title: 2018 Asian Games

text: Three new mascots were also unveiled: Bhin Bhin, a greater bird-of-paradise; Atung, a Bawean deer; and Kaka, a Javan rhinoceros. They represent the Eastern, Central, and Western regions of Indonesia, as well as strategy, speed and strength. The mascots' outfits reflect traditional textiles; Bhin Bhin wears a vest with Asmat pattern details, Atung wears a "sarong" with "batik tumpal" patterns, and Kaka wears a Palembang "songket" with floral patterns. Their names were derived from the national motto of Indonesia, "Bhinneka Tunggal Ika" ("Unity in Diversity").


title: 2018 Asian Games

text: The medal designs were unveiled in July 2018, featuring batik designs reflecting the unity and diversity of Indonesian and Asian culture.


title: 2018 Asian Games

text: On 18 August 2017, simultaneous events were held at Jakarta's National Monument and Palembang's Kuto Besak Fort to mark the one-year milestone before the Games. The event in Jakarta was attended by president Joko Widodo, and featured performances by Taeyeon and Kim Hyo-yeon of K-pop group Girls' Generation. Countdown clocks were unveiled at the Selamat Datang Monument and in front of Gelora Sriwijaya Stadium.


title: 2018 Asian Games

text: Several fun run events were held in Asian countries to promote the Games, beginning with Lahore, Pakistan in December 2017.


title: 2018 Asian Games

text: The games' merchandises and licensing products which included mascot stuffed toy were sold in host city Jakarta including the Super Store in the Gelora Bung Karno sports complex from end of 2017 until end of September 2018.


title: 2018 Asian Games

text: The official theme song of the Games, "Meraih Bintang" ("Reach for the Stars"), was performed by Via Vallen. The song was included in a larger released 13 July 2018, featuring various Indonesian musicians.


title: 2018 Asian Games

text: As part of an effort to appeal younger spectators, the Indonesian-Japanese idol group JKT48 was signed to perform at some of the Games' venues, such as jet ski, softball, and volleyball. They performed in select events between 19 August and 1 September in a group consisted of eight members from each of its teams.


title: 2018 Asian Games

text: As part of creating a memorable and collectible item, the Indonesian postal service Pos Indonesia issued a philatelic collectible items of the 2018 Asian Games series. These items were issued on 18 January 2018 and 18 August 2018.


title: 2018 Asian Games

text: The Games used a mix of new and existing venues in Jakarta and South Sumatra (with organizers aiming to reuse existing facilities and infrastructure, such as those built for the 2011 Southeast Asian Games, to help control costs), as well as Jakarta's neighbouring provinces of Banten, and West Java. Venues were divided into four clusters, with three in Jakarta and its surrounding areas and one in Palembang.


title: 2018 Asian Games

text: The Gelora Bung Karno Sports Complex in Jakarta hosted 13 events. The 56-year-old Main Stadium was refurbished for the Games, replacing its existing bleachers and seating with an all-seater design (reducing its capacity to 77,193), and adding new sound systems and LED lighting among other enhancements.


title: 2018 Asian Games

text: The Jakarta International Velodrome at Rawamangun in East Jakarta was rebuilt, at a cost of US$40 million for cycling, badminton, futsal, basketball, and wrestling. The Jakarta International Equestrian Park at Pulomas underwent a US$30.8 million renovation, with a capacity of 1,000, 100 stables, lodging for athletes, and other amenities.


title: 2018 Asian Games

text: Jakabaring Sport City was used as an event site. While there were preliminary plans to increase the size of Gelora Sriwijaya Stadium from 36,000 to 60,000 seats, they were shelved. The site still underwent some renovations, including also being converted from bleachers to an all-seater. A new 40-lane bowling alley was constructed on the complex, as well as eight additional tennis courts. The length of the canoeing and rowing venue in Jakabaring Lake was extended to 2,300 meters, while new rowing facilities and a tribune were built on the lake shore. Other existing venues which was used for Games were also renovated, including Ranau Sports Hall as sepak takraw venue.


title: 2018 Asian Games

text: A 10-hectare athletes' village was constructed in Jakarta's Kemayoran district, with 7,424 apartments in 10 towers, and a total capacity of 22,272. A second athletes' village was built at Jakabaring Sports City at Palembang, which housed 3,000 athletes and officials.


title: 2018 Asian Games

text: As part of the Games preparation, the construction of the Jakarta MRT and Jakarta LRT was accelerated, though neither were ready for general commercial operation at the time of the opening ceremonies. A line of Jakarta LRT connected the athletes' village at Kemayoran in Central Jakarta to the Velodrome at Rawamangun in East Jakarta. City bus operator TransJakarta added 416 buses to serve the officials, and also provide free rides on selected days during the Games.


title: 2018 Asian Games

text: Palembang upgraded their transportation facilities ahead for the Games by building 25 kilometres of the Palembang Light Rail Transit from Sultan Mahmud Badaruddin II International Airport to Jakabaring Sports City which opened for public use in July 2018. Sultan Mahmud Badaruddin II International Airport is expanding its existing arrival and departure terminals to increase its capacity and also connecting the airport with the light rail transit (LRT) terminal by building a skybridge. Other transportation facilities such as toll roads, flyovers, and bridges will be also built in and around the city.


title: 2018 Asian Games

text: The opening ceremony started at 19:00 Western Indonesian Time (UTC+7) on Saturday, 18 August 2018. Wishnutama, then-CEO of Indonesian TV network NET. was the creative director for the ceremony. The ceremony stage showcased a towering 26 meter-high mountain with a waterfall as its background, accompanied by Indonesian plants and flowers. North and South Korea delegates marched together under one unified flag of Korea, which marked the first time both countries did so in the Asian Games after 12 years.


title: 2018 Asian Games

text: In March 2017, the Olympic Council of Asia initially announced that the Games would feature 484 events in 42 sports, including the 28 permanent Olympic sports contested at the 2016 Summer Olympics, the five additional sports that will be contested at the 2020 Summer Olympics in Tokyo, as well as events in other non-Olympic sports. In April 2017, the OCA approved reductions in the programme in response to cost concerns; belt wrestling, cricket, kurash, skateboarding, sambo, and surfing were dropped from the programme, and there was to be a reduced number of competitions in bridge, jet ski, jujitsu, paragliding, sport climbing, taekwondo (in particular, all non-Olympic weight classes), and wushu. These changes reduced the total number of events to 431.


title: 2018 Asian Games

text: The final programme was unveiled in September 2017, increasing it to 462 events in 40 sports (61 disciplines) as the second-largest programme in Asian Games history. Additional disciplines being introduced at the 2020 Summer Olympics were also added, including 3x3 basketball and BMX freestyle.


title: 2018 Asian Games

text: For the first time in Asian Games history, esports and canoe polo were contested as demonstration sports in the Games. Six video game titles, most notably Pro Evolution Soccer 2018, were featured in the esports events.


title: 2018 Asian Games

text: All 45 members of the Olympic Council of Asia participated in the games. North Korea and South Korea competed as a unified team in some events under the name "Korea" (COR), and marched together under the Korean Unification Flag during the opening and closing ceremonies. With a gold medal in the Women's 500 metre dragon boat competition, a unified team won their first-ever gold medal in a multi-sport event.


title: 2018 Asian Games

text: Originally set to compete as Independent Asian Athletes, the Kuwaitis were allowed to compete under their own flags just two days before the opening ceremony.


title: 2018 Asian Games

text: Below is a list of all the participating NOCs. The number of competitors per delegation is indicated in brackets.


title: 2018 Asian Games

text: The closing ceremony started at 19:00 Western Indonesia Time (UTC+7) on Sunday, 2 September 2018 and ended at 21:25. In addition to local artists and a Chinese segment, the South Korean boybands Super Junior and iKon, and Indian singer Sidharth Slathia performed in the ceremony. Mayor of Hangzhou Xu Liyi received the Games flag for the 2022 Games from Jakarta Governor Anies Baswedan and South Sumatra Governor Alex Noerdin.


title: 2018 Asian Games

text: China led the medal table for the tenth consecutive time. Korea claimed their first gold medal at the Games in the canoeing women's traditional boat race 500 m event. A total of 37 NOCs won at least one medal, 29 NOCs won at least one gold medal and 9 NOCs failed to win any medal at the Games.


title: 2018 Asian Games

text: International Games Broadcast Services (IGBS), a joint venture between Host Broadcast Services (HBS) and IMG Media was appointed by the organiser in July 2017 to serve as the host broadcaster for these Games. Involving 37 live production units, it distributed 2,400 hours of the Games content to its international rights holders. The International Broadcast Centre was constructed in Jakarta Convention Center, with a smaller one constructed at the Palembang cluster.


title: 2018 Asian Games

text: Before the Games, authorities in Indonesia were confident both host cities would be ready for the Games although they have had only four years to prepare rather than the usual six after stepping in to fill the gap when Vietnam, whose city of Hanoi was originally chosen to host these Games by the Olympic Council of Asia, dropped out in 2014 citing concerns over costs. On top of that, work in both host cities was delayed throughout 2015 because government funding was not immediately available.


title: 2018 Asian Games

text: Various concerns from traffic congestion problems, series of terror attacks, which local police claimed is a pre-Asian Games crackdown on terror suspects and petty street criminals, and already-provoked Indonesian fans at the venue.


title: 2018 Asian Games

text: Jakarta struggled with air pollution and river pollution problems. River pollution revealed where authorities covered a foul-smelling river near the athletes' village with black nylon mesh over fears it will be an eyesore at the showpiece event. Governor of Jakarta at that time, and the Indonesian Environment and Forestry Ministry quickly relieved the issues by various solutions.


title: 2018 Asian Games

text: A doping case from a Turkmen wrestler was recorded, which jeopardize the sport's existence for future Games.


title: Nezha

text: Nezha (哪吒) is a protection deity in Chinese folk religion. His official Taoist name is "Marshal of the Central Altar" (). He was then given the title "Third Lotus Prince" () after he became a deity.


title: Nezha

text: According to Meir Shahar, Nezha is ultimately based on two figures from Hindu mythology. The first is a yaksha from the "Ramayana" named Nalakubar, the son of Yaksha King Kubera and nephew of the antagonist Ravana. The link to Nalakubar is established through variants in his Chinese name appearing in Buddhist sutras. The original variant Naluojiupoluo () changed to Naluojubaluo (), Nazhajuwaluo (), and finally Nazha (). The simple addition of the "mouth radical" () to Na () changes the name to the current form Nezha (). The second figure is the child god Krishna. Both Krishna and Nezha are powerful children that defeat mighty serpents, Kaliya in the case of the former and Ao Bing in the latter. The "Bhagavata Purana" describes how Nalakubar was rescued from imprisonment within a tree by Krishna. A 10th-century Tantric Buddhist sutra mentions a child god that seems to be an amalgam of Krishna and Nalakubar called Nana (). In addition, Nalakubar's father Kubera was eventually absorbed into the Buddhist pantheon as the Heavenly King Vaiśravaṇa. Shahar notes that Vaisravana was somehow connected to the historical Tang Dynasty general Li Jing. This explains the name and position of Nezha's father, the Pagoda-Bearing Heavenly King Li Jing.


title: Nezha

text: According to "Fengshen Yanyi", Nezha was born during the Shang dynasty in a military fortress at Chentang Pass. His father was military commander Li Jing, who later became the "Pagoda-wielding Heavenly King". Nezha's mother, Lady Yin, gave birth to a ball of flesh after gestating for three years and six months. Li Jing thought that his wife had given birth to a demon and attacked the ball with his sword. The ball split open, and Nezha jumped out as a boy instead of an infant. Nezha could speak and walk immediately after birth. He was later accepted by the immortal Taiyi Zhenren as a student. He had two older brothers, Jinzha, a disciple of Wenshu Guangfa Tianzun, and Muzha.


title: Nezha

text: One day, the people of Chentang Pass asked for rain and sacrificed much food to the East Sea Dragon King Ao Guang. The King rejected the food, instead, wanting girls and boys to eat. He sent Ye Sha to capture for him a girl and boy. Nezha and two other children were playing by the sea when Ye Sha appeared and captured one of Nezha's friends. Nezha then fought him and injured him severely, causing him to return to the King and beg for someone else to take care of Nezha. The Dragon King sent Ao Bing, his third son, whom Nezha then slew in combat. Ao Guang called for his brothers and confronted Nezha and his family. He threatened to flood Chentang Pass and report Nezha to the Jade Emperor. To save his family and the people, Nezha killed himself then, carving up his own flesh and dismembering his bones "returning" these to his parents in repayment for the debt of his birth. The Dragon Kings then hosted a huge celebration.


title: Nezha

text: Nezha then appeared in his mother's dream. In the dream, he asked her to build a temple for him, so that his soul would have a place to rest. This constitutes a link to Nezha's birth because the night before Nezha was born, Lady Yin had a dream where a Taoist put something into her bosom and told her to take this child. For both incidences, a dream was used to communicate a message.


title: Nezha

text: His mother then secretly built a temple for Nezha, and this temple later flourished. This temple became very well known and grew vastly because Nezha granted miracle cures to the sick and the crippled. However, Li Jing soon found out about this temple and burned it down because he was still angry at Nezha and felt that he had already caused too much trouble for their family.


title: Nezha

text: Li Jing burning the temple caused Nezha to desire his father's death. Thus, enmity between father and son grew. Nezha was later brought back to life by his teacher, Taiyi Zhenren, who used lotus roots to construct a human body for his soul and gave him two new weapons: the Wind Fire Wheels () and the Fire-tipped Spear (). With the reincarnation of Nezha by his master, Li Jing and Nezha fought many battles. However, Li Jing soon realized that his mortal body was no match for Nezha and so ran for his life. On the run, he met his second son, Muzha, who fought and was defeated by Nezha. At this, Li Jing attempted suicide but was stopped by Wenshu Guangfa Tianzun, who also contained Nezha. In the end, Nezha was forced to submit to his father by another deity, Randeng Daoren.


title: Nezha

text: Though according to "The Journey to the West", Nezha was born with the word "Ne" written on the palm of his left hand and "zha" on his right, hence his name. But after he caused havoc in the Eastern Ocean, the Heavenly King had been so worried about the disastrous consequences that he decided to kill Nezha.


title: Nezha

text: Thus Nezha returned his flesh and blood to his mother and bones to his father. Then Nezha's soul went to the Buddha, who resurrected him. After, he used his vast powers to subdue 96 caves of demons through dharma. After this Nezha wanted to kill his father because he had to remove every piece of flesh and every stain of blood to return his bones to his father. The heavenly king had no choice but to seek help from the Buddha. The Buddha gave him an intricately made golden pagoda, in each story of which were Buddhas radiant with splendor. The Buddha told Nezha to regard these Buddhas as his father. Thus ending the hatred between the father and the son. Thus earning Li Jing the title Pagoda-Bearing Heavenly king.


title: Nezha

text: Nezha is often depicted as a youth, instead of an adult. He is often shown flying in the sky riding on the Wind Fire Wheels (), has the Universe Ring () around his body (sometimes in his left hand), the Red Armillary Sash () around his shoulders and a Fire-tipped Spear () in his right hand. Sometimes, he is shown in his "three heads and six arms" form (). He has the ability to spit rainbows in some legends.


title: Nezha

text: Nezha has frequently appeared in Chinese mythology and ancient Chinese literature such as "Fengshen Yanyi" (or "Investiture of the Gods"), although the story of "Nezha Conquering the Sea" is the most well known among Chinese households.


title: Nezha

text: In "Journey to the West", Nezha was a general under his father, "Pagoda-wielding Heavenly King" Li Jing. He fought the Monkey King, Sun Wukong, when the latter rebelled against the Jade Emperor. They later became friends. Nezha made some appearances in the novel to help the four protagonists defeat powerful demons including his adoptive sister, Lady Earth Flow.


title: Nezha

text: Nezha has been the central character in over 20 different films and television programs, both live-action and animated, dating back at least as early as a pre-communist Chinese live-action feature film, "Nézhā Chūshì" (), produced by the Great Wall Film Company and premiered, according to differing accounts, in either 1927 or 1928., In addition, he has appeared as a supporting character in numerous others, chiefly among the many adaptations of "Investiture of the Gods" and "Journey to the West".


title: Nezha

text: The character increased in popularity in 1979 with the traditionally-animated feature film "Nezha Conquers the Dragon King", which was screened at the 1980 Cannes Film Festival and is considered one of the great classic works of Chinese animation. On 30 May 2014, Google paid homage to this film with an animated doodle on their Hong Kong search engine's homepage.


title: Nezha

text: In 2003, China Central Television began the broadcast of a new children's traditionally-animated series "The Legend of Nezha", which originally ran from 2003 to 2004, for a total of 52 episodes.


title: Nezha

text: On 26 July 2019, another stereoscopic, computer-animated feature film, "Ne Zha",() was more successful, setting numerous all-time records for box-office grosses, including third-highest-grossing of all films in China and highest-grossing animated film from outside the United States. A comment from www.fjsen.com says, Based on the previous work, the film is bold and innovative, which not only retains the essence of Chinese traditional culture, but also adds popular elements, making the culture to be displayed more easily accepted by the audience. It breaks the stereotype, emphasizes self-independence, and gives consideration to reality with metaphor. There are many scenes that reflect on the secular and ugly sides of society in the film. Compared with the truth, many people are more willing to believe in hearsay and then look at people with colored glasses. Nezha, a rebellious superhero, though seemingly rebellious to tradition, still retain the most positive passion and moving, still have the value orientation of Chinese culture of dedication and responsibility.


title: Nezha

text: The 2021 animated film "" depicts Nezha reincarnated in a cyberpunk setting 3,000 years after the original story.


title: Nezha

text: A 2017 announcement from the leadership of Hasbro confirmed that Nezha will appear in a crossover with the "Transformers" franchise. The series, titled "Nezha: Transformers", consists of 52 11-minute episodes and is produced by Allspark Animation and China Central Television.


title: Nezha

text: The character has also been evoked outside of the fantasy genre, in realistic contemporary and period drama films as a byword for a rebellious, nonconformist young person, such as in "Rebels of the Neon God" (, 1992), "Spin Kid" (, 2011), "Nezha" (, 2014) and "Operation Mekong" (2016, in which a character is codenamed Nezha).


title: Nezha

text: Nezha is a playable character in Koei Tecmo's crossover series "Warriors Orochi". The series features two different versions of Nezha, a cyborg and his human self.


title: Nezha

text: Ne Zha is a playable character in the multiplayer online battle arena video game "Smite", where he is a melee assassin.


title: Nezha

text: On December 16, 2015, third-person shooter action game "Warframe" released a playable character named and themed after Nezha.


title: Nezha

text: Nezha is also a collectible character in Netmarble's mobile RPG "Seven Knights" where the character, while depicted as female, retains much of Nezha's mythos and characteristics.


title: Nezha

text: Nezha is also a summonable Heroic Spirit in the popular Japanese hero collector game "Fate/Grand Order" where she can be summoned as a Lancer-class Servant. In the game, she is depicted as a female automaton. The game's second storyline, "Cosmos in the Lostbelt", explores an alternate history where Qin Shi Huang successfully becomes immortal by studying her body, allowing him to rule China into the present day.


title: Nezha

text: Nezha is also a summonable Astromon in the Korean creature capture game "Monster Super League" developed by 4:33 and Creative Labs. In the game, he is depicted as female.


title: Nezha

text: Nezha is also a summonable character in the Japanese LGBT+ Mobile Game "Tokyo Afterschool Summoners" developed by Lifewonders. Within the game he is depicted as non-binary.


title: Nezha

text: In 2022, the video game Overwatch (2016), made a skin based on Nezha for a female character as part of the Lunar New Year Event.


title: Nezha

text: Nezha, with his fire-tipped spear, is the title character in the casino game "The Third Prince." The game's battle feature draws on elements of Nezha's story.


title: Nezha

text: Nezha is worshipped in Chinese folk religion and is called "Marshal of the Central Altar" or "Prince Nezha", the "Third Prince".


title: Nezha

text: As in traditional folklore, Nezha flies around swiftly on his wind fire wheels, so he is also regarded as the tutelary god of many professional drivers, like trucks, taxis, or sightseeing bus drivers. They tend to place a small statue of Nezha in the vehicles for a safe drive.


title: Nezha

text: Nezha is also often regarded as the patron god of children and filial piety. Parents would make an offering to Nezha with the hope that their children would grow up strong, healthy, and be dutiful and respectful.


title: People's Republic of Bulgaria

text: The People's Republic of Bulgaria (PRB; , "Narodna Republika Balgariya, NRB") was the official name of Bulgaria, when it was a socialist republic from 1946 to 1990, ruled by the Bulgarian Communist Party (BCP) together with its coalition partner, the Bulgarian Agrarian People's Union. Bulgaria was closely allied with the Soviet Union during the Cold War, being part of Comecon as well as a member of the Warsaw Pact. The Bulgarian resistance movement during World War II deposed the Kingdom of Bulgaria administration in the Bulgarian coup d'état of 1944 which ended the country's alliance with the Axis powers and led to the People's Republic in 1946.


title: People's Republic of Bulgaria

text: The BCP modelled its policies after those of the Soviet Union, transforming the country over the course of a decade from an agrarian peasant society into an industrialized socialist society. In the mid-1950s and after the death of Stalin, the party's hardliners lost influence and a period of social liberalization and stability followed under Todor Zhivkov. Varying degrees of conservative or liberal influence followed. After a new energy and transportation infrastructure was constructed, by 1960 manufacturing became the dominant sector of the economy and Bulgaria became a major exporter of household goods and later of computer technologies, earning it the nickname of "Silicon Valley of the Eastern Bloc". The country's relatively high productivity levels and high scores on social development rankings made it a model for other socialist countries' administrative policies.


title: People's Republic of Bulgaria

text: In 1989, after a few years of liberal influence, political reforms were initiated and Todor Zhivkov, who had served as head of the party since 1954, was removed from office in a BCP congress. In 1990, under the leadership of Aleksandar Lilov, the BCP changed its name to the Bulgarian Socialist Party (BSP) and adopted social democracy and democratic socialism in place of Marxism–Leninism. Following the BSP victory in the 1990 election, which was the first openly contested multi-party election since 1931, the name of the state was changed to the Republic of Bulgaria. Geographically, the People's Republic of Bulgaria had the same borders as present-day Bulgaria and it bordered the Black Sea to the east; Romania to the north; Yugoslavia (via SRs Serbia and Macedonia) to the west and Greece and Turkey to the south.


title: People's Republic of Bulgaria

text: On 1 March 1941, the Kingdom of Bulgaria signed the Tripartite Pact, and officially became a member of the Axis. Following the German invasion of Yugoslavia and Greece in April, Bulgaria came to occupy large parts of those countries. In 1942, the Fatherland Front was formed from a mixture of Communists, Socialists, Agrarians and Zveno.


title: People's Republic of Bulgaria

text: In 1944, with the entry of the Red Army into Romania, the Kingdom of Bulgaria renounced the Axis and declared neutrality. On 5 September, the Soviet Union declared war on the kingdom and three days later the Red Army entered north-eastern Bulgaria, prompting the government to declare support in order to minimise military conflict. On 9 September, communist partisans launched a coup d'état which "de facto" ended the rule of the Bulgarian monarchy and its administration, after which a new government assumed power led by the Fatherland Front (FF), which itself was led by the Bulgarian Communist Party.


title: People's Republic of Bulgaria

text: After taking power, the FF formed a coalition led by former Prime Minister Kimon Georgiev, which included the Social Democrats and the Agrarians. Under the terms of the peace settlement, Bulgaria was allowed to keep Southern Dobruja, but formally renounced all claims to Greek and Yugoslav territory. 150,000 Bulgarians settled during the occupation were expelled from Western Thrace. The Communists deliberately took a minor role in the new government at first, while the Soviet representatives held the real power. A Communist-controlled People's Militia was set up, which harassed and intimidated non-Communist parties.


title: People's Republic of Bulgaria

text: On 1 February 1945, Regent Prince Kiril, former Prime Minister Bogdan Filov, and hundreds of other officials of the kingdom were arrested on charges of war crimes. By June, Kirill and the other Regents, twenty-two former ministers, and many others had been executed. The new government began to arrest Nazi collaborators. Thousands of people were charged with treason or participating in counter-revolutionary conspiracy and sentenced to either death or life in prison. As the war came to a halt, the government expanded its campaign of political revolution to attack economic elites in banking and private business. This strengthened when it became apparent that the United States and United Kingdom had little interest in Bulgaria. In November 1945, Communist Party leader Georgi Dimitrov returned to Bulgaria after 22 years in exile. He made a truculent speech that rejected cooperation with opposition groups. Elections held a few weeks later resulted in a large majority for the Fatherland Front.


title: People's Republic of Bulgaria

text: In September 1946, the monarchy was abolished by plebiscite, which resulted in 95.6 percent voting in favour of a republic, and young Tsar Simeon II was sent into exile. The Communists openly took power, and Bulgaria was declared a "People's Republic". Vasil Kolarov, the number-three man in the party, became president. Over the next year, the Communists consolidated their hold on power. Elections for a constituent assembly in October 1946 gave the Communists a majority. A month later, Dimitrov became prime minister.


title: People's Republic of Bulgaria

text: The Agrarians refused to co-operate with the authorities, and in June 1947 their leader Nikola Petkov was arrested, despite strong international protests. This marked the establishment of a Communist establishment in Bulgaria. In December 1947, the constituent assembly ratified a new constitution for the republic, referred to as the "Dimitrov Constitution". The constitution was drafted with the help of Soviet jurists using the 1936 Soviet Constitution as a model. By 1948, the remaining opposition parties were either realigned or dissolved; the Social Democrats merged with the Communists, while the Agrarian Union became a loyal partner of the Communists.


title: People's Republic of Bulgaria

text: During 1948–1949, Orthodox, Muslim, Protestant and Roman Catholic religious organizations were restrained or banned. The Orthodox Church of Bulgaria continued functioning but never regained the influence it held under the monarchy; many high roles within the church were assumed by communist functionaries.


title: People's Republic of Bulgaria

text: Dimitrov died in 1949 and for a time Bulgaria adopted collective leadership. Vulko Chervenkov led the Communist Party and Vasil Kolarov was prime minister. This broke down a year later, when Kolarov died and Chervenkov added prime minister to his titles. Chervenkov started a process of rapid industrialization modelled after the Soviet industrialisation led by Stalin in the 1930s and agriculture was collectivised.


title: People's Republic of Bulgaria

text: Chervenkov's support base even in the Communist Party was too narrow for him to survive long once his patron Stalin was gone. In March 1954, a year after Stalin's death, Chervenkov was deposed as Party Secretary with the approval of the new leadership in Moscow and replaced by Todor Zhivkov. Chervenkov stayed on as Prime Minister until April 1956, when he was finally dismissed and replaced by Anton Yugov.


title: People's Republic of Bulgaria

text: The government used force, threats and intimidation, branding opponents of the policy as fascists and chauvinists. Some were resettled as far as Vojvodina after they had been resettled from Pirin to SR Macedonia for unsuccessful Macedonization.


title: People's Republic of Bulgaria

text: Bulgaria adopted the Communist policy of closer rapprochement with Yugoslavia. Dimitrov then launched the initiative of a Balkan Federation that would range from Pirin to the Shar Mountains and reflect a Macedonian consciousness. For this purpose, he launched a policy of forced Macedonisation of the Bulgarian population in the Pirin region through conscious change of ethnic self-determination, held by means of administrative coercion and intensive propaganda.


title: People's Republic of Bulgaria

text: In December 1946, he conducted a census in Pirin. State authorities instructed the local population in the Pirin region to mark administrative records such as "Macedonian", including Pomaks, with the exception of those originating within the country. At its meeting on 21 December, the Regional Committee of the Workers' Party in Upper Cuma decided to accept a formula indicating 70% of residents were "Macedonians". As a result, among the 281,015 inhabitants, 169,444 were identified as ethnic Macedonians.


title: People's Republic of Bulgaria

text: In 1947, Bulgaria and Yugoslavia signed agreements whereby Pirin Macedonia became part of federal Yugoslavia, which proceeded to unify Pirin Macedonia with Vardar Macedonia and abolished visa regimes and removed customs services.


title: People's Republic of Bulgaria

text: Shortly thereafter – in 1948, due to the rupture in relations between Tito and Stalin, the contract was dissolved. For a while, BCP and the Bulgarian state held contradictory, protivobalgarska policy on the Macedonian issue. In 1963, at the March Plenum of the Central Committee of the Communist Party, Zhivkov declared that the population in Pirin Macedonia was part of Bulgaria that was forced by the Communist Party.


title: People's Republic of Bulgaria

text: According to declassified documents, Bulgaria planned on fomenting a crisis between Turkey and Greece in 1971. The operation was codenamed "Cross" and the plan was that Bulgarian secret agents would set fire in the Ecumenical Patriarchate of Constantinople and make it look like the work of Turks. The declassified documents state that "An intervention" in the religious entity would have "significantly damage[d] Turkish-Greek relations and force[d] the United States to choose one side in the ensuing crisis". In addition, the Bulgarians also planned to boost the effect of its operation against Greece and Turkey by conducting "active measures" "for putting the enemy in a position of delusion". The plan was developed by the 7th Department of the First Main Directorate of the DS (intelligence and secret police services of communist Bulgaria), and was affirmed by Deputy Head of the Directorate on 16 November 1970, and approved by its Head. The operation was supposed to be prepared by the middle of 1971 and then executed, but it was abandoned.


title: People's Republic of Bulgaria

text: In 1971, the new "Zhivkovskata" Constitution added so-called "Article 1", which grants the PA as the sole ruling a "leading force of society and the state". Zhivkov was promoted to Head of State (Chairman of the State Council) and Stanko Todorov became Prime Minister.


title: People's Republic of Bulgaria

text: Bulgaria signed the Helsinki Accords in 1975, which guaranteed human rights and fundamental freedoms, including freedom of movement, contacts, information, culture and education, right to work, and the rights to education and medical care. However, subsequent events regarding Bulgarian Turks in the 1980s were a direct violation of these commitments.


title: People's Republic of Bulgaria

text: In 1978, Bulgaria attracted international attention when dissident writer Georgi Markov was accosted on a London street by a stranger who rammed his leg with the tip of an umbrella. Markov died shortly afterwards of ricin poisoning. He was the victim of the Bulgarian secret service, as confirmed by KGB documents revealing that they had jointly planned the operation with Bulgaria.


title: People's Republic of Bulgaria

text: The Bulgarian People's Army sided with the Soviet Union and the Afghan communists during the Soviet–Afghan War in Afghanistan fighting the jihadist guerrillas from 1982 until its withdrawal in 1989.


title: People's Republic of Bulgaria

text: By the 1980s, the conservatives controlled the government. Some social and cultural liberalization and progress was led by Lyudmila Zhivkova, Todor's daughter, who became a source of strong disapproval and annoyance to the Communist Party due to her unorthodox lifestyle that included the practicing of Eastern religions. She died in 1981, approaching her 39th birthday.


title: People's Republic of Bulgaria

text: A campaign of forced assimilation was waged against the ethnic Turkish minority, who were forbidden to speak the Turkish language and were forced to adopt Bulgarian names took place in the winter of 1984. The issue strained Bulgaria's economic relations with the West. The 1989 expulsion of Turks from Bulgaria caused a significant drop in agricultural production in the southern regions due to the loss of around 300,000 workers.


title: People's Republic of Bulgaria

text: In the late 1980s, the Communists, like their leader, had grown too feeble to resist the demand for change. Liberal outcry at the breakup of an environmental demonstration in Sofia in October 1989 broadened into a general campaign for political reform. More moderate elements in the Communist leadership reacted by deposing Zhivkov and replacing him with foreign minister Petar Mladenov on 10 November 1989.


title: People's Republic of Bulgaria

text: This move gained a short respite for the Communist Party and prevented revolutionary change. Mladenov promised to open up the regime, stating that he supported multi-party elections. Demonstrations throughout the country led Mladenov to announce that the Communist Party would cede its monopoly over the political system. On 15 January 1990, the National Assembly formally amended the legal code to abolish the Communist Party's "leading role". In June 1990, the first multi-party elections since 1939 were held. Finally on 15 November 1990, the seventh Grand National Assembly voted to change the country's name to the Republic of Bulgaria and removed the Communist state emblem from the national flag.


title: People's Republic of Bulgaria

text: It is estimated that, beginning in 1944, between 50,000 and 100,000 people were killed in Bulgaria as part of agricultural collectivisation and political repression, although documentation is insufficient for a definitive judgement. According to one source, some 31,000 people were reported killed under the regime between 1944 and 1989. Figures for fatalities in forced labour camps also remain elusive.


title: People's Republic of Bulgaria

text: A 2009 poll conducted by the Pew Global Attitudes Project found that one-in-nine Bulgarians believe ordinary people are better off as a result of the transition to capitalism. Sixteen percent say the multi-party republic is run for the benefit of all people.


title: People's Republic of Bulgaria

text: The constitution was changed two times, with the Zhivkov Constitution lasting the longest. According to article 1, "The People's Republic of Bulgaria is a socialist state, headed by the working people of the village and the city. The leading force in society and politics is the Bulgarian Communist Party." The BCP created an extensive nomenklatura on each organizational level.


title: People's Republic of Bulgaria

text: The PRB functioned as a one-party people's republic, with People's Committees representing local governance. Their role was to exercise Party decisions in their respective areas and to otherwise defer to popular opinion in decision-making. In the late 1980s, the BCP had an estimated peak membership of 1,000,000 — more than 10% of the population.


title: People's Republic of Bulgaria

text: In 1946, the military rapidly adopted a Soviet military doctrine and organization. The country received large amounts of Soviet weaponry, and eventually established a domestic military vehicle production capability. By the year 1988, the Bulgarian People's Army ("Българска народна армия") numbered 152,000 men, serving in four different branches – Land Forces, Navy, Air and Air Defense Forces, and Missile Forces.


title: People's Republic of Bulgaria

text: The BPA operated 3,000 tanks, 2,000 armored vehicles, 2,500 large caliber artillery systems, over 500 combat aircraft, 33 combat vessels, as well as 67 Scud missile launchers, 24 SS-23 launchers and dozens of FROG-7 artillery rocket launchers.


title: People's Republic of Bulgaria

text: The PRB adopted a centrally planned economy, similar to those in other COMECON states. In the mid-1940s, when collectivisation began, Bulgaria was a primarily agrarian state, with some 80% of its population located in rural areas. Production facilities of all sectors were rapidly nationalised. Chervenkov finally ended all private economic activity.


title: People's Republic of Bulgaria

text: Bulgarian agricultural productivity increased rapidly after collectivisation. Large-scale mechanisation resulted in an immense growth in labour productivity. Government subsidies covered the large losses from the artificially low consumer prices.


title: People's Republic of Bulgaria

text: Chervenkov's Stalinist policy led to a massive industrialisation and development of the energy sector, which remained one of Bulgaria's most advanced economic sectors. His rule lasted from 1950 to 1956, and saw the construction of dozens of dams and hydroelectric powerplants, chemical works, the Elatsite gold and copper mine and many others. The war-time coupon system was abolished, and healthcare and education were government provided. All this was achieved with strict government control and organization. Labor came from prisoner brigades and the Bulgarian Brigadier Movement – a youth labor movement where young people worked on construction projects.


title: People's Republic of Bulgaria

text: Bulgaria was involved in computer construction, which earned it the nickname "Silicon Valley of the Eastern Bloc". Bulgarian engineers developed the first Bulgarian computer, the "Vitosha", as well as the Pravetz computers. Bulgaria is currently the only Balkan Country to operate a supercomputer, an IBM Blue Gene/P.


title: People's Republic of Bulgaria

text: In the 1960s, Zhivkov introduced reforms that had a positive effect on the country's economy. He put emphasis on light industry, agriculture, tourism, as well as on Information Technology in the 1970s and the 1980s. Surplus agricultural production could be sold freely, prices were lowered even more, and new equipment for light industrial production was imported. Bulgaria became the first Communist country to purchase a license from Coca-Cola in 1965.


title: People's Republic of Bulgaria

text: Despite relative stability, the economy shared the same drawbacks of other countries from Eastern Europe – it traded almost entirely with the Soviet Union (more than 60%) and planners could not take into account whether there were markets for the goods produced. This resulted in surpluses of certain products, while other commodities were in deficit.


title: People's Republic of Bulgaria

text: The other main trade partners were East Germany and Czechoslovakia, while countries such as Mongolia and various African countries were also large-scale importers of Bulgarian goods. The country enjoyed good trade relations with various non-Communist countries, most notably West Germany and Italy. In order to combat the low quality of many goods, a comprehensive State standard system was introduced in 1970, which included strict quality requirements for all sorts of products, machines and buildings.


title: People's Republic of Bulgaria

text: PRB had an average GDP per capita for an Eastern Bloc country. Average purchasing power was one of the lowest in the Eastern Bloc, mostly due to the larger availability of commodities than in other socialist countries. Workers employed abroad often received higher payments, thus could afford a wider range of goods to purchase. According to official figures, in 1988 100 out of 100 households had a television set, 95 out of 100 had a radio, 96 out of 100 had a refrigerator and 40 out of 100 had an automobile.


title: People's Republic of Bulgaria

text: Along the improvements on agriculture and the growth of productivity, the GDP per capita grew from $1.864 to $10.800 in 1989, peaking in 1984, when the GDP per capita was about $11.100.


title: People's Republic of Bulgaria

text: From the mid-1950s until the end of the 1970s the growth was constant, until the 1970s energy crisis hit the country. The crisis along with some structural problems, reduced the growth during the 1980s.


title: People's Republic of Bulgaria

text: Since 1965, Renault and Fiat chose Bulgaria to site their factories to make automobiles for sale in the Eastern Bloc partnership.


title: People's Republic of Bulgaria

text: Culture in PRB was strictly regulated by the government, although there were some periods of liberalization (meaning entrance in Bulgaria of Western literature, music, etc.). The thaw in intellectual life had continued from 1951 until the middle of the decade. Chervenkov's resignation and the literary and cultural flowering in the Soviet Union created expectations that the process would continue, but the Hungarian revolution of fall 1956 ended the experiment.


title: People's Republic of Bulgaria

text: Chervenkov was appointed minister of education and culture. In 1957 and 1958, he purged the leadership of the Bulgarian Writers' Union and dismissed liberal journalists and editors from their positions. His crackdowns effectively ended the "Bulgarian thaw" of independent writers and artists inspired by Khrushchev's 1956 speech against Stalinism. In July 1968, the 9th World Festival of Youth and Students took place in Sofia, attracting 20,000 people from 138 countries. The main holiday of the country was the Day of the People's Uprising of 9 September (Ден на народното въстание на 9 септември), celebrations of which were modelled on Soviet celebrations on October Revolution Day. The celebrarions saw the Bulgarian Politbureau greeting the masses from Georgi Dimitrov Mausoleum. However, in this case, there were no military parades. The yearlong 1300th anniversary celebrations of the Bulgarian State took place in 1981 to commemorate the establishment of the First Bulgarian State in 681. There were 23,000 events connected with the anniversary, including a military parade and the creation of a monument in Shumen.


title: Survival of the fittest

text: "Survival of the fittest" is a phrase that originated from Darwinian evolutionary theory as a way of describing the mechanism of natural selection. The biological concept of fitness is defined as reproductive success. In Darwinian terms, the phrase is best understood as "Survival of the form that will leave the most copies of itself in successive generations."


title: Survival of the fittest

text: Herbert Spencer first used the phrase, after reading Charles Darwin's "On the Origin of Species", in his "Principles of Biology" (1864), in which he drew parallels between his own economic theories and Darwin's biological ones: "This survival of the fittest, which I have here sought to express in mechanical terms, is that which Mr. Darwin has called 'natural selection', or the preservation of favoured races in the struggle for life."


title: Survival of the fittest

text: Darwin responded positively to Alfred Russel Wallace's suggestion of using Spencer's new phrase "survival of the fittest" as an alternative to "natural selection", and adopted the phrase in "The Variation of Animals and Plants Under Domestication" published in 1868. In "On the Origin of Species", he introduced the phrase in the fifth edition published in 1869, intending it to mean "better designed for an immediate, local environment".


title: Survival of the fittest

text: By his own account, Herbert Spencer described a concept similar to "survival of the fittest" in his 1852 "A Theory of Population". He first used the phrase – after reading Charles Darwin's "On the Origin of Species" – in his "Principles of Biology" of 1864 in which he drew parallels between his economic theories and Darwin's biological, evolutionary ones, writing, "This survival of the fittest, which I have here sought to express in mechanical terms, is that which Mr. Darwin has called 'natural selection', or the preservation of favored races in the struggle for life."


title: Survival of the fittest

text: In July 1866 Alfred Russel Wallace wrote to Darwin about readers thinking that the phrase "natural selection" personified nature as "selecting", and said this misconception could be avoided "by adopting Spencer's term" "Survival of the fittest". Darwin promptly replied that Wallace's letter was "as clear as daylight. I fully agree with all that you say on the advantages of H. Spencer's excellent expression of 'the survival of the fittest'. This however had not occurred to me till reading your letter. It is, however, a great objection to this term that it cannot be used as a substantive governing a verb". Had he received the letter two months earlier, he would have worked the phrase into the fourth edition of the "Origin" which was then being printed, and he would use it in his next book on "Domestic Animals etc.".


title: Survival of the fittest

text: Darwin wrote on page 6 of "The Variation of Animals and Plants Under Domestication" published in 1868, "This preservation, during the battle for life, of varieties which possess any advantage in structure, constitution, or instinct, I have called Natural Selection; and Mr. Herbert Spencer has well expressed the same idea by the Survival of the Fittest. The term 'natural selection' is in some respects a bad one, as it seems to imply conscious choice; but this will be disregarded after a little familiarity". He defended his analogy as similar to language used in chemistry, and to astronomers depicting the "attraction of gravity as ruling the movements of the planets", or the way in which "agriculturists speak of man making domestic races by his power of selection". He had "often personified the word Nature; for I have found it difficult to avoid this ambiguity; but I mean by nature only the aggregate action and product of many natural laws,—and by laws only the ascertained sequence of events."


title: Survival of the fittest

text: In the first four editions of "On the Origin of Species", Darwin had used the phrase "natural selection". In Chapter 4 of the 5th edition of "The Origin" published in 1869, Darwin implies again the synonym: "Natural Selection, or the Survival of the Fittest". By "fittest" Darwin meant "better adapted for the immediate, local environment", not the common modern meaning of "in the best physical shape" (think of a puzzle piece, not an athlete). In the introduction he gave full credit to Spencer, writing "I have called this principle, by which each slight variation, if useful, is preserved, by the term Natural Selection, in order to mark its relation to man's power of selection. But the expression often used by Mr. Herbert Spencer of the Survival of the Fittest is more accurate, and is sometimes equally convenient."


title: Survival of the fittest

text: In "The Man Versus The State", Spencer used the phrase in a postscript to justify a plausible explanation of how his theories would not be adopted by "societies of militant type". He uses the term in the context of societies at war, and the form of his reference suggests that he is applying a general principle.


title: Survival of the fittest

text: Though Spencer's conception of organic evolution is commonly interpreted as a form of Lamarckism, Herbert Spencer is sometimes credited with inaugurating Social Darwinism. The phrase "survival of the fittest" has become widely used in popular literature as a catchphrase for any topic related or analogous to evolution and natural selection. It has thus been applied to principles of unrestrained competition, and it has been used extensively by both proponents and opponents of Social Darwinism.


title: Survival of the fittest

text: Evolutionary biologists criticise the manner in which the term is used by non-scientists and the connotations that have grown around the term in popular culture. The phrase also does not help in conveying the complex nature of natural selection, so modern biologists prefer and almost exclusively use the term natural selection. The biological concept of fitness refers to both reproductive success (fecundity selection), as well as survival (viability selection), and is not prescriptive in the specific ways in which organisms can be more "fit" by having phenotypic characteristics that enhance survival and reproduction (which was the meaning that Spencer had in mind).


title: Survival of the fittest

text: While the phrase "survival of the fittest" is often used to mean "natural selection", it is avoided by modern biologists, because the phrase can be misleading. For example, survival is only one aspect of selection, and not always the most important. Another problem is that the word "fit" is frequently confused with a state of physical fitness. In the evolutionary meaning "fitness" is the rate of reproductive output among a class of genetic variants.


title: Survival of the fittest

text: The phrase can also be interpreted to express a theory or hypothesis: that "fit" as opposed to "unfit" individuals or species, in some sense of "fit", will survive some test. Nevertheless, when extended to individuals it is a conceptual mistake, the phrase is a reference to the "transgenerational" survival of the heritable attributes; particular individuals are quite irrelevant. This becomes more clear when referring to Viral quasispecies, in "survival of the flattest", which makes it clear "to survive" makes no reference to the question of even being alive itself; rather the functional capacity of proteins to carry out work.


title: Survival of the fittest

text: Interpretations of the phrase as expressing a theory are in danger of being tautological, meaning roughly "those with a propensity to survive have a propensity to survive"; to have content the theory must use a concept of fitness that is independent of that of survival.


title: Survival of the fittest

text: Interpreted as a theory of species survival, the theory that the fittest species survive is undermined by evidence that while direct competition is observed between individuals, populations and species, there is little evidence that competition has been the driving force in the evolution of large groups such as, for example, amphibians, reptiles, and mammals. Instead, these groups have evolved by expanding into empty ecological niches. In the punctuated equilibrium model of environmental and biological change, the factor determining survival is often not superiority over another in competition but ability to survive dramatic changes in environmental conditions, such as after a meteor impact energetic enough to greatly change the environment globally. The main land dwelling animals to survive the K-Pg impact 66 million years ago had the ability to live in tunnels, for example.


title: Survival of the fittest

text: In 2010 Sahney et al. argued that there is little evidence that intrinsic, biological factors such as competition have been the driving force in the evolution of large groups. Instead, they cited extrinsic, abiotic factors such as expansion as the driving factor on a large evolutionary scale. The rise of dominant groups such as amphibians, reptiles, mammals and birds occurred by opportunistic expansion into empty ecological niches and the extinction of groups happened due to large shifts in the abiotic environment.


title: Survival of the fittest

text: It has been claimed that "the survival of the fittest" theory in biology was interpreted by late 19th century capitalists as "an ethical precept that sanctioned cut-throat economic competition" and led to the advent of the theory of "social Darwinism" which was used to justify laissez-faire economics, war and racism. However, these ideas pre-date and commonly contradict Darwin's ideas, and indeed their proponents rarely invoked Darwin in support. The use of the term "social Darwinism" as a critique of capitalist ideologies was introduced in Richard Hofstadter's "Social Darwinism in American Thought" published in 1944.


title: Survival of the fittest

text: Russian zoologist and anarchist Peter Kropotkin viewed the concept of "survival of the fittest" as supporting co-operation rather than competition. In his book "" he set out his analysis leading to the conclusion that the fittest was not necessarily the best at competing individually, but often the community made up of those best at working together. He concluded that In the animal world we have seen that the vast majority of species live in societies, and that they find in association the best arms for the struggle for life: understood in its wide Darwinian sense – not as a struggle for the sheer means of existence, but as a struggle against all natural conditions unfavourable to the species. The animal species, in which individual struggle has been reduced to its narrowest limits, and the practice of mutual aid has attained the greatest development, are invariably the most numerous, the most prosperous, and the most open to further progress.


title: Survival of the fittest

text: Applying this concept to human society, Kropotkin presented mutual aid as one of the dominant factors of evolution, the other being self-assertion, and concluded that In the practice of mutual aid, which we can retrace to the earliest beginnings of evolution, we thus find the positive and undoubted origin of our ethical conceptions; and we can affirm that in the ethical progress of man, mutual support not mutual struggle – has had the leading part. In its wide extension, even at the present time, we also see the best guarantee of a still loftier evolution of our race.


title: Survival of the fittest

text: "Survival of the fittest" is sometimes claimed to be a tautology. The reasoning is that if one takes the term "fit" to mean "endowed with phenotypic characteristics which improve chances of survival and reproduction" (which is roughly how Spencer understood it), then "survival of the fittest" can simply be rewritten as "survival of those who are better equipped for surviving". Furthermore, the expression "does" become a tautology if one uses the most widely accepted definition of "fitness" in modern biology, namely reproductive success itself (rather than any set of characters conducive to this reproductive success). This reasoning is sometimes used to claim that Darwin's entire theory of evolution by natural selection is fundamentally tautological, and therefore devoid of any explanatory power.


title: Survival of the fittest

text: However, the expression "survival of the fittest" (taken on its own and out of context) gives a very incomplete account of the mechanism of natural selection. The reason is that it does not mention a key requirement for natural selection, namely the requirement of "heritability". It is true that the phrase "survival of the fittest", in and by itself, is a tautology if fitness is defined by survival and reproduction. Natural selection is the portion of variation in reproductive success that is caused by "heritable" characters (see the article on natural selection).


title: Survival of the fittest

text: If certain heritable characters increase or decrease the chances of survival and reproduction of their bearers, then it follows mechanically (by definition of "heritable") that those characters that improve survival and reproduction will increase in frequency over generations. This is precisely what is called "evolution by natural selection". On the other hand, if the characters which lead to differential reproductive success are not heritable, then no meaningful evolution will occur, "survival of the fittest" or not: if improvement in reproductive success is caused by traits that are not heritable, then there is no reason why these traits should increase in frequency over generations. In other words, natural selection does not simply state that "survivors survive" or "reproducers reproduce"; rather, it states that "survivors survive, reproduce and "therefore" propagate any "heritable" characters which have affected their survival and reproductive success". This statement is not tautological: it hinges on the testable hypothesis that such fitness-impacting heritable variations actually exist (a hypothesis that has been amply confirmed.)


title: Survival of the fittest

text: Momme von Sydow suggested further definitions of 'survival of the fittest' that may yield a testable meaning in biology and also in other areas where Darwinian processes have been influential. However, much care would be needed to disentangle tautological from testable aspects. Moreover, an "implicit shifting between a testable and an untestable interpretation can be an illicit tactic to immunize natural selection ... while conveying the impression that one is concerned with testable hypotheses".


title: Survival of the fittest

text: Skeptic Society founder and "Skeptic" magazine publisher Michael Shermer addresses the tautology problem in his 1997 book, "Why People Believe Weird Things", in which he points out that although tautologies are sometimes the beginning of science, they are never the end, and that scientific principles like natural selection are testable and falsifiable by virtue of their predictive power. Shermer points out, as an example, that population genetics accurately demonstrate when natural selection will and will not effect change on a population. Shermer hypothesizes that if hominid fossils were found in the same geological strata as trilobites, it would be evidence against natural selection.


title: Spherical harmonics

text: In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.


title: Spherical harmonics

text: Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).


title: Spherical harmonics

text: Spherical harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree formula_1 in formula_2 that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence formula_3 from the above-mentioned polynomial of degree formula_1; the remaining factor can be regarded as a function of the spherical angular coordinates formula_5 and formula_6 only, or equivalently of the orientational unit vector formula_7 specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. Notice, however, that spherical harmonics are "not" functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section "Higher dimensions" below).


title: Spherical harmonics

text: A specific set of spherical harmonics, denoted formula_8 or formula_9, are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.


title: Spherical harmonics

text: Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes.


title: Spherical harmonics

text: Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. In 1782, Pierre-Simon de Laplace had, in his "Mécanique Céleste", determined that the gravitational potential formula_10 at a point associated with a set of point masses located at points was given by


title: Spherical harmonics

text: Each term in the above summation is an individual Newtonian potential for a point mass. Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of and . He discovered that if then


title: Spherical harmonics

text: where is the angle between the vectors and . The functions formula_13 are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between and . (See Applications of Legendre polynomials in physics for a more detailed analysis.)


title: Spherical harmonics

text: In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their "Treatise on Natural Philosophy", and also first introduced the name of “spherical harmonics” for these functions. The solid harmonics were homogeneous polynomial solutions formula_10 of Laplace's equation


title: Spherical harmonics

text: By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. (See the section below, “Harmonic polynomial representation”.) The term “Laplace's coefficients” was employed by William Whewell to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics that had properly been introduced by Laplace and Legendre.


title: Spherical harmonics

text: The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. This could be achieved by expansion of functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre.


title: Spherical harmonics

text: The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. The (complex-valued) spherical harmonics formula_16 are eigenfunctions of the square of the orbital angular momentum operator


title: Spherical harmonics

text: Laplace's equation imposes that the Laplacian of a scalar field is zero. (Here the scalar field is understood to be complex, i.e. to correspond to a (smooth) function formula_18.) In spherical coordinates this is:


title: Spherical harmonics

text: Consider the problem of finding solutions of the form . By separation of variables, two differential equations result by imposing Laplace's equation:


title: Spherical harmonics

text: The second equation can be simplified under the assumption that has the form . Applying separation of variables again to the second equation gives way to the pair of differential equations


title: Spherical harmonics

text: for some number . A priori, is a complex constant, but because must be a periodic function whose period evenly divides , is necessarily an integer and is a linear combination of the complex exponentials . The solution function is regular at the poles of the sphere, where . Imposing this regularity in the solution of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter to be of the form for some non-negative integer with ; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial . Finally, the equation for has solutions of the form ; requiring the solution to be regular throughout forces .


title: Spherical harmonics

text: Here the solution was assumed to have the special form . For a given value of , there are independent solutions of this form, one for each integer with . These angular solutions formula_23 are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:


title: Spherical harmonics

text: Here formula_26 is called a spherical harmonic function of degree and order , formula_27 is an associated Legendre polynomial, is a normalization constant, and and represent colatitude and longitude, respectively. In particular, the colatitude , or polar angle, ranges from at the North Pole, to at the Equator, to at the South Pole, and the longitude , or azimuth, may assume all values with . For a fixed integer , every solution , formula_28, of the eigenvalue problem


title: Spherical harmonics

text: is a linear combination of formula_30. In fact, for any such solution, is the expression in spherical coordinates of a homogeneous polynomial formula_31 that is harmonic (see below), and so counting dimensions shows that there are linearly independent such polynomials.


title: Spherical harmonics

text: The general solution formula_18 to Laplace's equation formula_33 in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor ,


title: Spherical harmonics

text: where the formula_35 are constants and the factors are known as ("regular") solid harmonics formula_31. Such an expansion is valid in the ball


title: Spherical harmonics

text: For formula_38, the solid harmonics with negative powers of formula_39 (the "irregular" solid harmonics formula_40) are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about formula_41), instead of the Taylor series (about formula_42) used above, to match the terms and find series expansion coefficients formula_43.


title: Spherical harmonics

text: In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum


title: Spherical harmonics

text: The is conventional in quantum mechanics; it is convenient to work in units in which . The spherical harmonics are eigenfunctions of the square of the orbital angular momentum


title: Spherical harmonics

text: Laplace's spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis:


title: Spherical harmonics

text: These operators commute, and are densely defined self-adjoint operators on the weighted Hilbert space of functions "f" square-integrable with respect to the normal distribution as the weight function on R:


title: Spherical harmonics

text: for some real numbers "m" and "λ". Here "m" must in fact be an integer, for "Y" must be periodic in the coordinate "φ" with period a number that evenly divides 2"π". Furthermore, since


title: Spherical harmonics

text: Then and commute with , and the Lie algebra generated by , , is the special linear Lie algebra of order 2, formula_51, with commutation relations


title: Spherical harmonics

text: Thus (it is a "raising operator") and (it is a "lowering operator"). In particular, must be zero for "k" sufficiently large, because the inequality must hold in each of the nontrivial joint eigenspaces. Let be a nonzero joint eigenfunction, and let be the least integer such that


title: Spherical harmonics

text: The foregoing has been all worked out in the spherical coordinate representation, formula_56 but may be expressed more abstractly in the complete, orthonormal spherical ket basis.


title: Spherical harmonics

text: The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions formula_31. Specifically, we say that a (complex-valued) polynomial function formula_58 is "homogeneous" of degree formula_1 if


title: Spherical harmonics

text: For example, when formula_68, formula_69 is just the 3-dimensional space of all linear functions formula_31, since any such function is automatically harmonic. Meanwhile, when formula_71, we have a 5-dimensional space:


title: Spherical harmonics

text: For any formula_1, the space formula_74 of spherical harmonics of degree formula_1 is just the space of restrictions to the sphere formula_76 of the elements of formula_77. As suggested in the introduction, this perspective is presumably the origin of the term “spherical harmonic” (i.e., the restriction to the sphere of a harmonic function).


title: Spherical harmonics

text: defines a homogeneous polynomial of degree formula_1 with domain and codomain formula_31, which happens to be independent of formula_82. This polynomial is easily seen to be harmonic. If we write formula_63 in spherical coordinates formula_84 and then restrict to formula_85, we obtain


title: Spherical harmonics

text: After using the formula for the associated Legendre polynomial formula_88, we may recognize this as the formula for the spherical harmonic formula_89 (See the section below on special cases of the spherical harmonics.)


title: Spherical harmonics

text: Several different normalizations are in common use for the Laplace spherical harmonic functions formula_16. Throughout the section, we use the standard convention that for formula_91 (see associated Legendre polynomials)


title: Spherical harmonics

text: In acoustics, the Laplace spherical harmonics are generally defined as (this is the convention used in this article)


title: Spherical harmonics

text: where formula_95 are associated Legendre polynomials without the Condon–Shortley phase (to avoid counting the phase twice).


title: Spherical harmonics

text: where is the Kronecker delta and . This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e.,


title: Spherical harmonics

text: The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics formula_23 in terms of the real spherical harmonics formula_99 are


title: Spherical harmonics

text: The real spherical harmonics formula_99 are sometimes known as "tesseral spherical harmonics". These functions have the same orthonormality properties as the complex ones formula_23 above. The real spherical harmonics formula_103 with are said to be of cosine type, and those with of sine type. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as


title: Spherical harmonics

text: The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation.


title: Spherical harmonics

text: See here for a list of real spherical harmonics up to and including formula_105, which can be seen to be consistent with the output of the equations above.


title: Spherical harmonics

text: As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. However, the solutions of the non-relativistic Schrödinger equation without magnetic terms can be made real. This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Here, it is important to note that the real functions span the same space as the complex ones would.


title: Spherical harmonics

text: For example, as can be seen from the table of spherical harmonics, the usual functions (formula_106) are complex and mix axis directions, but the real versions are essentially just , , and .


title: Spherical harmonics

text: The complex spherical harmonics formula_107 give rise to the solid harmonics by extending from formula_76 to all of formula_109 as a homogeneous function of degree formula_1, i.e. setting


title: Spherical harmonics

text: It turns out that formula_112 is basis of the space of harmonic and homogeneous polynomials of degree formula_1. More specifically, it is the (unique up to normalization) Gelfand-Tsetlin-basis of this representation of the rotational group formula_114 and an explicit formula for formula_112 in cartesian coordinates can be derived from that fact.


title: Spherical harmonics

text: In naming this generating function after Herglotz, we follow , who credit unpublished notes by him for its discovery.


title: Spherical harmonics

text: Essentially all the properties of the spherical harmonics can be derived from this generating function. An immediate benefit of this definition is that if the vector formula_7 is replaced by the quantum mechanical spin vector operator formula_129, such that formula_130 is the operator analogue of the solid harmonic formula_131, one obtains a generating function for a standardized set of spherical tensor operators, formula_130:


title: Spherical harmonics

text: The parallelism of the two definitions ensures that the formula_134's transform under rotations (see below) in the same way as the formula_135's, which in turn guarantees that they are spherical tensor operators, formula_136, with formula_137 and formula_138, obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. They are, moreover, a standardized set with a fixed scale or normalization.


title: Spherical harmonics

text: The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of formula_139 and another of formula_140 and formula_141, as follows (Condon–Shortley phase):


title: Spherical harmonics

text: The factor formula_149 is essentially the associated Legendre polynomial formula_150, and the factors formula_151 are essentially formula_152.


title: Spherical harmonics

text: Using the expressions for formula_149, formula_154, and formula_155 listed explicitly above we obtain:


title: Spherical harmonics

text: Using the equations above to form the real spherical harmonics, it is seen that for formula_91 only the formula_159 terms (cosines) are included, and for formula_160 only the formula_161 terms (sines) are included:


title: Spherical harmonics

text: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation.


title: Spherical harmonics

text: The spherical harmonics have definite parity. That is, they are either even or odd with respect to inversion about the origin. Inversion is represented by the operator formula_174. Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with formula_7 being a unit vector,


title: Spherical harmonics

text: In terms of the spherical angles, parity transforms a point with coordinates formula_177 to formula_178. The statement of the parity of spherical harmonics is then


title: Spherical harmonics

text: Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic of degree changes the sign by a factor of .


title: Spherical harmonics

text: Consider a rotation formula_180 about the origin that sends the unit vector formula_7 to formula_182. Under this operation, a spherical harmonic of degree formula_1 and order formula_184 transforms into a linear combination of spherical harmonics of the same degree. That is,


title: Spherical harmonics

text: where formula_186 is a matrix of order formula_187 that depends on the rotation formula_180. However, this is not the standard way of expressing this property. In the standard way one writes,


title: Spherical harmonics

text: where formula_190 is the complex conjugate of an element of the Wigner D-matrix. In particular when formula_182 is a formula_192 rotation of the azimuth we get the identity,


title: Spherical harmonics

text: The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The formula_107's of degree formula_1 provide a basis set of functions for the irreducible representation of the group SO(3) of dimension formula_187. Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry.


title: Spherical harmonics

text: The Laplace spherical harmonics formula_26 form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions formula_198. On the unit sphere formula_76, any square-integrable function formula_200 can thus be expanded as a linear combination of these:


title: Spherical harmonics

text: This expansion holds in the sense of mean-square convergence — convergence in L of the sphere — which is to say that


title: Spherical harmonics

text: The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle Ω, and utilizing the above orthogonality relationships. This is justified rigorously by basic Hilbert space theory. For the case of orthonormalized harmonics, this gives:


title: Spherical harmonics

text: If the coefficients decay in "ℓ" sufficiently rapidly — for instance, exponentially — then the series also converges uniformly to "f".


title: Spherical harmonics

text: A square-integrable function formula_204 can also be expanded in terms of the real harmonics formula_99 above as a sum


title: Spherical harmonics

text: The convergence of the series holds again in the same sense, namely the real spherical harmonics formula_99 form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions formula_208. The benefit of the expansion in terms of the real harmonic functions formula_103 is that for real functions formula_204 the expansion coefficients formula_211 are guaranteed to be real, whereas their coefficients formula_212 in their expansion in terms of the formula_135 (considering them as functions formula_214) do not have that property.


title: Spherical harmonics

text: The total power of a function "f" is defined in the signal processing literature as the integral of the function squared, divided by the area of its domain. Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem (here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly different for orthonormal harmonics):


title: Spherical harmonics

text: is defined as the angular power spectrum (for Schmidt semi-normalized harmonics). In a similar manner, one can define the cross-power of two functions as


title: Spherical harmonics

text: is defined as the cross-power spectrum. If the functions and have a zero mean (i.e., the spectral coefficients and are zero), then and represent the contributions to the function's variance and covariance for degree , respectively. It is common that the (cross-)power spectrum is well approximated by a power law of the form


title: Spherical harmonics

text: When , the spectrum is "white" as each degree possesses equal power. When , the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when , the spectrum is termed "blue". The condition on the order of growth of is related to the order of differentiability of in the next section.


title: Spherical harmonics

text: One can also understand the differentiability properties of the original function in terms of the asymptotics of . In particular, if decays faster than any rational function of as , then is infinitely differentiable. If, furthermore, decays exponentially, then is actually real analytic on the sphere.


title: Spherical harmonics

text: The general technique is to use the theory of Sobolev spaces. Statements relating the growth of the to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Specifically, if


title: Spherical harmonics

text: then is in the Sobolev space . In particular, the Sobolev embedding theorem implies that is infinitely differentiable provided that


title: Spherical harmonics

text: A mathematical result of considerable interest and use is called the "addition theorem" for spherical harmonics. Given two vectors and , with spherical coordinates formula_84 and formula_223, respectively, the angle formula_224 between them is given by the relation


title: Spherical harmonics

text: in which the role of the trigonometric functions appearing on the right-hand side is played by the spherical harmonics and that of the left-hand side is played by the Legendre polynomials.


title: Spherical harmonics

text: where is the Legendre polynomial of degree . This expression is valid for both real and complex harmonics. The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by applying a rotation to the vector y so that it points along the "z"-axis, and then directly calculating the right-hand side.


title: Spherical harmonics

text: In the expansion (), the left-hand side is a constant multiple of the degree zonal spherical harmonic. From this perspective, one has the following generalization to higher dimensions. Let be an arbitrary orthonormal basis of the space of degree spherical harmonics on the -sphere. Then formula_227, the degree zonal harmonic corresponding to the unit vector , decomposes as


title: Spherical harmonics

text: Furthermore, the zonal harmonic formula_228 is given as a constant multiple of the appropriate Gegenbauer polynomial:


title: Spherical harmonics

text: Combining () and () gives () in dimension when and are represented in spherical coordinates. Finally, evaluating at gives the functional identity


title: Spherical harmonics

text: Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics


title: Spherical harmonics

text: Many of the terms in this sum are trivially zero. The values of formula_231 and formula_224 that result in non-zero terms in this sum are determined by the selection rules for the 3j-symbols.


title: Spherical harmonics

text: The Clebsch–Gordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. A variety of techniques are available for doing essentially the same calculation, including the Wigner 3-jm symbol, the Racah coefficients, and the Slater integrals. Abstractly, the Clebsch–Gordan coefficients express the tensor product of two irreducible representations of the rotation group as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities.


title: Spherical harmonics

text: The Laplace spherical harmonics formula_107 can be visualized by considering their "nodal lines", that is, the set of points on the sphere where formula_234, or alternatively where formula_235. Nodal lines of formula_107 are composed of "ℓ" circles: there are circles along longitudes and "ℓ"−|"m"| circles along latitudes. One can determine the number of nodal lines of each type by counting the number of zeros of formula_107 in the formula_5 and formula_6 directions respectively. Considering formula_107 as a function of formula_5, the real and imaginary components of the associated Legendre polynomials each possess "ℓ"−|"m"| zeros, each giving rise to a nodal 'line of latitude'. On the other hand, considering formula_107 as a function of formula_6, the trigonometric sin and cos functions possess 2|"m"| zeros, each of which gives rise to a nodal 'line of longitude'.


title: Spherical harmonics

text: When the spherical harmonic order "m" is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. Such spherical harmonics are a special case of zonal spherical functions. When (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. For the other cases, the functions checker the sphere, and they are referred to as tesseral.


title: Spherical harmonics

text: More general spherical harmonics of degree are not necessarily those of the Laplace basis formula_107, and their nodal sets can be of a fairly general kind.


title: Spherical harmonics

text: Analytic expressions for the first few orthonormalized Laplace spherical harmonics formula_23 that use the Condon–Shortley phase convention:


title: Spherical harmonics

text: The classical spherical harmonics are defined as complex-valued functions on the unit sphere formula_76 inside three-dimensional Euclidean space formula_109. Spherical harmonics can be generalized to higher-dimensional Euclidean space formula_251 as follows, leading to functions formula_252. Let P denote the space of complex-valued homogeneous polynomials of degree in real variables, here considered as functions formula_253. That is, a polynomial is in provided that for any real formula_61, one has


title: Spherical harmonics

text: These are the (regular) solid spherical harmonics. Let H denote the space of functions on the unit sphere


title: Spherical harmonics

text: An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian


title: Spherical harmonics

text: where "φ" is the axial coordinate in a spherical coordinate system on "S". The end result of such a procedure is


title: Spherical harmonics

text: where the indices satisfy and the eigenvalue is . The functions in the product are defined in terms of the Legendre function


title: Spherical harmonics

text: The space of spherical harmonics of degree is a representation of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). Indeed, rotations act on the two-dimensional sphere, and thus also on by function composition


title: Spherical harmonics

text: for a spherical harmonic and a rotation. The representation is an irreducible representation of SO(3).


title: Spherical harmonics

text: The elements of arise as the restrictions to the sphere of elements of : harmonic polynomials homogeneous of degree on three-dimensional Euclidean space . By polarization of , there are coefficients formula_271 symmetric on the indices, uniquely determined by the requirement


title: Spherical harmonics

text: The condition that be harmonic is equivalent to the assertion that the tensor formula_271 must be trace free on every pair of indices. Thus as an irreducible representation of , is isomorphic to the space of traceless symmetric tensors of degree .


title: Spherical harmonics

text: More generally, the analogous statements hold in higher dimensions: the space of spherical harmonics on the -sphere is the irreducible representation of corresponding to the traceless symmetric -tensors. However, whereas every irreducible tensor representation of and is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner.


title: Spherical harmonics

text: The special orthogonal groups have additional spin representations that are not tensor representations, and are "typically" not spherical harmonics. An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication.


title: Spherical harmonics

text: Spherical harmonics can be separated into two set of functions. One is hemispherical functions (HSH), orthogonal and complete on hemisphere. Another is complementary hemispherical harmonics (CHSH).


title: Spherical harmonics

text: The angle-preserving symmetries of the two-sphere are described by the group of Möbius transformations PSL(2,C). With respect to this group, the sphere is equivalent to the usual Riemann sphere. The group PSL(2,C) is isomorphic to the (proper) Lorentz group, and its action on the two-sphere agrees with the action of the Lorentz group on the celestial sphere in Minkowski space. The analog of the spherical harmonics for the Lorentz group is given by the hypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as is a subgroup of .


title: Spherical harmonics

text: More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in particular, hypergeometric series can be developed for any Lie group.


title: Jazmin Grace Grimaldi

text: Jazmin Grace Grimaldi (born March 4, 1992) is an American actress and singer. She is the daughter of Prince Albert II of Monaco and Tamara Rotolo.


title: Jazmin Grace Grimaldi

text: Prince Albert II publicly confirmed Grimaldi's paternity on June 1, 2006, claiming that he had wanted to protect her identity until she was an adult. Grimaldi is the older half-sister of Jacques, Hereditary Prince of Monaco and Princess Gabriella of Monaco. As Grimaldi's parents never married, she is not in the line of succession to the Monegasque throne. She is also the older half-sister of Alexandre Grimaldi-Coste.


title: Jazmin Grace Grimaldi

text: Grimaldi was born in Palm Springs, California, in the United States on March 4, 1992. Her mother, Tamara Jean Rotolo (born October 25, 1961, in California), later a real estate agent, was a waitress at the time she met Prince Albert of Monaco. Grimaldi was born when her mother's divorce proceedings to David Schumacher were not yet finalized. She did not meet her father until her visit to Monaco at the age of 11.


title: Jazmin Grace Grimaldi

text: During her youth, she was active in her middle school's basketball team and often performed in church choirs and school plays. When she was young, her mother would often show her photographs of her grandmother, Grace Kelly. Her fondest memories involving her grandmother were watching "High Society".


title: Jazmin Grace Grimaldi

text: Grimaldi grew up in Palm Desert, California and Orange County. She was educated at St. Margaret's Episcopal School where she graduated in 2006 and a Catholic high school, JSerra High School, from where she graduated in June 2010. She was an honor student and a soloist in a school choir that performed with singer Barry Manilow. Upon graduation, she received the JSerra Senior Faculty Award and the Fine Arts Award. She was also selected to be a lifetime member of the California Scholarship Federation, and has been since she was a sophomore, entitling her to the distinction of Sealbearer.


title: Jazmin Grace Grimaldi

text: After graduating from JSerra, she attended Fordham University where she studied theater and international business with an emphasis on humanitarian affairs. She graduated from Fordham in May 2014. She attended Stella Adler Studio of Acting for two years. After graduating from university, she worked as a consultant at United Nations' World Food Programme for two years.


title: Jazmin Grace Grimaldi

text: In February 2006, the French magazine "Voici" published photographs of Grimaldi and her mother during a visit to Monaco. The Prince has hired the security services of Gavin de Becker and Associates.


title: Jazmin Grace Grimaldi

text: On October 22, 2012, Grimaldi attended the 30th Anniversary of the Princess Grace Awards in New York City with her father and stepmother, Princess Charlene. In an article published in "Hello" magazine about the event, the Prince was described as protective of his daughter, who was said to admire and respect her father. Grimaldi's first published interview, in 2015 with "Harpers Bazaar" was conducted at the Prince's Palace of Monaco in Monte Carlo. Grimaldi talked about her grandmother Princess Grace, her father, her siblings and life in Monaco.


title: Jazmin Grace Grimaldi

text: On March 14, 2018, she traveled to Monaco to celebrate her father's 60th birthday, days before final preparations for 28th Rallye Aicha des Gazelles du Maroc in Morocco, as did the daughter of Princess Stephanie, Pauline Ducruet. Grimaldi took 3rd Place in her class (e-class), for the all-women's event.


title: Jazmin Grace Grimaldi

text: Grimaldi attend the 2019 Princess Grace Awards ceremony. In June 2020, during the COVID-19 pandemic, she tested positive for coronavirus, after showing mild symptoms for days. One month later, she announced on her Instagram that she had recovered from the virus.


title: Jazmin Grace Grimaldi

text: Grimaldi was cast in the third season of US web series "The Marvelous Mrs. Maisel". Grimaldi also starred in several films such as "Cicada", "The Scarlet M" and "Millie's Big Day" and cast in television dramas such as "Jay and Pluto". She also starred in several theatre play such as "The Glass Menagerie", "Uncle Vanya" and cast in" Jake and Diane".