title: Filial piety

text: In Chinese parentchild relations, the aspect of authority goes hand-in-hand with the aspect of benevolence. E.g. many Chinese parents support their children's education fully and do not allow their children to work during their studies, allowing them to focus on their studies. Because of the combination of benevolence and authoritarianism in such relations, children feel obliged to respond to parents' expectations, and internalize them. Ho found, however, that in Chinese parentchild relations, fear was also a contributing factor in meeting parents' filial expectations: children may not internalize their parents' expectations, but rather perform as good children in a detached way, through affectrole dissociation. Studying Korean family relations, scholar Dawnhee Yim argues that internalization of parents' obligations by children may lead to guilt, as well as suppression of hostile thoughts toward parents, leading to psychological problems. Jordan found that despite filial piety being asymmetrical in nature, Chinese interviewees felt that filial piety contained an element of reciprocity: "... it is easy to see the parent whom one serves today as the self who is served tomorrow." Furthermore, the practice of filial piety provides the pious child with a sense of adulthood and moral heroism.


title: Filial piety

text: The origins of filial piety in East Asia lie in ancestor worship, and can already be found in the pre-Confucian period. Epigraphical findings such as oracle bones contain references to filial piety; texts such as the "Classic of Changes" (10th4th century BCE) may contain early references to the idea of parallel conception of the filial son and the loyal minister.


title: Filial piety

text: In the T'ang dynasty (6th10th century), not performing filial piety was declared illegal, and even earlier, during the Han dynasty (2nd century BCE3rd century CE), this was already punished by beheading. Behavior regarded as unfilial such as mistreating or abandoning one's parents or grandparents, or refusing to complete the mourning period for them was punished by exile and beating, at best.


title: Filial piety

text: From the Han Dynasty onward, the practice of mourning rites came to be seen as the cornerstone of filial piety and was strictly practiced and enforced. This was a period of unrest, and the state promoted the practice of long-term mourning to reestablish their authority. Filial piety toward one's parents was expected to lead to loyalty to the ruler, expressed in the Han proverb "The Emperor rules all-under-heaven with filial piety". Government officials were expected to take leave for a mourning period of two years after their parents died. Local officials were expected to encourage filial piety to one's parentsand by extension, to the stateby behaving as an example of such piety. Indeed, the king himself would perform an exemplary role in expressing filial piety, through the ritual of "serving the elderly" (). Nearly all Han emperors had the word "xiào" in their temple name. The promotion of filial piety in this manner, as part of the idea of "li", was more an acceptable way to create order in society than resorting to law.


title: Filial piety

text: During the early Confucian period, the principles of filial piety were brought back by Japanese and Korean students to their respective homelands, where they became central to the education system. In Japan, rulers gave awards to people deemed to practice exemplary filial conduct.


title: Filial piety

text: During the Mongolian rule in the Yuan dynasty (13th14th century), the practice of filial piety was perceived to deteriorate. In the Ming dynasty (14th17th century), emperors and literati attempted to revive the customs of filial piety, though in that process, filial piety was reinterpreted, as rules and rituals were modified. Even on the grassroots level a revival was seen, as societies that provided vigilance against criminals started to promote Confucian values. A book that was composed by members of this movement was "The Twenty-four Cases of Filial Piety".


title: Filial piety

text: Filial piety is an important aspect of Buddhist ethics since early Buddhism, and was essential in the apologetics and texts of Chinese Buddhism. In the Early Buddhist Texts such as the Nikāyas and Āgamas, filial piety is prescribed and practiced in three ways: to repay the gratitude toward one's parents; as a good karma or merit; and as a way to contribute to and sustain the social order. In Buddhist scriptures, narratives are given of the Buddha and his disciples practicing filial piety toward their parents, based on the qualities of gratitude and reciprocity. Initially, scholars of Buddhism like Kenneth Ch'en saw Buddhist teachings on filial piety as a distinct feature of Chinese Buddhism. Later scholarship, led by people such as John Strong and Gregory Schopen, has come to believe that filial piety was part of Buddhist doctrine since early times. Strong and Schopen have provided epigraphical and textual evidence to show that early Buddhist laypeople, monks and nuns often displayed strong devotion to their parents, concluding filial piety was already an important part of the devotional life of early Buddhists.


title: Filial piety

text: When Buddhism was introduced in China, it had no organized celibacy. Confucianism emphasized filial piety to parents and loyalty to the emperor, and Buddhist monastic life was seen to go against its tenets. In the 3rd–5th century, as criticism of Buddhism increased, Buddhist monastics and lay authors responded by writing about and translating Buddhist doctrines and narratives that supported filiality, comparing them to Confucianism and thereby defending Buddhism and its value in society. The "Mouzi Lihuolun" referred to Confucian and Daoist classics, as well as historical precedents to respond to critics of Buddhism. The "Mouzi" stated that while on the surface the Buddhist monk seems to reject and abandon his parents, he is actually aiding his parents as well as himself on the path towards enlightenment. Sun Chuo (300–380) further argued that monks were working to ensure the salvation of all people and making their family proud by doing so, and Liu Xie stated that Buddhists practiced filial piety by sharing merit with their departed relatives. Buddhist monks were also criticized for not expressing their respect to the Chinese emperor by prostrating and other devotion, which in Confucianism was associated with the virtue of filial piety. Huiyuan (334–416) responded that although monks did not express such piety, they did pay homage in heart and mind; moreover, their teaching of morality and virtue to the public helped support imperial rule.


title: Filial piety

text: From the 6th century onward, Chinese Buddhists began to realize they had to stress Buddhism's own particular ideas about filial piety in order for Buddhism to survive. Śyāma, Sujāti, and other Buddhist stories of self-sacrifice spread a belief that a filial child should even be willing to sacrifice its own body. The "Ullambana Sūtra" introduced the idea of transfer of merit through the story of "Mulian Saves His Mother" and led to the establishment of the Ghost Festival. By this Buddhists attempted to show that filial piety also meant taking care of one's parents in the next life, not just this life. Furthermore, authors in China and Tibet, and to some extent Japanwrote that in Buddhism, all living beings have once been one's parents, and that practicing compassion to all living beings as though they were one's parents is the more superior form of filial piety. Another aspect emphasized was the great suffering a mother goes through when giving birth and raising a child. Chinese Buddhists described how difficult it is to repay the goodness of one's mother, and how many sins mothers often committed in raising her children. The mother became the primary source of well-being and indebtedness for the son, which was in contrast with pre-Buddhist perspectives emphasizing the father. Nevertheless, although some critics of Buddhism did not have much impact during this time, this changed in the period leading up to the Neo-Confucianist revival, when Emperor Wu Zong (841–845) started the Great Anti-Buddhist Persecution, citing lack of filial piety as one of his reasons for attacking Buddhist institutions.


title: Filial piety

text: Filial piety is still an important value in a number of Asian Buddhist cultures. In China, Buddhism continued to uphold a role in state rituals and mourning rites for ancestors, up until late imperial times (13th–20th century). Also, "sūtras" and narratives about filial piety are still widely used. The Ghost Festival is still popular in many Asian countries, especially those countries which are influenced by both Buddhism and Confucianism.


title: Filial piety

text: During the 17th century, some missionaries tried to prevent Chinese people from worshiping their ancestors. This was regarded as an assault on Chinese culture.


title: Filial piety

text: During the Qing dynasty, however, filial piety was redefined by the emperor Kangxi (16541722), who felt it more important that his officials were loyal to him than that they were filial sons: civil servants were often not allowed to go on extended leave to perform mourning rituals for their parents. The parallel conception of society therefore disappeared from Chinese society.


title: Filial piety

text: Unlike western societies, patriarchalism and its enactment in law grew more strict in late imperial China. The duties of the obedient child were much more precisely and rigidly prescribed, to the extent that legal scholar Hsu Dau-lin argued about this period that it "engendered a highly authoritarian spirit which was entirely alien to Confucius himself". Indeed, the late imperial Chinese held patriarchalism high as an organizing principle of society, as laws and punishments gradually became more strict and severe.


title: Filial piety

text: But during the same time, in Japan, a classic work about filial practices was compiled, called "Biographies of Japanese Filial Children" ().


title: Filial piety

text: During the rise of progressivism and communism in China in the early 20th century, Confucian values and family-centered living were discouraged by the state and intellectuals. During the New Culture Movement of 1911, Chinese intellectuals and foreign missionaries attacked the principle of filial piety, the latter considering it an obstruction of progress.


title: Filial piety

text: In Japan, filial piety was not regarded as an obstacle to modernization, though scholars are in disagreement as to why this was the case. Francis Hsu believed that "the human networks through which it found concrete expressions" were different in Japan, and there never was a movement against filial piety as there was in China.


title: Filial piety

text: The late imperial trend of increased patriarchalism made it difficult for the Chinese to build strong patrimonial groups that went beyond kin. Though filial piety was practiced much in both countries, the Chinese way was more limited to close kin than in Japan. When industrialization increased, filial piety was therefore criticized more in China than in Japan, because China felt it limited the way the country could meet the challenges from the West. For this reason, China developed a more critical stance towards filial piety and other aspects of Confucianism than other East Asian countries, including not only Japan, but also Taiwan.


title: Filial piety

text: In the 1950s, Mao Zedong's socialist measures led to the dissolution of family businesses and more dependence on the state instead; Taiwan's socialism did not go that far in state control.


title: Filial piety

text: Ethnographic evidence from the 19th and early 20th century shows that Chinese people still very much cared for their elders, and very often lived with one or more married sons.


title: Filial piety

text: In 21st-century Chinese societies, filial piety expectations and practice have decreased. One cause for this is the rise of the nuclear family without much coresidence with parents. Families are becoming smaller because of family planning and housing shortages. Other causes of decrease in practice are individualism, the loss of status of elderly, emigration of young people to cities and the independence of young people and women. To amplify this trend, the number of elderly people has increased quickly.


title: Filial piety

text: The relationship between husband and wife came to be more emphasized, and the extended family less and less. Kinship ties between the husband and wife's families have become more bilateral and equal. The way respect to elders is expressed is also changing. Communication with elders tends to become more reciprocal and less one-way, and kindness and courtesy is replacing obedience and subservience.


title: Filial piety

text: In modern Chinese societies, elder care has changed much. Studies have shown that there is a discrepancy between the parents' filial expectations and the actual behaviors of their children. The discrepancy with regard to respect shown by the children makes elderly people especially unhappy. Industrialization and urbanization have affected the practice of filial piety, with care being given more in financial ways rather than personal. But as of 2009, care-giving of the young to elderly people had not undergone any revolutionary changes in Mainland China, and family obligations still remained strong, still "almost automatic". Respect to elders remains a central value for East Asian people.


title: Filial piety

text: Comparing data from the 1990s from Taiwan and the PRC, sociologist Martin Whyte concluded that the elderly in Taiwan often received less support from the government, but more assistance from their children, than in China, despite the former being an economically more modern nation.


title: Filial piety

text: In mainland Chinese business culture, the culture of filial piety is decreasing in influence. As of 2003, western-style business practices and managerial style were promoted by the Chinese government to modernize the country. However, in Japan, employees usually regard their employer as a sort of father, to which they feel obliged to express filial devotion.


title: Filial piety

text: In some societies with large Chinese communities, legislation has been introduced to establish or uphold filial piety. In the 2000s, Singapore introduced a law that makes it an offense to refuse to support one's elderly parents; Taiwan has taken similar punitive measures. Hong Kong, on the other hand, has attempted to influence its population by providing incentives for fulfilling their obligations. For example, certain tax allowances are given to citizens that are willing to live with their elderly parents.


title: Filial piety

text: Some scholars have argued that medieval China's reliance on governance by filial piety formed a society that was better able to prevent crime and other misconduct than societies that did so only through legal means.


title: Filial piety

text: Chinese who immigrate to the United States generally continue to send money to their parents out of filial piety.


title: The Great Beauty

text: The Great Beauty ( ) is a 2013 art drama film co-written and directed by Paolo Sorrentino. Filming took place in Rome starting on 9 August 2012. It premiered at the 2013 Cannes Film Festival where it was screened in competition for the Palme d'Or. It was shown at the 2013 Toronto International Film Festival, the 2013 Tallinn Black Nights Film Festival (winning Grand Prix), and at the 2013 Reykjavik European Film Festival.


title: The Great Beauty

text: The film won Best Foreign Language Film at the 86th Academy Awards, as well as the Golden Globe and the BAFTA award in the same category. It is a co-production between the Italian Medusa Film and Indigo Film and the French Babe Films, with support from Banca Popolare di Vicenza, Pathé and France 2 Cinéma. With a production budget of €9.2 million, the film grossed over $24 million worldwide.


title: The Great Beauty

text: The film opens with a quote from Céline's novel "Journey to the End of the Night": "Travel is useful; it exercises the imagination. All the rest is disappointment and fatigue. Our journey is entirely imaginary. That is its strength. It goes from life to death. People, animals, cities, things – all are imagined. It's a novel, just a fictitious narrative. Littré says so, and he's never wrong. And besides, in the first place, anyone can do as much. You just have to close your eyes. It's on the other side of life."


title: The Great Beauty

text: Jep Gambardella is a 65-year-old seasoned journalist and theater critic, a fascinating man, mostly committed to wandering among the social events of a Rome immersed in the beauty of its history and in the superficiality of its inhabitants today, in a merciless contrast. He also ventured into creative writing in his youth: he is the author of only one work called "The Human Apparatus". Despite the appreciation and the many awards he received, Jep has not written other books, not only for his laziness but above all for a creative block from which he cannot escape. The purpose of his existence has been to become a "socialite", but not just any socialite, but "the king of society".


title: The Great Beauty

text: Jep is surrounded by several friends: Romano, a playwright who is perpetually on the leash of a young woman who exploits him; Lello, a mouthy and wealthy toy seller; Viola, a wealthy bourgeois and mother of a son with serious mental problems named Andrea who will commit suicide by crashing voluntarily with his car; Stefania, a self-centred radical chic writer; Dadina, the dwarf editor of the newspaper where Jep works.


title: The Great Beauty

text: One morning, he meets the husband of Elisa, a woman who has been Jep's first and probably only love: the man announces that Elisa has died, leaving behind only a diary in which the woman tells of her love for Jep; thus, her husband discovered that he had been a mere surrogate for 35 years, nothing more than "a good companion". Elisa's husband, now afflicted and grieved, will soon find consolation in the affectionate welcome of his foreign maid. After this episode, Jep begins a profound and melancholic reinterpretation of his life and a long meditation on himself and on the world around him. And, above all, he thinks about starting to write again.


title: The Great Beauty

text: During the following days, Jep meets Ramona, a stripper with painful secrets, and Cardinal Bellucci, in whom the passion for cooking is more alive than his Catholic faith; Jep is gradually convinced of the futility and uselessness of his existence. Soon his "vicious circle" also breaks down: Ramona, with whom he had established an innocent and profound relationship, dies of an incurable disease; Romano, disappointed by the deceptive attractiveness of Rome, leaves the city, farewelling only Jep; Stefania, humiliated by Jep, who had revealed her secrets and her lies to her face, left Jep's worldly circle; Viola, on the other hand, after the death of her son, donates all her possessions to the Church and becomes a missionary in Africa.


title: The Great Beauty

text: Just when hopes seem to abandon Jep once and for all, he is saved by a new episode: after a meeting, pushed by Dadina, who wants to get an interview with a "Saint", a Catholic missionary nun in the Third World, Jep goes to Giglio Island to report on the shipwreck of the "Costa Concordia". Right here, remembering his first meeting with Elisa in a flashback, a glimmer of hope rekindles in him: his next novel is finally ready to come to light.


title: The Great Beauty

text: The review aggregator website Rotten Tomatoes reported a 91% approval rating, based on 132 reviews, with a weighted average rating of 8.00/10. The website's critical consensus states, "Dazzlingly ambitious, beautifully filmed, and thoroughly enthralling, "The Great Beauty" offers virtuoso filmmaking from writer/director Paolo Sorrentino." The film holds a score of 86/100 on Metacritic based on 34 reviews, signifying "universal acclaim."


title: The Great Beauty

text: Robbie Collin at "The Daily Telegraph" awarded Sorrentino's film the maximum five stars and described it as "a shimmering coup de cinema". He likened it to Roberto Rossellini's "Rome, Open City" and Federico Fellini's "La Dolce Vita" in its ambition to record a period of Roman history on film. "Rossellini covered the Nazi occupation of 1944; Fellini the seductive, empty hedonism of the years that followed. Sorrentino's plan is to do the same for the Berlusconi era," he wrote. Deborah Young of "The Hollywood Reporter" stated "Sorrentino's vision of moral chaos and disorder, spiritual and emotional emptiness at this moment in time is even darker than Fellini's (though Ettore Scola's "The Terrace" certainly comes in somewhere)." Critics have also identified other purposefully explicit film homages: to "Roma", "8½", Scola's "Splendor", Michelangelo Antonioni's "La notte". Spanish film director Pedro Almodóvar named the film as one of the twelve best films of 2013, placing it second in his list. In 2016, the film was ranked among the 100 greatest films since 2000 in an international critics poll by 177 critics around the world. It is currently director Paolo Sorrentino's second highest rated film on Rotten Tomatoes.


title: Finger

text: A finger is a limb of the body and a type of digit, an organ of manipulation and found in the hands of most of the Tetrapods, so also with humans and other primates. Most land vertebrates have five fingers (Pentadactyly).


title: Finger

text: The five-rayed anterior limbs of terrestrial vertebrates can be derived phylogenetically from the pectoral fins of fish. Within the taxa of the terrestrial vertebrates, the basic pentadactyl plan, and thus also the fingers and phalanges, undergo many variations.


title: Finger

text: Morphologically the different fingers of terrestrial vertebrates are homolog. The wings of birds and those of bats are not homologous, they are analogue flight organs. However, the phalanges within them are homologous.


title: Finger

text: Chimpanzees have lower limbs that are specialized for manipulation, and (arguably) have fingers on their lower limbs as well. In the case of Primates in general, the digits of the hand are overwhelmingly referred to as "fingers". Primate fingers have both fingernails and fingerprints.


title: Finger

text: Research has been carried out on the embryonic development of domestic chickens showing that an interdigital webbing forms between the tissues that become the toes, which subsequently regresses by apoptosis. If apoptosis fails to occur, the interdigital skin remains intact. Many animals have developed webbed feet or skin between the fingers from this like the Wallace's flying frog.


title: Finger

text: Usually humans have five digits, the bones of which are termed phalanges, on each hand, although some people have more or fewer than five due to congenital disorders such as polydactyly or oligodactyly, or accidental or intentional amputations. The first digit is the thumb, followed by index finger, middle finger, ring finger, and little finger or pinkie. According to different definitions, the thumb can be called a finger, or not.


title: Finger

text: English dictionaries describe finger as meaning either one of the five digits including the thumb, or one of the four excluding the thumb (in which case they are numbered from 1 to 4 starting with the index finger closest to the thumb).


title: Finger

text: The palm has five bones known as metacarpal bones, one to each of the five digits. Human hands contain fourteen digital bones, also called phalanges, or phalanx bones: two in the thumb (the thumb has no middle phalanx) and three in each of the four fingers. These are the distal phalanx, carrying the nail, the middle phalanx, and the proximal phalanx.


title: Finger

text: Sesamoid bones are small ossified nodes embedded in the tendons to provide extra leverage and reduce pressure on the underlying tissue. Many exist around the palm at the bases of the digits; the exact number varies between different people.


title: Finger

text: The articulations are: interphalangeal articulations between phalangeal bones, and metacarpophalangeal joints connecting the phalanges to the metacarpal bones.


title: Finger

text: Each finger may flex and extend, abduct and adduct, and so also circumduct. Flexion is by far the strongest movement. In humans, there are two large muscles that produce flexion of each finger, and additional muscles that augment the movement. The muscle bulks that move each finger may be partly blended, and the tendons may be attached to each other by a net of fibrous tissue, preventing completely free movement. Although each finger seems to move independently, moving one finger also moves the other fingers slightly which is called finger interdependence or finger enslaving.


title: Finger

text: Fingers do not contain muscles (other than arrector pili). The muscles that move the finger joints are in the palm and forearm. The long tendons that deliver motion from the forearm muscles may be observed to move under the skin at the wrist and on the back of the hand.


title: Finger

text: The extrinsic muscles are the long flexors and extensors. They are called extrinsic because the muscle belly is located on the forearm.


title: Finger

text: The fingers have two long flexors, located on the underside of the forearm. They insert by tendons to the phalanges of the fingers. The deep flexor attaches to the distal phalanx, and the superficial flexor attaches to the middle phalanx. The flexors allow for the actual bending of the fingers. The thumb has one long flexor and a short flexor in the thenar muscle group. The human thumb also has other muscles in the thenar group (opponens and abductor brevis muscle), moving the thumb in opposition, making grasping possible.


title: Finger

text: The extensors are located on the back of the forearm and are connected in a more complex way than the flexors to the dorsum of the fingers. The tendons unite with the interosseous and lumbrical muscles to form the extensorhood mechanism. The primary function of the extensors is to straighten out the digits. The thumb has two extensors in the forearm; the tendons of these form the anatomical snuff box. Also, the index finger and the little finger have an extra extensor, used for instance for pointing. The extensors are situated within six separate compartments. The first compartment contains abductor pollicis longus and extensor pollicis brevis. The second compartment contains extensors carpi radialis longus and brevis. The third compartment contains extensor pollicis longus. The extensor digitorum indicis and extensor digitorum communis are within the fourth compartment. Extensor digiti minimi is in the fifth, and extensor carpi ulnaris is in the sixth.


title: Finger

text: The intrinsic muscle groups are the thenar and hypothenar muscles (thenar referring to the thumb, hypothenar to the small finger), the dorsal and palmar interossei muscles (between the metacarpal bones) and the lumbrical muscles. The lumbricals arise from the deep flexor (and are special because they have no bony origin) and insert on the dorsal extensor hood mechanism.


title: Finger

text: Aside from the genitals, the fingertips possess the highest concentration of touch receptors and thermoreceptors among all areas of the human skin, making them extremely sensitive to temperature, pressure, vibration, texture and moisture. Recent studies suggest fingers can feel nano-scale wrinkles on a seemingly smooth surface, a level of sensitivity not previously recorded. This makes the fingers commonly used sensory probes to ascertain properties of objects encountered in the world, making them prone to injury.


title: Finger

text: Although a common phenomenon, the underlying functions and mechanism of fingertip wrinkling following immersion in water are relatively unexplored. Originally it was assumed that the wrinkles were simply the result of the skin swelling in water, but it is now understood that the furrows are caused by the blood vessels constricting due to signalling by the sympathetic nervous system in response to water exposure. One hypothesis for why this occurs, the “rain tread” hypothesis, posits that the wrinkles may help the fingers grip things when wet, possibly being an adaption from a time when humans dealt with rain and dew in forested primate habitats. A 2013 study supporting this hypothesis found that the wrinkled fingertips provided better handling of wet objects but gave no advantage for handling dry objects. However, a 2014 study attempting to reproduce these results was unable to demonstrate any improvement of handling wet objects with wrinkled fingertips.


title: Finger

text: Fingertips, after having been torn off children, have been observed to regrow in less than 8 weeks. However, these fingertips do not look the same, although they do look more appealing than a skin graft or a sewn fingertip. No healing occurs if the tear happens below the nail. This works because the distal phalanges are regenerative in youth, and stem cells in the nails create new tissue that ends up as the fingertip.


title: Finger

text: Each finger has an orderly somatotopic representation on the cerebral cortex in the somatosensory cortex area 3b, part of area 1 and a distributed, overlapping representations in the supplementary motor area and primary motor area.


title: Finger

text: The somatosensory cortex representation of the hand is a dynamic reflection of the fingers on the external hand: in syndactyly people have a clubhand of webbed, shortened fingers. However, not only are the fingers of their hands fused, but the cortical maps of their individual fingers also form a club hand. The fingers can be surgically divided to make a more useful hand. Surgeons did this at the Institute of Reconstructive Plastic Surgery in New York to a 32-year-old man with the initials O. G.. They touched O. G.’s fingers before and after surgery while using MRI brain scans. Before the surgery, the fingers mapped onto his brain were fused close together; afterward, the maps of his individual fingers did indeed separate and take the layout corresponding to a normal hand.


title: Finger

text: A rare anatomical variation affects 1 in 500 humans, in which the individual has more than the usual number of digits; this is known as polydactyly. A human may also be born without one or more fingers or underdevelopment of some fingers such as symbrachydactyly. Extra fingers can be functional. One individual with seven fingers not only used them but claimed that they "gave him some advantages in playing the piano".


title: Finger

text: Phalanges are commonly fractured. A damaged tendon can cause significant loss of function in fine motor control, such as with a mallet finger. They can be damaged by cold, including frostbite and non-freezing cold injury (NFCI); and heat, including burns.


title: Finger

text: The fingers are commonly affected by diseases such as rheumatoid arthritis and gout. Diabetics often use the fingers to obtain blood samples for regular blood sugar testing. Raynaud's phenomenon and Paroxysmal hand hematoma are neurovascular disorders that affects the fingers.


title: Finger

text: Research has linked the ratio of lengths between the index and ring fingers to higher levels of testosterone, and to various physical and behavioral traits such as penis length and risk for development of alcohol dependence or video game addiction.


title: Finger

text: The English word "finger" stems from Old English "finger", ultimately from Proto-Germanic ' ('finger'). It is cognate with Gothic ', Old Norse ', or Old High German '. Linguists generally assume that ' is a "ro"-stem deriving from a previous form ', ultimately from Proto-Indo-European "" ('five').


title: Finger

text: The name pinkie derives from Dutch , of uncertain origin. In English only the digits on the hand are known as fingers. However, in some languages the translated version of fingers can mean either the digits on the hand or feet. In English a digit on a foot has the distinct name of toe.


title: Yaroslav Amosov

text: Yaroslav Amosov (born 9 September 1993) is a Ukrainian mixed martial artist, currently competing in the welterweight division of Bellator MMA, where he is the current Bellator Welterweight Champion. As of June 28, 2022, he is #4 in the Bellator men's pound-for-pound rankings.


title: Yaroslav Amosov

text: He is a four-time world sambo champion, a two-time European and Eurasia champion, and European Cup winner. Amosov is also the winner of many national and international competitions.


title: Yaroslav Amosov

text: In September 2012, though early into his Combat Sambo career, Amosov came finished in third place against Ikram Aliskerov in European Combat Sambo Federation sanctioned competition that was being hosted in Moldova. In December of that same year, Amosov again came in third place against Aliskerov in World Combat Sambo Federation sanctioned competition that was held in Moscow. So Ikram Aliskerov is one few combat athletes who have beaten Amosov on two separate occasions in Combat Sambo tournaments.


title: Yaroslav Amosov

text: In June 2016, after winning seven fights in a row in Tech-Krep FC, Amosov beat Croatian fighter Roberto Soldić to win the vacant Tech-Krep welterweight title.


title: Yaroslav Amosov

text: In 2017, Amosov made two successful defences of the Tech-Krep FC welterweight title against Diogo Cavalcanti and Nathan Oliveira. Amosov won both fights via submission in the first round.


title: Yaroslav Amosov

text: In November 2017. Bellator announced they had signed Yaroslav Amosov to the promotion following an undefeated run of 19 fights which only included two decisions.


title: Yaroslav Amosov

text: Amosov made his Bellator debut against Gerald Harris on 13 July 2018 at Bellator 202. Amosov won the fight via unanimous decision.


title: Yaroslav Amosov

text: Amosov defeated Rickels via brabo choke in the second round. This was Rickels' first submission loss in his career.


title: Yaroslav Amosov

text: Amosov then faced three-time NCAA Division I Wrestling Champion from Penn State Ed Ruth on 21 February 2020 at Bellator 239. Amosov won the fight via unanimous decision.


title: Yaroslav Amosov

text: Amosov faced Mark Lemminger on 21 August 2020 at Bellator 244. He won the fight via TKO between rounds 1 and 2 due to a cut rendering Lemminger unable to continue.


title: Yaroslav Amosov

text: Amosov faced four-time NCAA Division I All-American from Minnesota Logan Storley on November 12, 2020 at Bellator 252. He won the back-and-forth bout by split decision.


title: Yaroslav Amosov

text: Amosov fought for the Bellator Welterweight World Championship against current champ Douglas Lima at Bellator 260 on June 11, 2021. He won the fight by unanimous decision by outwrestling Lima throughout all 5 rounds.


title: Yaroslav Amosov

text: Amosov was scheduled to make his first title defense against Michael Page on May 13, 2022 at Bellator 281. However, Amosov was forced to pull out of the bout due to his participation in the war in Ukraine on the side of Ukraine, leaving Page to compete against Logan Storley for the interim title.


title: Yaroslav Amosov

text: After almost a year and a half away from the cage due to the war, Amosov is scheduled to defend his title in a rematch against Logan Storley on February 25, 2023 at Bellator 291.


title: Yaroslav Amosov

text: During the 2022 Russian invasion of Ukraine, after seeing his family evacuated to a safe zone, Amosov stayed in Ukraine and joined several other notable fighters, including former boxing heavyweight champion Wladimir Klitschko and his brother, former heavyweight champion Vitali Klitschko, current unified WBA (Super), IBF, WBO and IBO heavyweight champion Oleksandr Usyk, and former lightweight champion Vasyl Lomachenko, in defending Ukraine from the Russian invasion. During this time, Amosov was the Bellator Welterweight Champion. After the liberation of his hometown Irpin after the battle over the town, Amosov published of a video of himself rescuing the Bellator Welterweight Champion belt, which was hidden at his mother's home.


title: M-theory

text: M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution. Prior to Witten's announcement, string theorists had identified five versions of superstring theory. Although these theories initially appeared to be very different, work by many physicists showed that the theories were related in intricate and nontrivial ways. Physicists found that apparently distinct theories could be unified by mathematical transformations called S-duality and T-duality. Witten's conjecture was based in part on the existence of these dualities and in part on the relationship of the string theories to a field theory called eleven-dimensional supergravity.


title: M-theory

text: Although a complete formulation of M-theory is not known, such a formulation should describe two- and five-dimensional objects called branes and should be approximated by eleven-dimensional supergravity at low energies. Modern attempts to formulate M-theory are typically based on matrix theory or the AdS/CFT correspondence. According to Witten, M should stand for "magic", "mystery" or "membrane" according to taste, and the true meaning of the title should be decided when a more fundamental formulation of the theory is known.


title: M-theory

text: Investigations of the mathematical structure of M-theory have spawned important theoretical results in physics and mathematics. More speculatively, M-theory may provide a framework for developing a unified theory of all of the fundamental forces of nature. Attempts to connect M-theory to experiment typically focus on compactifying its extra dimensions to construct candidate models of the four-dimensional world, although so far none has been verified to give rise to physics as observed in high-energy physics experiments.


title: M-theory

text: One of the deepest problems in modern physics is the problem of quantum gravity. The current understanding of gravity is based on Albert Einstein's general theory of relativity, which is formulated within the framework of classical physics. However, nongravitational forces are described within the framework of quantum mechanics, a radically different formalism for describing physical phenomena based on probability. A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity.


title: M-theory

text: String theory is a theoretical framework that attempts to reconcile gravity and quantum mechanics. In string theory, the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how strings propagate through space and interact with each other. In a given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, and it can vibrate in different ways. On distance scales larger than the string scale, a string will look just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In this way, all of the different elementary particles may be viewed as vibrating strings. One of the vibrational states of a string gives rise to the graviton, a quantum mechanical particle that carries gravitational force.


title: M-theory

text: There are several versions of string theory: type I, type IIA, type IIB, and two flavors of heterotic string theory ( and ). The different theories allow different types of strings, and the particles that arise at low energies exhibit different symmetries. For example, the type I theory includes both open strings (which are segments with endpoints) and closed strings (which form closed loops), while types IIA and IIB include only closed strings. Each of these five string theories arises as a special limiting case of M-theory. This theory, like its string theory predecessors, is an example of a quantum theory of gravity. It describes a force just like the familiar gravitational force subject to the rules of quantum mechanics.


title: M-theory

text: In everyday life, there are three familiar dimensions of space: height, width and depth. Einstein's general theory of relativity treats time as a dimension on par with the three spatial dimensions; in general relativity, space and time are not modeled as separate entities but are instead unified to a four-dimensional spacetime, three spatial dimensions and one time dimension. In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime.


title: M-theory

text: In spite of the fact that the universe is well described by four-dimensional spacetime, there are several reasons why physicists consider theories in other dimensions. In some cases, by modeling spacetime in a different number of dimensions, a theory becomes more mathematically tractable, and one can perform calculations and gain general insights more easily. There are also situations where theories in two or three spacetime dimensions are useful for describing phenomena in condensed matter physics. Finally, there exist scenarios in which there could actually be more than four dimensions of spacetime which have nonetheless managed to escape detection.


title: M-theory

text: One notable feature of string theory and M-theory is that these theories require extra dimensions of spacetime for their mathematical consistency. In string theory, spacetime is "ten-dimensional" (nine spatial dimensions, and one time dimension), while in M-theory it is "eleven-dimensional" (ten spatial dimensions, and one time dimension). In order to describe real physical phenomena using these theories, one must therefore imagine scenarios in which these extra dimensions would not be observed in experiments.


title: M-theory

text: Compactification is one way of modifying the number of dimensions in a physical theory. In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles. In the limit where these curled-up dimensions become very small, one obtains a theory in which spacetime has effectively a lower number of dimensions. A standard analogy for this is to consider a multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling on the surface of the hose would move in two dimensions.


title: M-theory

text: Theories that arise as different limits of M-theory turn out to be related in highly nontrivial ways. One of the relationships that can exist between these different physical theories is called S-duality. This is a relationship which says that a collection of strongly interacting particles in one theory can, in some cases, be viewed as a collection of weakly interacting particles in a completely different theory. Roughly speaking, a collection of particles is said to be strongly interacting if they combine and decay often and weakly interacting if they do so infrequently. Type I string theory turns out to be equivalent by S-duality to the heterotic string theory. Similarly, type IIB string theory is related to itself in a nontrivial way by S-duality.


title: M-theory

text: Another relationship between different string theories is T-duality. Here one considers strings propagating around a circular extra dimension. T-duality states that a string propagating around a circle of radius is equivalent to a string propagating around a circle of radius in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, a string has momentum as it propagates around a circle, and it can also wind around the circle one or more times. The number of times the string winds around a circle is called the winding number. If a string has momentum and winding number in one description, it will have momentum and winding number in the dual description. For example, type IIA string theory is equivalent to type IIB string theory via T-duality, and the two versions of heterotic string theory are also related by T-duality.


title: M-theory

text: In general, the term "duality" refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be "dual" to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.


title: M-theory

text: Another important theoretical idea that plays a role in M-theory is supersymmetry. This is a mathematical relation that exists in certain physical theories between a class of particles called bosons and a class of particles called fermions. Roughly speaking, fermions are the constituents of matter, while bosons mediate interactions between particles. In theories with supersymmetry, each boson has a counterpart which is a fermion, and vice versa. When supersymmetry is imposed as a local symmetry, one automatically obtains a quantum mechanical theory that includes gravity. Such a theory is called a supergravity theory.


title: M-theory

text: A theory of strings that incorporates the idea of supersymmetry is called a superstring theory. There are several different versions of superstring theory which are all subsumed within the M-theory framework. At low energies, the superstring theories are approximated by supergravity in ten spacetime dimensions. Similarly, M-theory is approximated at low energies by supergravity in eleven dimensions.


title: M-theory

text: In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension , these are called -branes. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They can have mass and other attributes such as charge. A -brane sweeps out a -dimensional volume in spacetime called its "worldvolume". Physicists often study fields analogous to the electromagnetic field which live on the worldvolume of a brane. The word brane comes from the word "membrane" which refers to a two-dimensional brane.


title: M-theory

text: In string theory, the fundamental objects that give rise to elementary particles are the one-dimensional strings. Although the physical phenomena described by M-theory are still poorly understood, physicists know that the theory describes two- and five-dimensional branes. Much of the current research in M-theory attempts to better understand the properties of these branes.


title: M-theory

text: In the early 20th century, physicists and mathematicians including Albert Einstein and Hermann Minkowski pioneered the use of four-dimensional geometry for describing the physical world. These efforts culminated in the formulation of Einstein's general theory of relativity, which relates gravity to the geometry of four-dimensional spacetime.


title: M-theory

text: The success of general relativity led to efforts to apply higher dimensional geometry to explain other forces. In 1919, work by Theodor Kaluza showed that by passing to five-dimensional spacetime, one can unify gravity and electromagnetism into a single force. This idea was improved by physicist Oskar Klein, who suggested that the additional dimension proposed by Kaluza could take the form of a circle with radius around cm.


title: M-theory

text: The Kaluza–Klein theory and subsequent attempts by Einstein to develop unified field theory were never completely successful. In part this was because Kaluza–Klein theory predicted a particle (the radion), that has never been shown to exist, and in part because it was unable to correctly predict the ratio of an electron's mass to its charge. In addition, these theories were being developed just as other physicists were beginning to discover quantum mechanics, which would ultimately prove successful in describing known forces such as electromagnetism, as well as new nuclear forces that were being discovered throughout the middle part of the century. Thus it would take almost fifty years for the idea of new dimensions to be taken seriously again.


title: M-theory

text: New concepts and mathematical tools provided fresh insights into general relativity, giving rise to a period in the 1960s–70s now known as the golden age of general relativity. In the mid-1970s, physicists began studying higher-dimensional theories combining general relativity with supersymmetry, the so-called supergravity theories.


title: M-theory

text: General relativity does not place any limits on the possible dimensions of spacetime. Although the theory is typically formulated in four dimensions, one can write down the same equations for the gravitational field in any number of dimensions. Supergravity is more restrictive because it places an upper limit on the number of dimensions. In 1978, work by Werner Nahm showed that the maximum spacetime dimension in which one can formulate a consistent supersymmetric theory is eleven. In the same year, Eugène Cremmer, Bernard Julia, and Joël Scherk of the École Normale Supérieure showed that supergravity not only permits up to eleven dimensions but is in fact most elegant in this maximal number of dimensions.


title: M-theory

text: Initially, many physicists hoped that by compactifying eleven-dimensional supergravity, it might be possible to construct realistic models of our four-dimensional world. The hope was that such models would provide a unified description of the four fundamental forces of nature: electromagnetism, the strong and weak nuclear forces, and gravity. Interest in eleven-dimensional supergravity soon waned as various flaws in this scheme were discovered. One of the problems was that the laws of physics appear to distinguish between clockwise and counterclockwise, a phenomenon known as chirality. Edward Witten and others observed this chirality property cannot be readily derived by compactifying from eleven dimensions.


title: M-theory

text: In the first superstring revolution in 1984, many physicists turned to string theory as a unified theory of particle physics and quantum gravity. Unlike supergravity theory, string theory was able to accommodate the chirality of the standard model, and it provided a theory of gravity consistent with quantum effects. Another feature of string theory that many physicists were drawn to in the 1980s and 1990s was its high degree of uniqueness. In ordinary particle theories, one can consider any collection of elementary particles whose classical behavior is described by an arbitrary Lagrangian. In string theory, the possibilities are much more constrained: by the 1990s, physicists had argued that there were only five consistent supersymmetric versions of the theory.


title: M-theory

text: Although there were only a handful of consistent superstring theories, it remained a mystery why there was not just one consistent formulation. However, as physicists began to examine string theory more closely, they realized that these theories are related in intricate and nontrivial ways.


title: M-theory

text: In the late 1970s, Claus Montonen and David Olive had conjectured a special property of certain physical theories. A sharpened version of their conjecture concerns a theory called supersymmetric Yang–Mills theory, which describes theoretical particles formally similar to the quarks and gluons that make up atomic nuclei. The strength with which the particles of this theory interact is measured by a number called the coupling constant. The result of Montonen and Olive, now known as Montonen–Olive duality, states that supersymmetric Yang–Mills theory with coupling constant is equivalent to the same theory with coupling constant . In other words, a system of strongly interacting particles (large coupling constant) has an equivalent description as a system of weakly interacting particles (small coupling constant) and vice versa by spin-moment.


title: M-theory

text: In the 1990s, several theorists generalized Montonen–Olive duality to the S-duality relationship, which connects different string theories. Ashoke Sen studied S-duality in the context of heterotic strings in four dimensions. Chris Hull and Paul Townsend showed that type IIB string theory with a large coupling constant is equivalent via S-duality to the same theory with small coupling constant. Theorists also found that different string theories may be related by T-duality. This duality implies that strings propagating on completely different spacetime geometries may be physically equivalent.


title: M-theory

text: String theory extends ordinary particle physics by replacing zero-dimensional point particles by one-dimensional objects called strings. In the late 1980s, it was natural for theorists to attempt to formulate other extensions in which particles are replaced by two-dimensional supermembranes or by higher-dimensional objects called branes. Such objects had been considered as early as 1962 by Paul Dirac, and they were reconsidered by a small but enthusiastic group of physicists in the 1980s.


title: M-theory

text: Supersymmetry severely restricts the possible number of dimensions of a brane. In 1987, Eric Bergshoeff, Ergin Sezgin, and Paul Townsend showed that eleven-dimensional supergravity includes two-dimensional branes. Intuitively, these objects look like sheets or membranes propagating through the eleven-dimensional spacetime. Shortly after this discovery, Michael Duff, Paul Howe, Takeo Inami, and Kellogg Stelle considered a particular compactification of eleven-dimensional supergravity with one of the dimensions curled up into a circle. In this setting, one can imagine the membrane wrapping around the circular dimension. If the radius of the circle is sufficiently small, then this membrane looks just like a string in ten-dimensional spacetime. In fact, Duff and his collaborators showed that this construction reproduces exactly the strings appearing in type IIA superstring theory.


title: M-theory

text: In 1990, Andrew Strominger published a similar result which suggested that strongly interacting strings in ten dimensions might have an equivalent description in terms of weakly interacting five-dimensional branes. Initially, physicists were unable to prove this relationship for two important reasons. On the one hand, the Montonen–Olive duality was still unproven, and so Strominger's conjecture was even more tenuous. On the other hand, there were many technical issues related to the quantum properties of five-dimensional branes. The first of these problems was solved in 1993 when Ashoke Sen established that certain physical theories require the existence of objects with both electric and magnetic charge which were predicted by the work of Montonen and Olive.


title: M-theory

text: In spite of this progress, the relationship between strings and five-dimensional branes remained conjectural because theorists were unable to quantize the branes. Starting in 1991, a team of researchers including Michael Duff, Ramzi Khuri, Jianxin Lu, and Ruben Minasian considered a special compactification of string theory in which four of the ten dimensions curl up. If one considers a five-dimensional brane wrapped around these extra dimensions, then the brane looks just like a one-dimensional string. In this way, the conjectured relationship between strings and branes was reduced to a relationship between strings and strings, and the latter could be tested using already established theoretical techniques.


title: M-theory

text: Speaking at the string theory conference at the University of Southern California in 1995, Edward Witten of the Institute for Advanced Study made the surprising suggestion that all five superstring theories were in fact just different limiting cases of a single theory in eleven spacetime dimensions. Witten's announcement drew together all of the previous results on S- and T-duality and the appearance of two- and five-dimensional branes in string theory. In the months following Witten's announcement, hundreds of new papers appeared on the Internet confirming that the new theory involved membranes in an important way. Today this flurry of work is known as the second superstring revolution.


title: M-theory

text: One of the important developments following Witten's announcement was Witten's work in 1996 with string theorist Petr Hořava. Witten and Hořava studied M-theory on a special spacetime geometry with two ten-dimensional boundary components. Their work shed light on the mathematical structure of M-theory and suggested possible ways of connecting M-theory to real world physics.


title: M-theory

text: Initially, some physicists suggested that the new theory was a fundamental theory of membranes, but Witten was skeptical of the role of membranes in the theory. In a paper from 1996, Hořava and Witten wrote


title: M-theory

text: In the absence of an understanding of the true meaning and structure of M-theory, Witten has suggested that the "M" should stand for "magic", "mystery", or "membrane" according to taste, and the true meaning of the title should be decided when a more fundamental formulation of the theory is known. Years later, he would state, "I thought my colleagues would understand that it really stood for membrane. Unfortunately, it got people confused."


title: M-theory

text: In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mechanics.


title: M-theory

text: One important example of a matrix model is the BFSS matrix model proposed by Tom Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind in 1997. This theory describes the behavior of a set of nine large matrices. In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototype for a correct formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting.


title: M-theory

text: In geometry, it is often useful to introduce coordinates. For example, in order to study the geometry of the Euclidean plane, one defines the coordinates and as the distances between any point in the plane and a pair of axes. In ordinary geometry, the coordinates of a point are numbers, so they can be multiplied, and the product of two coordinates does not depend on the order of multiplication. That is, . This property of multiplication is known as the commutative law, and this relationship between geometry and the commutative algebra of coordinates is the starting point for much of modern geometry.


title: M-theory

text: Noncommutative geometry is a branch of mathematics that attempts to generalize this situation. Rather than working with ordinary numbers, one considers some similar objects, such as matrices, whose multiplication does not satisfy the commutative law (that is, objects for which is not necessarily equal to ). One imagines that these noncommuting objects are coordinates on some more general notion of "space" and proves theorems about these generalized spaces by exploiting the analogy with ordinary geometry.


title: M-theory

text: In a paper from 1998, Alain Connes, Michael R. Douglas, and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum field theory, a special kind of physical theory in which the coordinates on spacetime do not satisfy the commutativity property. This established a link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various physical theories.


title: M-theory

text: The application of quantum mechanics to physical objects such as the electromagnetic field, which are extended in space and time, is known as quantum field theory. In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields. Quantum field theories are also used throughout condensed matter physics to model particle-like objects called quasiparticles.


title: M-theory

text: One approach to formulating M-theory and studying its properties is provided by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. Proposed by Juan Maldacena in late 1997, the AdS/CFT correspondence is a theoretical result which implies that M-theory is in some cases equivalent to a quantum field theory. In addition to providing insights into the mathematical structure of string and M-theory, the AdS/CFT correspondence has shed light on many aspects of quantum field theory in regimes where traditional calculational techniques are ineffective.


title: M-theory

text: In the AdS/CFT correspondence, the geometry of spacetime is described in terms of a certain vacuum solution of Einstein's equation called anti-de Sitter space. In very elementary terms, anti-de Sitter space is a mathematical model of spacetime in which the notion of distance between points (the metric) is different from the notion of distance in ordinary Euclidean geometry. It is closely related to hyperbolic space, which can be viewed as a disk as illustrated on the left. This image shows a tessellation of a disk by triangles and squares. One can define the distance between points of this disk in such a way that all the triangles and squares are the same size and the circular outer boundary is infinitely far from any point in the interior.


title: M-theory

text: Now imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given time. The resulting geometric object is three-dimensional anti-de Sitter space. It looks like a solid cylinder in which any cross section is a copy of the hyperbolic disk. Time runs along the vertical direction in this picture. The surface of this cylinder plays an important role in the AdS/CFT correspondence. As with the hyperbolic plane, anti-de Sitter space is curved in such a way that any point in the interior is actually infinitely far from this boundary surface.


title: M-theory

text: This construction describes a hypothetical universe with only two space dimensions and one time dimension, but it can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and one can "stack up" copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space.


title: M-theory

text: An important feature of anti-de Sitter space is its boundary (which looks like a cylinder in the case of three-dimensional anti-de Sitter space). One property of this boundary is that, within a small region on the surface around any given point, it looks just like Minkowski space, the model of spacetime used in nongravitational physics. One can therefore consider an auxiliary theory in which "spacetime" is given by the boundary of anti-de Sitter space. This observation is the starting point for AdS/CFT correspondence, which states that the boundary of anti-de Sitter space can be regarded as the "spacetime" for a quantum field theory. The claim is that this quantum field theory is equivalent to the gravitational theory on the bulk anti-de Sitter space in the sense that there is a "dictionary" for translating entities and calculations in one theory into their counterparts in the other theory. For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory. In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 40 percent chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory would also have a 40 percent chance of colliding.


title: M-theory

text: One particular realization of the AdS/CFT correspondence states that M-theory on the product space is equivalent to the so-called (2,0)-theory on the six-dimensional boundary. Here "(2,0)" refers to the particular type of supersymmetry that appears in the theory. In this example, the spacetime of the gravitational theory is effectively seven-dimensional (hence the notation ), and there are four additional "compact" dimensions (encoded by the factor). In the real world, spacetime is four-dimensional, at least macroscopically, so this version of the correspondence does not provide a realistic model of gravity. Likewise, the dual theory is not a viable model of any real-world system since it describes a world with six spacetime dimensions.


title: M-theory

text: Nevertheless, the (2,0)-theory has proven to be important for studying the general properties of quantum field theories. Indeed, this theory subsumes many mathematically interesting effective quantum field theories and points to new dualities relating these theories. For example, Luis Alday, Davide Gaiotto, and Yuji Tachikawa showed that by compactifying this theory on a surface, one obtains a four-dimensional quantum field theory, and there is a duality known as the AGT correspondence which relates the physics of this theory to certain physical concepts associated with the surface itself. More recently, theorists have extended these ideas to study the theories obtained by compactifying down to three dimensions.


title: M-theory

text: In addition to its applications in quantum field theory, the (2,0)-theory has spawned important results in pure mathematics. For example, the existence of the (2,0)-theory was used by Witten to give a "physical" explanation for a conjectural relationship in mathematics called the geometric Langlands correspondence. In subsequent work, Witten showed that the (2,0)-theory could be used to understand a concept in mathematics called Khovanov homology. Developed by Mikhail Khovanov around 2000, Khovanov homology provides a tool in knot theory, the branch of mathematics that studies and classifies the different shapes of knots. Another application of the (2,0)-theory in mathematics is the work of Davide Gaiotto, Greg Moore, and Andrew Neitzke, which used physical ideas to derive new results in hyperkähler geometry.


title: M-theory

text: Another realization of the AdS/CFT correspondence states that M-theory on is equivalent to a quantum field theory called the ABJM theory in three dimensions. In this version of the correspondence, seven of the dimensions of M-theory are curled up, leaving four non-compact dimensions. Since the spacetime of our universe is four-dimensional, this version of the correspondence provides a somewhat more realistic description of gravity.


title: M-theory

text: The ABJM theory appearing in this version of the correspondence is also interesting for a variety of reasons. Introduced by Aharony, Bergman, Jafferis, and Maldacena, it is closely related to another quantum field theory called Chern–Simons theory. The latter theory was popularized by Witten in the late 1980s because of its applications to knot theory. In addition, the ABJM theory serves as a semi-realistic simplified model for solving problems that arise in condensed matter physics.


title: M-theory

text: In addition to being an idea of considerable theoretical interest, M-theory provides a framework for constructing models of real world physics that combine general relativity with the standard model of particle physics. Phenomenology is the branch of theoretical physics in which physicists construct realistic models of nature from more abstract theoretical ideas. String phenomenology is the part of string theory that attempts to construct realistic models of particle physics based on string and M-theory.


title: M-theory

text: Typically, such models are based on the idea of compactification. Starting with the ten- or eleven-dimensional spacetime of string or M-theory, physicists postulate a shape for the extra dimensions. By choosing this shape appropriately, they can construct models roughly similar to the standard model of particle physics, together with additional undiscovered particles, usually supersymmetric partners to analogues of known particles. One popular way of deriving realistic physics from string theory is to start with the heterotic theory in ten dimensions and assume that the six extra dimensions of spacetime are shaped like a six-dimensional Calabi–Yau manifold. This is a special kind of geometric object named after mathematicians Eugenio Calabi and Shing-Tung Yau. Calabi–Yau manifolds offer many ways of extracting realistic physics from string theory. Other similar methods can be used to construct models with physics resembling to some extent that of our four-dimensional world based on M-theory.


title: M-theory

text: Partly because of theoretical and mathematical difficulties and partly because of the extremely high energies (beyond what is technologically possible for the foreseeable future) needed to test these theories experimentally, there is so far no experimental evidence that would unambiguously point to any of these models being a correct fundamental description of nature. This has led some in the community to criticize these approaches to unification and question the value of continued research on these problems.


title: M-theory

text: In one approach to M-theory phenomenology, theorists assume that the seven extra dimensions of M-theory are shaped like a manifold. This is a special kind of seven-dimensional shape constructed by mathematician Dominic Joyce of the University of Oxford. These manifolds are still poorly understood mathematically, and this fact has made it difficult for physicists to fully develop this approach to phenomenology.


title: M-theory

text: For example, physicists and mathematicians often assume that space has a mathematical property called smoothness, but this property cannot be assumed in the case of a manifold if one wishes to recover the physics of our four-dimensional world. Another problem is that manifolds are not complex manifolds, so theorists are unable to use tools from the branch of mathematics known as complex analysis. Finally, there are many open questions about the existence, uniqueness, and other mathematical properties of manifolds, and mathematicians lack a systematic way of searching for these manifolds.


title: M-theory

text: Because of the difficulties with manifolds, most attempts to construct realistic theories of physics based on M-theory have taken a more indirect approach to compactifying eleven-dimensional spacetime. One approach, pioneered by Witten, Hořava, Burt Ovrut, and others, is known as heterotic M-theory. In this approach, one imagines that one of the eleven dimensions of M-theory is shaped like a circle. If this circle is very small, then the spacetime becomes effectively ten-dimensional. One then assumes that six of the ten dimensions form a Calabi–Yau manifold. If this Calabi–Yau manifold is also taken to be small, one is left with a theory in four-dimensions.


title: M-theory

text: Heterotic M-theory has been used to construct models of brane cosmology in which the observable universe is thought to exist on a brane in a higher dimensional ambient space. It has also spawned alternative theories of the early universe that do not rely on the theory of cosmic inflation.


title: Billboard Year-End Hot 100 singles of 1970

text: This is a list of "Billboard" magazine's Top Hot 100 songs of the year 1970. It covers from January 3 to November 28, 1970.


title: Shoshone

text: The Shoshone or Shoshoni ( or ) are a Native American tribe with four large cultural/linguistic divisions:


title: Shoshone

text: They traditionally speak the Shoshoni language, part of the Numic languages branch of the large Uto-Aztecan language family. The Shoshone were sometimes called the Snake Indians by neighboring tribes and early American explorers.


title: Shoshone

text: Their peoples have become members of federally recognized tribes throughout their traditional areas of settlement, often co-located with the Northern Paiute people of the Great Basin.


title: Shoshone

text: The name "Shoshone" comes from "Sosoni", a Shoshone word for high-growing grasses. Some neighboring tribes call the Shoshone "Grass House People," based on their traditional homes made from "sosoni". Shoshones call themselves "Newe", meaning "People".


title: Shoshone

text: The Shoshoni language is spoken by approximately 1,000 people today. It belongs to the Central Numic branch of the Uto-Aztecan language family. Speakers are scattered from central Nevada to central Wyoming.


title: Shoshone

text: The largest numbers of Shoshoni speakers live on the federally recognized Duck Valley Indian Reservation, located on the border of Nevada and Idaho; and Goshute Reservation in Utah. Idaho State University also offers Shoshoni-language classes.


title: Shoshone

text: The Shoshone are a Native American tribe, who originated in the western Great Basin and spread north and east into present-day Idaho and Wyoming. By 1500, some Eastern Shoshone had crossed the Rocky Mountains into the Great Plains. After 1750, warfare and pressure from the Blackfoot, Crow, Lakota, Cheyenne, and Arapaho pushed Eastern Shoshone south and westward. Some of them moved as far south as Texas, emerging as the Comanche by 1700.


title: Shoshone

text: As more European-American settlers migrated west, tensions rose with the indigenous people over competition for territory and resources. Wars occurred throughout the second half of the 19th century. The Northern Shoshone, led by Chief Pocatello, fought during the 1860s against settlers in Idaho (where the city Pocatello was named for him). As more settlers encroached on Shoshone hunting territory, the natives raided farms and ranches for food and attacked immigrants.


title: Shoshone

text: The warfare resulted in the Bear River Massacre (1863) when US forces attacked and killed an estimated 250 Northwestern Shoshone, who were at their winter encampment in present-day Franklin County, Idaho. A large number of the dead were non-combatants, including children, deliberately killed by the soldiers. This was the highest number of deaths which the Shoshone suffered at the hands of United States forces. 21 US soldiers were also killed.


title: Shoshone

text: During the American Civil War travelers continued to migrate westward along the Westward Expansion Trails. When the Shoshone, along with the Utes participated in attacks on the mail route that ran west out of Fort Laramie, the mail route had to be relocated south of the trail through Wyoming.


title: Shoshone

text: Allied with the Bannock, to whom they were related, the Shoshone fought against the United States in the Snake War from 1864 to 1868. They fought US forces together in 1878 in the Bannock War. In 1876, by contrast, the Shoshone fought alongside the U.S. Army in the Battle of the Rosebud against their traditional enemies, the Lakota and Cheyenne.


title: Shoshone

text: In 1879 a band of approximately 300 Eastern Shoshone (known as "Sheepeaters") became involved in the Sheepeater Indian War. It was the last Indian war fought in the Pacific Northwest region of the present-day United States.


title: Shoshone

text: In 1911 a small group of Bannock under a leader named Mike Daggett, also known as "Shoshone Mike," killed four ranchers in Washoe County, Nevada. The settlers formed a posse and went out after the Native Americans. They caught up with the Bannock band on February 25, 1911, and in a gun battle killed Mike Daggett and seven members of his band. They lost one man of the posse, Ed Hogle.in the Battle of Kelley Creek. The posse captured a baby, two children and a young woman. (The three older captives died of diseases within a year; the baby, Mary Jo Estep, died in 1992).


title: Shoshone

text: A rancher donated the partial remains of three adult males, two adult females, two adolescent males, and three children (believed to be Mike Daggett and his family, according to contemporary accounts) to the Smithsonian Institution in Washington, DC, for study. In 1994, the institution repatriated the remains to the Fort Hall Idaho Shoshone-Bannock Tribe.


title: Shoshone

text: In 2008 the Northwestern Band of the Shoshone Nation acquired the site of the Bear River Massacre and some surrounding land. They wanted to protect the holy land and to build a memorial to the massacre, the largest their nation had suffered. "In partnership with the American West Heritage Center and state leaders in Idaho and Utah, the tribe has developed public/private partnerships to advance tribal cultural preservation and economic development goals." They have become leaders in developing tribal renewable energy.


title: Shoshone

text: In 1845 the estimated population of Northern and Western Shoshone was 4,500, much reduced after they had suffered infectious disease epidemics and warfare. The completion of the First transcontinental railroad in 1869 was followed by European-American immigrants arriving in unprecedented numbers in the territory.


title: Shoshone

text: In 1937 the Bureau of Indian Affairs counted 3,650 Northern Shoshone and 1,201 Western Shoshone. As of the 2000 census, some 12,000 persons identified as Shoshone.


title: Shoshone

text: Shoshone people are divided into traditional bands based both on their homelands and primary food sources. These include:


title: Gray code

text: The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).


title: Gray code

text: For example, the representation of the decimal value "1" in binary would normally be "" and "2" would be "". In Gray code, these values are represented as "" and "". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two.


title: Gray code

text: Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems.


title: Gray code

text: Many devices indicate position by closing and opening switches. If that device uses natural binary codes, positions 3 and 4 are next to each other but all three bits of the binary representation differ:


title: Gray code

text: The problem with natural binary codes is that physical switches are not ideal: it is very unlikely that physical switches will change states exactly in synchrony. In the transition between the two states shown above, all three switches change state. In the brief period while all are changing, the switches will read some spurious position. Even without keybounce, the transition might look like — — — . When the switches appear to be in position , the observer cannot tell if that is the "real" position 1, or a transitional state between two other positions. If the output feeds into a sequential system, possibly via combinational logic, then the sequential system may store a false value.


title: Gray code

text: This problem can be solved by changing only one switch at a time, so there is never any ambiguity of position, resulting in codes assigning to each of a contiguous set of integers, or to each member of a circular list, a word of symbols such that no two code words are identical and each two adjacent code words differ by exactly one symbol. These codes are also known as "unit-distance", "single-distance", "single-step", "monostrophic" or "syncopic codes", in reference to the Hamming distance of 1 between adjacent codes.


title: Gray code

text: In principle, there can be more than one such code for a given word length, but the term Gray code was first applied to a particular binary code for non-negative integers, the "binary-reflected Gray code", or BRGC. Bell Labs researcher


title: Gray code

text: George R. Stibitz described such a code in a 1941 patent application, granted in 1943. Frank Gray introduced the term "reflected binary code" in his 1947 patent application, remarking that the code had "as yet no recognized name". He derived the name from the fact that it "may be built up from the conventional binary code by a sort of reflection process".


title: Gray code

text: In the standard encoding the least significant bit follows a repetitive pattern of 2 on, 2 off the next digit a pattern of 4 on, 4 off; the "n"th least significant bit a pattern of 2 on 2 off. The four-bit version of this is shown below:


title: Gray code

text: For decimal 15 the code rolls over to decimal 0 with only one switch change. This is called the cyclic or adjacency property of the code.


title: Gray code

text: In modern digital communications, Gray codes play an important role in error correction. For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise.


title: Gray code

text: Despite the fact that Stibitz described this code before Gray, the reflected binary code was later named after Gray by others who used it. Two different 1953 patent applications use "Gray code" as an alternative name for the "reflected binary code"; one of those also lists "minimum error code" and "cyclic permutation code" among the names. A 1954 patent application refers to "the Bell Telephone Gray code". Other names include "cyclic binary code", "cyclic progression code", "cyclic permuting binary" or "cyclic permuted binary" (CPB).


title: Gray code

text: The binary-reflected Gray code represents the underlying scheme of the classical Chinese rings puzzle, a sequential mechanical puzzle mechanism described by the French Louis Gros in 1872.


title: Gray code

text: It can serve as a solution guide for the Towers of Hanoi problem, based on a game by the French Édouard Lucas in 1883. Similarly, the so-called Towers of Bucharest and Towers of Klagenfurt game configurations yield ternary and pentary Gray codes.


title: Gray code

text: Martin Gardner wrote a popular account of the Gray code in his August 1972 Mathematical Games column in Scientific American.


title: Gray code

text: When the French engineer Émile Baudot changed from using a 6-unit (6-bit) code to 5-unit code for his printing telegraph system, in 1875 or 1876, he ordered the alphabetic characters on his print wheel using a reflected binary code, and assigned the codes using only three of the bits to vowels. With vowels and consonants sorted in their alphabetical order, and other symbols appropriately placed, the 5-bit character code has been recognized as a reflected binary code. This code became known as Baudot code and, with minor changes, was eventually adopted as International Telegraph Alphabet No. 1 (ITA1, CCITT-1) in 1932.


title: Gray code

text: About the same time, the German-Austrian demonstrated another printing telegraph in Vienna using a 5-bit reflected binary code for the same purpose, in 1874.


title: Gray code

text: Frank Gray, who became famous for inventing the signaling method that came to be used for compatible color television, invented a method to convert analog signals to reflected binary code groups using vacuum tube-based apparatus. Filed in 1947, the method and apparatus were granted a patent in 1953, and the name of Gray stuck to the codes. The "PCM tube" apparatus that Gray patented was made by Raymond W. Sears of Bell Labs, working with Gray and William M. Goodall, who credited Gray for the idea of the reflected binary code.


title: Gray code

text: Gray was most interested in using the codes to minimize errors in converting analog signals to digital; his codes are still used today for this purpose.


title: Gray code

text: Gray codes are used in linear and rotary position encoders (absolute encoders and quadrature encoders) in preference to weighted binary encoding. This avoids the possibility that, when multiple bits change in the binary representation of a position, a misread will result from some of the bits changing before others.


title: Gray code

text: For example, some rotary encoders provide a disk which has an electrically conductive Gray code pattern on concentric rings (tracks). Each track has a stationary metal spring contact that provides electrical contact to the conductive code pattern. Together, these contacts produce output signals in the form of a Gray code. Other encoders employ non-contact mechanisms based on optical or magnetic sensors to produce the Gray code output signals.


title: Gray code

text: Regardless of the mechanism or precision of a moving encoder, position measurement error can occur at specific positions (at code boundaries) because the code may be changing at the exact moment it is read (sampled). A binary output code could cause significant position measurement errors because it is impossible to make all bits change at exactly the same time. If, at the moment the position is sampled, some bits have changed and others have not, the sampled position will be incorrect. In the case of absolute encoders, the indicated position may be far away from the actual position and, in the case of incremental encoders, this can corrupt position tracking.


title: Gray code

text: In contrast, the Gray code used by position encoders ensures that the codes for any two consecutive positions will differ by only one bit and, consequently, only one bit can change at a time. In this case, the maximum position error will be small, indicating a position adjacent to the actual position.


title: Gray code

text: Due to the Hamming distance properties of Gray codes, they are sometimes used in genetic algorithms. They are very useful in this field, since mutations in the code allow for mostly incremental changes, but occasionally a single bit-change can cause a big leap and lead to new properties.


title: Gray code

text: Gray codes are also used in labelling the axes of Karnaugh maps since 1953 as well as in Händler circle graphs since 1958, both graphical methods for logic circuit minimization.


title: Gray code

text: Digital logic designers use Gray codes extensively for passing multi-bit count information between synchronous logic that operates at different clock frequencies. The logic is considered operating in different "clock domains". It is fundamental to the design of large chips that operate with many different clocking frequencies.


title: Gray code

text: If a system has to cycle through all possible combinations of on-off states of some set of controls, and the changes of the controls require non-trivial expense (e.g. time, wear, human work), a Gray code minimizes the number of setting changes to just one change for each combination of states. An example would be testing a piping system for all combinations of settings of its manually operated valves.


title: Gray code

text: A balanced Gray code can be constructed, that flips every bit equally often. Since bit-flips are evenly distributed, this is optimal in the following way: balanced Gray codes minimize the maximal count of bit-flips for each digit.


title: Gray code

text: George R. Stibitz utilized a reflected binary code in a binary pulse counting device in 1941 already.


title: Gray code

text: A typical use of Gray code counters is building a FIFO (first-in, first-out) data buffer that has read and write ports that exist in different clock domains. The input and output counters inside such a dual-port FIFO are often stored using Gray code to prevent invalid transient states from being captured when the count crosses clock domains.


title: Gray code

text: The updated read and write pointers need to be passed between clock domains when they change, to be able to track FIFO empty and full status in each domain. Each bit of the pointers is sampled non-deterministically for this clock domain transfer. So for each bit, either the old value or the new value is propagated. Therefore, if more than one bit in the multi-bit pointer is changing at the sampling point, a "wrong" binary value (neither new nor old) can be propagated. By guaranteeing only one bit can be changing, Gray codes guarantee that the only possible sampled values are the new or old multi-bit value. Typically Gray codes of power-of-two length are used.


title: Gray code

text: Sometimes digital buses in electronic systems are used to convey quantities that can only increase or decrease by one at a time, for example the output of an event counter which is being passed between clock domains or to a digital-to-analog converter. The advantage of Gray codes in these applications is that differences in the propagation delays of the many wires that represent the bits of the code cannot cause the received value to go through states that are out of the Gray code sequence. This is similar to the advantage of Gray codes in the construction of mechanical encoders, however the source of the Gray code is an electronic counter in this case. The counter itself must count in Gray code, or if the counter runs in binary then the output value from the counter must be reclocked after it has been converted to Gray code, because when a value is converted from binary to Gray code, it is possible that differences in the arrival times of the binary data bits into the binary-to-Gray conversion circuit will mean that the code could go briefly through states that are wildly out of sequence. Adding a clocked register after the circuit that converts the count value to Gray code may introduce a clock cycle of latency, so counting directly in Gray code may be advantageous.


title: Gray code

text: To produce the next count value in a Gray-code counter, it is necessary to have some combinational logic that will increment the current count value that is stored. One way to increment a Gray code number is to convert it into ordinary binary code, add one to it with a standard binary adder, and then convert the result back to Gray code. Other methods of counting in Gray code are discussed in a report by Robert W. Doran, including taking the output from the first latches of the master-slave flip flops in a binary ripple counter.


title: Gray code

text: As the execution of program code typically causes an instruction memory access pattern of locally consecutive addresses, bus encodings using Gray code addressing instead of binary addressing can reduce the number of state changes of the address bits significantly, thereby reducing the CPU power consumption in some low-power designs.


title: Gray code

text: The binary-reflected Gray code list for "n" bits can be generated recursively from the list for "n" − 1 bits by reflecting the list (i.e. listing the entries in reverse order), prefixing the entries in the original list with a binary , prefixing the entries in the reflected list with a binary , and then concatenating the original list with the reversed list. For example, generating the "n" = 3 list from the "n" = 2 list:


title: Gray code

text: The one-bit Gray code is "G" = (). This can be thought of as built recursively as above from a zero-bit Gray code "G" = ( Λ ) consisting of a single entry of zero length. This iterative process of generating "G" from "G" makes the following properties of the standard reflecting code clear:


title: Gray code

text: These characteristics suggest a simple and fast method of translating a binary value into the corresponding Gray code. Each bit is inverted if the next higher bit of the input value is set to one. This can be performed in parallel by a bit-shift and exclusive-or operation if they are available: the "n"th Gray code is obtained by computing formula_1. Prepending a bit leaves the order of the code words unchanged, prepending a bit reverses the order of the code words. If the bits at position formula_2 of codewords are inverted, the order of neighbouring blocks of formula_3 codewords is reversed. For example, if bit 0 is inverted in a 3 bit codeword sequence, the order of two neighbouring codewords is reversed


title: Gray code

text: Thus, performing an exclusive or on a bit formula_4 at position formula_2 with the bit formula_6 at position formula_7 leaves the order of codewords intact if formula_8, and reverses the order of blocks of formula_9 codewords if formula_10. Now, this is exactly the same operation as the reflect-and-prefix method to generate the Gray code.


title: Gray code

text: A similar method can be used to perform the reverse translation, but the computation of each bit depends on the computed value of the next higher bit so it cannot be performed in parallel. Assuming formula_11 is the formula_2th Gray-coded bit (formula_13 being the most significant bit), and formula_4 is the formula_2th binary-coded bit (formula_16 being the most-significant bit), the reverse translation can be given recursively: formula_17, and formula_18. Alternatively, decoding a Gray code into a binary number can be described as a prefix sum of the bits in the Gray code, where each individual summation operation in the prefix sum is performed modulo two.


title: Gray code

text: To construct the binary-reflected Gray code iteratively, at step 0 start with the formula_19, and at step formula_20 find the bit position of the least significant in the binary representation of formula_2 and flip the bit at that position in the previous code formula_22 to get the next code formula_23. The bit positions start 0, 1, 0, 2, 0, 1, 0, 3, ... See find first set for efficient algorithms to compute these values.


title: Gray code

text: The following functions in C convert between binary numbers and their associated Gray codes. While it may seem that Gray-to-binary conversion requires each bit to be handled one at a time, faster algorithms exist.


title: Gray code

text: // A more efficient version for Gray codes 32 bits or fewer through the use of SWAR (SIMD within a register) techniques.


title: Gray code

text: // A Four-bit-at-once variant changes a binary number (abcd)2 to (abcd)2 ^ (00ab)2, then to (abcd)2 ^ (00ab)2 ^ (0abc)2 ^ (000a)2.


title: Gray code

text: On newer processors, the number of ALU instructions in the decoding step can be reduced by taking advantage of the CLMUL instruction set. If MASK is the constant binary string of ones ended with a single zero digit, then carryless multiplication of MASK with the grey encoding of x will always give either x or its bitwise negation.


title: Gray code

text: Like BRGCs, each consists of a list of words, where each word differs from the next in only one digit (each word has a Hamming distance of 1 from the next word).


title: Gray code

text: It is possible to construct binary Gray codes with "n" bits with a length of less than , if the length is even. One possibility is to start with a balanced Gray code and remove pairs of values at either the beginning and the end, or in the middle. OEIS sequence A290772 gives the number of possible Gray sequences of length which include zero and use the minimum number of bits.


title: Gray code

text: There are many specialized types of Gray codes other than the binary-reflected Gray code. One such type of Gray code is the "n"-ary Gray code, also known as a non-Boolean Gray code. As the name implies, this type of Gray code uses non-Boolean values in its encodings.


title: Gray code

text: For example, a 3-ary (ternary) Gray code would use the values . The ("n", "k")-"Gray code" is the "n"-ary Gray code with "k" digits.


title: Gray code

text: The sequence of elements in the (3, 2)-Gray code is: . The ("n", "k")-Gray code may be constructed recursively, as the BRGC, or may be constructed iteratively. An algorithm to iteratively generate the ("N", "k")-Gray code is presented (in C):


title: Gray code

text: There are other Gray code algorithms for ("n","k")-Gray codes. The ("n","k")-Gray code produced by the above algorithm is always cyclical; some algorithms, such as that by Guan, lack this property when k is odd. On the other hand, while only one digit at a time changes with this method, it can change by wrapping (looping from "n" − 1 to 0). In Guan's algorithm, the count alternately rises and falls, so that the numeric difference between two Gray code digits is always one.


title: Gray code

text: Gray codes are not uniquely defined, because a permutation of the columns of such a code is a Gray code too. The above procedure produces a code in which the lower the significance of a digit, the more often it changes, making it similar to normal counting methods.


title: Gray code

text: See also Skew binary number system, a variant ternary number system where at most two digits change on each increment, as each increment can be done with at most one digit carry operation.