally cancelled due to safety concerns after the 2011 Tōhoku earthquake and tsunami, and the release of "Ten Desires" was also postponed. It was later rescheduled and held on May 8, 2011, with approximately 4,940 participating circles. In addition to events in Japan, there is another Reitaisai held in Taiwan that started in 2015. The 17th Reitaisai in 2020, which was originally going to run on March 22 before being postponed to May 17, was ultimately cancelled due to safety concerns over the COVID-19 pandemic, making it the first Reitaisai to be cancelled.


title: Corpse Husband

text: Corpse Husband (born August 8, 1997), commonly abbreviated as Corpse and stylized as CORPSE, is an American YouTuber and musician. Corpse is best known for his music and "faceless" work on YouTube. He is mostly known for his horror story narration and "Among Us" content. He also gained notable recognition for his deep low-pitched voice.


title: Corpse Husband

text: In 2015, Corpse began his career on YouTube by narrating horror stories on his channel, which he did consecutively until 2020. He made his musical debut in 2016, being featured on the single "Grim Grinning Ghost" by record producer The Living Tombstone and musical artist CrusherP.


title: Corpse Husband

text: In June 2020, Corpse released his debut single "Miss You!", which peaked at number 31 on the Hot Rock & Alternative Songs chart. His follow-up single "White Tee", which was released in the same month, reached number 32 on the same chart. In September 2020, Corpse began streaming and creating content on the video game "Among Us", which brought him much recognition, and has since gained over 7 million subscribers on YouTube. In the same month, he released the single "E-Girls Are Ruining My Life!", featuring vocals from rapper Savage Ga$p, which generated wide-spread commercial success for Corpse in 2020. The song charted at 28 in Bulgaria, 90 on the UK Singles Chart, and 24 in "Billboard"s Bubbling Under Hot 100 chart. It also reached more than 100 million streams on Spotify, and was ranked second on Spotify's Viral 50 songs chart. In 2022, it became his first track to be certified platinum by the Recording Industry Association of America.


title: Corpse Husband

text: In October 2020, he worked with U.S. representatives Alexandria Ocasio-Cortez and Ilhan Omar along with several other notable streamers including Disguised Toast, Cr1TiKaL and Pokimane for a session of "Among Us" as part of a get-out-the-vote initiative for the 2020 United States presidential election. The same month, he released the single "Agoraphobic", which debuted at number 21 on the Hot Rock & Alternative Songs chart four months later.


title: Corpse Husband

text: In March 2021, Corpse was featured on Machine Gun Kelly's song "DayWalker". On March 18, 2021, a music video for the song was released, featuring fellow YouTuber and streamer Valkyrae portraying Corpse. The song became Corpse's first "Billboard" Hot 100 entry, charting at number 88, and second on the UK Singles Chart, peaking at number 53.


title: Corpse Husband

text: In April 2021, Corpse participated in an hour-long charity "Among Us" stream on "The Tonight Show" with Jimmy Fallon, members of The Roots, fellow streamers Valkyrae and Sykkuno, and "Stranger Things" actors Gaten Matarazzo and Noah Schnapp, with proceeds going towards Feeding America.


title: Corpse Husband

text: In September 2021, Corpse released a single entitled "Hot Demon B!tches Near U!!!" in collaboration with Night Lovell.


title: Corpse Husband

text: In February 2022, Funimation announced Corpse's English dub debut voice role as Ōjirō Ōtori in the anime adaptation of Kazutaka Kodaka's "Tribe Nine".


title: Corpse Husband

text: Corpse has conditions such as fibromyalgia, thoracic outlet syndrome and GERD, the latter of which has partially caused his voice to become deeper. He has also stated that he frequently wears an eyepatch due to strain from the brightness of blue light screens.


title: Polynomial

text: In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is .


title: Polynomial

text: Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.


title: Polynomial

text: The word "polynomial" joins two diverse roots: the Greek "poly", meaning "many", and the Latin "nomen", or "name". It was derived from the term "binomial" by replacing the Latin root "bi-" with the Greek "poly-". That is, it means a sum of many terms (many monomials). The word "polynomial" was first used in the 17th century.


title: Polynomial

text: The "x" occurring in a polynomial is commonly called a "variable" or an "indeterminate". When the polynomial is considered as an expression, "x" is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers the function defined by the polynomial, then "x" represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably.


title: Polynomial

text: A polynomial "P" in the indeterminate "x" is commonly denoted either as "P" or as "P"("x"). Formally, the name of the polynomial is "P", not "P"("x"), but the use of the functional notation "P"("x") dates from a time when the distinction between a polynomial and the associated function was unclear. Moreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let "P"("x") be a polynomial" is a shorthand for "let "P" be a polynomial in the indeterminate "x"". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much simpler and easier to read if the name(s) of the indeterminate(s) do not appear at each occurrence of the polynomial.


title: Polynomial

text: The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials.


title: Polynomial

text: If "a" denotes a number, a variable, another polynomial, or, more generally, any expression, then "P"("a") denotes, by convention, the result of substituting "a" for "x" in "P". Thus, the polynomial "P" defines the function


title: Polynomial

text: Frequently, when using this notation, one supposes that "a" is a number. However, one may use it over any domain where addition and multiplication are defined (that is, any ring). In particular, if "a" is a polynomial then "P"("a") is also a polynomial.


title: Polynomial

text: More specifically, when "a" is the indeterminate "x", then the image of "x" by this function is the polynomial "P" itself (substituting "x" for "x" does not change anything). In other words,


title: Polynomial

text: A "polynomial expression" is an expression that can be built from constants and symbols called "variables" or "indeterminates" by means of addition, multiplication and exponentiation to a non-negative integer power. The constants are generally numbers, but may be any expression that do not involve the indeterminates, and represent mathematical objects that can be added and multiplied. Two polynomial expressions are considered as defining the same "polynomial" if they may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication. For example formula_3 and formula_4 are two polynomial expressions that represent the same polynomial; so, one has the equality formula_5.


title: Polynomial

text: where formula_7 are constants that are called the "coefficients" of the polynomial, and formula_8 is the indeterminate. The word "indeterminate" means that formula_8 represents no particular value, although any value may be substituted for it. The mapping that associates the result of this substitution to the substituted value is a function, called a "polynomial function".


title: Polynomial

text: That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number called the coefficient of the term and a finite number of indeterminates, raised to non-negative integer powers.


title: Polynomial

text: The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any term with nonzero coefficient. Because , the degree of an indeterminate without a written exponent is one.


title: Polynomial

text: A term with no indeterminates and a polynomial with no indeterminates are called, respectively, a constant term and a constant polynomial. The degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial 0 (which has no terms at all) is generally treated as not defined (but see below).


title: Polynomial

text: is a term. The coefficient is , the indeterminates are and , the degree of is two, while the degree of is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is .


title: Polynomial

text: It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.


title: Polynomial

text: Polynomials of small degree have been given specific names. A polynomial of degree zero is a "constant polynomial", or simply a "constant". Polynomials of degree one, two or three are respectively "linear polynomials," "quadratic polynomials" and "cubic polynomials". For higher degrees, the specific names are not commonly used, although "quartic polynomial" (for degree four) and "quintic polynomial" (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, the term in is a linear term in a quadratic polynomial.


title: Polynomial

text: The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞). The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero polynomial, , is the "x"-axis.


title: Polynomial

text: In the case of polynomials in more than one indeterminate, a polynomial is called "homogeneous" of if "all" of its non-zero terms have . The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. For example, is homogeneous of degree 5. For more details, see Homogeneous polynomial.


title: Polynomial

text: The commutative law of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of ", with the term of largest degree first, or in "ascending powers of ". The polynomial is written in descending powers of . The first term has coefficient , indeterminate , and exponent . In the second term, the coefficient . The third term is a constant. Because the "degree" of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.


title: Polynomial

text: Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined, using the distributive law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0. Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, a two-term polynomial is called a binomial, and a three-term polynomial is called a "trinomial". The term "quadrinomial" is occasionally used for a four-term polynomial.


title: Polynomial

text: A real polynomial is a polynomial with real coefficients. When it is used to define a function, the domain is not so restricted. However, a real polynomial function is a function from the reals to the reals that is defined by a real polynomial. Similarly, an integer polynomial is a polynomial with integer coefficients, and a complex polynomial is a polynomial with complex coefficients.


title: Polynomial

text: A polynomial in one indeterminate is called a "univariate polynomial", a polynomial in more than one indeterminate is called a multivariate polynomial. A polynomial with two indeterminates is called a bivariate polynomial. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance, when working with univariate polynomials, one does not exclude constant polynomials (which may result from the subtraction of non-constant polynomials), although strictly speaking, constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials as "bivariate", "trivariate", and so on, according to the maximum number of indeterminates allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It is also common to say simply "polynomials in , and ", listing the indeterminates allowed.


title: Polynomial

text: The evaluation of a polynomial consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions. For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using Horner's method:


title: Polynomial

text: Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the commutative law) and combining of like terms. For example, if


title: Polynomial

text: Polynomials can also be multiplied. To expand the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other. For example, if


title: Polynomial

text: Given a polynomial formula_24 of a single variable and another polynomial of any number of variables, the composition formula_25 is obtained by substituting each copy of the variable of the first polynomial by the second polynomial. For example, if formula_26 and formula_27 then


title: Polynomial

text: A composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials. The composition of two polynomials is another polynomial.


title: Polynomial

text: The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called "rational fractions", "rational expressions", or "rational functions", depending on context. This is analogous to the fact that the ratio of two integers is a rational number, not necessarily an integer. For example, the fraction is not a polynomial, and it cannot be written as a finite sum of powers of the variable .


title: Polynomial

text: For polynomials in one variable, there is a notion of Euclidean division of polynomials, generalizing the Euclidean division of integers. This notion of the division results in two polynomials, a "quotient" and a "remainder" , such that and . The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division.


title: Polynomial

text: When the denominator is monic and linear, that is, for some constant , then the polynomial remainder theorem asserts that the remainder of the division of by is the evaluation . In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division.


title: Polynomial

text: All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field of complex numbers, the irreducible factors are linear. Over the real numbers, they have the degree either one or two. Over the integers and the rational numbers the irreducible factors may have any degree. For example, the factored form of


title: Polynomial

text: The computation of the factored form, called "factorization" is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms are available in most computer algebra systems.


title: Polynomial

text: Calculating derivatives and integrals of polynomials is particularly simple, compared to other kinds of functions.


title: Polynomial

text: For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number , or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient understood to mean the sum of copies of . For example, over the integers modulo , the derivative of the polynomial is the polynomial .


title: Polynomial

text: A "polynomial function" is a function that can be defined by evaluating a polynomial. More precisely, a function of one argument from a given domain is a polynomial function if there exists a polynomial


title: Polynomial

text: that evaluates to formula_37 for all in the domain of (here, is a non-negative integer and are constant coefficients).


title: Polynomial

text: Generally, unless otherwise specified, polynomial functions have complex coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also restricted to the reals, the resulting function is a real function that maps reals to reals.


title: Polynomial

text: is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in


title: Polynomial

text: According to the definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions. An example is the expression formula_40 which takes the same values as the polynomial formula_41 on the interval formula_42, and thus both expressions define the same polynomial function on this interval.


title: Polynomial

text: A non-constant polynomial function tends to infinity when the variable increases indefinitely (in absolute value). If the degree is higher than one, the graph does not have any asymptote. It has two parabolic branches with vertical direction (one branch for positive "x" and one for negative "x").


title: Polynomial

text: When considering equations, the indeterminates (variables) of polynomials are also called unknowns, and the "solutions" are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in contrast to a "polynomial identity" like , where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality.


title: Polynomial

text: In elementary algebra, methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the cubic and quartic equations. For higher degrees, the Abel–Ruffini theorem asserts that there can not exist a general formula in radicals. However, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree.


title: Polynomial

text: The number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the complex solutions are counted with their multiplicity. This fact is called the fundamental theorem of algebra.


title: Polynomial

text: A "root" of a nonzero univariate polynomial is a value of such that . In other words, a root of is a solution of the polynomial equation or a zero of the polynomial function defined by . In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered.


title: Polynomial

text: A number is a root of a polynomial if and only if the linear polynomial divides , that is if there is another polynomial such that . It may happen that a power (greater than ) of divides ; in this case, is a "multiple root" of , and otherwise is a simple root of . If is a nonzero polynomial, there is a highest power such that divides , which is called the "multiplicity" of as a root of . The number of roots of a nonzero polynomial , counted with their respective multiplicities, cannot exceed the degree of , and equals this degree if all complex roots are considered (this is a consequence of the fundamental theorem of algebra).


title: Polynomial

text: Some polynomials, such as , do not have any roots among the real numbers. If, however, the set of accepted solutions is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra. By successively dividing out factors , one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.


title: Polynomial

text: There may be several meanings of "solving an equation". One may want to express the solutions as explicit numbers; for example, the unique solution of is . Unfortunately, this is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as algebraic expressions; for example, the golden ratio formula_45 is the unique positive solution of formula_46 In the ancient times, they succeeded only for degrees one and two. For quadratic equations, the quadratic formula provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), although much more complicated, are known for equations of degree three and four (see cubic equation and quartic equation). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, Niels Henrik Abel proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see Abel–Ruffini theorem). In 1830, Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it. This result marked the start of Galois theory and group theory, two important branches of modern algebra. Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).


title: Polynomial

text: When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to be useful, the unique way of solving it is to compute numerical approximations of the solutions. There are many methods for that; some are restricted to polynomials and others may apply to any continuous function. The most efficient algorithms allow solving easily (on a computer) polynomial equations of degree higher than 1,000 (see Root-finding algorithm).


title: Polynomial

text: For polynomials with more than one indeterminate, the combinations of values for the variables for which the polynomial function takes the value zero are generally called "zeros" instead of "roots". The study of the sets of zeros of polynomials is the object of algebraic geometry. For a set of polynomial equations with several unknowns, there are algorithms to decide whether they have a finite number of complex solutions, and, if this number is finite, for computing the solutions. See System of polynomial equations.


title: Polynomial

text: The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination.


title: Polynomial

text: A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation. Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any general algorithm for solving them, or even for deciding whether the set of solutions is empty (see Hilbert's tenth problem). Some of the most famous problems that have been solved during the last fifty years are related to Diophantine equations, such as Fermat's Last Theorem.


title: Polynomial

text: Polynomials where indeterminates are substituted for some other mathematical objects are often considered, and sometimes have a special name.


title: Polynomial

text: A trigonometric polynomial is a finite linear combination of functions sin("nx") and cos("nx") with "n" taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions.


title: Polynomial

text: If sin("nx") and cos("nx") are expanded in terms of sin("x") and cos("x"), a trigonometric polynomial becomes a polynomial in the two variables sin("x") and cos("x") (using List of trigonometric identities#Multiple-angle formulae). Conversely, every polynomial in sin("x") and cos("x") may be converted, with Product-to-sum identities, into a linear combination of functions sin("nx") and cos("nx"). This equivalence explains why linear combinations are called polynomials.


title: Polynomial

text: For complex coefficients, there is no difference between such a function and a finite Fourier series.


title: Polynomial

text: Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are also used in the discrete Fourier transform.


title: Polynomial

text: A matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial


title: Polynomial

text: A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices "A" in a specified matrix ring "M"("R").


title: Polynomial

text: A bivariate polynomial where the second variable is substituted for an exponential function applied to the first variable, for example , may be called an exponential polynomial.


title: Polynomial

text: A rational fraction is the quotient (algebraic fraction) of two polynomials. Any algebraic expression that can be rewritten as a rational fraction is a rational function.


title: Polynomial

text: While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is not zero.


title: Polynomial

text: The rational fractions include the Laurent polynomials, but do not limit denominators to powers of an indeterminate.


title: Polynomial

text: Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down (just like irrational numbers cannot), but the rules for manipulating their terms are the same as for polynomials. Non-formal power series also generalize polynomials, but the multiplication of two power series may not converge.


title: Polynomial

text: A "polynomial" over a commutative ring is a polynomial all of whose coefficients belong to . It is straightforward to verify that the polynomials in a given set of indeterminates over form a commutative ring, called the "polynomial ring" in these indeterminates, denoted formula_49 in the univariate case and formula_50 in the multivariate case.


title: Polynomial

text: The map from to sending to itself considered as a constant polynomial is an injective ring homomorphism, by which is viewed as a subring of . In particular, is an algebra over .


title: Polynomial

text: One can think of the ring as arising from by adding one new element "x" to "R", and extending in a minimal way to a ring in which satisfies no other relations than the obligatory ones, plus commutation with all elements of (that is ). To do this, one must add all powers of and their linear combinations as well.


title: Polynomial

text: Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring over the real numbers by factoring out the ideal of multiples of the polynomial . Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring (see modular arithmetic).


title: Polynomial

text: If is commutative, then one can associate with every polynomial in a "polynomial function" with domain and range equal to . (More generally, one can take domain and range to be any same unital associative algebra over .) One obtains the value by substitution of the value for the symbol in . One reason to distinguish between polynomials and polynomial functions is that, over some rings, different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where is the integers modulo ). This is not the case when is the real or complex numbers, whence the two concepts are not always distinguished in analysis. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like Euclidean division) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for .


title: Polynomial

text: If is an integral domain and and are polynomials in , it is said that "divides" or is a divisor of if there exists a polynomial in such that . If formula_52 then is a root of if and only formula_53 divides . In this case, the quotient can be computed using the polynomial long division.


title: Polynomial

text: and such that the degree of is smaller than the degree of (using the convention that the polynomial 0 has a negative degree). The polynomials and are uniquely determined by and . This is called "Euclidean division, division with remainder" or "polynomial long division" and shows that the ring is a Euclidean domain.


title: Polynomial

text: Analogously, "prime polynomials" (more correctly, "irreducible polynomials") can be defined as "non-zero polynomials which cannot be factorized into the product of two non-constant polynomials". In the case of coefficients in a ring, ""non-constant"" must be replaced by ""non-constant or non-unit"" (both definitions agree in the case of coefficients in a field). Any polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials. If the coefficients belong to a field or a unique factorization domain this decomposition is unique up to the order of the factors and the multiplication of any non-unit factor by a unit (and division of the unit factor by the same unit). When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials). These algorithms are not practicable for hand-written computation, but are available in any computer algebra system. Eisenstein's criterion can also be used in some cases to determine irreducibility.


title: Polynomial

text: In modern positional numbers systems, such as the decimal system, the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the radix or base, in this case, . As another example, in radix 5, a string of digits such as 132 denotes the (decimal) number = 42. This representation is unique. Let "b" be a positive integer greater than 1. Then every positive integer "a" can be expressed uniquely in the form


title: Polynomial

text: The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example in calculus is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone–Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Practical methods of approximation include polynomial interpolation and the use of splines.


title: Polynomial

text: Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. The chromatic polynomial of a graph counts the number of proper colourings of that graph.


title: Polynomial

text: The term "polynomial", as an adjective, can also be used for quantities or functions that can be written in polynomial form. For example, in computational complexity theory the phrase "polynomial time" means that the time it takes to complete an algorithm is bounded by a polynomial function of some variable, such as the size of the input.


title: Polynomial

text: Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, circa 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write .


title: Polynomial

text: The earliest known use of the equal sign is in Robert Recorde's "The Whetstone of Witte", 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's "Arithemetica integra", 1544. René Descartes, in "La géometrie", 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the 's denote constants and denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.


title: Great Plains

text: The Great Plains (), sometimes simply "the Plains", is a broad expanse of flatland in North America. It is located west of the Mississippi River and east of the Rocky Mountains, much of it covered in prairie, steppe, and grassland. It is the southern and main part of the Interior Plains, which also include the tallgrass prairie between the Great Lakes and Appalachian Plateau, and the Taiga Plains and Boreal Plains ecozones in Northern Canada. The term Western Plains is used to describe the ecoregion of the Great Plains, or alternatively the western portion of the Great Plains.


title: Great Plains

text: The term "Great Plains" usually refers specifically to the United States portion of the ecozone while the Canadian portion is known as the Canadian Prairies. In Canada it covers southeastern Alberta, southern Saskatchewan and a narrow band of southwestern Manitoba, these three provinces collectively known as the "Prairie Provinces". The entire region is known for supporting extensive cattle-ranching and dryland farming.


title: Great Plains

text: Grasslands are among the least protected biomes with vast areas having been converted for agricultural purposes and pastures.


title: Great Plains

text: The term "Great Plains" is used in the United States to describe a sub-section of the even more vast Interior Plains physiographic division, which covers much of the interior of North America. It also has currency as a region of human geography, referring to the Plains Indians or the Plains states.


title: Great Plains

text: In Canada the term is rarely used; Natural Resources Canada, the government department responsible for official mapping, treats the Interior Plains as one unit consisting of several related plateaus and plains. There is no region referred to as the "Great Plains" in the "Atlas of Canada". In terms of human geography, the term "prairie" is more commonly used in Canada, and the region is known as the Canadian Prairies, Prairie Provinces or simply "the Prairies".


title: Great Plains

text: The "North American Environmental Atlas", produced by the Commission for Environmental Cooperation, a NAFTA agency composed of the geographical agencies of the Mexican, American, and Canadian governments, uses the "Great Plains" as an ecoregion synonymous with predominant prairies and grasslands rather than as physiographic region defined by topography. The Great Plains ecoregion includes five sub-regions: Temperate Prairies, West-Central Semi-Arid Prairies, South-Central Semi-Arid Prairies, Texas Louisiana Coastal Plains, and Tamaulipas-Texas Semi-Arid Plain, which overlap or expand upon other Great Plains designations.


title: Great Plains

text: The region is about east to west and north to south. Much of the region was home to American bison herds until they were hunted to near extinction during the mid/late-19th century. It has an area of approximately . Current thinking regarding the geographic boundaries of the Great Plains is shown by this map at the Center for Great Plains Studies, University of Nebraska–Lincoln. This definition, however, is primarily ecological, not physiographic. The Boreal Plains of Western Canada are physiographically the same, but differentiated by their tundra and forest (rather than grassland) appearance.


title: Great Plains

text: The term "Great Plains", for the region west of about the 96th and east of the Rocky Mountains, was not generally used before the early 20th century. Nevin Fenneman's 1916 study "Physiographic Subdivision of the United States" brought the term Great Plains into more widespread usage. Before that the region was almost invariably called the High Plains, in contrast to the lower Prairie Plains of the Midwestern states. Today the term "High Plains" is used for a subregion of the Great Plains. The term still remains little-used in Canada compared to the more common, "prairie".


title: Great Plains

text: The Great Plains are the westernmost portion of the vast North American Interior Plains, which extend east to the Appalachian Plateau. The United States Geological Survey divides the Great Plains in the United States into ten physiographic subdivisions:


title: Great Plains

text: Further to this can be added Canadian physiographic sub-regions such as the Alberta Plain, Cypress Hills, Manitoba Escarpment (eastward), Manitoba Plain, Missouri Coteau (shared), Rocky Mountain Foothills (eastward), and Saskatchewan Plain.


title: Great Plains

text: The Great Plains consist of a broad stretch of country underlain by nearly horizontal strata extending westward from the 97th meridian west to the base of the Rocky Mountains, a distance of . It extends northward from the Mexican boundary far into Canada. Although the altitude of the plains increases gradually from 600 or on the east to 4,000–5,000 or near the mountains, the local relief is generally small. The semi-arid climate excludes tree growth and opens far-reaching views.


title: Great Plains

text: The plains are by no means a simple unit. They are of diverse structure and of various stages of erosional development. They are occasionally interrupted by buttes and escarpments. They are frequently broken by valleys. Yet on the whole, a broadly extended surface of moderate relief so often prevails that the name, Great Plains, for the region as a whole is well-deserved.


title: Great Plains

text: The western boundary of the plains is usually well-defined by the abrupt ascent of the mountains. The eastern boundary of the plains (in the United States) is more climatic than topographic. The line of of annual rainfall trends a little east of northward near the 97th meridian. If a boundary must be drawn where nature presents only a gradual transition, this rainfall line may be taken to divide the drier plains from the moister prairies. However, in Canada the eastern boundary of the plains is well defined by the presence of the Canadian Shield to the northeast.


title: Great Plains

text: The plains (within the United States) may be described in northern, intermediate, central and southern sections, in relation to certain peculiar features. In Canada, no such division is used: the climatic and vegetation regions are more impactful on human settlement than mere topography, and therefore the region is split into (from north to south), the taiga plains, boreal plains, aspen parkland, and prairie ecoregion regions.


title: Great Plains

text: The northern section of the Great Plains, north of latitude 44°, includes eastern Montana, eastern Wyoming, most of North Dakota and South Dakota, southwestern Minnesota and portions of the Canadian provinces including southeastern Alberta, southern Saskatchewan and southwestern Manitoba. The strata here are Cretaceous or early Tertiary, lying nearly horizontal. The surface is shown to be a plain of degradation by a gradual ascent here and there to the crest of a ragged escarpment, the escarpment-remnant of a resistant stratum. There are also the occasional lava-capped mesas and dike formed ridges, surmounting the general level by or more and manifestly demonstrating the widespread erosion of the surrounding plains. All these reliefs are more plentiful towards the mountains in central Montana. The peneplain is no longer in the cycle of erosion that witnessed its production. It appears to have suffered a regional uplift or increase in elevation, for the upper Missouri River and its branches no longer flow on the surface of the plain, but in well graded, maturely opened valleys, several hundred feet below the general level. A significant exception to the rule of mature valleys occurs, however, in the case of the Missouri, the largest river, which is broken by several falls on hard sandstones about east of the mountains. This peculiar feature is explained as the result of displacement of the river from a better graded preglacial valley by the Pleistocene ice sheet. Here, the ice sheet overspread the plains from the moderately elevated Canadian highlands far on the north-east, instead of from the much higher mountains nearby on the west. The present altitude of the plains near the mountain base is .


title: Great Plains

text: The northern plains are interrupted by several small mountain areas. The Black Hills, chiefly in western South Dakota, are the largest group. They rise like a large island from the sea, occupying an oval area of about north-south by east-west. At Black Elk Peak, they reach an altitude of and have an effective relief over the plains of 2000 or This mountain mass is of flat-arched, dome-like structure, now well dissected by radiating consequent streams. The weaker uppermost strata have been eroded down to the level of the plains where their upturned edges are evenly truncated. The next following harder strata have been sufficiently eroded to disclose the core of underlying igneous and metamorphic crystalline rocks in about half of the domed area.


title: Great Plains

text: In the intermediate section of the plains, between latitudes 44° and 42°, including southern South Dakota and northern Nebraska, the erosion of certain large districts is peculiarly elaborate. Known as the Badlands, it is a minutely dissected form with a relief of a few hundred feet. This is due to several causes:


title: Great Plains

text: The central section of the Great Plains, between latitudes 42° and 36°, occupying eastern Colorado and western Kansas, is mostly a dissected fluviatile plain. That is, this section was once smoothly covered with a gently sloping plain of gravel and sand that had been spread far forward on a broad denuded area as a piedmont deposit by the rivers which issued from the mountains. Since then, it has been more or less dissected by the erosion of valleys. The central section of the plains thus presents a marked contrast to the northern section.


title: Great Plains

text: While the northern section owes its smoothness to the removal of local gravels and sands from a formerly uneven surface by the action of degrading rivers and their inflowing tributaries, the southern section owes its smoothness to the deposition of imported gravels and sands upon a previously uneven surface by the action of aggrading rivers and their outgoing distributaries. The two sections are also alike in that residual eminences still here and there surmount the peneplain of the northern section, while the fluviatile plain of the central section completely buried the pre-existent relief. An exception to this statement must be made for the southwest, close to the mountains in southern Colorado, where some lava-capped mesas (Mesa de Maya, Raton Mesa) stand several thousand feet above the general plain level, and thus testify to the widespread erosion of this region before it was aggraded.


title: Great Plains

text: The southern section of the Great Plains, between latitudes 35.5° and 25.5°, lies in western Texas, eastern New Mexico, and western Oklahoma. Like the central section, it is for the most part a dissected fluviatile plain. However, the lower lands which surround it on all sides place it in such strong relief that it stands up as a table-land, known from the time of Mexican occupation as the Llano Estacado. It measures roughly east-west and north-south. It is of very irregular outline, narrowing to the south. Its altitude is at the highest western point, nearest the mountains whence its gravels were supplied. From there, it slopes southeastward at a decreasing rate, first about , then about 7 ft per mile (1.3 m/km), to its eastern and southern borders, where it is in altitude. Like the High Plains farther north, it is extraordinarily smooth.


title: Great Plains

text: It is very dry, except for occasional shallow and temporary water sheets after rains. Llano is separated from the plains on the north by the mature consequent valley of the Canadian River, and from the mountains on the west by the broad and probably mature valley of the Pecos River. On the east, it is strongly undercut by the retrogressive erosion of the headwaters of the Red, Brazos, and Colorado rivers of Texas and presents a ragged escarpment approximately high, overlooking the central denuded area of that state. There, between the Brazos and Colorado rivers, occurs a series of isolated outliers capped by limestone that underlies both the Llano Uplift on the west and the Grand Prairies escarpment on the east. The southern and narrow part of the table-land, called the Edwards Plateau, is more dissected than the rest, and falls off to the south in a frayed-out fault scarp. This scarp overlooks the coastal plain of the Rio Grande embayment. The central denuded area, east of the Llano, resembles the east-central section of the plains in exposing older rocks. Between these two similar areas, in the space limited by the Canadian and Red Rivers, rise the subdued forms of the Wichita Mountains in Oklahoma, the westernmost member of the Ouachita system.


title: Great Plains

text: In general, the Great Plains have a wide range of weather, with very cold and harsh winters and very hot and humid summers. Wind speeds are often very high, especially in winter.


title: Great Plains

text: The 100th meridian roughly corresponds with the line that divides the Great Plains into an area that receives or more of rainfall per year and an area that receives less than . In this context, the High Plains, as well as Southern Alberta, south-western Saskatchewan and Eastern Montana are mainly semi arid steppe land and are generally characterised by rangeland or marginal farmland. The region (especially the High Plains) is periodically subjected to extended periods of drought; high winds in the region may then generate devastating dust storms. The eastern Great Plains near the eastern boundary falls in the humid subtropical climate zone in the southern areas, and the northern and central areas fall in the humid continental climate.


title: Great Plains

text: Many thunderstorms occur in the plains in the spring through summer. The southeastern portion of the Great Plains is the most tornado active area in the world and is sometimes referred to as Tornado Alley.


title: Great Plains

text: The Great Plains are part of the floristic North American Prairies Province, which extends from the Rocky Mountains to the Appalachians.


title: Great Plains

text: Mammals: Although the American bison ("Bison bison") historically ranged throughout much of North America (from New York to Oregon and Canada to northern Mexico), they are strongly associated with the Great Plains where they once roamed in immense herds. Pronghorn ("Antilocapra americana") range into western areas of the region. The black-tailed prairie dog ("Cynomys ludovicianus") is another iconic species among several rodents that are linked to the region including the thirteen-lined ground squirrel ("Ictidomys tridecemlineatus"), spotted ground squirrel ("Xerospermophilus spilosoma"), Franklin's ground squirrel ("Poliocitellus franklinii"), plains pocket gopher ("Geomys bursarius"), hispid pocket mouse ("Chaetodipus hispidus"), olive-backed pocket mouse ("Perognathus fasciatus"), plains pocket mouse ("Perognathus flavescens"), and plains harvest mouse ("Reithrodontomys montanus"), Two carnivores associated with the Great Plains include the swift fox ("Vulpes velox") and the endangered black-footed ferret ("Mustela nigripes").


title: Great Plains

text: Birds: The lesser prairie chicken ("Tympanuchus pallidicinctus") is endemic to the Great Plains and the distribution of the greater prairie chicken ("Tympanuchus cupido") predominantly occurs in the region, although the latter historically ranged further eastward. The Harris's sparrow ("Zonotrichia querula") spends winter months in southern areas of the region. Other species migrate from the south in the spring and spend their breeding season on the plains, including the white-faced ibis ("Plegadis chihi"), mountain plover ("Charadrius montanus"), marbled godwit ("Limosa fedoa"), Sprague's pipit ("Anthus spragueii"), Cassin's sparrow ("Peucaea cassinii"), Baird's sparrow ("Centronyx bairdii"), lark bunting ("Calamospiza melanocorys"), chestnut-collared longspur ("Calcarius ornatus"), thick-billed longspur or McCown's longspur ("Rhynchophanes mccownii"), and dickcissel ("Spiza americana").


title: Great Plains

text: Reptiles: The prairie rattlesnake ("Crotalus viridis") ranges throughout much of the Great Plains and into the valleys and lower elevations of the eastern Rocky Mountains and portions of the American southwest. Other snakes include the plains hog-nosed snake ("Heterodon nasicus"), western milksnake ("Lampropeltis gentilis"), great plains ratsnake ("Pantherophis emoryi"), bullsnake ("Pituophis catenifer sayi"), plains black-headed snake ("Tantilla nigriceps"), plains gartersnake ("Thamnophis radix"), and lined snake ("Tropidoclonion lineatum"). Reptile diversity increases significantly in southern regions of the Great Plains. The ornate box turtle ("Terrapene ornata") and great plains skink ("Plestiodon obsoletus") occur in southern areas.


title: Great Plains

text: Amphibians: Although few salamanders are strongly associated with region, the western tiger salamander ("Ambystoma mavortium") ranges through much of the Great Plains and the Rocky Mountains, as does the rocky mountain toad ("Anaxyrus w. woodhousi"). Other anurans related to region include the great plains toad ("Anaxyrus cognatus"), plains leopard frog ("Lithobates blairi"), and plains spadefoot ("Spea bombifrons").


title: Great Plains

text: Fish: Some species predominately associated with various river basins in the Great Plains include sturgeon chub ("Macrhybopsis gelida"), peppered chub ("Macrhybopsis tetranema"), prairie chub ("Macrhybopsis australis"), western silvery minnow ("Hybognathus argyritis"), plains minnow ("Hybognathus placitus"), smalleye shiner ("Notropis buccula"), Arkansas River shiner ("Notropis girardi"), Red River shiner ("Notropis bairdi"), Topeka shiner ("Notropis topeka"), plains topminnow ("Fundulus sciadicus"), plains killifish ("Fundulus zebrinus"), Red River pupfish ("Cyprinodon rubrofluviatilis"), and Arkansas darter ("Etheostoma cragini").


title: Great Plains

text: During the Cretaceous Period (145–66 million years ago), the Great Plains were covered by a shallow inland sea called the Western Interior Seaway. However, during the Late Cretaceous to the Paleocene (65–55 million years ago), the seaway had begun to recede, leaving behind thick marine deposits and a relatively flat terrain which the seaway had once occupied.


title: Great Plains

text: During the Cenozoic era, specifically about 25 million years ago during the Miocene and Pliocene epochs, the continental climate became favorable to the evolution of grasslands. Existing forest biomes declined and grasslands became much more widespread. The grasslands provided a new niche for mammals, including many ungulates and glires, that switched from browsing diets to grazing diets. Traditionally, the spread of grasslands and the development of grazers have been strongly linked. However, an examination of mammalian teeth suggests that it is the open, gritty habitat and not the grass itself which is linked to diet changes in mammals, giving rise to the "grit, not grass" hypothesis.


title: Great Plains

text: Paleontological finds in the area have yielded bones of mammoths, saber-toothed cats and other ancient animals, as well as dozens of other megafauna (large animals over ) – such as giant sloths, horses, mastodons, and American lion – that dominated the area of the ancient Great Plains for thousands to millions of years. The vast majority of these animals became extinct in North America at the end of the Pleistocene (around 13,000 years ago).


title: Great Plains

text: A number of significant fossil sites are located in the Great Plains including Agate Fossil Beds National Monument (Nebraska), Ashfall Fossil Beds (Nebraska), Clayton Lake State Park (New Mexico), Dinosaur Valley State Park (Texas), Hudson-Meng Bison Kill (Nebraska), Makoshika State Park (Montana), and The Mammoth Site (South Dakota).


title: Great Plains

text: Public and protected lands in the Great Plains include National Parks and National Monuments, administers by the National Park Service with the responsibility of preserving ecological and historical places and making them available to the public. The U.S. Fish & Wildlife Service manages the National Wildlife Refuges, with the primary responsibility of conserving and protecting fish, wildlife, plants, and habitat in the public trust. Both are agencies of the Department of the Interior.


title: Great Plains

text: In contrast, U.S. Forest Service, an agency of the U. S. Department of Agriculture, administers the National Forests and National Grasslands, under a multiple-use concept. By law, the U.S. Forest Service must consider all resources, with no single resource emphasized to the detriment of others, including water, soil, grazing, timber harvesting, and minerals (mining and drilling), as well as recreation and conservation of fish and wildlife. Each individual state also administers state lands, typically smaller areas, for various purposes including conservation and recreation.


title: Great Plains

text: Grasslands are among the least protected biomes. Humans have converted much of the prairies for agricultural purposes or to create pastures. Several of the protected lands in the region are centered around aberrant and uncharacteristic features of the region, such as mountains, outcrops, and canyons (e.g. Devil's Tower National Monument, Wind Cave National Park, Scotts Bluff National Monument), and as splendid and worthy as they are, they are not primarily focused on conserving the plains and prairies.


title: Great Plains

text: The first Peoples (Paleo-Indians) arrived on the Great Plains thousands of years ago. Historically, the Great Plains were the range of the Blackfoot, Crow, Sioux, Cheyenne, Arapaho, Comanche, and others. Eastern portions of the Great Plains were inhabited by tribes who lived at Etzanoa and in semi-permanent villages of earth lodges, such as the Arikara, Mandan, Pawnee, and Wichita. The introduction of corn around 800 CE allowed the development of the mound-building Mississippian culture along rivers that crossed the Great Plains and that included trade networks west to the Rocky Mountains. Mississippians settled the Great Plains at sites now in Oklahoma and South Dakota.


title: Great Plains

text: Siouan language speakers may have originated in the lower Mississippi River region. They were agriculturalists and may have been part of the Mound Builder civilization during the 9th–12th centuries. Wars with the Ojibwe and Cree peoples pushed the Lakota (Teton Sioux) west onto the Great Plains in the mid- to late-17th century. The Shoshone originated in the western Great Basin and spread north and east into present-day Idaho and Wyoming. By 1500, some Eastern Shoshone had crossed the Rocky Mountains into the Great Plains. After 1750, warfare and pressure from the Blackfoot, Crow, Lakota, Cheyenne, and Arapaho pushed Eastern Shoshone south and westward. Some of them moved as far south as Texas, emerging as the Comanche by 1700.


title: Great Plains

text: The first known contact between Europeans and Indians in the Great Plains occurred in what is now Texas, Kansas, and Nebraska from 1540 to 1542 with the arrival of Francisco Vázquez de Coronado, a Spanish conquistador. In that same period, Hernando de Soto crossed a west-northwest direction in what is now Oklahoma and Texas which is now known as the De Soto Trail. The Spanish thought that the Great Plains were the location of the mythological "Quivira and Cíbola", a place said to be rich in gold.


title: Great Plains

text: People in the southwest began to acquire horses in the 16th century by trading or stealing them from Spanish colonists in New Mexico. As horse culture moved northward, the Comanche were among the first to commit to a fully mounted nomadic lifestyle. This occurred by the 1730s, when they had acquired enough horses to put all their people on horseback.


title: Great Plains

text: The real beginning of the horse culture of the plains began with the Pueblo Revolt of 1680 in New Mexico and the capture of thousands of horses and other livestock. In 1683 a Spanish expedition into Texas found horses among Native people. In 1690, a few horses were found by the Spanish among the Indians living at the mouth of the Colorado River of Texas and the Caddo of eastern Texas had a sizeable number.


title: Great Plains

text: The French explorer Claude Charles Du Tisne found 300 horses among the Wichita on the Verdigris River in 1719, but they were still not plentiful. Another Frenchman, Bourgmont, could only buy seven at a high price from the Kaw in 1724, indicating that horses were still scarce among tribes in Kansas. By 1770, that Plains Indians culture was mature, consisting of mounted buffalo-hunting nomads from Saskatchewan and Alberta southward nearly to the Rio Grande.


title: Great Plains

text: The milder winters of the southern Plains favored a pastoral economy by the Indians. On the northeastern Plains of Canada, the Indians were less favored, with families owning fewer horses, remaining more dependent upon dogs for transporting goods, and hunting bison on foot. The scarcity of horses in the north encouraged raiding and warfare in competition for the relatively small number of horses that survived the severe winters.


title: Great Plains

text: Comanche power peaked in the 1840s when they conducted large-scale raids hundreds of miles into Mexico proper, while also warring against the Anglo-Americans and Tejanos who had settled in independent Texas.


title: Great Plains

text: The fur trade brought thousands of colonial settlers into the Great Plains over the next 100 years. Fur trappers made their way across much of the region, making regular contacts with Indians. The United States acquired the Louisiana Purchase in 1803 and conducted the Lewis and Clark Expedition in 1804–1806, and more information became available concerning the Plains, and various pioneers entered the areas. Fur trading posts were often the basis of later settlements. Through the 19th century, more settlers migrated to the Great Plains as part of a vast westward expansion of population, and new settlements became dotted across the Great Plains.


title: Great Plains

text: The settlers also brought diseases against which the Indians had no resistance. Between a half and two-thirds of the Plains Indians are thought to have died of smallpox by the time of the Louisiana Purchase. The 1837 Great Plains smallpox epidemic spread across the Great Plains, killing many thousands between 1837 and 1840. In the end, it is estimated that two-thirds of the Blackfoot population died, along with half of the Assiniboines and Arikaras, a third of the Crows, and a quarter of the Pawnees.


title: Great Plains

text: Beginning in 1821, the Santa Fe Trail ran from the Missouri River to New Mexico, skirting north of Comancheria. Beginning in the 1830s, the Oregon Trail led from the Missouri River across the Great Plains.


title: Great Plains

text: After 1870, the new railroads across the Plains brought hunters who killed off almost all the bison for their hides. The railroads offered attractive packages of land and transportation to American farmers, who rushed to settle the land. They also took advantage of the homestead laws to obtain farms. Land speculators and local boosters identified many potential towns, and those reached by the railroad had a chance, while the others became ghost towns. Towns flourished if they were favored by proximity to the railroad.


title: Great Plains

text: Much of the Great Plains became open range where cattle roamed free, hosting ranching operations where anyone was free to run cattle. In the spring and fall, ranchers held roundups where their cowboys branded new calves, treated animals, and sorted the cattle for sale. Such ranching began in Texas and gradually moved northward. Between 1866 and 1895, cowboys herded 10 million cattle north to rail heads such as Dodge City, Kansas and Ogallala, Nebraska; from there, cattle were shipped east.


title: Great Plains

text: The U.S. passed the Homestead Acts of 1862 to encourage agricultural development of the Great Plains and house a growing population. It allowed a settler to claim up to of land, provided that he lived on it for a period of five years and cultivated it. The provisions were expanded under the Kinkaid Act of 1904 to include a homestead of an entire section. Hundreds of thousands of people claimed such homesteads, sometimes building houses out of the very turf of the land. Many of them were not skilled farmers, and failures were frequent. The Dominion Lands Act of 1871 served a similar function for establishing homesteads on the prairies in Canada.


title: Great Plains

text: The railroads opened up the Great Plains for settlement, making it possible to ship wheat and other crops at low cost to the urban markets in the East and overseas. Homestead land was free for American settlers. Railroads sold their land at cheap rates to immigrants in the expectation that they would generate traffic as soon as farms were established. Immigrants poured in, especially from Germany and Scandinavia. On the plains, very few single men attempted to operate a farm or ranch by themselves; they understood the need for a hard-working wife and numerous children to handle the many responsibilities. During the early years of settlement, farm women played an integral role in assuring family survival by working outdoors. After approximately one generation, women increasingly left the fields, thus redefining their roles within the family. New technology encouraged women to turn to domestic roles, including sewing and washing machines. Media and government extension agents promoted the "scientific housekeeping" movement, along with county fairs which featured achievements in home cookery and canning, advice columns for women regarding farm book keeping, and home economics courses in the schools.


title: Great Plains

text: The eastern image of farm life in the prairies emphasized the isolation of the lonely farmer and wife, yet plains residents created busy social lives for themselves. They often sponsored activities which combined work, food, and entertainment, such as barn raisings, corn huskings, quilting bees, Grange meetings, church activities and school functions. Women organized shared meals and potluck events, as well as extended visits among families.


title: Great Plains

text: The region roughly centered on the Oklahoma Panhandle was known as the Dust Bowl during the late 1920s and early 1930s, including southeastern Colorado, southwestern Kansas, the Texas Panhandle, and extreme northeastern New Mexico. The effects of an extended drought, inappropriate cultivation, and financial crises of the Great Depression forced many farmers off the land throughout the Great Plains.


title: Great Plains

text: From the 1950s on, many areas of the Great Plains have become productive crop-growing areas because of extensive irrigation on large land-holdings. The United States is a major exporter of agricultural products. The southern portion of the Great Plains lies over the Ogallala Aquifer, a huge underground layer of water-bearing strata. Center pivot irrigation is used extensively in drier sections of the Great Plains, resulting in aquifer depletion at a rate that is greater than the ground's ability to recharge.


title: Great Plains

text: The rural Plains have lost a third of their population since 1920. Several hundred thousand square miles of the Great Plains have fewer than , the density standard that Frederick Jackson Turner used to declare the American frontier "closed" in 1893. Many have fewer than . There are more than 6,000 ghost towns in Kansas alone, according to Kansas historian Daniel Fitzgerald. This problem is often exacerbated by the consolidation of farms and the difficulty of attracting modern industry to the region. In addition, the smaller school-age population has forced the consolidation of school districts and the closure of high schools in some communities. The continuing population loss has led some to suggest that the current use of the drier parts of the Great Plains is not sustainable, and there has been a proposal to return approximately of these drier parts to native prairie land as a Buffalo Commons.


title: Great Plains

text: The Great Plains contributes substantially to wind power in the United States. T. Boone Pickens developed wind farms after a career as a petroleum executive, and he called for the U.S. to invest $1 trillion to build an additional 200,000 MW of wind power in the Plains as part of his Pickens Plan. He cited Sweetwater, Texas, as an example of economic revitalization driven by wind power development.


title: Woman on top

text: Woman on top is any sex position in which the woman is on top of her sexual partner during sexual activity. The position most commonly associated with the woman on top is often called the cowgirl or riding position, which derives its name from the image of the woman "riding" the man as a cowgirl rides a bucking horse. In that position, a man typically lies on his back with his legs closed, while the female partner straddles him, usually in a kneeling position facing either forward or back, and either the man or woman inserts the man's erect penis into the woman's vagina or anus. The cowgirl position is commonly cited as one of the more popular sex positions, especially by women, because it gives them control over the rhythm and pace of vaginal stimulation and the extent and duration of penetration and because of its ability to adequately stimulate the clitoris. There are other positions in which the woman may be on top, including the 69 position and the pompoir sex position.


title: Woman on top

text: In any of the woman on top positions, the woman is usually the active partner during sexual activity, and in addition to satisfying herself can stimulate or massage the man's scrotum, especially if his legs are spread. The woman may take the position in the course of sexual activity, especially if the man has attained orgasm while she has not; and in this position she or her partner may perform fingering for her to achieve orgasm.


title: Woman on top

text: Initially, the man typically lies on his back with his legs together, while the female partner straddles him in a kneeling position. With the woman's legs open, either the man or woman would insert the man's erect penis into the woman's vagina. Once the woman has attained sufficient vaginal lubrication, she feels comfortable in thrusting up and down the penis and is in a position to vary her position for variety and enhanced pleasure. She can, for example, tilt her body forward toward a full lying position or backwards, or rock sideways. The woman has control over the rhythm and speed of her thrusting. She can also spread the man's legs and position herself between them. In this position, she can tighten her legs at will, to increase the pressure on the man's penis and on her crotch, and the man can also tighten his legs to increase the pressure. In this position, the man's legs are free to raise, as high as the Amazon position, with his legs rolled up towards her chest, but this also should be done with care. These changes can alter the depth and angle of penetration.


title: Woman on top

text: The woman may also be on top in a precursor to the lateral coital position. In this position, the woman will usually prefer to kneel, as a squat will usually place a significantly greater stress on her legs.


title: Woman on top

text: In the cowgirl position, the man lies on his back or sits, with the woman straddling him across his pelvis facing forward, either in a kneeling or squatting position. The woman will align her vagina with the man's erect penis, and lower herself onto the penis with either the man or the woman guiding the penis into the woman's vagina for penetration. With the erect penis inside the woman, the woman can glide up and down the man's penis, controlling the rhythm and pace of vaginal stimulation and the extent and duration of penetration. The woman is free to change her position; possibilities range from leaning back to stretching out on her partner's chest to rocking from side to side or in a circular motion. Each of those changes would alter the angle and depth of penetration and which part of the vaginal area and sex organs are stimulated. The man's hands are free to reach and touch the woman's breasts, clitoris, buttocks, face etc., and the partners can maintain eye contact. With the woman in the sitting or leaning back positions, the man may also stimulate her nipples and clitoris with his fingers, or suck on her nipples if she inclines towards him or he raises his torso towards her.


title: Woman on top

text: In this position, a woman can continue with the up and down motion after the man has achieved orgasm until she reaches orgasm. Afterwards, she may collapse onto the man's chest, with her vagina being very sensitive to further stimulation. On the other hand, if the woman reaches orgasm before the man, the woman may raise her pelvis to enable the man to actively thrust upwards or the couple may change to a position which enables the man to continue thrusting, such as the missionary position.


title: Woman on top

text: When the woman lies on her partner's chest, this is also called the reverse missionary position. Stretching out with her legs on the man puts more pressure on the woman's mons pubis and clitoris. If the man separates his legs, the woman can put her legs between his. In this position, she can increase the tightness on the man's penis by pressing her thighs together, or the man may press the woman's thighs together. This increases vaginal friction, but she can still control the pace and rhythm of her thrusts. As most women approach orgasm, they experience acute vaginal contractions, which also increases the man's sexual stimulation by increasing tightness on his penis.


title: Woman on top

text: In the reverse cowgirl position, the woman straddles the man facing the man's feet in a kneeling or squatting position. After either she or the man has guided the penis into the woman's vagina, the woman can keep her torso upright, or she can lean forward or back to lie on the man's chest or move sideways at will. The man's legs can be flat on the bed or arched. The woman also has control over the pace, rhythm, depth, and duration of penetration. By changing her position and thrusts, the woman can change the level of stimulation for the man, while simultaneously stimulating her clitoris or an area termed "the G-spot".


title: Woman on top

text: An advantage of the woman-on-top positions is that the man's weight is not on the woman. These positions are ideal during pregnancy as there is no pressure on the woman's abdomen, and it is easy for her to control the depth and speed of penetration. They are also useful if the man is recovering from a serious illness or surgery or if the woman has recently given birth.


title: Woman on top

text: Sex with the woman on top has been known in most cultures in all times, and has been depicted in art:


title: Woman on top

text: Among adults, a female sitting on the lap of another person usually indicates an intimate or romantic relationship between the two, or may be regarded as flirtatious. In public, such behaviour may be regarded as a public display of affection. In private, sitting on a man's lap or being pulled by the man onto his lap, especially if she straddles his legs or waist, when accompanied with a smile, kiss or other flirtatious gesture, may be regarded as showing either's interest in sexual activity or as foreplay. It is rare for a man to sit on a woman's lap, though not for him to lay his head there.


title: Woman on top

text: The erotic significance in sitting on a lap is a factor in the erotic activity in strip clubs such as a lap dance, where one person (usually a woman) straddles the lap of the other and gyrates their lower extremities in a provocative manner.


title: Aron Ralston

text: Aron Lee Ralston (born October 27, 1975) is an American mountaineer, mechanical engineer, and motivational speaker, known for surviving a canyoneering accident by cutting off part of his right arm.


title: Aron Ralston

text: On April 26, 2003, during a solo descent of Bluejohn Canyon in southeastern Utah, he dislodged a boulder, pinning his right wrist to the side of the canyon wall. After five days, he had to break his forearm in order to amputate it with a dull pocket knife to break free, make his way through the rest of the canyon, rappel down a drop, and hike to safety.


title: Aron Ralston

text: The incident is documented in Ralston's autobiography "Between a Rock and a Hard Place" and is the subject of the 2010 film "127 Hours" where he is portrayed by James Franco.


title: Aron Ralston

text: After the accident he continued mountaineering and became the first person to ascend all of Colorado's fourteeners solo in winter.


title: Aron Ralston

text: Aron Ralston was born on October 27, 1975, in Marion, Ohio. He and his family moved to Denver when he was 12, where he attended Cherry Creek High School and learned to ski and backpack. He received his college degree from Carnegie Mellon University in Pittsburgh, finishing with degrees in mechanical engineering and French, with a minor in piano. At Carnegie Mellon, he served as a resident assistant, studied abroad, and was an active intramural sports participant. He also worked as a rafting guide during the summer.


title: Aron Ralston

text: Ralston worked as a mechanical engineer with Intel in Ocotillo, Tacoma and Albuquerque for five years, but found himself burned out by working in a large corporation. During his time as an engineer he had built up skills in mountaineering, and in 2002 he quit in order to climb Denali. He moved to Aspen, Colorado in order to pursue a life of climbing mountains.


title: Aron Ralston

text: Ralston began working towards his goal of climbing all of Colorado's "fourteeners" — peaks over altitude, of which there are 59 — solo and during winter (a feat that had never been recorded before). In 2003, he was caught in a Grade 5 avalanche on Resolution Peak, Colorado with his skiing partners Mark Beverly and Chadwick Spencer. No one was seriously injured, but his friends did not speak to him in the aftermath, causing him to reevaluate his approach to risk management.


title: Aron Ralston

text: On April 26, 2003, Aron Ralston was canyoneering alone through Bluejohn Canyon, in eastern Wayne County, Utah, just south of the Horseshoe Canyon unit of Canyonlands National Park. While he was descending the lower stretches of the slot canyon, a suspended boulder dislodged while he was climbing down from it. The boulder first smashed his left hand, and then crushed his right hand against the canyon wall. Ralston had not informed anyone of his hiking plans, nor did he have any way to call for help.