title: Frank Costello

text: From May 1950 to May 1951, the U.S. Senate conducted a large-scale investigation of organized crime, commonly known as the Kefauver Hearings, chaired by Senator Estes Kefauver of Tennessee. Costello was convicted of contempt of the Senate and sentenced to 18 months in prison. Senator Kefauver concluded that politician Carmine DeSapio was assisting the activities of Costello, and that Costello had become influential in decisions made by the Tammany Hall council. DeSapio admitted to having met Costello several times, but insisted that "politics was never discussed". In 1952, the government began proceedings to strip Costello of his U.S. citizenship, and he was indicted for evasion of $73,417 in income taxes between 1946 and 1949. He was sentenced to five years in prison and fined $20,000. In 1954, Costello appealed the conviction and was released on $50,000 bail; from 1952 to 1961 he was in and out of half a dozen federal and local prisons and jails, his confinement interrupted by periods when he was out on bail pending determination of appeals.


title: Frank Costello

text: In 1956, Adonis, a powerful Costello ally, chose deportation to Italy over a long prison sentence. His departure left Costello weakened, but Genovese still had to neutralize one more powerful Costello ally, Anastasia, who had taken over the Mangano crime family after the disappearance of boss Vincent Mangano and the murder of brother Philip Mangano on April 14, 1951.


title: Frank Costello

text: In early 1957, Genovese decided to move on Costello. Genovese ordered Vincent Gigante to murder Costello, and on May 2, 1957, Gigante shot and wounded Costello outside his apartment building. The altercation persuaded Costello to relinquish power to Genovese and retire. Genovese then controlled what is now called the Genovese crime family. A doorman identified Gigante as the gunman, but in 1958 Costello testified that he was unable to recognize his assailant; Gigante was acquitted of attempted murder.


title: Frank Costello

text: On October 25, 1957, Anastasia was murdered at the barber shop of the Park Sheraton Hotel at 56th Street and 7th Avenue in Manhattan. Carlo Gambino was expected to be proclaimed boss of Anastasia's family at the November 14, 1957, Apalachin Meeting. Genovese called to discuss the future of "Cosa Nostra" in light of his takeover. When police raided the meeting, to the detriment of Genovese's reputation, Gambino's appointment was postponed to a later meeting in New York City. In 1959, Genovese was convicted of selling a large quantity of heroin. On April 17, 1959, Genovese was sentenced to 15 years in the Atlanta Federal Penitentiary in Atlanta, Georgia.


title: Frank Costello

text: During his retirement, Costello was still known as "The Prime Minister of the Underworld". He still retained power and influence in New York's Mafia and remained busy throughout his final years. "Cosa Nostra" bosses and old associates such as Gambino and Lucchese still paid visits to Costello at his Waldorf Astoria penthouse, seeking advice on important Mafia affairs. Costello's old friend Meyer Lansky also kept in touch. Costello occupied himself with gardening and displayed some of his flowers at local horticultural shows.


title: Frank Costello

text: On February 20, 1961, the United States Supreme Court upheld a lower court order that stripped Costello of his US citizenship. But on February 17, 1964, the same court set aside a deportation order for Costello, citing a legal technicality.


title: Frank Costello

text: In early February 1973, Costello suffered a heart attack at his Manhattan home and was rushed to Doctors Hospital in Manhattan, where he died on February 18. Costello's sedate memorial service at a Manhattan funeral home was attended by 50 relatives, friends, and law enforcement agents. Costello is interred in a private mausoleum in St. Michael's Cemetery in East Elmhurst, Queens. In 1974, after his enemy Carmine Galante was released from prison, he allegedly ordered the bombing of the doors to Costello's mausoleum.


title: Equinox

text: A solar equinox is a moment in time when the Sun crosses the Earth's equator, which is to say, appears directly above the equator, rather than north or south of the equator. On the day of the equinox, the Sun appears to rise "due east" and set "due west". This occurs twice each year, around 20 March and 23 September.


title: Equinox

text: More precisely, an equinox is traditionally defined as the time when the plane of Earth's equator passes through the geometric center of the Sun's disk. Equivalently, this is the moment when Earth's rotation axis is directly perpendicular to the Sun-Earth line, tilting neither toward nor away from the Sun. In modern times, since the Moon (and to a lesser extent the planets) causes Earth's orbit to vary slightly from a perfect ellipse, the equinox is officially defined by the Sun's more regular ecliptic longitude rather than by its declination. The instants of the equinoxes are currently defined to be when the apparent geocentric longitude of the Sun is 0° and 180°.


title: Equinox

text: The word is derived from the Latin ', from ' (equal) and ' (genitive ') (night). On the day of an equinox, daytime and nighttime are of approximately equal duration all over the planet. They are not exactly equal, however, because of the angular size of the Sun, atmospheric refraction, and the rapidly changing duration of the length of day that occurs at most latitudes around the equinoxes. Long before conceiving this equality, primitive equatorial cultures noted the day when the Sun rises due east and sets due west, and indeed this happens on the day closest to the astronomically defined event. As a consequence, according to a properly constructed and aligned sundial, the daytime duration is 12 hours.


title: Equinox

text: In the Northern Hemisphere, the March equinox is called the vernal or spring equinox while the September equinox is called the autumnal or fall equinox. In the Southern Hemisphere, the reverse is true. During the year, equinoxes alternate with solstices. Leap years and other factors cause the dates of both events to vary slightly.


title: Equinox

text: Hemisphere-neutral names are "northward equinox" for the March equinox, indicating that at that moment the solar declination is crossing the celestial equator in a northward direction, and "southward equinox" for the September equinox, indicating that at that moment the solar declination is crossing the celestial equator in a southward direction.


title: Equinox

text: Systematically observing the sunrise, people discovered that it occurs between two extreme locations at the horizon and eventually noted the midpoint between the two. Later it was realized that this happens on a day when the duration of the day and the night are practically equal and the word "equinox" comes from Latin "aequus", meaning "equal", and "nox", meaning "night".


title: Equinox

text: In the northern hemisphere, the "vernal equinox" (March) conventionally marks the beginning of spring in most cultures and is considered the start of the New Year in the Assyrian calendar, Hindu, and the Persian or Iranian calendars, while the "autumnal equinox" (September) marks the beginning of autumn. Ancient Greek calendars too had the beginning of the year either at the autumnal or vernal equinox and some at solstices. The Antikythera mechanism predicts the equinoxes and solstices.


title: Equinox

text: The equinoxes are the only times when the solar terminator (the "edge" between night and day) is perpendicular to the equator. As a result, the northern and southern hemispheres are equally illuminated.


title: Equinox

text: For the same reason, this is also the time when the Sun rises for an observer at one of Earth's rotational poles and sets at the other. For a brief period lasting approximately four days, both North and South Poles are in daylight. For example, in 2021 sunrise on the North Pole is 18 March 07:09 UTC, and sunset on the South Pole is 22 March 13:08 UTC. Also in 2021, sunrise on the South Pole is 20 September 16:08 UTC, and sunset on the North Pole is 24 September 22:30 UTC.


title: Equinox

text: In other words, the equinoxes are the only times when the subsolar point is on the equator, meaning that the Sun is exactly overhead at a point on the equatorial line. The subsolar point crosses the equator moving northward at the March equinox and southward at the September equinox.


title: Equinox

text: When Julius Caesar established the Julian calendar in 45 BC, he set 25 March as the date of the spring equinox; this was already the starting day of the year in the Persian and Indian calendars. Because the Julian year is longer than the tropical year by about 11.3 minutes on average (or 1 day in 128 years), the calendar "drifted" with respect to the two equinoxes – so that in 300 AD the spring equinox occurred on about 21 March, and by the 1580s AD it had drifted backwards to 11 March.


title: Equinox

text: This drift induced Pope Gregory XIII to establish the modern Gregorian calendar. The Pope wanted to continue to conform with the edicts of the Council of Nicaea in 325 AD concerning the date of Easter, which means he wanted to move the vernal equinox to the date on which it fell at that time (21 March is the day allocated to it in the Easter table of the Julian calendar), and to maintain it at around that date in the future, which he achieved by reducing the number of leap years from 100 to 97 every 400 years. However, there remained a small residual variation in the date and time of the vernal equinox of about ±27 hours from its mean position, virtually all because the distribution of 24 hour centurial leap-days causes large jumps (see ).


title: Equinox

text: The dates of the equinoxes change progressively during the leap-year cycle, because the Gregorian calendar year is not commensurate with the period of the Earth's revolution about the Sun. It is only after a complete Gregorian leap-year cycle of 400 years that the seasons commence at approximately the same time. In the 21st century the earliest March equinox will be 19 March 2096, while the latest was 21 March 2003. The earliest September equinox will be 21 September 2096 while the latest was 23 September 2003 (Universal Time).


title: Equinox

text: Day is usually defined as the period when sunlight reaches the ground in the absence of local obstacles. On the date of the equinox, the center of the Sun spends a roughly equal amount of time above and below the horizon at every location on the Earth, so night and day are about the same length. Sunrise and sunset can be defined in several ways, but a widespread definition is the time that the top limb of the Sun is level with the horizon. With this definition, the day is longer than the night at the equinoxes:


title: Equinox

text: In sunrise/sunset tables, the atmospheric refraction is assumed to be 34 arcminutes, and the assumed semidiameter (apparent radius) of the Sun is 16 arcminutes. (The apparent radius varies slightly depending on time of year, slightly larger at perihelion in January than aphelion in July, but the difference is comparatively small.) Their combination means that when the upper limb of the Sun is on the visible horizon, its centre is 50 arcminutes below the geometric horizon, which is the intersection with the celestial sphere of a horizontal plane through the eye of the observer.


title: Equinox

text: These effects make the day about 14 minutes longer than the night at the equator and longer still towards the poles. The real equality of day and night only happens in places far enough from the equator to have a seasonal difference in day length of at least 7 minutes, actually occurring a few days towards the winter side of each equinox.


title: Equinox

text: The times of sunset and sunrise vary with the observer's location (longitude and latitude), so the dates when day and night are equal also depend upon the observer's location.


title: Equinox

text: A third correction for the visual observation of a sunrise (or sunset) is the angle between the apparent horizon as seen by an observer and the geometric (or sensible) horizon. This is known as the dip of the horizon and varies from 3 arcminutes for a viewer standing on the sea shore to 160 arcminutes for a mountaineer on Everest. The effect of a larger dip on taller objects (reaching over 2½° of arc on Everest) accounts for the phenomenon of snow on a mountain peak turning gold in the sunlight long before the lower slopes are illuminated.


title: Equinox

text: The date on which the day and night are exactly the same is known as an "equilux"; the neologism, believed to have been coined in the 1980s, achieved more widespread recognition in the 21st century. At the most precise measurements, a true equilux is rare, because the lengths of day and night change more rapidly than any other time of the year around the equinoxes. In the mid-latitudes, daylight increases or decreases by about three minutes per day at the equinoxes, and thus adjacent days and nights only reach within one minute of each other. The date of the closest approximation of the equilux varies slightly by latitude; in the mid-latitudes, it occurs a few days before the spring equinox and after the fall equinox in each respective hemisphere.


title: Equinox

text: In the half-year centered on the June solstice, the Sun rises north of east and sets north of west, which means longer days with shorter nights for the northern hemisphere and shorter days with longer nights for the southern hemisphere. In the half-year centered on the December solstice, the Sun rises south of east and sets south of west and the durations of day and night are reversed.


title: Equinox

text: Also on the day of an equinox, the Sun rises everywhere on Earth (except at the poles) at about 06:00 and sets at about 18:00 (local solar time). These times are not exact for several reasons:


title: Equinox

text: Some of the statements above can be made clearer by picturing the day arc (i.e., the path along which the Sun appears to move across the sky). The pictures show this for every hour on equinox day. In addition, some 'ghost' suns are also indicated below the horizon, up to 18° below it; the Sun in such areas still causes twilight. The depictions presented below can be used for both the northern and the southern hemispheres. The observer is understood to be sitting near the tree on the island depicted in the middle of the ocean; the green arrows give cardinal directions.


title: Equinox

text: The March equinox occurs about when the Sun appears to cross the celestial equator northward. In the Northern Hemisphere, the term "vernal point" is used for the time of this occurrence and for the precise direction in space where the Sun exists at that time. This point is the origin of some celestial coordinate systems, which are usually rooted to an astronomical epoch since it gradually varies (precesses) over time:


title: Equinox

text: The modern definition of equinox is the instant when the Sun's apparent geocentric ecliptic longitude is 0° (northward equinox) or 180° (southward equinox). Note that at that moment, its latitude will not be exactly zero, since Earth is not exactly in the plane of the ecliptic. Its declination will also not be exactly zero, so the scientific definition is slightly different than the traditional one. The "mean" ecliptic is defined by the barycenter of Earth and the Moon combined, to minimize the fact that the orbital inclination of the Moon causes the Earth to wander slightly above and below the ecliptic. See the adjacent diagram.


title: Equinox

text: Because of the precession of the Earth's axis, the position of the vernal point on the celestial sphere changes over time, and the equatorial and the ecliptic coordinate systems change accordingly. Thus when specifying celestial coordinates for an object, one has to specify at what time the vernal point and the celestial equator are taken. That reference time can either be a conventional time (like J2000), or an arbitrary point in time, as for the equinox of date.


title: Equinox

text: The upper culmination of the vernal point is considered the start of the sidereal day for the observer. The hour angle of the vernal point is, by definition, the observer's sidereal time.


title: Equinox

text: Using the current official IAU constellation boundaries – and taking into account the variable precession speed and the rotation of the celestial equator – the equinoxes shift through the constellations as follows (expressed in astronomical year numbering when the etc.):


title: Equinox

text: The equinoxes are sometimes regarded as the start of spring and autumn. A number of traditional harvest festivals are celebrated on the date of the equinoxes.


title: Equinox

text: Religious architecture is often determined by the equinox; the Angkor Wat Equinox during which the sun rises in a perfect alignment over Angkor Wat in Cambodia is one such example.


title: Equinox

text: Catholic churches, since the recommendations of Charles Borromeo, have chosen the equinox as their reference point for the orientation of churches.


title: Equinox

text: One effect of equinoctial periods is the temporary disruption of communications satellites. For all geostationary satellites, there are a few days around the equinox when the Sun goes directly behind the satellite relative to Earth (i.e. within the beam-width of the ground-station antenna) for a short period each day. The Sun's immense power and broad radiation spectrum overload the Earth station's reception circuits with noise and, depending on antenna size and other factors, temporarily disrupt or degrade the circuit. The duration of those effects varies but can range from a few minutes to an hour. (For a given frequency band, a larger antenna has a narrower beam-width and hence experiences shorter duration "Sun outage" windows.)


title: Equinox

text: Satellites in geostationary orbit also experience difficulties maintaining power during the equinox because they have to travel through Earth's shadow and rely only on battery power. Usually, a satellite travels either north or south of the Earth's shadow because Earth's axis is not directly perpendicular to a line from the Earth to the Sun at other times. During the equinox, since geostationary satellites are situated above the Equator, they are in Earth's shadow for the longest duration all year.


title: Equinox

text: Equinoxes are defined on any planet with a tilted rotational axis. A dramatic example is Saturn, where the equinox places its ring system edge-on facing the Sun. As a result, they are visible only as a thin line when seen from Earth. When seen from above – a view seen during an equinox for the first time from the "Cassini" space probe in 2009 – they receive very little sunshine; indeed, they receive more planetshine than light from the Sun. This phenomenon occurs once every 14.7 years on average, and can last a few weeks before and after the exact equinox. Saturn's most recent equinox was on 11 August 2009, and its next will take place on 6 May 2025.


title: Equinox

text: Mars's most recent equinoxes were on 7 February 2021 (northern spring), and on 24 February 2022 (northern autumn).


title: Fall Guys

text: Fall Guys (formerly known as Fall Guys: Ultimate Knockout) is a free-to-play platform battle royale game developed by Mediatonic. The game involves up to 60 players who control jellybean-like creatures and compete against each other in a series of randomly selected mini-games, such as obstacle courses or tag. Players are eliminated as the rounds progress until, eventually, the last remaining player is crowned the winner. The game draws inspiration from game shows like "Takeshi's Castle", "It's a Knockout", "Wipeout", and playground games like tag and British Bulldog.


title: Fall Guys

text: The game was released by Devolver Digital for Microsoft Windows and PlayStation 4 on 4 August 2020. Following their acquisition of Mediatonic, the publishing rights were transferred to Epic Games. Subsequently, the game was made free-to-play on 21 June 2022 and released on additional platforms including Nintendo Switch, PlayStation 5, Xbox One and Xbox Series X/S, with full cross-platform play support among all platforms. As part of the transition, the game adapted a seasonal battle pass system for its monetization, offering avatar customizations to players.


title: Fall Guys

text: "Fall Guys" received positive reviews from critics for its chaotic gameplay and visual appearance. The game was a commercial success, selling more than 10 million copies and attracting more than 50 million players after the game went free-to-play. Since the game's launch, Mediatonic has released numerous updates for the game. The ninth and current season is underwater themed.


title: Fall Guys

text: Up to 60 players compete in matches with battle royale-style gameplay. Players, represented as jellybean-like figures, move around a three-dimensional playing field, with additional moves such as jumping, grabbing/climbing, or diving to assist gameplay. The aim is to qualify for subsequent rounds by successfully completing each of the randomly selected mini-games with randomly selected variations. Certain mini-games involve running towards a finish line at the end of the map, surviving, and playing tag with other players, while others add elements of teamwork. On every mini-game, obstacles appear around the map for added complexity. Players who are too slow, who fall into pink slime, or who fail certain requirements for a mini-game, are eliminated. In the final round, the remaining few players compete in a final match with a randomised mini-game designed for a smaller player size. The winner of the match is the last player standing or the first to finish.


title: Fall Guys

text: Using an in-game currency, "Kudos", players can purchase single cosmetics, nameplates, nametags, patterns, faceplates, colors, and emotes for their character to show off in-game. Players obtain Kudos by doing daily challenges and get crowns by winning a match. Crowns were a premium currency until 21 June 2022, whereas they now only count towards Crown Rank. Any Crowns that users didn't spend before the game went free to play were converted into Kudos.


title: Fall Guys

text: "Fall Guys" regularly hosts new seasons approximately every 2 to 6 months, hosting a new theme and additional content every season.


title: Fall Guys

text: Season 4 added the "Crown Shards" currency, earned by playing in Squad Mode, where a team of four other players cooperates to earn points to progress to the next round, or by completing daily challenges. Crown Shards will be automatically converted for additional Crowns.


title: Fall Guys

text: Season 5 added limited time events, where players need to do certain challenges to earn exclusive costumes and other types of cosmetics. In game events usually last between one and seven days and can sometimes be based on different franchises, such as Ratchet & Clank, Aloy from Horizon Zero Dawn, Jungle Book, etc.


title: Fall Guys

text: Sweet Thieves was added in Season 6, where one team is invisible (only being visible when moving fast) while the other team is not. The invisible team (Thieves) needs to bring sweets to the goals while the other team (Guardians) is trying to prevent them by using the grabbing mechanic to capture the Thieves and put them in a jail. A button can be pressed every 30 seconds to remove the jail barrier.


title: Fall Guys

text: Free to play Season 1 added Show Bucks, a premium in game currency that can be used to purchase full costumes, premium colors, patterns, faceplates, emotes, and the current paid Season Pass. Players are only able to obtain Show Bucks through the Season Pass or by purchasing it via microtransactions.


title: Fall Guys

text: The same season added Bean Hill Zone, the first-ever limited time collaboration round with Sega. The round came with a limited time event and its own show. In Season 3: Sunken Secrets the round was added to Solo Show.


title: Fall Guys

text: New costumes are regularly added, usually for a limited time. Some are of characters from different video games and media franchises, such as "Half-Life", "Portal", Godzilla, "Team Fortress 2", "Sonic the Hedgehog", "Doom", "Hotline Miami", "Ratchet & Clank", "Cuphead", "Among Us," "Little Big Planet", "Astro's Playroom", "Assassin's Creed", Hatsune Miku, "SpongeBob SquarePants", and so on.


title: Fall Guys

text: The conception of what became "Fall Guys" began when Mediatonic was discussing another project in January 2018. Lead designer Joe Walsh said that it reminded him of game shows such as "Takeshi's Castle" and "Total Wipeout". He drew from that inspiration to create a pitch document for what would become "Fall Guys". Originally titled "Fools' Gauntlet", (also later under development using the name "Stumble Chums"), Walsh's pitch featured 100 players competing in a battle royale composed of physical challenges. Creative director Jeff Tanton, while initially sceptical that creating another battle royale game would be successful, was quickly convinced of the game's potential, and forwarded Walsh's pitch to Mediatonic's founders.


title: Fall Guys

text: Tanton and Walsh worked on a pitch deck, and principal concept artist Dan Hoang created images featuring colourful, bean-shaped characters racing on an obstacle course in the sky. Tanton explained that Hoang's character designs helped shift the focus of the game away from the obstacle course itself, to the characters. With the pitch deck completed, Tanton pitched the game to 10 different publishers at the 2018 Game Developers Conference. Six months after Devolver Digital agreed to publish the game, development began.


title: Fall Guys

text: "Fall Guys" began its initial prototyping process with a small team, growing to 30 people during development. Initial progress on individual minigames was slow which caused the team to worry that there wouldn't be enough content for launch. A turning point came when the team made a group of pillars that "Took the opinions of people out of the occasion" and allowed the developers to "kill ideas faster". Such pillars include ensuring a minigame was "50-50 chaos and skill" and that a level had to be "different every time". Inspired by game shows, and differentiating from first-person shooter battle royale games, Mediatonic focused on gameplay variety. By presenting the player with several, randomised rounds of game modes, Mediatonic hoped to recreate the experience of being on a game show. To help keep the "spirit of playground games and game shows", Mediatonic created an internal rule that game modes needed to be explained in three words. Over time, the game underwent numerous other changes. The player count was decreased from 100 to 60, because overpopulated games "stopped being readable or fun". In testing, Mediatonic noted that players overestimated the number of players, stating "we didn't need as many players as we thought to create the crowds we wanted the gameplay to include."


title: Fall Guys

text: "It's a Knockout", a game show forcing contestants to dress up in oversized costumes, inspired the idea that the characters should "have that element of being completely uniquely, badly designed for the task that we were gonna put them through". The ragdoll physics is deliberate, to avoid "hyper athletic "Ninja Warrior" characters" and because "falling over is funny". According to Walsh, striking the right balance between funny ragdoll collisions and game performance was critical, because "as soon as you lose the ragdoll-ness of the character, you lose the comedy". The character designs were inspired by the look of vinyl toys.


title: Fall Guys

text: Mediatonic began working on season 2 before the game's release with level designer Joseph Juson, noting that it was more of an "expansion" compared to season 3 where the team tried to bring new ideas.


title: Fall Guys

text: "Fall Guys: Ultimate Knockout" was announced at E3 in June 2019 and was released on 4 August 2020 for Microsoft Windows and PlayStation 4. Several updates ensued for the first season and on 8 October 2020, the medieval-themed second season was released. The success prompted more hiring.


title: Fall Guys

text: The Chinese company Bilibili announced in August 2020 that it would be developing a mobile version for Android and iOS that would be exclusive to China.


title: Fall Guys

text: In February 2021, "Fall Guys" was announced for Nintendo Switch, Xbox One, and Xbox Series X, scheduled for mid 2021. However, in April 2021, these versions of the game were delayed to 2022, citing greater focus on cross-compatibility among all versions of the game.


title: Fall Guys

text: In March 2021, Mediatonic and its parent company, Tonic Games Group, were bought by Epic Games. Epic Games later announced that they will focus on adding cross-platform compatibility for all versions of "Fall Guys", utilising resources from games such as "Fortnite" that became available after acquisition by Epic Games.


title: Fall Guys

text: On 18 November 2021, Mediatonic announced that Epic Games Accounts will be required in season 6 allowing cross-platform progression between PS4 and Steam and the ability to have custom names for Steam players. On 22 February 2022, Mediatonic released an update adding Lobbies and Friends lists. This uses the Epic Overlay system that games like "Fortnite" or "Rocket League" have to invite anyone on any platform via Epic Games Accounts.


title: Fall Guys

text: On 15 March 2022, Mediatonic announced that starting 5 April 2022 there would be a new launcher that all PlayStation players would need to download to keep playing the game on PlayStation. On 9 May 2022, the old PlayStation launcher shut down and became unusable. Both launchers were the same besides the new one being published by Epic Games instead of Devolver Digital. People who downloaded the new launcher would be able to redeem a free DLC Pink Shark costume that was able to be redeemed until the end of season 6.


title: Fall Guys

text: Mediatonic and Epic Games transitioned "Fall Guys" to a free-to-play system alongside its 21 June release on Xbox One, Xbox Series X/S and Nintendo Switch. Additionally, a dedicated PlayStation 5 version also released, featuring enhanced visuals and DualSense effects. The game was also rebranded as simply "Fall Guys", without the "Ultimate Knockout" subtitle. To support the free-to-play model, the game included a new season pass system with both a free and premium tier of rewards, the latter available to those that purchase the premium pass with real-world funds. For Windows, the game was removed from the Steam storefront and made available on Epic Games Store, though the Steam version will continue to be updated in sync with the Epic Games Store version. The free-to-play versions of "Fall Guys" also added support for cross-platform play with other platforms. Legacy players who had purchased "Fall Guys" before this transition were given a "Legacy Pack" that includes cosmetics, costumes, and the Season 1: Free for All premium season pass for free.


title: Fall Guys

text: In July 2022, a bug was discovered that unintentionally bought items from the in-game store instead of previewing them. Initially deemed by Mediatonic as "not an issue", they later acknowledged it as a bug after complaints mounted. On 1 November 2022 a update was issued to include a hold-to-purchase mechanic for all in-game purchases similar to the mechanic in Fortnite.


title: Fall Guys

text: On November 2022, Mediatonic announced on twitter of a new feature called Vaulting where some rounds will be temporarily removed to increase stability across all platforms.


title: Fall Guys

text: "Fall Guys" received "generally favourable" reviews, according to review aggregator website Metacritic.


title: Fall Guys

text: Tom Wiggins of "Stuff" magazine praised the game, calling it ""Super Monkey Ball" for the "Fortnite" generation". "Mercury News" praised the "controlled chaos" gameplay of "Fall Guys", stating it to be utilizing a combination of the elements from the battle royale game "Fortnite" and the party game "Mario Party" in such a way that made it "tailored to the coronavirus age". Ollie Toms from "Rock, Paper, Shotgun" praised "Fall Guys" for its creative approach as a battle royale game incorporating family-friendly elements rather than resorting to violent themes. He praised the simplicity of the game and the diversity of content added with each season. He argues that the ragdoll physics can be both exciting and frustrating to play: "What really makes and breaks this game is the imprecision of your actions. It's not something many competitive games can get away with in the way "Fall Guys" does."


title: Fall Guys

text: The weekend before release during a closed beta, "Fall Guys" briefly became the most-watched game on Twitch as well as the sixth-best-selling Steam game where it was available for pre-order.


title: Fall Guys

text: "Fall Guys" has been frequently compared to "Among Us", in that both are online games which grew in popularity during the COVID-19 pandemic.


title: Fall Guys

text: The addition of Squad Mode in season 4 has been positively received. "Kotaku" says it evokes a new perspective on intense gameplay, indicating the importance of teamwork. The addition of Sweet Thieves has been positively received. The website "Destructoid" says "This is the most fun I've had with "Fall Guys" in forever." and "If you care about trophies, Sweet Thieves is worth checking out. But it stands on its own, too."


title: Fall Guys

text: On 11 October 2022 the hashtag #savefallguys started trending on Twitter. This was to get Mediatonic's attention and ask the company to fix numerous problems that fans have with the game including lack of round variety, bugs, sub par events, prices in the in-game shop, how skill based matchmaking is implemented, and more. On 15 November 2022 the Fall Guys twitter stated that in order for the game to stay stable some rounds will be "vaulted" and rounds will be rotated in and out of the vault. Mediatonic has yet to respond about skill-based matchmaking or shop prices.


title: Fall Guys

text: Within 24 hours of release, the game had an active player base of more than 1.5 million players. On 10 August 2020, Devolver Digital announced 2 million copies sold on Steam. During the first day of release, the servers for "Fall Guys" were unexpectedly overloaded due to popularity.


title: Fall Guys

text: The game's popularity brought several brand collaborations for custom content. Shortly after release Mediatonic announced a fundraiser by which the brand that donated the most money to the charity SpecialEffect would have their custom skin featured in the game.


title: Fall Guys

text: By 26 August 2020, more than 7 million copies of "Fall Guys" had been sold on Steam, and it was the most downloaded monthly PS Plus game of all time. By November 2020, the game had reached more than 10 million sales on Steam. According to firm Superdata the game's online revenue for PC in its first month was from players, making it the biggest launch on that platform since "Overwatch" by those metrics. Multiple outlets described the game's popularity as a "phenomenon". In December 2020, Mediatonic confirmed sales of more than 11 million copies on PC.


title: Fall Guys

text: In June 2022, within two weeks of going free-to-play and its release on multiple consoles, "Fall Guys" player count rose to over 50 million.


title: Deity

text: A deity or god is a supernatural being who is considered divine or sacred. The "Oxford Dictionary of English" defines deity as a god or goddess, or anything revered as divine. C. Scott Littleton defines a deity as "a being with powers greater than those of ordinary humans, but who interacts with humans, positively or negatively, in ways that carry humans to new levels of consciousness, beyond the grounded preoccupations of ordinary life".


title: Deity

text: Religions can be categorized by how many deities they worship. Monotheistic religions accept only one deity (predominantly referred to as "God"), whereas polytheistic religions accept multiple deities. Henotheistic religions accept one supreme deity without denying other deities, considering them as aspects of the same divine principle. Nontheistic religions deny any supreme eternal creator deity, but may accept a pantheon of deities which live, die and may be reborn like any other being.


title: Deity

text: Although most monotheistic religions traditionally envision their god as omnipotent, omnipresent, omniscient, omnibenevolent, and eternal, none of these qualities are essential to the definition of a "deity" and various cultures have conceptualized their deities differently. Monotheistic religions typically refer to their god in masculine terms, while other religions refer to their deities in a variety of ways—male, female, hermaphroditic, or genderless.


title: Deity

text: Historically, many ancient cultures—including the ancient Mesopotamians, Egyptians, Greeks, Romans, and Germanic peoples—personified natural phenomena, variously as either deliberate causes or effects. Some Avestan and Vedic deities were viewed as ethical concepts. In Indian religions, deities were envisioned as manifesting within the temple of every living being's body, as sensory organs and mind. Deities were envisioned as a form of existence (Saṃsāra) after rebirth, for human beings who gain merit through an ethical life, where they become guardian deities and live blissfully in heaven, but are also subject to death when their merit is lost.


title: Deity

text: The English language word "deity" derives from Old French "deité," the Latin "deitatem" or "divine nature", coined by Augustine of Hippo from ' ("god"). Deus is related through a common Proto-Indo-European (PIE) origin to "*deiwos". This root yields the ancient Indian word "Deva" meaning "to gleam, a shining one", from *div- "to shine", as well as Greek ' "divine" and Zeus; and Latin ' "god" (Old Latin "deivos"). Deva is masculine, and the related feminine equivalent is devi. Etymologically, the cognates of "Devi" are Latin ' and Greek "". In Old Persian, "daiva-" means "demon, evil god", while in Sanskrit it means the opposite, referring to the "heavenly, divine, terrestrial things of high excellence, exalted, shining ones".


title: Deity

text: The closely linked term "god" refers to "supreme being, deity", according to Douglas Harper, and is derived from Proto-Germanic "*guthan", from PIE ', which means "that which is invoked". "Guth" in the Irish language means "voice". The term ' is also the source of Old Church Slavonic ("to call"), Sanskrit "" ("invoked", an epithet of Indra), from the root ("to call, invoke."),


title: Deity

text: An alternate etymology for the term "god" comes from the Proto-Germanic Gaut, which traces it to the PIE root ' ("poured"), derived from the root ' ("to pour, pour a libation"). The term ' is also the source of the Greek ' "to pour". Originally the German root was a neuter noun. The gender of the monotheistic God shifted to masculine under the influence of Christianity. In contrast, all ancient Indo-European cultures and mythologies recognized both masculine and feminine deities.


title: Deity

text: There is no universally accepted consensus on what a deity is, and concepts of deities vary considerably across cultures. Huw Owen states that the term "deity or god or its equivalent in other languages" has a bewildering range of meanings and significance. It has ranged from "infinite transcendent being who created and lords over the universe" (God), to a "finite entity or experience, with special significance or which evokes a special feeling" (god), to "a concept in religious or philosophical context that relates to nature or magnified beings or a supra-mundane realm", to "numerous other usages".


title: Deity

text: A deity is typically conceptualized as a supernatural or divine concept, manifesting in ideas and knowledge, in a form that combines excellence in some or all aspects, wrestling with weakness and questions in other aspects, heroic in outlook and actions, yet tied up with emotions and desires. In other cases, the deity is a principle or reality such as the idea of "soul". The Upanishads of Hinduism, for example, characterize Atman (soul, self) as "deva" (deity), thereby asserting that the "deva" and eternal supreme principle (Brahman) is part of every living creature, that this soul is spiritual and divine, and that to realize self-knowledge is to know the supreme.


title: Deity

text: Theism is the belief in the existence of one or more deities. Polytheism is the belief in and worship of multiple deities, which are usually assembled into a pantheon of gods and goddesses, with accompanying rituals. In most polytheistic religions, the different gods and goddesses are representations of forces of nature or ancestral principles, and can be viewed either as autonomous or as aspects or emanations of a creator God or transcendental absolute principle (monistic theologies), which manifests immanently in nature. Henotheism accepts the existence of more than one deity, but considers all deities as equivalent representations or aspects of the same divine principle, the highest. Monolatry is the belief that many deities exist, but that only one of these deities may be validly worshipped.


title: Deity

text: Monotheism is the belief that only one deity exists. A monotheistic deity, known as "God", is usually described as omnipotent, omnipresent, omniscient, omnibenevolent and eternal. However, not all deities have been regarded this way and an entity does not need to be almighty, omnipresent, omniscient, omnibenevolent or eternal to qualify as a deity.


title: Deity

text: Deism is the belief that only one deity exists, who created the universe, but does not usually intervene in the resulting world. Deism was particularly popular among western intellectuals during the eighteenth and nineteenth centuries. Pantheism is the belief that the universe itself is God or that everything composes an all-encompassing, immanent deity. Pandeism is an intermediate position between these, proposing that the creator became a pantheistic universe. Panentheism is the belief that divinity pervades the universe, but that it also transcends the universe. Agnosticism is the position that it is impossible to know for certain whether a deity of any kind exists. Atheism is the non-belief in the existence of any deity.


title: Deity

text: Scholars infer the probable existence of deities in the prehistoric period from inscriptions and prehistoric arts such as cave drawings, but it is unclear what these sketches and paintings are and why they were made. Some engravings or sketches show animals, hunters or rituals. It was once common for archaeologists to interpret virtually every prehistoric female figurine as a representation of a single, primordial goddess, the ancestor of historically attested goddesses such as Inanna, Ishtar, Astarte, Cybele, and Aphrodite; this approach has now generally been discredited. Modern archaeologists now generally recognize that it is impossible to conclusively identify any prehistoric figurines as representations of any kind of deities, let alone goddesses. Nonetheless, it is possible to evaluate ancient representations on a case-by-case basis and rate them on how likely they are to represent deities. The Venus of Willendorf, a female figurine found in Europe and dated to about 25,000 BCE has been interpreted by some as an exemplar of a prehistoric female deity. A number of probable representations of deities have been discovered at 'Ain Ghazal and the works of art uncovered at Çatalhöyük reveal references to what is probably a complex mythology.


title: Deity

text: Diverse African cultures developed theology and concepts of deities over their history. In Nigeria and neighboring West African countries, for example, two prominent deities (locally called "Òrìṣà") are found in the Yoruba religion, namely the god Ogun and the goddess Osun. Ogun is the primordial masculine deity as well as the archdivinity and guardian of occupations such as tools making and use, metal working, hunting, war, protection and ascertaining equity and justice. Osun is an equally powerful primordial feminine deity and a multidimensional guardian of fertility, water, maternal, health, social relations, love and peace. Ogun and Osun traditions were brought into the Americas on slave ships. They were preserved by the Africans in their plantation communities, and their festivals continue to be observed.


title: Deity

text: In Southern African cultures, a similar masculine-feminine deity combination has appeared in other forms, particularly as the Moon and Sun deities. One Southern African cosmology consists of "Hieseba" or "Xuba" (deity, god), "Gaune" (evil spirits) and "Khuene" (people). The "Hieseba" includes "Nladiba" (male, creator sky god) and "Nladisara" (females, Nladiba's two wives). The Sun (female) and the Moon (male) deities are viewed as offspring of "Nladiba" and two "Nladisara". The Sun and Moon are viewed as manifestations of the supreme deity, and worship is timed and directed to them. In other African cultures the Sun is seen as male, while the Moon is female, both symbols of the godhead. In Zimbabwe, the supreme deity is androgynous with male-female aspects, envisioned as the giver of rain, treated simultaneously as the god of darkness and light and is called "Mwari Shona". In the Lake Victoria region, the term for a deity is "Lubaale", or alternatively "Jok".


title: Deity

text: Ancient Egyptian culture revered numerous deities. Egyptian records and inscriptions list the names of many whose nature is unknown and make vague references to other unnamed deities. Egyptologist James P. Allen estimates that more than 1,400 deities are named in Egyptian texts, whereas Christian Leitz offers an estimate of "thousands upon thousands" of Egyptian deities. Their terms for deities were "nṯr" (god), and feminine "nṯrt" (goddess); however, these terms may also have applied to any being – spirits and deceased human beings, but not demons – who in some way were outside the sphere of everyday life. Egyptian deities typically had an associated cult, role and mythologies.


title: Deity

text: Around 200 deities are prominent in the Pyramid texts and ancient temples of Egypt, many zoomorphic. Among these, were "Min" (fertility god), "Neith" (creator goddess), "Anubis", "Atum", "Bes", "Horus", "Isis", "Ra", "Meretseger", "Nut", "Osiris", "Shu", "Sia" and "Thoth". Most Egyptian deities represented natural phenomenon, physical objects or social aspects of life, as hidden immanent forces within these phenomena. The deity "Shu", for example represented air; the goddess "Meretseger" represented parts of the earth, and the god "Sia" represented the abstract powers of perception. Deities such as "Ra" and "Osiris" were associated with the judgement of the dead and their care during the afterlife. Major gods often had multiple roles and were involved in multiple phenomena.


title: Deity

text: The first written evidence of deities are from early 3rd millennium BCE, likely emerging from prehistoric beliefs. However, deities became systematized and sophisticated after the formation of an Egyptian state under the Pharaohs and their treatment as sacred kings who had exclusive rights to interact with the gods, in the later part of the 3rd millennium BCE. Through the early centuries of the common era, as Egyptians interacted and traded with neighboring cultures, foreign deities were adopted and venerated.


title: Deity

text: The ancient Canaanites were polytheists who believed in a pantheon of deities, the chief of whom was the god El, who ruled alongside his consort Asherah and their seventy sons. Baal was the god of storm, rain, vegetation and fertility, while his consort Anat was the goddess of war and Astarte, the West Semitic equivalent to Ishtar, was the goddess of love. The people of the Kingdoms of Israel and Judah originally believed in these deities, alongside their own national god Yahweh. El later became syncretized with Yahweh, who took over El's role as the head of the pantheon, with Asherah as his divine consort and the "sons of El" as his offspring. During the later years of the Kingdom of Judah, a monolatristic faction rose to power insisting that only Yahweh was fit to be worshipped by the people of Judah. Monolatry became enforced during the reforms of King Josiah in 621 BCE. Finally, during the national crisis of the Babylonian captivity, some Judahites began to teach that deities aside from Yahweh were not just unfit to be worshipped, but did not exist. The "sons of El" were demoted from deities to angels.


title: Deity

text: Ancient Mesopotamian culture in southern Iraq had numerous "dingir" (deities, gods and goddesses). Mesopotamian deities were almost exclusively anthropomorphic. They were thought to possess extraordinary powers and were often envisioned as being of tremendous physical size. They were generally immortal, but a few of them, particularly Dumuzid, Geshtinanna, and Gugalanna were said to have either died or visited the underworld. Both male and female deities were widely venerated.


title: Deity

text: In the Sumerian pantheon, deities had multiple functions, which included presiding over procreation, rains, irrigation, agriculture, destiny, and justice. The gods were fed, clothed, entertained, and worshipped to prevent natural catastrophes as well as to prevent social chaos such as pillaging, rape, or atrocities. Many of the Sumerian deities were patron guardians of city-states.


title: Deity

text: The most important deities in the Sumerian pantheon were known as the Anunnaki, and included deities known as the "seven gods who decree": An, Enlil, Enki, Ninhursag, Nanna, Utu and Inanna. After the conquest of Sumer by Sargon of Akkad, many Sumerian deities were syncretized with East Semitic ones. The goddess Inanna, syncretized with the East Semitic Ishtar, became popular, with temples across Mesopotamia.


title: Deity

text: The Mesopotamian mythology of the first millennium BCE treated Anšar (later Aššur) and Kišar as primordial deities. Marduk was a significant god among the Babylonians. He rose from an obscure deity of the third millennium BCE to become one of the most important deities in the Mesopotamian pantheon of the first millennium BCE. The Babylonians worshipped Marduk as creator of heaven, earth and humankind, and as their national god. Marduk's iconography is zoomorphic and is most often found in Middle Eastern archaeological remains depicted as a "snake-dragon" or a "human-animal hybrid".


title: Deity

text: In Germanic languages, the terms cognate with 'god' such as and were originally neuter but became masculine, as in modern Germanic languages, after Christianisation due their use in referring to the Christian god.


title: Deity

text: In Norse mythology, (singular or ) are the principal group of gods, while the term (singular ) refers specifically to the female . These terms, states John Lindow, may be ultimately rooted in the Indo-European root for "breath" (as in "life giving force"), and are cognate with (a heathen god) and Gothic: "anses".


title: Deity

text: Another group of deities found in Norse mythology are termed as "Vanir", and are associated with fertility. The "Æsir" and the "Vanir" went to war, according to the Nordic sources. The account in "Ynglinga saga" describes the Æsir–Vanir War ending in truce and ultimate reconciliation of the two into a single group of gods, after both sides chose peace, exchanged ambassadors (hostages), and intermarried.


title: Deity

text: The Norse mythology describes the cooperation after the war, as well as differences between the "Æsir" and the "Vanir" which were considered scandalous by the other side. The goddess Freyja of the "Vanir" taught magic to the "Æsir", while the two sides discover that while "Æsir" forbid mating between siblings, "Vanir" accepted such mating.


title: Deity

text: Temples hosting images of Germanic deities (such as Thor, Odin and Freyr), as well as pagan worship rituals, continued in Nordic countries into the 12th century, according to historical records. It has been proposed that over time, Christian equivalents were substituted for the Germanic deities to help suppress paganism, during Christianisation. Worship of the Germanic gods has been revived in the modern period as part of the new religious movement of Heathenry.


title: Deity

text: The ancient Greeks revered both gods and goddesses. These continued to be revered through the early centuries of the common era, and many of the Greek deities inspired and were adopted as part of much larger pantheon of Roman deities. The Greek religion was polytheistic, but had no centralized church, nor any sacred texts. The deities were largely associated with myths and they represented natural phenomena or aspects of human behavior.


title: Deity

text: Several Greek deities probably trace back to more ancient Indo-European traditions, since the gods and goddesses found in distant cultures are mythologically comparable and are cognates. Eos, the Greek goddess of the dawn, for instance, is cognate to Indic "Ushas", Roman "Aurora" and Latvian "Auseklis". Zeus, the Greek king of gods, is cognate to Latin "Iūpiter", Old German "Ziu", and Indic "Dyaus", with whom he shares similar mythologies. Other deities, such as Aphrodite, originated from the Near East.


title: Deity

text: Greek deities varied locally, but many shared panhellenic themes, celebrated similar festivals, rites, and ritual grammar. The most important deities in the Greek pantheon were the Twelve Olympians: Zeus, Hera, Poseidon, Athena, Apollo, Artemis, Aphrodite, Hermes, Demeter, Dionysus, Hephaestus, and Ares. Other important Greek deities included Hestia, Hades and Heracles. These deities later inspired the "Dii Consentes" galaxy of Roman deities.


title: Deity

text: Besides the Olympians, the Greeks also worshipped various local deities. Among these were the goat-legged god Pan (the guardian of shepherds and their flocks), Nymphs (nature spirits associated with particular landforms), Naiads (who dwelled in springs), Dryads (who were spirits of the trees), Nereids (who inhabited the sea), river gods, satyrs (a class of lustful male nature spirits), and others. The dark powers of the underworld were represented by the Erinyes (or Furies), said to pursue those guilty of crimes against blood-relatives.


title: Deity

text: The Greek deities, like those in many other Indo-European traditions, were anthropomorphic. Walter Burkert describes them as "persons, not abstractions, ideas or concepts". They had fantastic abilities and powers; each had some unique expertise and, in some aspects, a specific and flawed personality. They were not omnipotent and could be injured in some circumstances. Greek deities led to cults, were used politically and inspired votive offerings for favors such as bountiful crops, healthy family, victory in war, or peace for a loved one recently deceased.


title: Deity

text: The Roman pantheon had numerous deities, both Greek and non-Greek. The more famed deities, found in the mythologies and the 2nd millennium CE European arts, have been the anthropomorphic deities syncretized with the Greek deities. These include the six gods and six goddesses: Venus, Apollo, Mars, Diana, Minerva, Ceres, Vulcan, Juno, Mercury, Vesta, Neptune, Jupiter (Jove, Zeus); as well Bacchus, Pluto and Hercules. The non-Greek major deities include Janus, Fortuna, Vesta, Quirinus and Tellus (mother goddess, probably most ancient). Some of the non-Greek deities had likely origins in more ancient European culture such as the ancient Germanic religion, while others may have been borrowed, for political reasons, from neighboring trade centers such as those in the Minoan or ancient Egyptian civilization.


title: Deity

text: The Roman deities, in a manner similar to the ancient Greeks, inspired community festivals, rituals and sacrifices led by "flamines" (priests, pontifs), but priestesses (Vestal Virgins) were also held in high esteem for maintaining sacred fire used in the votive rituals for deities. Deities were also maintained in home shrines ("lararium"), such as Hestia honored in homes as the goddess of fire hearth. This Roman religion held reverence for sacred fire, and this is also found in Hebrew culture (Leviticus 6), Vedic culture's Homa, ancient Greeks and other cultures.


title: Deity

text: Ancient Roman scholars such as Varro and Cicero wrote treatises on the nature of gods of their times. Varro stated, in his "Antiquitates Rerum Divinarum", that it is the superstitious man who fears the gods, while the truly religious person venerates them as parents. Cicero, in his "Academica", praised Varro for this and other insights. According to Varro, there have been three accounts of deities in the Roman society: the mythical account created by poets for theatre and entertainment, the civil account used by people for veneration as well as by the city, and the natural account created by the philosophers. The best state is, adds Varro, where the civil theology combines the poetic mythical account with the philosopher's. The Roman deities continued to be revered in Europe through the era of Constantine, and past 313 CE when he issued the Edict of Toleration.


title: Deity

text: The Inca culture has believed in "Viracocha" (also called "Pachacutec") as the creator deity. "Viracocha" has been an abstract deity to Inca culture, one who existed before he created space and time. All other deities of the Inca people have corresponded to elements of nature. Of these, the most important ones have been "Inti" (sun deity) responsible for agricultural prosperity and as the father of the first Inca king, and "Mama Qucha" the goddess of the sea, lakes, rivers and waters. "Inti" in some mythologies is the son of "Viracocha" and "Mama Qucha".


title: Deity

text: Inca people have revered many male and female deities. Among the feminine deities have been "Mama Kuka" (goddess of joy), "Mama Ch'aska" (goddess of dawn), "Mama Allpa" (goddess of harvest and earth, sometimes called "Mama Pacha" or "Pachamama"), Mama Killa (moon goddess) and "Mama Sara" (goddess of grain). During and after the imposition of Christianity during Spanish colonialism, the Inca people retained their original beliefs in deities through syncretism, where they overlay the Christian God and teachings over their original beliefs and practices. The male deity "Inti" became accepted as the Christian God, but the Andean rituals centered around Inca deities have been retained and continued thereafter into the modern era by the Inca people.


title: Deity

text: In Maya culture, "Kukulkan" has been the supreme creator deity, also revered as the god of reincarnation, water, fertility and wind. The Maya people built step pyramid temples to honor "Kukulkan", aligning them to the Sun's position on the spring equinox. Other deities found at Maya archaeological sites include "Xib Chac" – the benevolent male rain deity, and "Ixchel" – the benevolent female earth, weaving and pregnancy goddess. The Maya calendar had 18 months, each with 20 days (and five unlucky days of "Uayeb"); each month had a presiding deity, who inspired social rituals, special trading markets and community festivals.


title: Deity

text: A deity with aspects similar to "Kulkulkan" in the Aztec culture has been called "Quetzalcoatl". However, states Timothy Insoll, the Aztec ideas of deity remain poorly understood. What has been assumed is based on what was constructed by Christian missionaries. The deity concept was likely more complex than these historical records. In Aztec culture, there were hundred of deities, but many were henotheistic incarnations of one another (similar to the avatar concept of Hinduism). Unlike Hinduism and other cultures, Aztec deities were usually not anthropomorphic, and were instead zoomorphic or hybrid icons associated with spirits, natural phenomena or forces. The Aztec deities were often represented through ceramic figurines, revered in home shrines.


title: Deity

text: The Polynesian people developed a theology centered on numerous deities, with clusters of islands having different names for the same idea. There are great deities found across the Pacific Ocean. Some deities are found widely, and there are many local deities whose worship is limited to one or a few islands or sometimes to isolated villages on the same island.


title: Deity

text: The Māori people, of what is now New Zealand, called the supreme being as "Io", who is also referred elsewhere as "Iho-Iho", "Io-Mataaho", "Io Nui", "Te Io Ora", "Io Matua Te Kora" among other names. The "Io" deity has been revered as the original uncreated creator, with power of life, with nothing outside or beyond him.Other deities in the Polynesian pantheon include "Tangaloa" (god who created men), "La'a Maomao" (god of winds), "Tu-Matauenga" or "Ku" (god of war), "Tu-Metua" (mother goddess), "Kane" (god of procreation) and "Rangi" (sky god father).


title: Deity

text: The Polynesian deities have been part of a sophisticated theology, addressing questions of creation, the nature of existence, guardians in daily lives as well as during wars, natural phenomena, good and evil spirits, priestly rituals, as well as linked to the journey of the souls of the dead.


title: Deity

text: Christianity is a monotheistic religion in which most mainstream congregations and denominations accept the concept of the Holy Trinity. Modern orthodox Christians believe that the Trinity is composed of three equal, cosubstantial persons: God the Father, God the Son, and the Holy Spirit. The first person to describe the persons of the Trinity as "homooúsios" (ὁμοούσιος; "of the same substance") was the Church Father Origen. Although most early Christian theologians (including Origen) were Subordinationists, who believed that the Father was superior to the Son and the Son superior to the Holy Spirit, this belief was condemned as heretical by the First Council of Nicaea in the fourth century, which declared that all three persons of the Trinity are equal. Christians regard the universe as an element in God's actualization and the Holy Spirit is seen as the divine essence that is "the unity and relation of the Father and the Son". According to George Hunsinger, the doctrine of the Trinity justifies worship in a Church, wherein Jesus Christ is deemed to be a full deity with the Christian cross as his icon.


title: Deity

text: The theological examination of Jesus Christ, of divine grace in incarnation, his non-transferability and completeness has been a historic topic. For example, the Council of Chalcedon in 451 CE declared that in "one person Jesus Christ, fullness of deity and fullness of humanity are united, the union of the natures being such that they can neither be divided nor confused". Jesus Christ, according to the New Testament, is the self-disclosure of the one, true God, both in his teaching and in his person; Christ, in Christian faith, is considered the incarnation of God.


title: Deity

text: Ilah, ' (; plural: '), is an Arabic word meaning "god". It appears in the name of the monotheistic god of Islam as Allah ('). which literally means "the god" in Arabic. Islam is strictly monotheistic and the first statement of the "shahada", or Muslim confession of faith, is that "there is no ' (deity) but "" (God)", who is perfectly unified and utterly indivisible.


title: Deity

text: The term Allah is used by Muslims for God. The Persian word "Khuda" (Persian: خدا) can be translated as god, lord or king, and is also used today to refer to God in Islam by Persian, Urdu, Tat and Kurdish speakers. The Turkic word for god is Tengri; it exists as "Tanrı" in Turkish.


title: Deity

text: Judaism affirms the existence of one God (Yahweh, or YHWH), who is not abstract, but He who revealed himself throughout Jewish history particularly during the Exodus and the Exile. Judaism reflects a monotheism that gradually arose, was affirmed with certainty in the sixth century "Second Isaiah", and has ever since been the axiomatic basis of its theology.


title: Deity

text: The classical presentation of Judaism has been as a monotheistic faith that rejected deities and related idolatry. However, states Breslauer, modern scholarship suggests that idolatry was not absent in biblical faith, and it resurfaced multiple times in Jewish religious life. The rabbinic texts and other secondary Jewish literature suggest worship of material objects and natural phenomena through the medieval era, while the core teachings of Judaism maintained monotheism.


title: Deity

text: According to Aryeh Kaplan, God is always referred to as "He" in Judaism, "not to imply that the concept of sex or gender applies to God", but because "there is no neuter in the Hebrew language, and the Hebrew word for God is a masculine noun" as he "is an active rather than a passive creative force".


title: Deity

text: In Mandaeism, "Hayyi Rabbi" (lit=The Great Life), or 'The Great Living God', is the supreme God from which all things emanate. He is also known as 'The First Life', since during the creation of the material world, Yushamin emanated from Hayyi Rabbi as the "Second Life." "The principles of the Mandaean doctrine: the belief of the only one great God, Hayyi Rabbi, to whom all absolute properties belong; He created all the worlds, formed the soul through his power, and placed it by means of angels into the human body. So He created Adam and Eve, the first man and woman." Mandaeans recognize God to be the eternal, creator of all, the one and only in domination who has no partner.


title: Deity

text: Anitism, composed of a diverse array of indigenous religions from the Philippines, has multiple pantheon of deities, with each ethnic group having their own. The most notable deities are almost always the deity or deities considered by specific ethnic groups as their supreme deity or deities.


title: Deity

text: Bathala is the Tagalog supreme deity, while Mangechay is the Kapampangan supreme deity. The Sambal supreme deity is Malayari, the Blaan supreme deity is Melu, the Bisaya supreme deity is Kaptan, and so on. There are more than a hundred different ethnic groups in the Philippines, each having their own supreme deity or deities. Each supreme deity or deities normally rules over a pantheon of deities, contributing to the sheer diversity of deities in Anitism.


title: Deity

text: Buddhists do not believe in a creator deity. However, deities are an essential part of Buddhist teachings about cosmology, rebirth, and saṃsāra. Buddhist deities (such as "devas" and "bodhisattvas") are believed to reside in a pleasant, heavenly realm within Buddhist cosmology, which is typically subdivided into twenty six sub-realms.


title: Deity

text: Devas are numerous, but they are still mortal; they live in the heavenly realm, then die and are reborn like all other beings. A rebirth in the heavenly realm is believed to be the result of leading an ethical life and accumulating very good karma. A "deva" does not need to work, and is able to enjoy in the heavenly realm all pleasures found on Earth. However, the pleasures of this realm lead to attachment ("upādāna"), lack of spiritual pursuits, and therefore no nirvana. The vast majority of Buddhist lay people in countries practicing Theravada, states Kevin Trainor, have historically pursued Buddhist rituals and practices because they are motivated by their potential rebirth into the "deva" realm. The "deva" realm in Buddhist practice in Southeast Asia and East Asia, states Keown, include gods found in Hindu traditions such as Indra and Brahma, and concepts in Hindu cosmology such as Mount Meru.


title: Deity

text: Mahayana Buddhism also includes different kinds of deities, such as numerous Buddhas, bodhisattvas and fierce deities.


title: Deity

text: The concept of God varies in Hinduism, it being a diverse system of thought with beliefs spanning henotheism, monotheism, polytheism, panentheism, pantheism and monism among others.


title: Deity

text: In the ancient Vedic texts of Hinduism, a deity is often referred to as Deva (god) or Devi (goddess). The root of these terms mean "heavenly, divine, anything of excellence". Deva is masculine, and the related feminine equivalent is devi. In the earliest Vedic literature, all supernatural beings are called Asuras. Over time, those with a benevolent nature become deities and are referred to as "Sura", Deva or Devi.


title: Deity

text: Devas or deities in Hindu texts differ from Greek or Roman theodicy, states Ray Billington, because many Hindu traditions believe that a human being has the potential to be reborn as a "deva" (or "devi"), by living an ethical life and building up saintly "karma". Such a "deva" enjoys heavenly bliss, till the merit runs out, and then the soul (gender neutral) is reborn again into Saṃsāra. Thus deities are henotheistic manifestations, embodiments and consequence of the virtuous, the noble, the saint-like living in many Hindu traditions.


title: Deity

text: Like many ancient Indian traditions, Jainism does not believe in a creator, omnipotent, omniscient, eternal God; however, the cosmology of Jainism incorporates a meaningful causality-driven reality, and includes four realms of existence ("gati"), and one of them for "deva" (celestial beings, gods). A human being can choose and live an ethical life (karma), such as being non-violent (ahimsa) against all living beings, thereby gain merit and be reborn as "deva".


title: Deity

text: Jain texts reject a trans-cosmic God, one who stands outside of the universe and lords over it, but they state that the world is full of "devas" who are in human-image with sensory organs, with the power of reason, conscious, compassionate and with finite life. Jainism believes in the existence of the soul (Self, atman) and considers it to have "god-quality", whose knowledge and liberation is the ultimate spiritual goal in both religions. Jains also believe that the spiritual nobleness of perfected souls (Jina) and "devas" make them worship-worthy beings, with powers of guardianship and guidance to better "karma". In Jain temples or festivals, the Jinas and Devas are revered.


title: Deity

text: Ahura Mazda (); is the Avestan name for the creator and sole God of Zoroastrianism. The literal meaning of the word "" is "mighty" or "lord" and " " is "wisdom". Zoroaster, the founder of Zoroastrianism, taught that Ahura Mazda is the most powerful being in all of the existence and the only deity who is worthy of the highest veneration. Nonetheless, Ahura Mazda is not omnipotent because his evil twin brother Angra Mainyu is nearly as powerful as him. Zoroaster taught that the "daevas" were evil spirits created by Angra Mainyu to sow evil in the world and that all people must choose between the goodness of Ahura Mazda and the evil of Angra Mainyu. According to Zoroaster, Ahura Mazda will eventually defeat Angra Mainyu and good will triumph over evil once and for all. Ahura Mazda was the most important deity in the ancient Achaemenid Empire. He was originally represented anthropomorphically, but, by the end of the Sasanian Empire, Zoroastrianism had become fully aniconic.


title: Deity

text: Attempts to rationally explain belief in deities extend all the way back to ancient Greece. The Greek philosopher Democritus argued that the concept of deities arose when human beings observed natural phenomena such as lightning, solar eclipses, and the changing of the seasons. Later, in the third century BCE, the scholar Euhemerus argued in his book "Sacred History" that the gods were originally flesh-and-blood mortal kings who were posthumously deified, and that religion was therefore the continuation of these kings' mortal reigns, a view now known as Euhemerism. Sigmund Freud suggested that God concepts are a projection of one's father.


title: Deity

text: A tendency to believe in deities and other supernatural beings may be an integral part of the human consciousness. Children are naturally inclined to believe in supernatural entities such as gods, spirits, and demons, even without being introduced into a particular religious tradition. Humans have an overactive agency detection system, which has a tendency to conclude that events are caused by intelligent entities, even if they really are not. This is a system which may have evolved to cope with threats to the survival of human ancestors: in the wild, a person who perceived intelligent and potentially dangerous beings everywhere was more likely to survive than a person who failed to perceive actual threats, such as wild animals or human enemies. Humans are also inclined to think teleologically and ascribe meaning and significance to their surroundings, a trait which may lead people to believe in a creator-deity. This may have developed as a side effect of human social intelligence, the ability to discern what other people are thinking.


title: Deity

text: Stories of encounters with supernatural beings are especially likely to be retold, passed on, and embellished due to their descriptions of standard ontological categories (person, artifact, animal, plant, natural object) with counterintuitive properties (humans that are invisible, houses that remember what happened in them, etc.). As belief in deities spread, humans may have attributed anthropomorphic thought processes to them, leading to the idea of leaving offerings to the gods and praying to them for assistance, ideas which are seen in all cultures around the world.


title: Deity

text: Sociologists of religion have proposed that the personality and characteristics of deities may reflect a culture's sense of self-esteem and that a culture projects its revered values into deities and in spiritual terms. The cherished, desired or sought human personality is congruent with the personality it defines to be gods. Lonely and fearful societies tend to invent wrathful, violent, submission-seeking deities, while happier and secure societies tend to invent loving, non-violent, compassionate deities. Émile Durkheim states that gods represent an extension of human social life to include supernatural beings. According to Matt Rossano, God concepts may be a means of enforcing morality and building more cooperative community groups.


title: Missi Pyle

text: Missi Pyle (born November 16, 1972) is an American actress and singer. She has appeared in a number of successful films, including "Galaxy Quest" (1999), "Big Fish" (2003), "Bringing Down the House" (2003), "" (2004), "Charlie and the Chocolate Factory" (2005), "Harold & Kumar Escape from Guantanamo Bay" (2008), "The Artist" (2011), "Gone Girl" (2014), "Captain Fantastic" (2016), and "Ma" (2019).


title: Missi Pyle

text: Pyle has also appeared in various television series, such as "Cleaners" (2013–2014), "Jennifer Falls" (2014), "Another Period" (2015–2018), "Bordertown" (2016), "The Soul Man" (2016), "Mom" (2017), "Impulse" (2018), "The Mentalist" (2010) and "Dirty John" (2020).


title: Missi Pyle

text: In addition to acting, Pyle has also worked as a singer. With actress Shawnee Smith, she served as half of Smith & Pyle, a country rock duo.


title: Missi Pyle

text: Pyle attended the North Carolina School of the Arts in Winston-Salem, North Carolina, and graduated in 1995. For her achievements, Pyle was honored by the Poplar Pike Playhouse at her alma mater Germantown High School in Germantown, Tennessee.


title: Missi Pyle

text: Pyle has guest starred on many television shows, including "Heroes", "Mad About You", "Boston Legal", "Frasier", "The Sarah Silverman Program", and "2 Broke Girls". She started her film career with a minor role in "As Good as It Gets", starring Helen Hunt and Jack Nicholson. After her breakout role in "Galaxy Quest", she had supporting roles in "Bringing Down the House" (for which she and Queen Latifah were nominated for an MTV Movie Award for Best Fight), "Josie and the Pussycats", "Home Alone 4", "Exposed", "Big Fish", "Along Came Polly", "Soul Plane", "Stormbreaker", and "Charlie and the Chocolate Factory". She was also the female lead in "BachelorMan". She also appears in "". She also had a brief appearance in "50 First Dates" and starred in "".


title: Missi Pyle

text: Pyle played Jake's elementary school teacher Ms. Pasternak on "Two and a Half Men". The role was recast in 2009 for undisclosed reasons, however it is believed that it was to show the emotional change she went through after Charlie dumped and hurt her [source IMDB]. Pyle returned to the series in the same role in 2011 and again in the series finale "Of Course He's Dead" in 2015.


title: Missi Pyle

text: In 2008, Missi starred in the Broadway play "Boeing-Boeing" opposite Christine Baranski, Mark Rylance, Greg Germann, Paige Davis and Rebecca Gayheart. The play closed in January 2009.


title: Missi Pyle

text: Pyle was scheduled to ring the closing bell for the trading day at the New York Stock Exchange on September 29, 2008, to promote "Boeing-Boeing", but backed out minutes before the close of trading in order to not associate herself with ending a calamitous day of financial turmoil which resulted in the Dow Jones Industrial Average losing 778 points, the largest point loss ever in a session in the wake of the financial crisis of 2007–2008. She watched from the floor as an NYSE staffer pushed the button and gaveled the end of trading instead.


title: Missi Pyle

text: She was part of a country music group with actress Shawnee Smith called Smith & Pyle. The two actresses met while filming an ABC comedy pilot titled "Traveling in Packs". The band started after Smith invited Pyle to join her in attending the Coachella Valley Music and Arts Festival. While stuck in traffic, Pyle talked about her dream to be a rock star and Smith agreed to form a band with her. Their first album, "It's OK to Be Happy", was released digitally through iTunes and Amazon.com in July 2008. The debut album was recorded in Joshua Tree, California at Rancho De La Luna and was produced by Chris Goss. The two actresses have also become business partners and formed their own record label called Urban Prairie Records, under which "It's OK to be Happy" was released. According to an interview Smith did with pretty-scary.net in August 2008, a Smith & Pyle television or webisode series might be in the works. Smith also mentioned the idea of a series on Fangoria radio with Dee Snider. Currently there are three videos posted on YouTube that show the making of the record. They also have a 10-minute "making of" video on Vimeo called "Smith & Pyle: Desert Sessions".


title: Missi Pyle

text: Pyle married naturalist Casey Anderson on September 12, 2008. The wedding was country-western themed and took place in Montana. Wedding guests included bandmate Shawnee Smith and comedian Steve Agee. Anderson's pet grizzly bear, Brutus, served as his best man in the ceremony. Pyle and Anderson appeared in "Grizzly Dogs", the October 22, 2010, episode of "Dog Whisperer", in which they sought the help of Cesar Millan to rehabilitate their dog. Pyle confirmed their breakup in 2013. She has one daughter.


title: Singular value decomposition

text: In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any formula_1 matrix. It is related to the polar decomposition.


title: Singular value decomposition

text: Specifically, the singular value decomposition of an formula_1 complex matrix is a factorization of the form formula_3 where is an formula_4 complex unitary matrix, formula_5 is an formula_1 rectangular diagonal matrix with non-negative real numbers on the diagonal, is an formula_7 complex unitary matrix, and formula_8 is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted formula_9


title: Singular value decomposition

text: The diagonal entries formula_10 of formula_5 are uniquely determined by and are known as the singular values of . The number of non-zero singular values is equal to the rank of . The columns of and the columns of are called left-singular vectors and right-singular vectors of , respectively. They form two sets of orthonormal bases and , and if they are sorted so that the singular values formula_12 with value zero are all in the highest-numbered columns (or rows), the singular value decomposition can be written as formula_13 where formula_14 is the rank of .


title: Singular value decomposition

text: The SVD is not unique. It is always possible to choose the decomposition so that the singular values formula_15 are in descending order. In this case, formula_16 (but not and ) is uniquely determined by .


title: Singular value decomposition

text: The term sometimes refers to the compact SVD, a similar decomposition formula_17 in which formula_5 is square diagonal of size formula_19, where formula_14 is the rank of , and has only the non-zero singular values. In this variant, is an formula_21 semi-unitary matrix and formula_22 is an formula_23 semi-unitary matrix, such that formula_24


title: Singular value decomposition

text: Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix. The SVD is also extremely useful in all areas of science, engineering, and statistics, such as signal processing, least squares fitting of data, and process control.


title: Singular value decomposition

text: In the special case when is an real square matrix, the matrices and can be chosen to be real matrices too. In that case, "unitary" is the same as "orthogonal". Then, interpreting both unitary matrices as well as the diagonal matrix, summarized here as , as a linear transformation of the space , the matrices and represent rotations or reflection of the space, while formula_16 represents the scaling of each coordinate by the factor . Thus the SVD decomposition breaks down any linear transformation of into a composition of three geometrical transformations: a rotation or reflection (), followed by a coordinate-by-coordinate scaling (formula_16), followed by another rotation or reflection ().


title: Singular value decomposition

text: In particular, if has a positive determinant, then and can be chosen to be both rotations with reflections, or both rotations without reflections. If the determinant is negative, exactly one of them will have a reflection. If the determinant is zero, each can be independently chosen to be of either type.


title: Singular value decomposition

text: If the matrix is real but not square, namely with , it can be interpreted as a linear transformation from to . Then and can be chosen to be rotations/reflections of and , respectively; and formula_16, besides scaling the first formula_28 coordinates, also extends the vector with zeros, i.e. removes trailing coordinates, so as to turn into .


title: Singular value decomposition

text: As shown in the figure, the singular values can be interpreted as the magnitude of the semiaxes of an ellipse in 2D. This concept can be generalized to -dimensional Euclidean space, with the singular values of any square matrix being viewed as the magnitude of the semiaxis of an -dimensional ellipsoid. Similarly, the singular values of any matrix can be viewed as the magnitude of the semiaxis of an -dimensional ellipsoid in -dimensional space, for example as an ellipse in a (tilted) 2D plane in a 3D space. Singular values encode magnitude of the semiaxis, while singular vectors encode direction. See below for further details.


title: Singular value decomposition

text: Since and are unitary, the columns of each of them form a set of orthonormal vectors, which can be regarded as basis vectors. The matrix maps the basis vector to the stretched unit vector . By the definition of a unitary matrix, the same is true for their conjugate transposes and , except the geometric interpretation of the singular values as stretches is lost. In short, the columns of , and are orthonormal bases. When the formula_29 is a positive-semidefinite Hermitian matrix, and are both equal to the unitary matrix used to diagonalize formula_29. However, when formula_29 is not positive-semidefinite and Hermitian but still diagonalizable, its eigendecomposition and singular value decomposition are distinct.


title: Singular value decomposition

text: Because and are unitary, we know that the columns of yield an orthonormal basis of and the columns of yield an orthonormal basis of (with respect to the standard scalar products on these spaces).


title: Singular value decomposition

text: The geometric content of the SVD theorem can thus be summarized as follows: for every linear map one can find orthonormal bases of and such that maps the -th basis vector of to a non-negative multiple of the -th basis vector of , and sends the left-over basis vectors to zero. With respect to these bases, the map is therefore represented by a diagonal matrix with non-negative real diagonal entries.


title: Singular value decomposition

text: To get a more visual flavor of singular values and SVD factorization – at least when working on real vector spaces – consider the sphere of radius one in . The linear map maps this sphere onto an ellipsoid in . Non-zero singular values are simply the lengths of the semi-axes of this ellipsoid. Especially when , and all the singular values are distinct and non-zero, the SVD of the linear map can be easily analyzed as a succession of three consecutive moves: consider the ellipsoid and specifically its axes; then consider the directions in sent by onto these axes. These directions happen to be mutually orthogonal. Apply first an isometry sending these directions to the coordinate axes of . On a second move, apply an endomorphism diagonalized along the coordinate axes and stretching or shrinking in each direction, using the semi-axes lengths of as stretching coefficients. The composition then sends the unit-sphere onto an ellipsoid isometric to . To define the third and last move, apply an isometry to this ellipsoid to obtain . As can be easily checked, the composition coincides with .


title: Singular value decomposition

text: The scaling matrix formula_16 is zero outside of the diagonal (grey italics) and one diagonal element is zero (red bold). Furthermore, because the matrices and are unitary, multiplying by their respective conjugate transposes yields identity matrices, as shown below. In this case, because and are real valued, each is an orthogonal matrix.


title: Singular value decomposition

text: A non-negative real number is a singular value for if and only if there exist unit-length vectors formula_41 in "K" and formula_42 in "K" such that


title: Singular value decomposition

text: The vectors formula_41 and formula_42 are called left-singular and right-singular vectors for , respectively.


title: Singular value decomposition

text: the diagonal entries of formula_16 are equal to the singular values of . The first columns of and are, respectively, left- and right-singular vectors for the corresponding singular values. Consequently, the above theorem implies that:


title: Singular value decomposition

text: A singular value for which we can find two left (or right) singular vectors that are linearly independent is called "degenerate". If formula_48 and formula_49 are two left-singular vectors which both correspond to the singular value σ, then any normalized linear combination of the two vectors is also a left-singular vector corresponding to the singular value σ. The similar statement is true for right-singular vectors. The number of independent left and right-singular vectors coincides, and these singular vectors appear in the same columns of and corresponding to diagonal elements of formula_16 all with the same value "σ".


title: Singular value decomposition

text: As an exception, the left and right-singular vectors of singular value 0 comprise all unit vectors in the cokernel and kernel, respectively, of , which by the rank–nullity theorem cannot be the same dimension if . Even if all singular values are nonzero, if then the cokernel is nontrivial, in which case is padded with orthogonal vectors from the cokernel. Conversely, if , then is padded by orthogonal vectors from the kernel. However, if the singular value of 0 exists, the extra columns of or already appear as left or right-singular vectors.


title: Singular value decomposition

text: Non-degenerate singular values always have unique left- and right-singular vectors, up to multiplication by a unit-phase factor "e" (for the real case up to a sign). Consequently, if all singular values of a square matrix are non-degenerate and non-zero, then its singular value decomposition is unique, up to multiplication of a column of by a unit-phase factor and simultaneous multiplication of the corresponding column of by the same unit-phase factor.


title: Singular value decomposition

text: In general, the SVD is unique up to arbitrary unitary transformations applied uniformly to the column vectors of both and spanning the subspaces of each singular value, and up to arbitrary unitary transformations on vectors of and spanning the kernel and cokernel, respectively, of .


title: Singular value decomposition

text: The singular value decomposition is very general in the sense that it can be applied to any matrix, whereas eigenvalue decomposition can only be applied to diagonalizable matrices. Nevertheless, the two decompositions are related.


title: Singular value decomposition

text: The right-hand sides of these relations describe the eigenvalue decompositions of the left-hand sides. Consequently:


title: Singular value decomposition

text: In the special case that is a normal matrix, which by definition must be square, the spectral theorem says that it can be unitarily diagonalized using a basis of eigenvectors, so that it can be written for a unitary matrix and a diagonal matrix with complex elements along the diagonal. When is positive semi-definite, will be non-negative real numbers so that the decomposition is also a singular value decomposition. Otherwise, it can be recast as an SVD by moving the phase of each to either its corresponding or . The natural connection of the SVD to non-normal matrices is through the polar decomposition theorem: , where is positive semidefinite and normal, and is unitary.


title: Singular value decomposition

text: Thus, except for positive semi-definite matrices, the eigenvalue decomposition and SVD of , while related, differ: the eigenvalue decomposition is , where is not necessarily unitary and is not necessarily positive semi-definite, while the SVD is , where formula_16 is diagonal and positive semi-definite, and and are unitary matrices that are not necessarily related except through the matrix . While only non-defective square matrices have an eigenvalue decomposition, any formula_54 matrix has a SVD.


title: Singular value decomposition

text: The singular value decomposition can be used for computing the pseudoinverse of a matrix. (Various authors use different notation for the pseudoinverse; here we use .) Indeed, the pseudoinverse of the matrix with singular value decomposition is


title: Singular value decomposition

text: where is the pseudoinverse of , which is formed by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix. The pseudoinverse is one way to solve linear least squares problems.


title: Singular value decomposition

text: A set of homogeneous linear equations can be written as for a matrix and vector . A typical situation is that is known and a non-zero is to be determined which satisfies the equation. Such an belongs to 's null space and is sometimes called a (right) null vector of . The vector can be characterized as a right-singular vector corresponding to a singular value of that is zero. This observation means that if is a square matrix and has no vanishing singular value, the equation has no non-zero as a solution. It also means that if there are several vanishing singular values, any linear combination of the corresponding right-singular vectors is a valid solution. Analogously to the definition of a (right) null vector, a non-zero satisfying , with denoting the conjugate transpose of , is called a left null vector of .


title: Singular value decomposition

text: A total least squares problem seeks the vector that minimizes the 2-norm of a vector under the constraint . The solution turns out to be the right-singular vector of corresponding to the smallest singular value.


title: Singular value decomposition

text: Another application of the SVD is that it provides an explicit representation of the range and null space of a matrix . The right-singular vectors corresponding to vanishing singular values of span the null space of and the left-singular vectors corresponding to the non-zero singular values of span the range of . For example, in the above example the null space is spanned by the last two rows of and the range is spanned by the first three columns of .


title: Singular value decomposition

text: As a consequence, the rank of equals the number of non-zero singular values which is the same as the number of non-zero diagonal elements in formula_16. In numerical linear algebra the singular values can be used to determine the "effective rank" of a matrix, as rounding error may lead to small but non-zero singular values in a rank deficient matrix. Singular values beyond a significant gap are assumed to be numerically equivalent to zero.


title: Singular value decomposition

text: Some practical applications need to solve the problem of approximating a matrix with another matrix formula_56, said to be truncated, which has a specific rank . In the case that the approximation is based on minimizing the Frobenius norm of the difference between and formula_56 under the constraint that formula_58, it turns out that the solution is given by the SVD of , namely


title: Singular value decomposition

text: where formula_60 is the same matrix as formula_16 except that it contains only the largest singular values (the other singular values are replaced by zero). This is known as the Eckart–Young theorem, as it was proved by those two authors in 1936 (although it was later found to have been known to earlier authors; see ).


title: Singular value decomposition

text: The SVD can be thought of as decomposing a matrix into a weighted, ordered sum of separable matrices. By separable, we mean that a matrix can be written as an outer product of two vectors , or, in coordinates, formula_62. Specifically, the matrix can be decomposed as


title: Singular value decomposition

text: Here and are the -th columns of the corresponding SVD matrices, are the ordered singular values, and each is separable. The SVD can be used to find the decomposition of an image processing filter into separable horizontal and vertical filters. Note that the number of non-zero is exactly the rank of the matrix.


title: Singular value decomposition

text: Separable models often arise in biological systems, and the SVD factorization is useful to analyze such systems. For example, some visual area V1 simple cells' receptive fields can be well described by a Gabor filter in the space domain multiplied by a modulation function in the time domain. Thus, given a linear filter evaluated through, for example, reverse correlation, one can rearrange the two spatial dimensions into one dimension, thus yielding a two-dimensional filter (space, time) which can be decomposed through SVD. The first column of in the SVD factorization is then a Gabor while the first column of represents the time modulation (or vice versa). One may then define an index of separability


title: Singular value decomposition

text: which is the fraction of the power in the matrix M which is accounted for by the first separable matrix in the decomposition.


title: Singular value decomposition

text: It is possible to use the SVD of a square matrix to determine the orthogonal matrix closest to . The closeness of fit is measured by the Frobenius norm of . The solution is the product . This intuitively makes sense because an orthogonal matrix would have the decomposition where is the identity matrix, so that if then the product amounts to replacing the singular values with ones. Equivalently, the solution is the unitary matrix of the Polar Decomposition in either order of stretch and rotation, as described above.


title: Singular value decomposition

text: A similar problem, with interesting applications in shape analysis, is the orthogonal Procrustes problem, which consists of finding an orthogonal matrix which most closely maps to . Specifically,


title: Singular value decomposition

text: The Kabsch algorithm (called Wahba's problem in other fields) uses SVD to compute the optimal rotation (with respect to least-squares minimization) that will align a set of points with a corresponding set of points. It is used, among other applications, to compare the structures of molecules.


title: Singular value decomposition

text: The SVD and pseudoinverse have been successfully applied to signal processing, image processing and big data (e.g., in genomic signal processing).


title: Singular value decomposition

text: The SVD is also applied extensively to the study of linear inverse problems and is useful in the analysis of regularization methods such as that of Tikhonov. It is widely used in statistics, where it is related to principal component analysis and to correspondence analysis, and in signal processing and pattern recognition. It is also used in output-only modal analysis, where the non-scaled mode shapes can be determined from the singular vectors. Yet another usage is latent semantic indexing in natural-language text processing.


title: Singular value decomposition

text: In general numerical computation involving linear or linearized systems, there is a universal constant that characterizes the regularity or singularity of a problem, which is the system's "condition number" formula_67. It often controls the error rate or convergence rate of a given computational scheme on such systems.


title: Singular value decomposition

text: The SVD also plays a crucial role in the field of quantum information, in a form often referred to as the Schmidt decomposition. Through it, states of two quantum systems are naturally decomposed, providing a necessary and sufficient condition for them to be entangled: if the rank of the formula_16 matrix is larger than one.