Split (1)

train · 2.1M rows

title stringlengths 1 149 ⌀	section stringlengths 1 1.9k ⌀	text stringlengths 13 73.5k
Momoiro Clover Z	History	2019–2021: Self-Titled Album On May 17, 2019, Momoiro Clover Z released their self-titled fifth studio album, their first studio album to not feature Momoka Ariyasu and their first as a four-member group. In 2021, they performed the theme song for the Sailor Moon Eternal movie. 2021–2022: Shukuten 2023–present: Upcoming Seventh album studio
Momoiro Clover Z	Music style	The band's songs are intentionally ridiculous "hyperactive J-pop numbers". Their live performances are heavily choreographed and feature acrobatic stunts. The group is noted for their "anarchic energy" that is similar to that of punk bands. The response from the audience has been characterised as "seismic".Some of Momoiro Clover's works are quite complex, switching from one musical style to another during one song and connecting "seemingly unconnected melodies". The group has worked with many noted songwriters and musicians, belonging to different genres of music, from pop to punk and heavy metal. Overall, the group and its music has been noted as progressive and forward-thinking. Ian Martin from The Japan Times dubbed Momoiro Clover "a pop group who provoke squealing, teenage admiration from punks, indie kids, noise musicians and heavy-psychedelic longhairs throughout the Japanese underground music scene". Momoiro Clover "is known for upbeat tunes, eccentric choreography and the members' costumes". A music critic from The Japan Times cites Momoiro Clover as an example of "a seamless integration of personality, image, and music, with each element mutually complementary".
Momoiro Clover Z	Discography	Battle and Romance (2011) 5th Dimension (2013) Amaranthus (2016) Hakkin no Yoake (2016) Momoiro Clover Z (2019) Shukuten (2022)
Momoiro Clover Z	Collaboration	Momoiro Clover Z have collaborated with overseas artists. Kiss released a collaboration single with Momoiro Clover Z, titled "Yume no Ukiyo ni Saitemina" (January 2015). Lady Gaga designated Momoiro Clover Z for an opening act of her concert (August 2014). Marty Friedman participated as a guitarist in "Mōretsu Uchū Kōkyōkyoku Dai 7 Gakushō "Mugen no Ai"" (March 2012) and "Moon Pride" (July 2014). Yngwie Malmsteen participated as a guitarist in "Mōretsu Uchū Kōkyōkyoku Dai 7 Gakushō "Mugen no Ai" -Emperor Style-" (June 2014).The group sings the theme music for the following anime.
Momoiro Clover Z	Collaboration	Yosuga no Sora - "Pinky Jones" (November 2010) Dragon Crisis! - "Mirai Bowl" (January 2011) Bodacious Space Pirates - "Mōretsu Uchū Kōkyōkyoku Dai 7 Gakushō "Mugen no Ai‍"" (March 2012) Pokémon - "Mite Mite Kocchichi" (June 2012) Joshiraku - "Nippon Egao Hyakkei" (June 2012) in collaboration with Yoshida Brothers Pretty Guardian Sailor Moon Crystal - "Moon Pride", "Moon Rainbow" (月虹, Gekkō) (July 2014)Pretty Guardian Sailor Moon Crystal Season III - "Fall in Love with a New Moon" (ニュームーンに恋して, Nyū Mūn ni Koishite) (June 2016) Pretty Guardian Sailor Moon Eternal: The Movie - "Moon Color Chainon" (月色Chainon, Tsukiiro Chainon) (January 2021, with main voice actresses: Kotono Mitsuishi, Hisako Kanemoto, Rina Sato, Ami Koshimizu, and Shizuka Ito) Dragon Ball Z: Resurrection 'F' - "Z no Chikai (April 2015) Crayon Shin-chan: Burst Serving! Kung Fu Boys ~Ramen Rebellion~ - "Xiao Yi Xiao" (April 2018).
Momoiro Clover Z	Overseas performances	Japan Media Arts Festival 2011 in Dortmund, Germany (September 9) Hari Belia Negara 2012 in Putrajaya, Malaysia (May 26) Japan Expo 2012 in Paris, France (July 5) Anime Expo 2015 in Los Angeles, California (July 2) Japan SAKURA Festival 2016 in Hanoi, Vietnam (April 16, 17) Bilibili Macro Link 2016 in Shanghai, China (July 23) Trans America Ultra Live 2016 in Hawaii, Los Angeles and New York (November 15–19)
Momoiro Clover Z	Awards	In 2012, their first album Battle and Romance won the CD Shop Award as the best CD of the previous year as voted by music shop salesclerks from all over Japan. It was the first time an idol (group) got this prize.
Momoiro Clover Z	Filmography	Shirome (シロメ) - August 2010Horror film. During filming, the girls were reportedly led to believe they were participating in a documentary about an urban legend and that everything happening was genuine.The Citizen Police 69 (市民ポリス69) - March 2011 Ninifuni - February 2012 Momodora (ももドラ momo+dra) - February 20125-episode internet drama omnibus film.Maku ga Agaru (幕が上がる) - February 2015The five members played leading roles and later won Japan Academy Prize.
Priest (tool)	Priest (tool)	A priest (poacher's, game warden's or angler's "priest"), sometimes called a fish bat, or "persuader" is a tool for killing game or fish. The name "priest" comes from the notion of administering the "last rites" to the fish or game. Anglers often use priests to quickly kill fish.
Priest (tool)	Description	Priests usually come in the form of a heavy metal head attached to a metal or wooden stick. The small baton is a blunt instrument used for quickly killing fish or game. Early versions are made of lignum vitae (Latin for "wood of life"), the densest hardwood. One example is described as "Lead filled head. Brass ring to handle. With large Head for dispatching Game. Size overall 14 inches long".
Priest (tool)	In culture	Identified as a "keeper's priest" the tool is a featured murder weapon in Series 12 of the BBC's Dalziel and Pascoe, Episodes 2 and 3, "Under Dark Stars", which left a round bruised mark on impact. Used as the murder weapon of convenience in Series 8 of the BBC's Father Brown, Episode 7, "The River Corrupted," thereby framing the owner of the tool. Tommy "Fishpriest" Barth (portrayed by Ethan Hawk) the main character in the "Fishpriest" podcast series, is nicknamed after his weapon of choice. "Poacher's Priest" is the name of a 2023 novel by Samuel Mills. In the story, the protagonist Odilio Brimble uses a priest to club a salmon in front of his horrified daughter.
Quadray coordinates	Quadray coordinates	Quadray coordinates, also known as caltrop, tetray or Chakovian coordinates, were developed by Darrel Jarmusch and others, as another take on simplicial coordinates, a coordinate system using a simplex or tetrahedron as its basis polyhedron.
Quadray coordinates	Geometric definition	The four basis (but not necessarily unit) vectors stem from the center of a regular tetrahedron and go to its four corners. Their coordinate addresses are (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) respectively. These may be positively scaled without rotation (e.g. negation) and linearly combined to span conventional XYZ space, with at least one of the four coordinates unneeded (set to zero).
Quadray coordinates	Pedagogical significance	A typical application might set the edges of the basis tetrahedron as unit. The tetrahedron itself may also be defined as the unit of volume (see below). The four quadrays may be linearly combined to provide integer coordinates for the inverse tetrahedron (0,1,1,1), (1,0,1,1), (1,1,0,1), (1,1,1,0), and for the cube, octahedron, rhombic dodecahedron and cuboctahedron of volumes 3, 4, 6 and 20 respectively, given the starting tetrahedron of unit volume.
Quadray coordinates	Pedagogical significance	For example, given A, B, C, D as (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) respectively, the vertices of an octahedron with the same edge length and volume four would be A + B, A + C, A + D, B + C, B + D, C + D or all eight permutations of {1,1,0,0}. The 12 permutations of {2,1,1,0} define the vertices of the volume 20 cuboctahedron centered at (0,0,0,0). These vectors point from any given sphere to its 12 surrounding neighbors in the cubic close packing (CCP), equivalently the IVM (isotropic vector matrix) in Synergetics. Therefore CCP ball centers all have non-negative integer coordinates.
Quadray coordinates	Pedagogical significance	If one now calls this volume "4D" as in "four-dimensional" or "four-directional" we have primed the pump for an understanding of R. Buckminster Fuller's "4D geometry," or Synergetics. In this American transcendentalist philosophy, the regular tetrahedron of edges one, as defined by four intertangent uni-radius balls, is taken as unit of volume. A set of familiar convex polyhedra, termed "the concentric hierarchy" is nested around it, per the above table, such that the cube has volume 3, the octahedron volume 4, rhombic dodecahedron volume 6, and cuboctahedron volume 20.
Wolfgang Straub	Wolfgang Straub	Wolfgang Straub (born 1969, in Waiblingen) is a Swiss lawyer and photographer.
Wolfgang Straub	Photographic works	His series of still lifes ‘Le dictionnaire des analphabètes’ deal with visual evidence of the paradoxical.His 'Enchanted Gardens' series deals with conveying an emotional content by means of altering forms of expression.Straub's works are present in several public and private collections.
Wolfgang Straub	Photographic publications	Enchanted Gardens, Wyss Bern/Museum Franz Gertsch Burgdorf 2010, ISBN 978-3-033-02263-8 and ISBN 978-3-7285-2010-4
Wolfgang Straub	Solo exhibitions	2009 Museum Franz Gertsch, Burgdorf 2010 Leica Gallery Switzerland, Biel
Types of cheese	Types of cheese	There are many different types of cheese. Cheeses can be grouped or classified according to criteria such as length of fermentation, texture, methods of production, fat content, animal milk, and country or region of origin. The method most commonly and traditionally used is based on moisture content, which is then further narrowed down by fat content and curing or ripening methods. The criteria may either be used singly or in combination, with no single method being universally used.The combination of types produces around 51 different varieties recognized by the International Dairy Federation, over 400 identified by Walter and Hargrove, over 500 by Burkhalter, and over 1,000 by Sandine and Elliker. Some attempts have been made to rationalise the classification of cheese; a scheme was proposed by Pieter Walstra that uses the primary and secondary starter combined with moisture content, and Walter and Hargrove suggested classifying by production methods. This last scheme results in 18 types, which are then further grouped by moisture content.
Types of cheese	Fresh and whey cheeses	The main factor in categorizing these cheeses is age. Fresh cheeses without additional preservatives can spoil in a matter of days.For these simplest cheeses, milk is curdled and drained, with little other processing. Examples include cottage cheese, cream cheese, curd cheese, farmer cheese, caș, chhena, fromage blanc, queso fresco, paneer, fresh goat's milk chèvre, Breingen-Tortoille, Irish Mellieriem Rochers and Belgian Mellieriem Rochers. Such cheeses are often soft and spreadable, with a mild flavour.Whey cheeses are fresh cheeses made from whey, a by-product from the process of producing other cheeses which would otherwise be discarded. Corsican brocciu, Italian ricotta, Romanian urda, Greek mizithra, Croatian skuta, Cypriot anari cheese, Himalayan chhurpi and Norwegian Brunost are examples. Brocciu is mostly eaten fresh, and is as such a major ingredient in Corsican cuisine, but it can also be found in an aged form.Some fresh cheeses such as fromage blanc and fromage frais (the latter differing from the former in that it contains live cultures) are commonly sold and consumed as desserts.
Types of cheese	Stretched curd cheeses	Stretched curd, for which the Italian term pasta filata is often used, is a group of cheeses where the hot curd is stretched, today normally mechanically, producing various effects. Many traditional pasta filata cheeses such as the Italian mozzarella and halloumi from the Eastern Mediterranean also fall into the fresh cheese category. Fresh curds are stretched and kneaded in hot water to form a ball of mozzarella, which in southern Italy is usually eaten within a few hours of being made. Stored in brine, it can easily be shipped, and it is known worldwide for its use on pizza. But not all stretch-curd cheeses are fresh; the Italian provolone, Ragusano, caciocavallo and many others are hard or semi-hard, and aged. Oaxaca cheese from Mexico is semi-hard, but not aged. Like the pressed cooked cheeses (below), all these are made using thermophilic lactic fermentation starters. Many of the various types of string cheese are made this way.
Types of cheese	Cooked pressed cheeses	Swiss-type cheeses, also known as Alpine cheeses, are a group of hard or semi-hard cheeses with a distinct character, whose origins lie in the Alps of Europe, although they are now eaten and imitated in most cheesemaking parts of the world. They are classified as "cooked", meaning made using thermophilic lactic fermentation starters, incubating the curd with a period at a high temperature of 45°C or more. Since they are later pressed to expel excess moisture, the group are also described as "'cooked pressed cheeses'", fromages à pâte pressée cuite in French. Their distinct character arose from the requirements of cheese made in the summer on high Alpine grasslands (alpage in French), and then transported with the cows down to the valleys in the winter, in the historic culture of Alpine transhumance. Traditionally the cheeses were made in large rounds or "wheels" with a hard rind, and were robust enough for both keeping and transporting.The best known cheeses of the type, all made from cow's milk, include the Swiss Emmental, Gruyère and Appenzeller, as well as the French Beaufort and Comté (from the Jura Mountains, near the Alps). Both countries have many other traditional varieties, as do the Alpine regions of Austria (Alpkäse) and Italy (Asiago), though these have not achieved the same degree of intercontinental fame. Most global modern production is industrial, and usually made in rectangular blocks, and by wrapping in plastic no rind is allowed to form. Historical production was all with "raw" milk, although the periods of high heat in making largely controlled unwelcome bacteria, but modern production may use thermized or pasteurized milk.The general eating characteristics of the Alpine cheeses are a firm but still elastic texture, flavor that is not sharp, acidic or salty, but rather nutty and buttery. When melted, which they often are in cooking, they are "gooey", and "slick, stretchy and runny".Another related group of cooked pressed cheeses is the very hard Italian "grana" cheeses; the best known are Parmesan and Grana Padano. Although their origins lie in the flat and (originally) swampy Po Valley, they share the broad Alpine cheesemaking process, and began after local monasteries initiated drainage programmes from the 11th century onwards. These were Benedictine and Cistercian monasteries, both with sister-houses benefiting from Alpine cheesemaking. They seem to have borrowed their techniques from them, but produced very different cheeses, using much more salt, and less heating, which suited the local availability of materials.
Types of cheese	Moisture: soft to hard	Categorizing cheeses by moisture content or firmness is a common but inexact practice. The lines between soft, semi-soft, semi-hard and hard are arbitrary, and many types of cheese are made in softer or firmer variants. The factor that controls cheese hardness is moisture content, which depends on the pressure with which it is packed into molds, and upon aging time.
Types of cheese	Moisture: soft to hard	Soft cheese Cream cheeses are not matured. Brie and Neufchâtel are soft-type cheeses that mature for no more than a month. Neufchâtel can be sold after 10 days of maturation. Semi-soft cheese Semi-soft cheeses, and the sub-group Monastery cheeses, have a high moisture content and tend to be mild-tasting. Well-known varieties include Havarti, Munster, Port Salut and Butterkäse.
Types of cheese	Moisture: soft to hard	Medium-hard cheese Cheeses that range in texture from semi-soft to firm include Swiss-style cheeses such as Emmental and Gruyère. The same bacteria that give such cheeses their eyes also contribute to their aromatic and sharp flavours. Other semi-soft to firm cheeses include Gouda, Edam, Jarlsberg, Cantal, and Kashkaval/Cașcaval. Cheeses of this type are ideal for melting and are often served on toast for quick snacks or simple meals.
Types of cheese	Moisture: soft to hard	Semi-hard cheese Harder cheeses have a lower moisture content than softer cheeses. They are generally packed into molds under more pressure and aged for a longer time than the soft cheeses. Cheeses that are classified as semi-hard to hard include the familiar Cheddar, originating in the village of Cheddar in England but now used as a generic term for this style of cheese, of which varieties are imitated worldwide and are marketed by strength or the length of time they have been aged.
Types of cheese	Moisture: soft to hard	Cheddar is one of a family of semi-hard or hard cheeses (including Cheshire and Gloucester), whose curd is cut, gently heated, piled, and stirred before being pressed into forms. Colby and Monterey Jack are similar but milder cheeses; their curd is rinsed before it is pressed, washing away some acidity and calcium. A similar curd-washing takes place when making the Dutch cheeses Edam and Gouda.
Types of cheese	Moisture: soft to hard	Hard cheese Hard cheeses—grating cheeses such as Grana Padano, Parmesan or Pecorino—are quite firmly packed into large forms and aged for months or years.
Types of cheese	Source of milk	Some cheeses are categorized by the source of the milk used to produce them or by the added fat content of the milk from which they are produced. While most of the world's commercially available cheese is made from cow's milk, many parts of the world also produce cheese from goats and sheep. Examples include Roquefort (produced in France) and Pecorino (produced in Italy) from ewe's milk. One farm in Sweden also produces cheese from moose's milk. Sometimes cheeses marketed under the same name are made from milk of different species—feta cheeses, for example, are made from sheep's milk in Greece. Pule cheese are made from Balkan donkey milk and goat's milk (produced in Serbia).
Types of cheese	Source of milk	Double cream cheeses are soft cheeses of cows' milk enriched with cream so that their fat in dry matter (FDM or FiDM) content is 60–75%; triple cream cheeses are enriched to at least 75%.
Types of cheese	Mold	There are three main categories of cheese in which the presence of mold is an important feature: soft-ripened cheeses, washed-rind cheeses and blue cheeses.
Types of cheese	Mold	Soft-ripened Soft-ripened cheeses begin firm and rather chalky in texture, but are aged from the exterior inwards by exposing them to mold. The mold may be a velvety bloom of P. camemberti that forms a flexible white crust and contributes to the smooth, runny, or gooey textures and more intense flavours of these aged cheeses. Brie and Camembert, the most famous of these cheeses, are made by allowing white mold to grow on the outside of a soft cheese for a few days or weeks. Goat's milk cheeses are often treated in a similar manner, sometimes with white molds (Chèvre-Boîte) and sometimes with blue.
Types of cheese	Mold	Washed-rind Washed-rind cheeses are soft in character and ripen inwards like those with white molds; however, they are treated differently. Washed-rind cheeses are periodically cured in a solution of saltwater brine or mold-bearing agents that may include beer, wine, brandy and spices, making their surfaces amenable to a class of bacteria (Brevibacterium linens, the reddish-orange smear bacteria) that impart pungent odors and distinctive flavours and produce a firm, flavourful rind around the cheese. Washed-rind cheeses can be soft (Limburger), semi-hard, or hard (Appenzeller). The same bacteria can also have some effect on cheeses that are simply ripened in humid conditions, like Camembert. The process requires regular washings, particularly in the early stages of production, making it quite labor-intensive compared to other methods of cheese production.
Types of cheese	Mold	Smear-ripened S-rind cheeses are also smear-ripened with solutions of bacteria or fungi (most commonly Brevibacterium linens, Debaryomyces hansenii or Geotrichum candidum), which usually gives them a stronger flavor as the cheese matures. In some cases, older cheeses are smeared on young cheeses to transfer the microorganisms. Many, but not all, of these cheeses have a distinctive pinkish or orange coloring of the exterior. Unlike with other washed-rind cheeses, the washing is done to ensure uniform growth of desired bacteria or fungi and to prevent the growth of undesired molds. Examples of smear-ripened cheeses include Munster and Port Salut.
Types of cheese	Mold	Blue So-called blue cheese is created by inoculating a cheese with Penicillium roqueforti or Penicillium glaucum. This is done while the cheese is still in the form of loosely pressed curds, and may be further enhanced by piercing a ripening block of cheese with skewers in an atmosphere in which the mold is prevalent. The mold grows within the cheese as it ages. These cheeses have distinct blue veins, which gives them their name and, often, assertive flavours. The molds range from pale green to dark blue, and may be accompanied by white and crusty brown molds. Their texture can be soft or firm. Some of the most renowned cheeses in this type include Roquefort, Gorgonzola and Stilton.
Types of cheese	Granular	Granular cheese is a type of cheese produced by repeatedly stirring and draining a mixture of curd and whey. It can refer to a wide variety of cheeses, including the grana cheeses such as Parmigiano-Reggiano.
Types of cheese	Brined	Brined or pickled cheese is matured in a solution of brine in an airtight or semi-permeable container. This process gives the cheese good stability, inhibiting bacterial growth even in hot environments. Brined cheeses may be soft or hard, varying in moisture content, and in color and flavor, according to the type of milk used. All will be rindless, and generally taste clean, salty and acidic when fresh, developing some piquancy when aged, and most will be white. Varieties of brined cheese include bryndza, feta, halloumi, sirene, and telemea. Brined cheese is the main type of cheese produced and eaten in the Middle East and Mediterranean areas.
Types of cheese	Processed	Processed cheese is made from traditional cheese and emulsifying salts, often with the addition of milk, more salt, preservatives, and food coloring. Its texture is consistent, and it melts smoothly. It is sold packaged and either pre-sliced or unsliced, in several varieties. Some are sold as sausage-like logs and chipolatas (mostly in Germany and the US), and some are molded into the shape of animals and objects. It is also available as "Easy Cheese", a product distributed by Mondelez International, that is packaged in aerosol cans and available in some countries.Some, if not most, varieties of processed cheese are made using a combination of real cheese waste (which is steam-cleaned, boiled and further processed), whey powders, and various mixtures of vegetable oils, palm oils or fats. Some processed-cheese slices contain as little as two to six percent cheese; some have smoke flavours added.
Multiplex (juggling)	Multiplex (juggling)	Multiplexing is a juggling trick or form of toss juggling where more than one ball is in the hand at the time of the throw. The opposite, a squeeze catch, is when more than one ball is caught in the hand simultaneously on the same beat. If a multiplex throw were time-reversed, it would be a squeeze catch.
Multiplex (juggling)	Terminology	Number of props Multiplex throws are given different names depending on the number of balls used, for example a one-ball throw (with one ball held) would be called a uniplex, a two-ball throw would be called a duplex, and a three-ball throw, a triplex. A four and a five-ball throw would be called a quadruplex and a quintuplex, respectively. Throw types Multiplex throws are generally grouped into different categories: Stack, Split, Cut, and Slice. Stacked multiplex throws involve throwing both balls from one hand and catching them both in the same or other hand. Split multiplex throws, as the name suggests, involve throwing both balls from one hand, "splitting" them in the air, and catching them in separate hands. Cut multiplex throws involve throwing both balls to the same or other hand like a stacked multiplex but in a staggered fashion so the bottom ball of the duplex is caught, and re-thrown before the top ball is caught. These are used in the Shower Explosion family of multiplex tricks.
Multiplex (juggling)	Terminology	Sliced multiplex throws involve throwing both balls with one ball going directly to the opposite hand as a pass. This throw is usually made with the catching hand directly above the throwing hand so that when the throw is made, one ball goes straight up into the catching hand, with little to no air time, while the remaining ball is caught on a later beat.
Multiplex (juggling)	Terminology	In the case of triplexes, a split can result in one or two balls being caught in the opposite hand. An 'inside' split triplex denotes one ball being caught in the opposite hand, due to the single ball being on the inside of the triplex, and an 'outside' split triplex denotes two balls being caught in the opposite hand. A cut and split multiplex can be combined in a triplex and this is referred to as a cut-split triplex indicating that both types of throw are involved.
Multiplex (juggling)	Terminology	Air position Multiplex throws can be further classified depending upon the position of the balls in the air after the throw is made. In the case of duplexes, two balls side by side is a horizontal duplex, and two balls one above the other is a vertical duplex. This terminology can be applied to either the stack, split or cut duplex types of throw. So, a vertical stacked duplex refers to two balls being thrown together, one above the other in the air, and caught together in the same or other hand. A horizontal split duplex refers to two balls being thrown together, side by side in the air, and caught in separate hands. In the case of triplexes, three balls one above the other is a vertical triplex and three balls in a triangle pattern is a triangle triplex.
Multiplex (juggling)	Notation	Siteswap notation is a way of writing down a key feature of juggling patterns: the order in which the balls are thrown. Multiplex throws are notated inside square brackets [ ]. For example, two balls held in the hand for a beat is notated [22] while [54] represents a split duplex where one ball is rethrown five beats later and the other ball four beats later.
Multiplex (juggling)	Notation	When working out the average of a multiplex siteswap, to determine the number of balls in the pattern, the throws inside the brackets are added together but treated as one throw. So, [43]23 = [4+3]+2+3 = 12. 12 / 3 (throws) = a 4-ball pattern. It is possible to combine two siteswaps to make a new trick. The three-ball siteswap '423' and the two-ball siteswap '330' combined give the five-ball siteswap [43][32]3. Since siteswaps can be rotated, 330 can also be read as '033' and '303' and thus, when combined with 423, give the five-ball siteswaps 4[32][33] and [43]2[33] respectively. If two siteswaps of differing lengths are combined, for instance 423 and 31, the length of the new siteswap can be determined by multiplying the two lengths together. Using the aforementioned siteswaps as an example, 423 (length 3) and 31 (length 2) give the siteswap [43][21][33][41][32][31] (length 6).
Multiplex (juggling)	Styles	Claymotion Claymotion is a style of multiplex juggling that was developed by British juggler Richard Clay in the early 1990s and was first given the name 'Claymotion' by Erica Kelch-Slesnick in 1997. Claymotion juggling is a sub-category of multiplex juggling that has a start-stop rhythm to it, not unlike cigar box juggling, so there are times when there are no balls in the air. Emphasis is placed on the graceful movements of the arms and so throws are typically low and controlled.
Life support (aviation)	Life support (aviation)	Life support, or aircrew life support, in aviation, is the field centered on, and related technologies used in, ensuring the safety of aircrew, particularly military aviation. This includes safety equipment capable of helping them survive in the case of a crash, accident, or malfunction. Life support functions and technology are also prominent in the field of human spaceflight.
Audio Interchange File Format	Audio Interchange File Format	Audio Interchange File Format (AIFF) is an audio file format standard used for storing sound data for personal computers and other electronic audio devices. The format was developed by Apple Inc. in 1988 based on Electronic Arts' Interchange File Format (IFF, widely used on Amiga systems) and is most commonly used on Apple Macintosh computer systems.
Audio Interchange File Format	Audio Interchange File Format	The audio data in most AIFF files is uncompressed pulse-code modulation (PCM). This type of AIFF file uses much more disk space than lossy formats like MP3—about 10 MB for one minute of stereo audio at a sample rate of 44.1 kHz and a bit depth of 16 bits. There is also a compressed variant of AIFF known as AIFF-C or AIFC, with various defined compression codecs.
Audio Interchange File Format	Audio Interchange File Format	In addition to audio data, AIFF can include loop point data and the musical note of a sample, for use by hardware samplers and musical applications. The file extension for the standard AIFF format is .aiff or .aif. For the compressed variants it is supposed to be .aifc, but .aiff or .aif are accepted as well by audio applications supporting the format.
Audio Interchange File Format	AIFF on macOS	With the development of the OS X operating system now known as macOS, Apple created a new type of AIFF which is, in effect, an alternative little-endian byte order format.Because the AIFF architecture has no provision for alternative byte order, Apple used the existing AIFF-C compression architecture, and created a "pseudo-compressed" codec called sowt (twos spelled backwards). The only difference between a standard AIFF file and an AIFF-C/sowt file is the byte order; there is no compression involved at all.Apple uses this new little-endian AIFF type as its standard on macOS. When a file is imported to or exported from iTunes in "AIFF" format, it is actually AIFF-C/sowt that is being used. When audio from an audio CD is imported by dragging to the macOS Desktop, the resulting file is also an AIFF-C/sowt. In all cases, Apple refers to the files simply as "AIFF", and uses the ".aiff" extension.
Audio Interchange File Format	AIFF on macOS	For the vast majority of users this technical situation is completely unnoticeable and irrelevant. The sound quality of standard AIFF and AIFF-C/sowt are identical, and the data can be converted back and forth without loss. Users of older audio applications, however, may find that an AIFF-C/sowt file will not play, or will prompt the user to convert the format on opening, or will play as static.
Audio Interchange File Format	AIFF on macOS	All traditional AIFF and AIFF-C files continue to work normally on macOS, and many third-party audio applications as well as hardware continue to use the standard AIFF big-endian byte order.
Audio Interchange File Format	AIFF Apple Loops	Apple has also created another recent extension to the AIFF format in the form of Apple Loops used by GarageBand and Logic Pro, which allows the inclusion of data for pitch and tempo shifting by an application in the more common variety, and MIDI-sequence data and references to GarageBand playback instruments in another variety. Apple Loops use either the .aiff (or .aif) or .caf extension regardless of type.
Audio Interchange File Format	Data format	An AIFF file is divided into a number of chunks. Each chunk is identified by a chunk ID more broadly referred to as FourCC. Types of chunks found in AIFF files: Common Chunk (required) Sound Data Chunk (required) Marker Chunk Instrument Chunk Comment Chunk Name Chunk Author Chunk Copyright Chunk Annotation Chunk Audio Recording Chunk MIDI Data Chunk Application Chunk ID3 Chunk
Audio Interchange File Format	Metadata	AIFF files can store metadata in Name, Author, Comment, Annotation, and Copyright chunks. An ID3v2 tag chunk can also be embedded in AIFF files, as well as an Application Chunk with Extensible Metadata Platform (XMP) data in it.
Audio Interchange File Format	Common compression types	AIFF supports only uncompressed PCM data. AIFF-C also supports compressed audio formats, which can be specified in the "COMM" chunk. The compression type is "NONE" for PCM audio data. The compression type is accompanied by a printable name. Common compression types and names include, but are not limited to:
Fault tree analysis	Fault tree analysis	Fault tree analysis (FTA) is a type of failure analysis in which an undesired state of a system is examined. This analysis method is mainly used in safety engineering and reliability engineering to understand how systems can fail, to identify the best ways to reduce risk and to determine (or get a feeling for) event rates of a safety accident or a particular system level (functional) failure. FTA is used in the aerospace, nuclear power, chemical and process, pharmaceutical, petrochemical and other high-hazard industries; but is also used in fields as diverse as risk factor identification relating to social service system failure. FTA is also used in software engineering for debugging purposes and is closely related to cause-elimination technique used to detect bugs.
Fault tree analysis	Fault tree analysis	In aerospace, the more general term "system failure condition" is used for the "undesired state" / top event of the fault tree. These conditions are classified by the severity of their effects. The most severe conditions require the most extensive fault tree analysis. These system failure conditions and their classification are often previously determined in the functional hazard analysis.
Fault tree analysis	Usage	Fault tree analysis can be used to: understand the logic leading to the top event / undesired state. show compliance with the (input) system safety / reliability requirements. prioritize the contributors leading to the top event- creating the critical equipment/parts/events lists for different importance measures monitor and control the safety performance of the complex system (e.g., is a particular aircraft safe to fly when fuel valve x malfunctions? For how long is it allowed to fly with the valve malfunction?). minimize and optimize resources. assist in designing a system. The FTA can be used as a design tool that helps to create (output / lower level) requirements. function as a diagnostic tool to identify and correct causes of the top event. It can help with the creation of diagnostic manuals / processes.
Fault tree analysis	History	Fault tree analysis (FTA) was originally developed in 1962 at Bell Laboratories by H.A. Watson, under a U.S. Air Force Ballistics Systems Division contract to evaluate the Minuteman I Intercontinental Ballistic Missile (ICBM) Launch Control System. The use of fault trees has since gained widespread support and is often used as a failure analysis tool by reliability experts. Following the first published use of FTA in the 1962 Minuteman I Launch Control Safety Study, Boeing and AVCO expanded use of FTA to the entire Minuteman II system in 1963–1964. FTA received extensive coverage at a 1965 System Safety Symposium in Seattle sponsored by Boeing and the University of Washington. Boeing began using FTA for civil aircraft design around 1966.Subsequently, within the U.S. military, application of FTA for use with fuses was explored by Picatinny Arsenal in the 1960s and 1970s. In 1976 the U.S. Army Materiel Command incorporated FTA into an Engineering Design Handbook on Design for Reliability. The Reliability Analysis Center at Rome Laboratory and its successor organizations now with the Defense Technical Information Center (Reliability Information Analysis Center, and now Defense Systems Information Analysis Center) has published documents on FTA and reliability block diagrams since the 1960s. MIL-HDBK-338B provides a more recent reference.In 1970, the U.S. Federal Aviation Administration (FAA) published a change to 14 CFR 25.1309 airworthiness regulations for transport category aircraft in the Federal Register at 35 FR 5665 (1970-04-08). This change adopted failure probability criteria for aircraft systems and equipment and led to widespread use of FTA in civil aviation. In 1998, the FAA published Order 8040.4, establishing risk management policy including hazard analysis in a range of critical activities beyond aircraft certification, including air traffic control and modernization of the U.S. National Airspace System. This led to the publication of the FAA System Safety Handbook, which describes the use of FTA in various types of formal hazard analysis.Early in the Apollo program the question was asked about the probability of successfully sending astronauts to the moon and returning them safely to Earth. A risk, or reliability, calculation of some sort was performed and the result was a mission success probability that was unacceptably low. This result discouraged NASA from further quantitative risk or reliability analysis until after the Challenger accident in 1986. Instead, NASA decided to rely on the use of failure modes and effects analysis (FMEA) and other qualitative methods for system safety assessments. After the Challenger accident, the importance of probabilistic risk assessment (PRA) and FTA in systems risk and reliability analysis was realized and its use at NASA has begun to grow and now FTA is considered as one of the most important system reliability and safety analysis techniques.Within the nuclear power industry, the U.S. Nuclear Regulatory Commission began using PRA methods including FTA in 1975, and significantly expanded PRA research following the 1979 incident at Three Mile Island. This eventually led to the 1981 publication of the NRC Fault Tree Handbook NUREG–0492, and mandatory use of PRA under the NRC's regulatory authority.
Fault tree analysis	History	Following process industry disasters such as the 1984 Bhopal disaster and 1988 Piper Alpha explosion, in 1992 the United States Department of Labor Occupational Safety and Health Administration (OSHA) published in the Federal Register at 57 FR 6356 (1992-02-24) its Process Safety Management (PSM) standard in 19 CFR 1910.119. OSHA PSM recognizes FTA as an acceptable method for process hazard analysis (PHA).
Fault tree analysis	History	Today FTA is widely used in system safety and reliability engineering, and in all major fields of engineering.
Fault tree analysis	Methodology	FTA methodology is described in several industry and government standards, including NRC NUREG–0492 for the nuclear power industry, an aerospace-oriented revision to NUREG–0492 for use by NASA, SAE ARP4761 for civil aerospace, MIL–HDBK–338 for military systems, IEC standard IEC 61025 is intended for cross-industry use and has been adopted as European Norm EN 61025. Any sufficiently complex system is subject to failure as a result of one or more subsystems failing. The likelihood of failure, however, can often be reduced through improved system design. Fault tree analysis maps the relationship between faults, subsystems, and redundant safety design elements by creating a logic diagram of the overall system.
Fault tree analysis	Methodology	The undesired outcome is taken as the root ('top event') of a tree of logic. For instance, the undesired outcome of a metal stamping press operation being considered might be a human appendage being stamped. Working backward from this top event it might be determined that there are two ways this could happen: during normal operation or during maintenance operation. This condition is a logical OR. Considering the branch of the hazard occurring during normal operation, perhaps it is determined that there are two ways this could happen: the press cycles and harms the operator, or the press cycles and harms another person. This is another logical OR. A design improvement can be made by requiring the operator to press two separate buttons to cycle the machine—this is a safety feature in the form of a logical AND. The button may have an intrinsic failure rate—this becomes a fault stimulus that can be analyzed. When fault trees are labeled with actual numbers for failure probabilities, computer programs can calculate failure probabilities from fault trees. When a specific event is found to have more than one effect event, i.e. it has impact on several subsystems, it is called a common cause or common mode. Graphically speaking, it means this event will appear at several locations in the tree. Common causes introduce dependency relations between events. The probability computations of a tree which contains some common causes are much more complicated than regular trees where all events are considered as independent. Not all software tools available on the market provide such capability.
Fault tree analysis	Methodology	The tree is usually written out using conventional logic gate symbols. A cut set is a combination of events, typically component failures, causing the top event. If no event can be removed from a cut set without failing to cause the top event, then it is called a minimal cut set.
Fault tree analysis	Methodology	Some industries use both fault trees and event trees (see Probabilistic Risk Assessment). An event tree starts from an undesired initiator (loss of critical supply, component failure etc.) and follows possible further system events through to a series of final consequences. As each new event is considered, a new node on the tree is added with a split of probabilities of taking either branch. The probabilities of a range of 'top events' arising from the initial event can then be seen.
Fault tree analysis	Methodology	Classic programs include the Electric Power Research Institute's (EPRI) CAFTA software, which is used by many of the US nuclear power plants and by a majority of US and international aerospace manufacturers, and the Idaho National Laboratory's SAPHIRE, which is used by the U.S. Government to evaluate the safety and reliability of nuclear reactors, the Space Shuttle, and the International Space Station. Outside the US, the software RiskSpectrum is a popular tool for fault tree and event tree analysis, and is licensed for use at more than 60% of the world's nuclear power plants for probabilistic safety assessment. Professional-grade free software is also widely available; SCRAM is an open-source tool that implements the Open-PSA Model Exchange Format open standard for probabilistic safety assessment applications.
Fault tree analysis	Graphic symbols	The basic symbols used in FTA are grouped as events, gates, and transfer symbols. Minor variations may be used in FTA software.
Fault tree analysis	Graphic symbols	Event symbols Event symbols are used for primary events and intermediate events. Primary events are not further developed on the fault tree. Intermediate events are found at the output of a gate. The event symbols are shown below: The primary event symbols are typically used as follows: Basic event – failure or error in a system component or element (example: switch stuck in open position) External event – normally expected to occur (not of itself a fault) Undeveloped event – an event about which insufficient information is available, or which is of no consequence Conditioning event – conditions that restrict or affect logic gates (example: mode of operation in effect)An intermediate event gate can be used immediately above a primary event to provide more room to type the event description.
Fault tree analysis	Graphic symbols	FTA is a top-to-bottom approach. Gate symbols Gate symbols describe the relationship between input and output events. The symbols are derived from Boolean logic symbols: The gates work as follows: OR gate – the output occurs if any input occurs. AND gate – the output occurs only if all inputs occur (inputs are independent from the source). Exclusive OR gate – the output occurs if exactly one input occurs. Priority AND gate – the output occurs if the inputs occur in a specific sequence specified by a conditioning event. Inhibit gate – the output occurs if the input occurs under an enabling condition specified by a conditioning event. Transfer symbols Transfer symbols are used to connect the inputs and outputs of related fault trees, such as the fault tree of a subsystem to its system. NASA prepared a complete document about FTA through practical incidents.
Fault tree analysis	Basic mathematical foundation	Events in a fault tree are associated with statistical probabilities or Poisson-Exponentially distributed constant rates. For example, component failures may typically occur at some constant failure rate λ (a constant hazard function). In this simplest case, failure probability depends on the rate λ and the exposure time t: P=1−e−λt where: P≈λt if 0.001 A fault tree is often normalized to a given time interval, such as a flight hour or an average mission time. Event probabilities depend on the relationship of the event hazard function to this interval.
Fault tree analysis	Basic mathematical foundation	Unlike conventional logic gate diagrams in which inputs and outputs hold the binary values of TRUE (1) or FALSE (0), the gates in a fault tree output probabilities related to the set operations of Boolean logic. The probability of a gate's output event depends on the input event probabilities.
Fault tree analysis	Basic mathematical foundation	An AND gate represents a combination of independent events. That is, the probability of any input event to an AND gate is unaffected by any other input event to the same gate. In set theoretic terms, this is equivalent to the intersection of the input event sets, and the probability of the AND gate output is given by: P (A and B) = P (A ∩ B) = P(A) P(B)An OR gate, on the other hand, corresponds to set union: P (A or B) = P (A ∪ B) = P(A) + P(B) - P (A ∩ B)Since failure probabilities on fault trees tend to be small (less than .01), P (A ∩ B) usually becomes a very small error term, and the output of an OR gate may be conservatively approximated by using an assumption that the inputs are mutually exclusive events: P (A or B) ≈ P(A) + P(B), P (A ∩ B) ≈ 0An exclusive OR gate with two inputs represents the probability that one or the other input, but not both, occurs: P (A xor B) = P(A) + P(B) - 2P (A ∩ B)Again, since P (A ∩ B) usually becomes a very small error term, the exclusive OR gate has limited value in a fault tree.
Fault tree analysis	Basic mathematical foundation	Quite often, Poisson-Exponentially distributed rates are used to quantify a fault tree instead of probabilities. Rates are often modeled as constant in time while probability is a function of time. Poisson-Exponential events are modelled as infinitely short so no two events can overlap. An OR gate is the superposition (addition of rates) of the two input failure frequencies or failure rates which are modeled as Poisson point processes. The output of an AND gate is calculated using the unavailability (Q1) of one event thinning the Poisson point process of the other event (λ2). The unavailability (Q2) of the other event then thins the Poisson point process of the first event (λ1). The two resulting Poisson point processes are superimposed according to the following equations.
Fault tree analysis	Basic mathematical foundation	The output of an AND gate is the combination of independent input events 1 and 2 to the AND gate: Failure Frequency = λ1Q2 + λ2Q1 where Q = 1 - eλt ≈ λt if λt < 0.001 Failure Frequency ≈ λ1λ2t2 + λ2λ1t1 if λ1t1 < 0.001 and λ2t2 < 0.001In a fault tree, unavailability (Q) may be defined as the unavailability of safe operation and may not refer to the unavailability of the system operation depending on how the fault tree was structured. The input terms to the fault tree must be carefully defined.
Fault tree analysis	Analysis	Many different approaches can be used to model a FTA, but the most common and popular way can be summarized in a few steps. A single fault tree is used to analyze one and only one undesired event, which may be subsequently fed into another fault tree as a basic event. Though the nature of the undesired event may vary dramatically, a FTA follows the same procedure for any undesired event; be it a delay of 0.25 ms for the generation of electrical power, an undetected cargo bay fire, or the random, unintended launch of an ICBM.
Fault tree analysis	Analysis	FTA analysis involves five steps: Define the undesired event to study.
Fault tree analysis	Analysis	Definition of the undesired event can be very hard to uncover, although some of the events are very easy and obvious to observe. An engineer with a wide knowledge of the design of the system is the best person to help define and number the undesired events. Undesired events are used then to make FTAs. Each FTA is limited to one undesired event.
Fault tree analysis	Analysis	Obtain an understanding of the system.
Fault tree analysis	Analysis	Once the undesired event is selected, all causes with probabilities of affecting the undesired event of 0 or more are studied and analyzed. Getting exact numbers for the probabilities leading to the event is usually impossible for the reason that it may be very costly and time-consuming to do so. Computer software is used to study probabilities; this may lead to less costly system analysis. System analysts can help with understanding the overall system. System designers have full knowledge of the system and this knowledge is very important for not missing any cause affecting the undesired event. For the selected event all causes are then numbered and sequenced in the order of occurrence and then are used for the next step which is drawing or constructing the fault tree.
Fault tree analysis	Analysis	Construct the fault tree. After selecting the undesired event and having analyzed the system so that we know all the causing effects (and if possible their probabilities) we can now construct the fault tree. Fault tree is based on AND and OR gates which define the major characteristics of the fault tree. Evaluate the fault tree.
Fault tree analysis	Analysis	After the fault tree has been assembled for a specific undesired event, it is evaluated and analyzed for any possible improvement or in other words study the risk management and find ways for system improvement. A wide range of qualitative and quantitative analysis methods can be applied. This step is as an introduction for the final step which will be to control the hazards identified. In short, in this step we identify all possible hazards affecting the system in a direct or indirect way.
Fault tree analysis	Analysis	Control the hazards identified. This step is very specific and differs largely from one system to another, but the main point will always be that after identifying the hazards all possible methods are pursued to decrease the probability of occurrence.
Fault tree analysis	Comparison with other analytical methods	FTA is a deductive, top-down method aimed at analyzing the effects of initiating faults and events on a complex system. This contrasts with failure mode and effects analysis (FMEA), which is an inductive, bottom-up analysis method aimed at analyzing the effects of single component or function failures on equipment or subsystems. FTA is very good at showing how resistant a system is to single or multiple initiating faults. It is not good at finding all possible initiating faults. FMEA is good at exhaustively cataloging initiating faults, and identifying their local effects. It is not good at examining multiple failures or their effects at a system level. FTA considers external events, FMEA does not. In civil aerospace the usual practice is to perform both FTA and FMEA, with a failure mode effects summary (FMES) as the interface between FMEA and FTA.
Fault tree analysis	Comparison with other analytical methods	Alternatives to FTA include dependence diagram (DD), also known as reliability block diagram (RBD) and Markov analysis. A dependence diagram is equivalent to a success tree analysis (STA), the logical inverse of an FTA, and depicts the system using paths instead of gates. DD and STA produce probability of success (i.e., avoiding a top event) rather than probability of a top event.
Eugeroic	Eugeroic	Eugeroics (originally "eugrégorique" or "eugregoric"), also known as wakefulness-promoting agents and wakefulness-promoting drugs, are a class of drugs that promote wakefulness and alertness. They are medically indicated for the treatment of certain sleep disorders including excessive daytime sleepiness (EDS) in narcolepsy or obstructive sleep apnea (OSA). Eugeroics are also often prescribed off-label for the treatment of EDS in idiopathic hypersomnia. In contrast to classical psychostimulants, such as methylphenidate and amphetamine, which are also used in the treatment of these disorders, eugeroics typically do not produce euphoria, and, consequently, have a lower addictive potential.Modafinil and armodafinil are each thought to act as selective, weak, atypical dopamine reuptake inhibitors (DRI), whereas adrafinil acts as a prodrug for modafinil. Other eugeroics include solriamfetol, which acts as a norepinephrine–dopamine reuptake inhibitor (NDRI), and pitolisant, which acts as a histamine 3 (H3) receptor antagonist/inverse agonist.
Eugeroic	Examples	Marketed Armodafinil (Nuvigil) Modafinil (Provigil) Pitolisant (Wakix) Solriamfetol (Sunosi) Discontinued Adrafinil Never marketed Flmodafinil (CRL-40,940) Fluorafinil (CRL-40,941) Fluorenol Methylbisfluoromodafinil 2-Phenyl-3-aminobutane In development Selective orexin receptor agonists (two are currently under development by Takeda, danavorexton and TAK-994) CE-123 is under patent by Red Bull.
Plane partition	Plane partition	In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers πi,j (with positive integer indices i and j) that is nonincreasing in both indices. This means that πi,j≥πi,j+1 and πi,j≥πi+1,j for all i and j.Moreover, only finitely many of the πi,j may be nonzero. Plane partitions are a generalization of partitions of an integer.
Plane partition	Plane partition	A plane partition may be represented visually by the placement of a stack of πi,j unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture. The image has matrix form 4432143113211 Plane partitions are also often described by the positions of the unit cubes. From this point of view, a plane partition can be defined as a finite subset P of positive integer lattice points (i, j, k) in N3 , such that if (r, s, t) lies in P and if (i,j,k) satisfies 1≤i≤r , 1≤j≤s , and 1≤k≤t , then (i, j, k) also lies in P The sum of a plane partition is n=∑i,jπi,j.
Plane partition	Plane partition	The sum describes the number of cubes of which the plane partition consists. Much interest in plane partitions concerns the enumeration of plane partitions in various classes. The number of plane partitions with sum n is denoted by PL(n). For example, there are six plane partitions with sum 3 32111121111111 so PL(3) = 6. Plane partitions may be classified by how symmetric they are. Many symmetric classes of plane partitions are enumerated by simple product formulas.
Plane partition	Generating function of plane partitions	The generating function for PL(n) is PL 13 24 x5+⋯ (sequence A000219 in the OEIS).It is sometimes referred to as the MacMahon function, as it was discovered by Percy A. MacMahon.
Plane partition	Generating function of plane partitions	This formula may be viewed as the 2-dimensional analogue of Euler's product formula for the number of integer partitions of n. There is no analogous formula known for partitions in higher dimensions (i.e., for solid partitions). The asymptotics for plane partitions were first calculated by E. M. Wright. One obtains, for large n , that PL 36 12 25 36 exp ⁡(3ζ(3)1/3(n2)2/3+ζ′(−1)).
Plane partition	Generating function of plane partitions	Evaluating numerically yields ln PL 2.00945 0.69444 ln 1.4631.
Plane partition	Generating function of plane partitions	Plane partitions in a box Around 1896, MacMahon set up the generating function of plane partitions that are subsets of the r×s×t box B(r,s,t)={(i,j,k)\|1≤i≤r,1≤j≤s,1≤k≤t} in his first paper on plane partitions. The formula is given by A proof of this formula can be found in the book Combinatory Analysis written by MacMahon. MacMahon also mentions the generating functions of plane partitions. The formula for the generating function can be written in an alternative way, which is given by Multiplying each component by 1−q1−q , and setting q = 1 in the formulas above yields that the total number N1(r,s,t) of plane partitions that fit in the r×s×t box B(r,s,t) is equal to the following product formula: The planar case (when t = 1) yields the binomial coefficients: B(r,s,1)=(r+sr).
Plane partition	Special plane partitions	Special plane partitions include symmetric, cyclic and self-complementary plane partitions, and combinations of these properties. In the subsequent sections, the enumeration of special sub-classes of plane partitions inside a box are considered. These articles use the notation Ni(r,s,t) for the number of such plane partitions, where r, s, and t are the dimensions of the box under consideration, and i is the index for the case being considered.
Plane partition	Special plane partitions	Action of S2, S3 and C3 on plane partitions S2 is the group of permutations acting on the first two coordinates of a point. This group contains the identity, which sends (i, j, k) to itself, and the transposition (i, j, k) → (j, i, k). The number of elements in an orbit η is denoted by \|η\| . B/S2 denotes the set of orbits of elements of B under the action of S2 . The height of an element (i, j, k) is defined by The height increases by one for each step away from the back right corner. For example, the corner position (1, 1, 1) has height 1 and ht(2, 1, 1) = 2. The height of an orbit is defined to be the height of any element in the orbit. This notation of the height differs from the notation of Ian G. Macdonald.There is a natural action of the permutation group S3 on a Ferrers diagram of a plane partition—this corresponds to simultaneously permuting the three coordinates of all nodes. This generalizes the conjugation operation for integer partitions. The action of S3 can generate new plane partitions starting from a given plane partition. Below there are shown six plane partitions of 4 that are generated by the S3 action. Only the exchange of the first two coordinates is manifest in the representation given below.
Plane partition	Special plane partitions	313121121111111111 C3 is called the group of cyclic permutations and consists of and (i,j,k)→(k,i,j).

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.