Club career.

Danubio.

Born in Toledo, Giménez made his professional debut for Danubio F.C. in the Uruguayan Primera División on 17 November 2012 (when the manager was Juan Ramón Carrasco) against River Plate in which he started and played the full 90 minutes as Danubio lost the match 2–0.

Atlético Madrid.

On 25 April 2013, it was confirmed that Giménez had signed for €900,000 with Spanish side Atlético Madrid, and would join the club in the pre-season of the 2013–14 season.  On 14 September, Giménez made his "Atleti" – and La Liga – debut, starting in a 4–2 home win over UD Almería.  Giménez scored his first goal for the club on 6 December 2014, opening a 2–0 win away to Elche CF to move Atlético into second place. He also scored a header against their biggest rival Real Madrid in Atlético's 2–0 win in the Copa del Rey in January 2015.  On 20 February 2019, Gimenez scored the opener in 2-0 home win against Italian champions Juventus to give Atlético the advantage in the first leg of the Round of 16 fixture in the UEFA Champions League.  Ahead of 2019-20 season, he was made third captain of the team on 16 August 2019, behind Koke and Oblak. Gimenez has been injured for most of the 19/20 season

International career.

Giménez participated for Uruguay at the 2013 FIFA U-20 World Cup as the team finished as runner-up to France.  He debuted for the Uruguay senior team in a 2014 FIFA World Cup qualifier against Colombia on 10 September 2013.  On 2 June 2014, Giménez was named in Uruguay's squad for the 2014 FIFA World Cup finals. The 19-year-old defender made his tournament debut against England in the team's second group match, deputising for the injured captain Diego Lugano in a 2–1 victory for "La Celeste".  (with 19 years and 149 days old, he is the youngest Uruguayan player to debut at a world cup). He went on to start in the final group match – a 1–0 win over Italy – and the 2–0 round of 16 loss to Colombia.  Giménez scored his first international goal in a 1–0 friendly win against South Korea in September 2014.  In May 2015, he was named in Uruguay's squad for the 2015 Copa América by coach Óscar Tabárez. On 20 June, Giménez scored Uruguay's goal in a 1–1 draw with Paraguay which saw both teams progress to the knockout stage.  In May 2018 he was named in Uruguay's provisional 26 man squad for the 2018 FIFA World Cup in Russia. On 15 June, with the score tied at 0–0, he scored the winning goal in the final minute of Uruguay's opening game of the World Cup against Egypt.  In March 2019, manager Tabárez included Giménez in the final 23-man Uruguay squad for the 2019 Copa América in Brazil.

Honours.

Atlético Madrid  Uruguay U20  Individual

José María Giménez de Vargas (; born 20 January 1995) is a Uruguayan professional footballer who plays as a central defender for Spanish club Atlético Madrid and the Uruguay national team.

荷西·玛利亚·希梅内斯·德·巴尔加斯 (,1995年-1月20日),乌拉圭职业足球员,目前效力西甲球队马德里竞技,司职中坚。


Appearance.

The blade of the sword is 65.5 cm and the tang is 9.4 cm long. There is no hole on the tang to fasten the sword with a hilt. The sword is broken at the top of the tang. Analysis of the broken surface shows the material of the sword is forged mild steel. As the ‘branches’ appear to be quite delicate, and their functionality in melee combat doubtful, it is unlikely that the Seven-Branched Sword was used as a military weapon. Instead, it probably had a ceremonial function.  The sword has been stored in the Isonokami Shrine since ancient times. The inscription on the blade had been hidden by rust and was rediscovered in the 1870s by Masatomo Kan, a Shinto priest at the shrine. There is a two-sided inscription on the sword which is inlaid in gold. This sword appears to have been mentioned in the "Nihon Shoki". Many scholars have engaged in study to interpret the vague inscription. Closeup pictures of the sword taken with X-ray were published in 1996.

Origins.

Analysis and archeology have suggested that the sword's origins lie in Jin Dynasty China in 369. However, the sword's peculiar design—with the tip of the blade counting as the "seventh" branch—is indicative of contemporary Korean tree-motifs. Other examples of this motif include the Baekje Crown and the Silla Crown. If the weapon had indeed been produced in Korea, it would have represented one of the prime artistic accomplishments of Baekje swordsmithery.

Description in Nihon Shoki.

The sword is mentioned in the biography of Empress Jingū, a legendary Japanese empress in the ancient era. The following is the original Classical Chinese text;  In English;

Inscription on the sword.

The inscription states:  In original Chinese characters:  Characters in parentheses are ambiguous. Characters represented with black blocks are entirely unreadable.  In English:

Interpretation of the inscription.

2nd character on the first side, and when the sword was made: The first four characters are generally decoded as "4th year of Taihe" (the era name of Emperor Fei of the Eastern Jin dynasty in China), but the second one is ambiguous. Taihe 4 corresponds to year 369 CE. Kim Sok Hyong, a North Korean scholar proposed a theory that the first two characters instead refer to a local era name of the Baekje, but this theory has been challenged since no other archaeological discovery reveals the existence of unique era names assigned by the Baekje. Hong Sung-Hwa, a scholar at Korea University, argues that "十(一)月十六日(the sixteenth day of the eleventh month)" suggests the year 408, because if the 6th, November is 日干支 (日干支 is a day by the sexagenary cycle) of 丙午 in 408, the year is the 4 years by King Jun-ji (腆支王) in Baekje. From this we can speculate that Baekje assigned era names autonomously (Goguryeo and Silla had their own era names). In 409, Wei's envoy visited Baekje, and King Jun-ji extended hospitality to him, so that sword was formed by 408.  Middle of the first side: The characters show the sword was made of steel and can repel an enemy. The subsequent characters are the most controversial part of the inscription. Kim notes that the sword uses the term "候王" translated as "enfeoffed lord," and thus claimed that the Wa king was subservient to the Baekje ruler. The majority of Japanese scholars do not agree with Kim's theory. They note that the meaning of the term "候王" varied in the different periods. After the Han Dynasty, the term was used fluently and always just as an honorific.  End of the first side: Although four of the five last characters are undecodable, the last character indicates that the previous ones were either the name of author or a prayer phrase such as "永年大吉祥"(Have Great Fortunes Forever"). In both cases, the phrase should generally indicate the end of inscription, and that the text on the second side is not a continuation. There is also a theory that the second side was written by different person, or at different time.  11th to 13th characters on the second side, and who presented the sword: the 11th to 13th characters may be decodable as "王世子"(Crown Prince), causing some scholars to regard that the sword was presented by the Crown Prince of Baekje, who eventually ascended as King Geungusu. However, as this segment includes ambiguous characters, it is not entirely clear who in Baekje presented the sword.  17th character on the second side: The character is regarded to be either "音"(Sound) or "晉"(Jin Dynasty). The former interpretation indicates that the phrase "奇生聖音" has a Buddhist or Taoist nuance, and that the bestower has "lived under august (holy) sounds". Other scholars suggest that the phrase means "born coincidentally on august (holy) Jin Dynasty".  19th to 23rd characters on the second side, and the presentee: The phrase, "爲倭王旨造", is translated in various ways through different interpretations of the 22nd character "".  Treating it as a personal name leads to the Baekje-centric idea that Baekje's presenter boldly writes the name of the King of Wa, and thus regards him lower. Treating it as "order" leads to the Japan-centric idea that Baekje presented the sword because the King of Wa ordered him to do so. Therefore, the interpretation of this character tends to be controversial. Ueda Masaaki (quoted by Saeki, 1977) is rather an exception among Japanese historians because he “has maintained that the Seven-branched sword was ‘bestowed’ on the Wa ruler by the king of Baekche.” Ueda “based his interpretation on the argument that the term ‘koo’ [howang] appearing in the inscription denotes a ruler in vassalage to the Paekche king and that the inscription is written in the commanding tone of a superior addressing an inferior, exemplified by the sentence reading ‘Hand down [this sword] to [your] posterity." However, Saeki (1977) argues that one can not interpret the inscription to mean either “to bestow” the sword on the King in vassalage or “to respectfully present” to the emperor, as many Japanese scholars have maintained since the Meiji period. Saeki seems to be inclined to take Hirano’s argument that the inscription simply indicates the fact there was a respectful and sincere relationship between the rulers of Baekche and Wa.  Yet another theory proposed by Kōsaku Hamada of Kyushu University suggests that the original seven-branched sword was created by Eastern Jin in 369 (泰和四年) for a vassal lord with the first inscription. In 372, King Geunchogo of Baekje sent an embassy to arrive at the court of Eastern Jin in 372, and then a Jin envoy was sent to the Paekche court, granting the title of “General Stabilizing the East and Governor of Le-lang" (鎭東將軍). The sword was given to the king around this time. The king of Baekje ordered the creation of a replica of the sword with the second inscription and sent it to Wa for an alliance as peers under Eastern Jin. Thus no vassalage relationships are involved between Baekje and Wa. He claims that this explains the commanding tone of the first inscription and the respect paid to Jin (owes his life to august Jin) in the second inscription.  While the inscription of the sword is controversial and is used by many nationalists to support their own agendas, the sword does prove, at the very least, that there were very close ties between the Baekje and the Wa, and the opening of friendly relations between two countries probably dates to the year 372.  In connection with the date of fabrication, Hong Sung-Hwa, a scholar of Korea University, says in 396-409, Baekje came under attack by Goguryeo, so Baekje needed to form an alliance with the Wei; King Jun-ji of Baekje gave King of Wei the sword.

In popular media.

In popular media, the seven branched sword has appeared in:

The is a sword of continental manufacture believed to be identical with the artifact of that name, a gift of the king of Baekje that was bestowed upon a Yamato ruler as a gift who is mentioned in the Nihon Shoki in the fifty-second year of the reign of the semi-mythical Empress Jingū. It is a 74.9 cm (29.5 in) long iron sword with six branch-like protrusions along the central blade. The original sword has been conserved since antiquity in the Isonokami Shrine in Nara Prefecture, Japan and is not on public display. An inscription on the side of the blade is an important source for understanding the relationships between kingdoms of the Korean peninsula and Japan in that period.

七支刀是保存在日本奈良县天理市石上神宫的铁剑。这就是《日本书纪》神功皇后五十二年条所记载的七支刀。全长约75厘米。刀身两侧各交错伸出三把双刃枝刀,非实用品。刀身两面有60多个字的金嵌铭文,叙述百济王命人制造七支刀进贡与日本之经过,是探讨东亚各国关系的宝贵史料。关于铭文的解释尚无定论,有369年百济王为倭王制造的百济上献品说、百济王对倭王的下赐品说、东晋通过百济王下赐倭王说以及5世纪后半叶的百济刀之说等等。但都严厉地批判把《日本书纪》和《古事记》中关于上贡倭王的记载与铭文直接联系起来的传统说法。七支刀是日本的国宝。


亚历山德拉公主 (卢森堡)

Early life and education.

Princess Alexandra was born in 16 February 1991 at Grand Duchess Charlotte Maternity Hospital in Luxembourg, She is the fourth child and only daughter of Grand Duke Henri and Grand Duchess Maria Teresa.  Princess Alexandra attended primary school at Angelsberg and finished her secondary education at the Lycée Vauban in Luxembourg. She received her baccalauréat with honors in the literature stream. She then attended university in the United States at Franciscan University of Steubenville. After studying psychology and social sciences in the United States, she continued her studies in Paris where she received a bachelor's degree in philosophy, with special interest for ethics and anthropology. Princess Alexandra has interests in politics and religious studies. Since 2017, she holds a master's degree in interreligious studies from the Irish School of Ecumenics with a specialization in conflict resolution.  The Princess has worked in the field of journalism in the Middle East. She has also acquired experience in international relations while doing an internship at the Security Council of the United Nations in New York during the period when Luxembourg was a non-permanent member. She works as a volunteer to help refugees.

Line of succession.

Excluded from the line of succession from birth until 2011, when absolute primogeniture was adopted in respect to Grand Duke Henri's descendants, she is currently sixth in the line. Prince Louis gave up his place in the line of succession when he married Tessy Antony.

Activities.

In 2017 she accompanied her father during the state visit to Japan while her mother was absent.  Princess Alexandra is a volunteer in helping refugees. Princess Alexandra is also a patron to the "Lëtzebuerger Déiereschutzliga" and the "Lëtzebuerger Blannevereenegung" Foundation.

Princess Alexandra of Luxembourg ("Alexandra Joséphine Teresa Charlotte Marie Wilhelmine"; born 16 February 1991) is the fourth child and only daughter of Grand Duke Henri and Grand Duchess Maria Teresa. She has three older brothers: Hereditary Grand Duke Guillaume, Prince Félix, and Prince Louis, and one younger: Prince Sébastien.

亚历山德拉·约瑟芬·特丽莎·夏洛特·玛丽·威廉敏娜(:1991年-2月16日),出生于卢森堡大公爵夫人夏洛特妇产医院。卢森堡公主,拿骚公主,波旁-帕尔玛公主。是亨利大公唯一的女儿和第四个孩子。公主的教父母是第14代利涅亲王米歇尔和玛丽亚-安娜·加利齐纳王妃。卢森堡王位第五顺位继承人。


Early life and background.

Kapoor was born on 28 September 1982 in Bombay (now Mumbai) to Rishi and Neetu, both actors of the Hindi film industry. He is the great-grandson of Prithviraj Kapoor and the grandson of actor-director Raj. His elder sister, Riddhima (born 1980), is an interior and fashion designer. The actresses Karisma and Kareena Kapoor are his first cousins. Kapoor was educated at the Bombay Scottish School in Mahim. As a student, he found little interest in academics and would rank low among his peers. However, he has stated that he performed better in sports, particularly football. Kapoor has been vocal about how his parent's troubled marriage affected him as a child: "Sometimes the fights would get really bad. I would be sitting on the steps, my head between my knees, till five or six in the morning, waiting for them to stop". These experiences led to a "reservoir of emotions building up inside [him]", which he said compelled him to develop an interest in film. In his early years, Kapoor was close to his mother, but had a dysfunctional relationship with his father. After completing his tenth standard examinations, he worked as an assistant director to his father on the film "Aa Ab Laut Chalen" (1999), during which he developed a closer bond with him.  After completing his pre-university education from the H.R. College of Commerce and Economics, Kapoor relocated to New York City to learn film-making at the School of Visual Arts, and subsequently pursued method acting at the Lee Strasberg Theatre and Film Institute. In film school, Kapoor directed and starred in two short films, entitled "Passion to Love" and "India 1964". The loneliness of living alone in New York City coupled with his experience in film school, which he described as "useless", inspired him to pursue a career in Bollywood. Upon returning to Mumbai, Kapoor was hired as an assistant director to Sanjay Leela Bhansali on the 2005 film "Black". He described the experience: "I was getting beaten up, abused, doing everything from cleaning the floor to fixing the lights from 7 am to 4 am, but I was learning every day." He later remarked that his motive for working on "Black" was to get Bhansali to offer him an acting job.

Career.

Debut and initial success (2007–2010).

Following the release of "Black", Bhansali cast Kapoor as the protagonist of his 2007 tragic romance "Saawariya", alongside Sonam Kapoor and Rani Mukerji. The film tells the story of a tramp, played by him, who falls obsessively in love with a woman awaiting the return of her lover. In an interview with the news and entertainment portal Rediff.com, Kapoor stated that his character was written as a tribute to his grandfather's iconic roles as a tramp. "Saawariya" was the first Indian film to be produced by a Hollywood studio (Sony Pictures Entertainment), and was a highly anticipated release. However, film critics were disappointed with the picture with BBC's Jaspreet Pandohar calling it a "misfire on a massive scale". CNN-IBN's Rajeev Masand considered it "contrived and fake", but was impressed by Kapoor's "affable charm" and wrote that "he’s got that star quality to him which is so rare to find." At the box office, "Saawariya" failed to earn profits. However, at the annual Filmfare Awards ceremony, Kapoor was awarded a Best Male Debut trophy.  Despite the commercial failure of "Saawariya", Kapoor was contracted by Yash Raj Films for a primary role in the Siddharth Anand-directed romantic comedy "Bachna Ae Haseeno" (2008). The film was his first commercial success, in which his role was that of a womaniser who is romantically involved with three women, played by Bipasha Basu, Minissha Lamba, and Deepika Padukone, at different stages of his life. Rachel Saltz of "The New York Times" expressed mixed views on his performance, but predicted that his "puppy-dog sweetness" would "serve him well as a Bollywood leading man".  In 2009, Kapoor had three film releases. In Dharma Productions's "Wake Up Sid", a coming of age film from director Ayan Mukerji, Kapoor portrayed Siddharth "Sid" Mehra, a rich, lazy teenager whose life undergoes a series of changes after interacting with an ambitious journalist (played by Konkana Sen Sharma). When Mukerji narrated the then-untitled script of the film to him, Kapoor came up with the title himself. The media expressed doubt on the film's financial prospect as it depicted a romantic relationship between a younger man and an older woman. It eventually emerged as a sleeper hit and garnered acclaim from the critics. Taran Adarsh of Bollywood Hungama reviewed that Kapoor's performance in the film proved that he was "amongst the best in the business today".  Kapoor next starred opposite Katrina Kaif in "Ajab Prem Ki Ghazab Kahani", a slapstick comedy from director Rajkumar Santoshi, that emerged as the fourth highest-grossing Bollywood film of 2009. Film critic Gaurav Malani praised Kapoor's flair for comedy, but criticised his "over-excited husky baritone". Kapoor's final release that year was the Shimit Amin-directed "", a drama about a sardar who aspires to be a salesman. Film critic Mayank Shekhar praised the film and found Kapoor's performance to be "astonishingly sincere", but the film earned little at the box office. Kapoor later professed to being highly disappointed by the film's commercial failure. At the 55th Filmfare Awards, Kapoor was awarded the Filmfare Critics Award for Best Actor for his performances in all three of his 2009 releases, and he also received two Best Actor nominations at the ceremony for "Wake Up Sid" and "Ajab Prem Ki Ghazab Kahani".  Prakash Jha's big-budget ensemble political drama "Raajneeti" was Kapoor's first release of 2010. The film, which starred Nana Patekar, Ajay Devgn, Arjun Rampal, Manoj Bajpayee, Katrina Kaif and Sarah Thompson in prominent roles, was inspired by the Indian epic "The Mahabharata" and Mario Puzo's 1969 novel "The Godfather". Kapoor's role was of Samar Pratap (based on the characters of Arjuna and Michael Corleone), the youngest heir of an Indian political dynasty, who is reluctantly drawn to politics after the assassination of his father. Kapoor described it as his first complex role and considered it a departure from the "lover boy roles" that he had previously played. Nikhat Kazmi of "The Times of India" reviewed: "The film finally belongs to Ranbir Kapoor who perfects the art of minimalismand literally grows before your eyesas the simmering volcano that cannot be held back, once it erupts." Robert Abele of the "Los Angeles Times", however, was more critical of his performance which he considered "stony rather than calculatingespecially jarring compared to the histrionic turns around him". Indian trade journalists were apprehensive of "Raajneeti" recovering its investment. The film, however, proved to be a major commercial success with worldwide earnings of over. Kapoor received a third Best Actor nomination at Filmfare for the film.  Later that same year, Kapoor collaborated with Priyanka Chopra on Anand's "Anjaana Anjaani", a comedy-drama involving two strangers who vow to commit suicide on New Year's Eve. The film was a moderate financial success, but garnered little praise from the critics. Rajeev Masand noted that Kapoor "struggles with a badly-defined role" and NDTV's Anupama Chopra concluded: "Ranbir tries hard to salvage the film, dropping his shirt several times but even his lovingly shot chest can’t save the film."

Critical acclaim (2011–2013).

Following an item number in the children's film "Chillar Party" (2011), Kapoor took on the role of Janardhan "Jordan" Jakhar in Imtiaz Ali's "Rockstar", a drama that follows the journey of an aspiring musician from a humble background to international stardom. In preparation for the role, Kapoor lived with a Jat family in Pitam Pura and studied their mannerisms. He additionally learned to play the guitar and practiced extensively at A.R. Rahman's (the film's music composer) music studio. As part of the film's promotional activity, Kapoor performed at a live concert in Mumbai. Film critics were polarised on their view of the film, but were unanimous in their praise for Kapoor; Aniruddha Guha of "Daily News and Analysis" was particularly impressed with the film and thought that Kapoor's portrayal was one of "Hindi cinema’s most accomplished performances by a lead actor". For the role, Kapoor won both the Best Actor and Best Actor (Critics) trophies at the 57th Filmfare Awards ceremony, along with Best Actor awards at Screen and IIFA. With a gross revenue of, "Rockstar" was one of the top-grossing Hindi films of the year.  The 2012 romantic comedy "Barfi!" was Kapoor's first release to earn over at the domestic box office. Directed by Anurag Basu, the film (set in the 1970s) tells the story of its titular protagonist (a deaf and mute man, played by Kapoor) who falls in love with a woman who is already engaged (played by Ileana D'Cruz) and later, an autistic girl (played by Priyanka Chopra). In preparation, Kapoor observed the work of actors Roberto Benigni, Charlie Chaplin, and his grandfather. "Barfi!" received praise from the critics, and the performances of the three lead actors were acclaimed. Ronnie Scheib of "Variety" praised Kapoor for successfully channeling Chaplin in "tone and affect", and Raja Sen of Rediff.com wrote that "he does very strongly indeed with this Chaplin-tribute role, bestowing his character with heart every step of the way". The film was submitted as India's official entry for the 85th Academy Awards, and was screened at the Marrakech and Busan International Film Festivals. Kapoor won second consecutive Best Actor awards at the Filmfare, Screen, and IIFA Award ceremonies.  Kapoor achieved further success in 2013 when he reunited with director Ayan Mukerji for the romantic comedy "Yeh Jawaani Hai Deewani" co-starring Deepika Padukone, Kalki Koechlin, and Aditya Roy Kapoor. He was cast as Kabir "Bunny" Thapar, a commitment-phobic photographer, a character Kapoor found to be an extension of himself. His pairing with Padukone, after their highly publicised break-up, led to hype surrounding the film's release. "Yeh Jawaani Hai Deewani" emerged as one of the highest-grossing Indian films of all time with earnings of, proving to be Kapoor's third consecutive commercial success in three years and earned him another Best Actor nomination at Filmfare. Film critics found the film to be "riddled with clichés", but praised both Kapoor and Padukone, with "Daily News and Analysis" Tushar Joshi labelling their on-screen chemistry as "unsurpassable". Kapoor's second release of 2013 was the action-comedy film "Besharam" in which he played a petty thief, alongside Pallavi Sharda and his parents. The film met with an overwhelming negative reception and emerged as a failure; Sudhish Kamath of "The Hindu" described it as "Kapoor's hall of shame".

Commercial struggles and resurgence (2014–present).

After a year-long absence from the screen, Kapoor featured as a mysterious thief in "Roy" (2015), a romantic thriller which critic Sarita Tanwar described as a "boring, exhausting and pretentious" film. The series of financial failures continued with his next release, the Anurag Kashyap-directed crime drama "Bombay Velvet", based on the historian Gyan Prakash's book "Mumbai Fables", which also starred Anushka Sharma and Karan Johar. Kapoor portrayed an ambitious boxer in the 1960s who is drawn towards a life of crime; he based his "tapori" diction on Johnny Walker's character in "Mr. & Mrs. '55" (1955). Made on a budget of, the film opened to meager box-office collections and mixed reviews from critics. Ritika Bhatia of "Business Standard" found Kapoor's part to be a departure from his previous roles, writing that he "does test the waters with methodical sincerity but his portrayal lacks depth on occasion". "Bombay Velvet" was eventually screened at the Locarno and Bucheon film festivals.  Kapoor next reunited with Deepika Padukone in "Tamasha" (2015), a romantic drama from Imtiaz Ali. He played the role of Ved Sahni, a man who desires a career in the arts but settles for a life of monotony as an engineer. Once again, the film failed commercially and received mixed reviews from critics. Kapoor's performance was praised; Lisa Tsering of "The Hollywood Reporter" wrote, "Kapoor is beautifully cast, his trademark mix of assertive theatricality and subtle emotion well suited to the role of a man who seemingly has it all under control, until he doesn’t." Kapoor's career prospect briefly improved in 2016 when he took on the lead role of an aspiring musician in Karan Johar's "Ae Dil Hai Mushkil" (2016). The romantic drama, co-starring Anushka Sharma and Aishwarya Rai, tells the story of unrequited love, and proved to be one of the highest-grossing Bollywood films of the year. Critical reception was mixed; Joe Leydon of "Variety" thought that "Kapoor [is] often erring on the side of annoying while trying to balance vulnerability and self-centeredness". Both "Tamasha" and "Ae Dil Hai Mushkil" earned Kapoor Best Actor nominations at Filmfare.  In a collaboration with director Anurag Basu, Kapoor formed a production company named Picture Shuru Productions, whose first release was the musical comedy-mystery film "Jagga Jasoos" (2017), which tells the story of a gifted teenager's search for his missing stepfather. Production on the project was plagued with difficultiesprincipal photography began in 2014, but the release date was pushed back several times owing to changes in the script, multiple reshoots, and rumours of animosity between Kapoor and co-star Katrina Kaif (the couple had broken-up before they finished filming). Anna M. M. Vetticad of "Firstpost" considered the film to be "a laudable experiment that zigzags off course", and while praising Kapoor's dedication to his part, criticised the decision to cast him as a teenage schoolboy. It did not perform well commercially which led Kapoor to regret his venture into production.  In 2018, Kapoor portrayed the actor Sanjay Dutt in Rajkumar Hirani's biopic "Sanju". It deals with Dutt's troubled relationship with his father, his drug addiction, and arrest for illegal possession of firearms. Kapoor was keen to avoid a hagiographic narrative and did not want to mimic Dutt's mannerisms. He interacted extensively with Dutt and took a month's time to transform himself physically before filming each phase of the actor's life. Saibal Chatterjee of NDTV thought that despite dramatic liberties, the film had successfully portrayed Dutt's complexity, and wrote that Kapoor "pulls out the stops in astonishingly effective ways, subsuming his personality into that of the protagonist". Uday Bhatia of "Mint" was more critical for making Dutt too sympathetic despite his misdeeds, and wrote that Kapoor's "imitation (astonishing at times) takes over the performance". "Sanju"s strong financial performance re-established Kapoor's stardom. With earnings of over, it emerged as one of Indian cinema's biggest earners and Kapoor's highest-grossing release. He won another Best Actor award at Filmfare.

Upcoming projects.

As of March 2019, Kapoor has three upcoming projects. He will feature opposite Alia Bhatt in Ayan Mukerji's fantasy film trilogy, the first of which is named "Brahmastra". He has committed to play a dacoit in Karan Malhotra's period adventure film "Shamshera", alongside Vaani Kapoor and Sanjay Dutt, and will star with Ajay Devgn in Luv Ranjan's as-yet untitled action thriller.

Personal life.

Kapoor has been vocal about his personal life, and has stated that his parents' marriage taught him how complex a relationship can be. He had his first serious relationship while in the seventh grade, and suffered from depression when it ended. While filming "Bachna Ae Haseeno" in 2008, Kapoor began dating his co-star Deepika Padukone. The relationship attracted substantial media coverage in India and they speculated on an impending engagement. However, the couple broke up a year later. Kapoor maintained that the split was amicable, though the media widely reported that the split was due to infidelity on Kapoor's part. Kapoor later confessed: "Yes, I have, out of immaturity, out of inexperience, out of taking advantage of certain temptations, out of callousness." Later in 2015, Kapoor stated that they both had resolved the conflict and had moved on with their lives. Since the split, he has been reticent to publicly discuss his personal life.  Rumours of an affair with Katrina Kaif first emerged during the production of "Ajab Prem Ki Ghazab Kahani" in 2009. In August 2013, a set of paparazzi photographs of Kapoor and Kaif at a beach in Spain were published by "Stardust". Although Kapoor initially declined to speak of the relationship, he admitted to it in 2015: "Both of us are sure about our relationship and if we don't open about it now, it would be showing disrespect to the relationship." As of February 2016, the media reported that they had broken up. In 2018, he began dating Alia Bhatt, his co-star in "Brahmastra" (2019). He also admitted to being addicted to smoking and drinking.

Off-screen work.

In addition to acting, Kapoor is a football enthusiast and supports charities and organisations. He is the vice-captain of the All Stars Football Club, a celebrity football club that raises money for charity. In March 2013, he played the game to raise funds for the Magic Funds Organisation, an NGO for underprivileged children. Along with the chartered accountant Bimal Parekh, Kapoor acquired ownership rights in 2014 for the Mumbai-based football team of the Indian Super League, named Mumbai City FC. Also that year, Kapoor announced his participation with the digital music company Saavn as a content and programming adviser. In 2016, he featured in the documentary series "Girls with Goals", to create awareness about and raise funds for YUWA, an all-girls football team in the state of Jharkhand.  Kapoor supports the empowerment of the girl child and is the goodwill ambassador of Shabana Azmi's Mijwan Welfare Society, an NGO which helps empower women. He supports environmental charities, and in 2011 donated money to the Community Water Initiative, a charitable organisation owned by PepsiCo. In 2012, he appeared alongside other celebrities in a short film by Zoya Akhtar to generate awareness on breast cancer. He is the campaign ambassador for NDTV's Marks for Sports, a nationwide initiative to promote fitness and health in India. In 2013, Kapoor participated in an auction on eBay, where the highest bidder gets an opportunity to interact with him; the proceeds were donated to Save the Children, a non-profit organisation that raised funds for flood affected households in Uttarakhand. That same year, he appeared alongside other celebrities in a commercial, produced by the National Film Development Corporation of India, to create awareness about children's education. In December 2014, Kapoor again took part in an eBay auction; the "phiran" he wore in "Rockstar" was sold, with proceeds going to the redevelopment of the flood-ravaged states of Kashmir and Assam. Kapoor had also fronted a campaign to collect donations for the victims of the April 2015 Nepal earthquake. In 2015, he presented 2,000 raincoats to the Mumbai traffic police as an appreciation for their service during the year's heavy monsoon. In 2018, Kapoor collaborated with Aamir Khan's Paani Foundation to help local farmers and villagers suffering from drought in parts of Maharastra.

In the media.

Born into a family of popular actors, Kapoor faced the media spotlight from a young age; "Hindustan Times" published that "he was always a star kid from whom everyone had great expectations". Despite the failure of his first film ("Saawariya"), IANS reported that he "rose like [a] meteor on film firmament by giving compelling performances in films like "Raajneeti", "Rockstar" and [..] "Barfi!"" Discussing the commercial viability of Kapoor, Apoorva Mehta (the COO of Dharma Productions) noted in 2013, "In a short career span of 10 films, Ranbir Kapoor has achieved a tremendous jump in the business done by his films." Also that year, "The Economic Times" credited him as "the most bankable actor of his generation". However, following the success of "Yeh Jawaani Hai Deewani", each of Kapoor's releases under-performed at the box-office. This led trade journalists to criticise his choice of films, noting that his inclination towards experimental projects negatively impacted his commercial appeal.  Nationally, Kapoor is one of the most popular and high-profile celebrities. In 2012 and 2013 "Forbes" featured him among the top twenty in India's Celebrity 100, a list based on the income and popularity of the country's celebrities. For the next two years, he was ranked 11th with an estimated annual earning of and respectively, making him one of the highest-paid actors in the country. Kapoor has frequently featured in Rediff.com's annual listing of "Bollywood's Best Actors"; he was ranked second in 2009, first in 2011, third in 2012, and held the sixth position in 2015.  Kapoor has been cited as one of the most attractive Indian celebrities by the media. He has featured on "The Times of India"s listing of the 'Most Desirable Man' from 2010 to 2015, ranking among the top ten each year. In 2009 "People" magazine listed him as the "Sexiest Man Alive" in India, and in 2013 he topped "Filmfare"s poll of the "Most Stylish Young Actor". Also in 2013, he was one of the recipients of the "People of the Year" award by the Limca Book of Records. In 2010, he was voted the "Sexiest Asian Man" in a poll conducted by the magazine "Eastern Eye". Kapoor continued to feature among the top ten of the list from 20112014. Kapoor is also the celebrity endorser for various brands and services, including Pepsi, Panasonic, Renault India, Lenovo and the Spanish football club FC Barcelona.

Awards and nominations.

Kapoor has been the recipient of six Filmfare Awards: Best Male Debut for "Saawariya" (2007), Critics Award for Best Actor for "Wake Up Sid" (2009), "Ajab Prem Ki Ghazab Kahani" (2009), and "Rocket Singh: Salesman of the Year" (2009) (jointly for the three films), and "Rockstar" (2011), and Best Actor awards for "Rockstar", "Barfi!" (2012), and "Sanju" (2018).

Ranbir Kapoor (; born 28 September 1982) is an Indian film actor. He is one of the highest-paid actors of Hindi cinema and has featured in "Forbes India"s Celebrity 100 list since 2012. Kapoor is the recipient of several awards, including six Filmfare Awards.

兰比尔·卡浦尔(,英语:Ranbir Kapoor,1982年-9月28日)生于印度孟买,是印度宝莱坞男演员。卡浦尔是现今印度其中一位最成功及收入最高的演员之一并曾5次荣获印度电影观众奖的奖项。


Beginnings.

The first clergyman of any denomination to set foot on Antarctica was Arnold Spencer-Smith (1883–1916), an Anglican priest who was chaplain and photographer for the Ross Sea Party of Shackleton's Imperial Trans Antarctic Expedition. Spencer-Smith set up a chapel in Polding's darkroom in Scott's Hut at Cape Evans. He arranged an altar with cross and candlesticks and an aumbry where he reserved the Blessed Sacrament; he made a lamp to hang by the aumbry to indicate the real presence. In his diary, Spencer-Smith records when he celebrated Eucharist and how many were present. He also records when he heard confession on the continent. Spencer-Smith was the first clergyman to land in Antarctica, the first to celebrate the Eucharist and the first to die and be buried there.  A cross on Wind Vane Hill, Cape Evans, was erected by the Ross Sea Party, led by Captain Aeneas Mackintosh, of Sir Ernest Shackleton’s Imperial Trans-Antarctic Expedition of 1914–1917, in memory of three members of the party who died in the vicinity in 1916. The cross has been designated a Historic Site or Monument (HSM 17), following a proposal by New Zealand and the United Kingdom to the Antarctic Treaty Consultative Meeting.  The first Roman Catholic service in Antarctica was performed in 1947 by William Menster (1913–2007), Lieutenant Commander of the United States Navy during Operation Highjump. During a Catholic service held in a tent set up on land, he consecrated Antarctica. Ministering to approximately 2,000 men from a variety of Christian denominations gave him experience in leading ecumenical services.  Although they are used mostly for Christian worship, the Chapel of the Snows has also been used for Buddhist and Bahá'í ceremonies. Some of the buildings are among historically important sites and under protection, such as the Chilean Captain Arturo Prat Base's wooden cross and a Statue of Our Lady of the Carmel.

History (including the subantarctic islands).

Around 1900, whaling stations and expedition camps were erected in the larger area. After World War II, some military expeditions explored the region. The International Geophysical Year (1957–1958) marked the end of a long period during the Cold War when scientific interchange between East and West had been seriously interrupted and has since contributed to civil scientific exploration. Since the 1950s, most stations in Antarctica have been constructed exclusively for scientific research. Extended stays in the region can be an extremely stressful experience for the researchers who often have been separated from their families for months at a time. Jesuits, which have had a long tradition of geophysical research in Antarctica, contributed as well to the early Antarctic missions. Notable Jesuit seismicity scholars like Edward A. Bradley (compare Bradley Nunatak), Henry F. Birkenhauer (compare Birkenhauer Island) and J. Joseph Lynch and Daniel J. Linehan have been among those.  The first churches south of the Antarctic Convergence and north of 60° S latitude (and thus not part of the Antarctic Treaty System) are Notre-Dame des Vents at Port-aux-Français on the main island of Kerguelen and the Norwegian Lutheran Church (Grytviken, South Georgia), a Lutheran chapel in Grytviken, South Georgia (since 1913). After years of abandonment and weathering the harsh elements of the region (roof damaged in 1994), the Grytviken church was renovated by the keepers of South Georgia Museum and volunteers in 1996–1998 and now serves for occasional church services and marriage ceremonies.  Some churches north of the Antarctic Convergence serve Antarctic territories, such as the Christ Church Cathedral in Stanley, which is the southernmost Anglican cathedral in the world. It serves as the parish church not only for the Falkland Islands, but also South Georgia and the British Antarctic Territory. Punta Arenas in the southern tip of the South American mainland (just north of Tierra del Fuego) also has a Roman Catholic cathedral, which serves the Chilean Antarctic Territory.  Christians have increasingly turned toward Internet communications for fellowship in the 21st century.

Notable buildings.

Out of nearly 90 stations in Antarctica, half are only used in summer months. Most research stations have a small meeting room dual-purposed for religious assemblies. Larger stations and communities use a separate room, often a makeshift steel container for religious purposes.  The Chapel of the Snows was erected in 1956 as a Christian chapel used by several denominations, at McMurdo Station, Ross Island. The Chapel offers various Protestant and Catholic services, but also allows for meetings of other religions, such as Latter-day Saints, Bahá'í and Buddhists and is used as well for secular groups (e.g. Alcoholics Anonymous). The chapel had been rebuilt after a fire in 1978 and was reconsecrated in 1989.

Catholic sites.

The majority of Catholic Antarctic sites exist due to the Argentine and Chilean presence on the Continent. The Worldwide Antarctic Program proposes building a Catholic chapel at Mario Zucchelli Station, Terra Nova Bay, Antarctica; while the first Catholic chapel (named after Saint Francis of Assisi) was built in 1976 at the Argentine Esperanza Base. The southernmost Christian chapel (in fact, the southernmost place of worship of any religion) lies at the Argentine Belgrano II Base at Coats Land. It is a permanent Catholic chapel made entirely of ice.  Further dedicated religious buildings are the Chapel of Santisima Virgen de Lujan, Antarctica, a Catholic chapel at Marambio Base, Seymour Island, San Francisco de Asis Chapel, a Catholic chapel at Esperanza Base, Antarctic Peninsula and Santa Maria Reina de la Paz Church, a Catholic church at the Villa Las Estrellas, South Shetland Islands. Santa María Reina de la Paz is a repurposed container allowing for 36 persons to convene.  There is a small Catholic shrine near McMurdo Station called "Our Lady of the Snows". Popularly known as "Roll Cage Mary", the shrine was dedicated to a United States Navy Seabee who was killed during the construction of the station. It has been maintained by Carmelite nuns in New Zealand.

Eastern Orthodox Christian revival.

Religious activities were halted during the Soviet Antarctic Expedition. However, this changed with the fall of the Soviet Union and the Communism within Central and Eastern Europe.  The St. Ivan Rilski Chapel, a Bulgarian Orthodox chapel at St. Kliment Ohridski Base, South Shetland Islands was built in 2003 and is the first Eastern Orthodox edifice in Antarctica, and was the southernmost Eastern Orthodox building of worship in the world till 2011 when St Volodymyr (Vladimir the Great) Chapel at Ukrainian Vernadsky Research Base was erected.  The Buromsky Island cemetery belongs among the most important monuments in the region. A Russian orthodox cross has been erected in the meanwhile to mark the site. The island holds a cemetery for about 60 citizens of the Soviet Union, Czechoslovakia, the German Democratic Republic and Switzerland who died in the performance of their duties while serving as members of Soviet and Russian Antarctic expeditions. It has been designated a Historic Site or Monument (HSM 9) following a proposal by Russia to the Antarctic Treaty Consultative Meeting.  In 2002 Patriarch Alexy II of Moscow asked for a "Temple for Antarctica" (Храм Антарктиде) and initiated the Trinity Church, Antarctica, a Russian Orthodox church at Bellingshausen Station, South Shetland Islands. The Church is being positioned as a widely visible landmark and is maintained by the Moscow Patriarch. It was first maintained by a single priest. In the meanwhile two priests maintain the services, which are being exchanged annually. They both contribute to the maintenance of the overall station. January 29, 2007, saw the first church wedding on Antarctica, between a Chilean and Russian. The first baptism (for adults) also took place there.

Islam.

While the Pakistani program at Jinnah Antarctic Station brought Muslims to Antarctica in 1991, there are no mosques on the continent or on any of the outlying islands, as the best ways of observing Islamic religious customs there are still being determined.

Hinduism.

The bases of India namely Maitri, Dakshin Gangotri and Bharati have brought hindus and other indian religious people to Antarctica.

Christian buildings are the only religious buildings on the continent of Antarctica. Although they are used mostly for Christian worship, the Chapel of the Snows has also been used for Buddhist and Bahá'í ceremonies. Some of the early religious buildings are now protected as important historical monuments.

目前,南极洲上仅有基督宗教的宗教建筑物。南极洲共有至少7个用作宗教崇拜的教堂,当中包括:


Early life and education.

Michael Levitt was born in Pretoria, South Africa, to a Jewish family from Plungė, Lithuania; his father was from Lithuania and his mother from the Czech Republic. He attended Sunnyside Primary School and then Pretoria Boys High School between 1960 and 1962. The family moved to England when he was 15. Levitt spent 1963 studying applied mathematics at the University of Pretoria. He attended King's College London, graduating with a first-class honours degree in Physics in 1967.  In 1967, he visited Israel for the first time. Together with his Israeli wife, Rina, a multimedia artist, he left to study at Cambridge, where their three children were born. Levitt was a PhD student in Computational biology at Gonville and Caius College, Cambridge, and was based at the Laboratory of Molecular Biology from 1968 to 1972, where he developed a computer program for studying the conformations of molecules that underpinned much of his later work.

Career and research.

In 1979, he returned to Israel and conducted research at the Weizmann Institute of Science in Rehovot, becoming an Israeli citizen in 1980. He served in the Israel Defense Forces for six weeks in 1985. In 1986, he began teaching at Stanford University, and since then has split his time between Israel and California. He went on to gain a research fellowship at Gonville and Caius College, Cambridge.  From 1980 to 1987, he was Professor of Chemical Physics at the Weizmann Institute of Science, Rehovot. Thereafter, he served as Professor of Structural biology, at Stanford University, California.  Levitt was one of the first researchers to conduct molecular dynamics simulations of DNA and proteins and developed the first software for this purpose. He is currently well known for developing approaches to predict macromolecular structures, having participated in many Critical Assessment of Techniques for Protein Structure Prediction (CASP) competitions, where he criticised molecular dynamics for inability to refine protein structures. He has also worked on simplified representations of protein structure for analysing folding and packing, as well as developing scoring systems for large-scale sequence-structure comparisons. He has mentored many successful scientists, including Mark Gerstein and Ram Samudrala. Cyrus Chothia was one of his colleagues.

Industrial collaboration.

Levitt has served on the Scientific Advisory Boards of the following companies: Dupont Merck Pharmaceuticals, AMGEN, Protein Design Labs, Affymetrix, Molecular Applications Group, 3D Pharmaceuticals, Algodign, Oplon Ltd, Cocrystal Discovery, InterX, and StemRad, Ltd.

Covid-19.

He has done research on Covid-19  In March, he successfully predicted when the rate of increase in the number of deaths in China would slow down He predicted that there would be a slowdown in covid cases  He accurately predicted the initial trajectory of the pandemic, and when China would peak. He says that the lockdown saved no lives, and may have cost more lives. He said to Neil Ferguson that there was an overestimation in potential deaths.

Awards and honours.

Levitt was elected an EMBO Member in 1983, a Fellow of the Royal Society (FRS) in 2001, and a member of the National Academy of Sciences in 2002, and received the 2013 Nobel Prize in Chemistry, together with Martin Karplus and Arieh Warshel, "for the development of multiscale models for complex chemical systems". He received the DeLano Award for Computational Biosciences in 2014. He was elected an ISCB Fellow by the International Society for Computational Biology in 2015.

Personal life.

Levitt holds American, British and Israeli citizenship.  His wife Rina died on 23 January 2017.  He is the sixth Israeli to be awarded the Nobel Prize in Chemistry in under a decade.

Michael Levitt, (; born 9 May 1947) is an American-British-Israeli-South African biophysicist and a professor of structural biology at Stanford University, a position he has held since 1987. Levitt received the 2013 Nobel Prize in Chemistry, together with Martin Karplus and Arieh Warshel, for "the development of multiscale models for complex chemical systems".

迈可·列维特(英语:Michael Levitt,1947年-5月9日),FRS,出生于南非,是一名犹太裔美国、英国及以色列籍生物物理学家,现任美国史丹佛大学结构生物学教授。列维特的研究专长为计算生物学,也是美国国家科学院院士。


History.

The Ubuntu Touch project was started in 2011. Mark Shuttleworth announced on that by Ubuntu 14.04, the goal was that Ubuntu would support smartphones, tablets, smart TVs and other smart screens (such as car head units and smartwatches), but to date has only been supported by vendors on a few smartphones, one tablet and a number of third-party devices which hobbyists have ported the operating system to. The initial goal set by Shuttleworth for Ubuntu was to reach full convergence (same platform and libraries on all devices). The Ubuntu platform for phones was unveiled on. The Ubuntu Touch Developer Preview was released on. Canonical released Ubuntu Touch 1.0, the first developer/partner version on 17 October 2013, along with Ubuntu 13.10 that "primarily supports the Galaxy Nexus and Nexus 4 phones, though there are images available for other phones and tablets", and released a "relatively'stable' build for wider testing and feedback" on 17 April 2014, along with Ubuntu 14.04. A preview version of the software is available for installation on certain additional Android handsets including the Samsung Galaxy S4 Google Edition as a Developer Preview as of 21 February 2013. Developers have access to all of the source code under a license allowing modification and redistribution of the software.  Ubuntu Touch was released to manufacturers on 16 September 2014. BQ Aquaris E4.5 Ubuntu Edition, the world's first Ubuntu-based smartphone went on sale in Europe on 9 February 2015.  In April 2016, the world's first Ubuntu-based tablet, the BQ Aquaris M10 Ubuntu Edition, was released.  In August 2016, OTA-13 was announced to support Android 6.0 BSP.  In August 2018, UBPorts released its OTA-4, upgrading the Ubuntu Touch's base from the Canonical's starting Ubuntu 15.04 "Vivid Vervet" to the nearest, current long-term support version Ubuntu 16.04 LTS "Xenial Xerus".

Ubuntu for Android.

Ubuntu for Android was a variant of Ubuntu designed to run on Android phones. It was expected to come pre-loaded on several phones. An "Ubuntu for Android" mock-up was shown at Mobile World Congress 2012. A Developer Preview was released in February 2013 but as of April 2014, this project is no longer under active development by Canonical.  It would contain different graphical interfaces: when the device is connected to a desktop monitor, it features a standard Ubuntu Desktop interface (Unity). When the device is connected to a TV, the interface featured is the Ubuntu TV experience. It would have the ability to run standard Ubuntu Desktop applications and also the ability to run Android applications on the Ubuntu Desktop. which includes apps to make and receive calls and SMS messages directly from the desktop. Developers will be able to create one app, with two interfaces: a smartphone UI, and, when docked, a desktop UI.  A phone running Ubuntu for Android has to meet several requirements such as a dual-core 1 GHz CPU, video acceleration through a shared kernel driver with associated X driver; OpenGL, ES/EGL, 2 GB storage, HDMI for video-out with secondary frame buffer device, USB host mode and 512 MB RAM.

Ubuntu Mobile.

Ubuntu Mobile Internet Device Edition is a discontinued Ubuntu distribution planned to run on the Intel Mobile Internet Device platform, x86 mobile computers based on the Intel Atom processor. It was planned to use the GNOME framework Hildon as the basis for its GUI. In June 2008, Ubuntu Mobile 8.04 was released. Ubuntu Mobile ended active development in 2009 after 9.10 Alpha 6.  Equipment producers would have been able to customize their distributions, including options such as Flash, Java, or custom interfaces.  According to Canonical, Ubuntu Mobile would provide an "uncompromised Web 2.0 experience". It was to include features such as Web browsing, email, media, camera, VoIP, instant messaging, GPS, blogging, digital TV, games, contacts, and calendars, with regular software updates.

Features.

Ubuntu Touch uses the Qt 5-based touch user interface and various software frameworks originally developed for Maemo and MeeGo such as oFono as telephony stack, "accounts-sso" for single sign-on, and Maliit for input. Using libhybris the system can often be used with Linux kernels used in Android, which makes it easily ported to most recent Android smartphones.  Libertine is Ubuntu's project to run traditional desktop X applications.

Lock screen.

When Ubuntu Touch is turned on no lock screen immediately appears, as applications will prompt the user to unlock if necessary when they are opened. The centre of the "Welcome Screen" is a visualisation of activity on the device. It shows the user's status and recent events on the welcome screen, completed with a design around the circle which reflects activity on the phone over the preceding month.

Included applications.

Ubuntu Touch includes core applications such a calculator, an e-mail client, an alarm clock, a file manager, and even a terminal among others. Twelve or more core applications are currently being developed. Several Ubuntu Touch applications work on the desktop as well, including Browser, Calendar, Clocks, Gallery, Notes, Reminders, Terminal, and Weather.

Side stage.

Side stage was introduced in 2013 and allows users to run both "tablet apps" and "phone apps" side by side, resizing each on the top as and when you need to see more of them. It aims to "go even further" with the idea of multitasking, allowing screen space to be divided in this manner. Examples shown in the announcement video included a notes app being used alongside a web browser, and a user swiping from the right edge to bring a mobile version of the Facebook app into view over a playing video.

Design.

Users can access the whole system by swiping from the edges of the screen. A short swipe from the left edge allows for instant access to applications pinned to the launcher, while swiping all the way across reveals the home scope, which can be set by the user. This menu is available from the home screen and any running app.  Ubuntu Touch's multitasking is accessed by swiping the finger from the right edge of the screen to the left, which switches to the previous application (short swipe) or shows all open apps (long swipe). Swiping up from the bottom is used to show or hide tools specific to the app being used, which gives Ubuntu Phone the ability to run applications with a large, uncluttered canvas by default.

Target market.

Mark Shuttleworth, the founder of the company Canonical Ltd., believed that Ubuntu for phones will first find a niche in countries where Ubuntu is well known; more specifically, developing markets such as India and China where computers have Ubuntu pre-installed. However, the success of Ubuntu Phone in these markets is difficult to predict. Despite Ubuntu's popularity among open source developers, penetrating the legacy-bound business market will continue to be somewhat challenging for Ubuntu. Companies employing the "bring your own device" (BYOD) method have already adapted to using Android and iOS devices and the benefits posed by Ubuntu may not be adequately considered.

Requirements.

Hardware.

Ubuntu Touch requires that a system's CPU support certain hardware features.

Software.

Ubuntu touch can be installed on a number of phones that originally appeared with Android. It is necessary for these devices to offer an open source tree, so that the drivers can be recompiled for the new OS.

Reception.

Adrian Covert, writing for CNN on, predicted that the operating system will not gain wide use, stating, "carving out a niche in the seemingly unshakable mobile space—ruled by the Android-and-Apple duopoly—still requires a critical mass of users and a lively ecosystem of app developers. Realistically speaking, the chances of this even upstaging Windows Phone or BlackBerry 10 are slim. At best, Ubuntu seems like a sandbox for the most enthusiastic early adopters and a cheap enterprise solution for companies on a tight budget."  Joey Sneddon of "OMG! Ubuntu" disagreed with Covert's assessment, writing, "commentators like Covert are missing the point. See, Ubuntu Phones aren't really going to claw much market share away from Apple or Google. And this neat 'dock your phone and use it as a desktop' feature, whilst innovative, won't be the main lure for many [...]. During his keynote address earlier this week, Mark Shuttleworth continually referred to 'emerging' markets as the battleground on which an Ubuntu Phone would fight it out for impact [...]. It's this sector, the low-end, that the battle for the hearts, minds and hands of the less tech-savvy will take place."  Rich Trenholm writing for CNET on, "[...] But on first impression I'm hugely taken with Ubuntu Touch. It's elegant, thoughtful, and versatile, while remaining beautifully straightforward. [...] it's by far the strongest potential rival to Android, iOS, and Windows Phone. In fact, I prefer it to iOS, which long ago lost its shine, and heck, maybe even to Android, too. Fingers crossed that manufacturers and phone carriers get behind it, because I'd happily lay down my own cash for an Ubuntu Touch phone."  Jason Jenkins, writing for CNET on, MWC Awards 2013, "[...] Lots was said about the impressive number of carriers and manufacturers Firefox OS has lined up behind it. But once put to a vote, Ubuntu Touch was the clear winner, with Firefox OS the runner-up. The team thought that Ubuntu Touch, the tablet version of which we got our hands-on for the first time at MWC, feels more like the complete package at this point. [...]"  Jesse Smith from DistroWatch Weekly reviewed the Ubuntu Phone in Meizu Pro 5 lauding the price, interface, responsiveness and frequency of updates, calling it a "pleasant phone and communications experience" and distinguished it as a user-oriented device as opposed to Android's application-oriented nature and noted that it would be more likely to appeal to those interested in technology—Linux fans in particular—but was too new to appeal to the public at large. This was due, in part, to the limited selection of phone apps, partially due to the small size of the market, but stated that LibreOffice and GIMP were available. It took him a while to understand what differentiated scopes from applications. He also noted that the platform was free of advertisements.

Ubuntu Touch (also known as Ubuntu Phone) is a mobile version of the Ubuntu operating system, being developed by the UBports community. Written in Qt, it is designed primarily for touchscreen mobile devices such as smartphones and tablet computers, but the original goal of convergence was intended to bring Ubuntu Touch to laptops, desktops, IOT devices, TVs and smart watches for a complete unified user experience.

英语:Ubuntu Touch(又称Ubuntu Phone)是Canonical公司与Ubuntu社群所开发的移动设备操作系统,专为触控萤幕行动设备所设计,例如:智能手机和平板电脑。现时,Ubuntu Touch 因为 Canonical 公司的马克·沙特尔沃思在 2017 年 4 月 5 日宣布该项目因为市场原因停止开发后由一个社区组织 UBports 接手并进行开发。


Plot.

By its eleventh expansion pack, the massively multiplayer online role-playing game (MMORPG) has become a global success, with a user base of millions of players. However, during the release of its twelfth expansion pack:, thirty thousand Japanese gamers who are logged on at the time of the update suddenly find themselves transported into the virtual game world and donning their in-game avatars. In the midst of the event, a socially awkward gamer named Shiroe, along with his friends, Naotsugu and Akatsuki, decide to team up so that they may face this world, which unfortunately has now become their reality, along with the challenges and obstacles ahead of them.

Media.

Light novels.

"Log Horizon" began as a light novel written by Mamare Touno. It initially appeared in installments on the user-generated content site "Shōsetsuka ni Narō" ("So You Want to be a Novelist") starting on April 13, 2010, and subsequently published in book form by Enterbrain since March 2011. Yen Press obtained the rights on its light novel imprint to release the novels in English from 2015.

Manga.

The novel series has received four manga adaptations, all written by Mamare Touno. The first adaptation is illustrated by Motoya Matsu and titled, "Log Horizon Gaiden: Honey Moon Logs". It began serialization on January 27, 2012, and is published by ASCII Media Works in the "Dengeki Daioh" magazine. The second adaptation is illustrated by Kazuhiro Hara and titled "Log Horizon". It began serialization on May 18, 2012, and was published by Enterbrain in the "Famitsu Comic Clear" web magazine. Yen Press obtained this manga for an English release. The third adaptation is illustrated by Koyuki and titled, "Log Horizon: The West Wind Brigade". It began serialization on July 9, 2012, ended on March 9, 2018, and was published by Fujimi Shobo in the "Age Premium" magazine. Another manga illustrated by Sōchū and titled, "Log Horizon Gaiden: Nyanta-honcho Shiawase no Recipe" began serialization on December 21, 2012, ended on March 31, 2018, and was published by Enterbrain in the "Comic B's LOG" magazine an collected in six volumes. A manga "Log Horizon: Kanami, Go! East!" illustrated by Kou was serialized in "Comic B's LOG" from October 1, 2015, to December 1, 2016, and compiled in two volumes.

Anime.

A 25-episode anime adaptation produced by Satelight aired on NHK Educational TV from October 5, 2013 to March 22, 2014. The series was streamed as a simulcast by Crunchyroll in North America and other select parts of the world.  A 25-episode second season produced by Studio Deen aired from October 4, 2014 to March 28, 2015. Both seasons have been licensed by Sentai Filmworks in North America for digital and home video release. For both seasons, the opening theme is "database" by Man with a Mission featuring Takuma. The ending theme for the first season is "Your song*" and for the second season it is "Wonderful Wonder World*", both performed by Yunchi.  A 12-episode third season titled "Log Horizon: Destruction of the Round Table" (Japanese:ログ・ホライズン 円卓崩壊 Hepburn:Rogu Horaizun Entaku Hōkai) has been announced, and it was originally to premiere in October 2020, but it was delayed to January 2021 due to the COVID-19 pandemic. The third season is named after the title of Volume 12 of the web novel series and the official acronym is DORT. The staff and cast will reprise their roles from the second season.

Reception.

Rebecca Silverman of the Anime News Network noted that the series has "its own unique take on what has become a subgenre of fantasy" when compared to the previous "Sword Art Online" and the earlier ".hack" series. With respect to the anime adaptation, Silverman noted that one of the major drawbacks was the artistry. She held the designs of the characters with small regard, calling them "somewhat bland and generic in design, which is a bit of an accomplishment given how many character creation options there apparently are". She used the design of Akatsuki as an example, explaining that the character's beauty has to be constantly reminded to the audience despite the fact that "she's one of the less striking female characters on screen". Another issue Silverman pointed out was the apparent overuse of Naotsugu's perverted nature for comedic relief and finding that for some viewers he may be "line-crossingly obnoxious". Despite these drawbacks, Silverman remarked that the "show shouldn't be dismissed as 'just another ripoff' before giving it a chance" since "it has the potential to expand rather than rehash the basic premise of players trapped in a game".

The novel has received four manga adaptations, with one based on the main story and the other three revolving around characters in the series. An anime television series adaptation animated by Satelight aired on NHK Educational TV between October 5, 2013 and March 22, 2014. A second season by Studio Deen aired between October 4, 2014 and March 28, 2015. A third season by Studio Deen was originally to premiere in October 2020, but it was delayed to January 2021 due to the COVID-19 pandemic.

《记录的地平线》()是橙乃真希所创作,由原和弘负责插画的日本轻小说作品。已改编两期电视动画,第一期于2013年秋季播放,2014年3月22日第一期最后一集放送完后同时宣布第二期于同年秋季播放。2015年《这本WEB小说真厉害!》票选排名第四名。2020年1月22日宣布第三期于同年秋季播放,后因2019冠状病毒病疫情影响,延后至2021年1月播放。


Definitions and concepts.

Let "V" be a vector space over a field F. For instance, suppose "V" is R or C, the standard "n"-dimensional space of column vectors over the real or complex numbers, respectively. In this case, the idea of representation theory is to do abstract algebra concretely by using "n" × "n" matrices of real or complex numbers.  There are three main sorts of algebraic objects for which this can be done: groups, associative algebras and Lie algebras.  This generalizes to any field F and any vector space "V" over F, with linear maps replacing matrices and composition replacing matrix multiplication: there is a group GL("V",F) of automorphisms of "V", an associative algebra End("V") of all endomorphisms of "V", and a corresponding Lie algebra gl("V",F).

Definition.

There are two ways to say what a representation is. The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication. A representation of a group "G" or (associative or Lie) algebra "A" on a vector space "V" is a map with two properties. First, for any "g" in "G" (or "a" in "A"), the map is linear (over F). Second, if we introduce the notation "g" · "v" for formula_3 ("g", "v"), then for any "g", "g" in "G" and "v" in "V": where "e" is the identity element of "G" and "g""g" is the product in "G". The requirement for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation (1) is ignored. Equation (2) is an abstract expression of the associativity of matrix multiplication. This doesn't hold for the matrix commutator and also there is no identity element for the commutator. Hence for Lie algebras, the only requirement is that for any "x", "x" in "A" and "v" in "V": where ["x", "x"] is the Lie bracket, which generalizes the matrix commutator "MN" − "NM".  The second way to define a representation focuses on the map "φ" sending "g" in "G" to a linear map "φ"("g"): "V" → "V", which satisfies and similarly in the other cases. This approach is both more concise and more abstract. From this point of view:

Terminology.

The vector space "V" is called the representation space of "φ" and its dimension (if finite) is called the dimension of the representation (sometimes "degree", as in ). It is also common practice to refer to "V" itself as the representation when the homomorphism "φ" is clear from the context; otherwise the notation ("V","φ") can be used to denote a representation.  When "V" is of finite dimension "n", one can choose a basis for "V" to identify "V" with F, and hence recover a matrix representation with entries in the field F.  An effective or faithful representation is a representation ("V","φ"), for which the homomorphism "φ" is injective.

Equivariant maps and isomorphisms.

If "V" and "W" are vector spaces over F, equipped with representations "φ" and "ψ" of a group "G", then an equivariant map from "V" to "W" is a linear map "α": "V" → "W" such that for all "g" in "G" and "v" in "V". In terms of "φ": "G" → GL("V") and "ψ": "G" → GL("W"), this means for all "g" in "G", that is, the following diagram commutes:  Equivariant maps for representations of an associative or Lie algebra are defined similarly. If "α" is invertible, then it is said to be an isomorphism, in which case "V" and "W" (or, more precisely, "φ" and "ψ") are "isomorphic representations", also phrased as "equivalent representations". An equivariant map is often called an "intertwining map" of representations. Also, in the case of a group, it is on occasion called a -map.  Isomorphic representations are, for practical purposes, "the same"; they provide the same information about the group or algebra being represented. Representation theory therefore seeks to classify representations up to isomorphism.

Subrepresentations, quotients, and irreducible representations.

If ("V","ψ") is a representation of (say) a group "G", and "W" is a linear subspace of "V" that is preserved by the action of "G" in the sense that "g" · "w" ∈ "W" for all "w" ∈ "W" (Serre calls these "W" "stable under G"), then "W" is called a "subrepresentation": by defining "φ"("g") to be the restriction of "ψ"("g") to "W", ("W", "φ") is a representation of "G" and the inclusion of "W" into "V" is an equivariant map. The quotient space "V"/"W" can also be made into a representation of "G".  If "V" has exactly two subrepresentations, namely the trivial subspace {0} and "V" itself, then the representation is said to be "irreducible"; if "V" has a proper nontrivial subrepresentation, the representation is said to be "reducible".  The definition of an irreducible representation implies Schur's lemma: an equivariant map "α": "V" → "W" between irreducible representations is either the zero map or an isomorphism, since its kernel and image are subrepresentations. In particular, when "V" = "W", this shows that the equivariant endomorphisms of "V" form an associative division algebra over the underlying field F. If F is algebraically closed, the only equivariant endomorphisms of an irreducible representation are the scalar multiples of the identity.  Irreducible representations are the building blocks of representation theory: if a representation "V" is not irreducible then it is built from a subrepresentation and a quotient that are both "simpler" in some sense; for instance, if "V" is finite-dimensional, then both the subrepresentation and the quotient have smaller dimension.

Direct sums and indecomposable representations.

If ("V","φ") and ("W","ψ") are representations of (say) a group "G", then the direct sum of "V" and "W" is a representation, in a canonical way, via the equation  The direct sum of two representations carries no more information about the group "G" than the two representations do individually. If a representation is the direct sum of two proper nontrivial subrepresentations, it is said to be decomposable. Otherwise, it is said to be indecomposable.

Complete reducibility.

In favorable circumstances, every finite-dimensional representation is a direct sum of irreducible representations: such representations are said to be semisimple. In this case, it suffices to understand only the irreducible representations. Examples where this "complete reducibility" phenomenon occur include finite and compact groups, and semisimple Lie algebras.  In cases where complete reducibility does not hold, one must understand how indecomposable representations can be built from irreducible representations as extensions of a quotient by a subrepresentation.

Tensor products of representations.

Suppose formula_11 and formula_12 are representations of a group formula_13. Then we can form a representation formula_14 of G acting on the tensor product vector space formula_15 as follows: If formula_17 and formula_18 are representations of a Lie algebra, then the correct formula to use is  In general, the tensor product of irreducible representations is "not" irreducible; the process of decomposing a tensor product as a direct sum of irreducible representations is known as Clebsch–Gordan theory.  In the case of the representation theory of the group SU(2) (or equivalently, of its complexified Lie algebra formula_20), the decomposition is easy to work out. The irreducible representations are labeled by a parameter formula_21 that is a non-negative integer or half integer; the representation then has dimension formula_22. Suppose we take the tensor product of the representation of two representations, with labels formula_23 and formula_24 where we assume formula_25. Then the tensor product decomposes as a direct sum of one copy of each representation with label formula_21, where formula_21 ranges from formula_28 to formula_29 in increments of 1. If, for example, formula_30, then the values of formula_21 that occur are 0, 1, and 2. Thus, the tensor product representation of dimension formula_32 decomposes as a direct sum of a 1-dimensional representation formula_33 a 3-dimensional representation formula_34 and a 5-dimensional representation formula_35.

Branches and topics.

Representation theory is notable for the number of branches it has, and the diversity of the approaches to studying representations of groups and algebras. Although, all the theories have in common the basic concepts discussed already, they differ considerably in detail. The differences are at least 3-fold:

Finite groups.

Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to geometry and crystallography. Representations of finite groups exhibit many of the features of the general theory and point the way to other branches and topics in representation theory.  Over a field of characteristic zero, the representation of a finite group "G" has a number of convenient properties. First, the representations of "G" are semisimple (completely reducible). This is a consequence of Maschke's theorem, which states that any subrepresentation "V" of a "G"-representation "W" has a "G"-invariant complement. One proof is to choose any projection "π" from "W" to "V" and replace it by its average "π" defined by "π" is equivariant, and its kernel is the required complement.  The finite-dimensional "G"-representations can be understood using character theory: the character of a representation "φ": "G" → GL("V") is the class function "χ": "G" → F defined by where formula_38 is the trace. An irreducible representation of "G" is completely determined by its character.  Maschke's theorem holds more generally for fields of positive characteristic "p", such as the finite fields, as long as the prime "p" is coprime to the order of "G". When "p" and |"G"| have a common factor, there are "G"-representations that are not semisimple, which are studied in a subbranch called modular representation theory.  Averaging techniques also show that if F is the real or complex numbers, then any "G"-representation preserves an inner product formula_39 on "V" in the sense that for all "g" in "G" and "v", "w" in "W". Hence any "G"-representation is unitary.  Unitary representations are automatically semisimple, since Maschke's result can be proven by taking the orthogonal complement of a subrepresentation. When studying representations of groups that are not finite, the unitary representations provide a good generalization of the real and complex representations of a finite group.  Results such as Maschke's theorem and the unitary property that rely on averaging can be generalized to more general groups by replacing the average with an integral, provided that a suitable notion of integral can be defined. This can be done for compact topological groups (including compact Lie groups), using Haar measure, and the resulting theory is known as abstract harmonic analysis.  Over arbitrary fields, another class of finite groups that have a good representation theory are the finite groups of Lie type. Important examples are linear algebraic groups over finite fields. The representation theory of linear algebraic groups and Lie groups extends these examples to infinite-dimensional groups, the latter being intimately related to Lie algebra representations. The importance of character theory for finite groups has an analogue in the theory of weights for representations of Lie groups and Lie algebras.  Representations of a finite group "G" are also linked directly to algebra representations via the group algebra F["G"], which is a vector space over F with the elements of "G" as a basis, equipped with the multiplication operation defined by the group operation, linearity, and the requirement that the group operation and scalar multiplication commute.

Modular representations.

Modular representations of a finite group "G" are representations over a field whose characteristic is not coprime to |"G"|, so that Maschke's theorem no longer holds (because |"G"| is not invertible in F and so one cannot divide by it). Nevertheless, Richard Brauer extended much of character theory to modular representations, and this theory played an important role in early progress towards the classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2-subgroups were "too small".  As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.

Unitary representations.

A unitary representation of a group "G" is a linear representation "φ" of "G" on a real or (usually) complex Hilbert space "V" such that "φ"("g") is a unitary operator for every "g" ∈ "G". Such representations have been widely applied in quantum mechanics since the 1920s, thanks in particular to the influence of Hermann Weyl, and this has inspired the development of the theory, most notably through the analysis of representations of the Poincaré group by Eugene Wigner. One of the pioneers in constructing a general theory of unitary representations (for any group "G" rather than just for particular groups useful in applications) was George Mackey, and an extensive theory was developed by Harish-Chandra and others in the 1950s and 1960s.  A major goal is to describe the "unitary dual", the space of irreducible unitary representations of "G". The theory is most well-developed in the case that "G" is a locally compact (Hausdorff) topological group and the representations are strongly continuous. For "G" abelian, the unitary dual is just the space of characters, while for "G" compact, the Peter–Weyl theorem shows that the irreducible unitary representations are finite-dimensional and the unitary dual is discrete. For example, if "G" is the circle group "S", then the characters are given by integers, and the unitary dual is Z.  For non-compact "G", the question of which representations are unitary is a subtle one. Although irreducible unitary representations must be "admissible" (as Harish-Chandra modules) and it is easy to detect which admissible representations have a nondegenerate invariant sesquilinear form, it is hard to determine when this form is positive definite. An effective description of the unitary dual, even for relatively well-behaved groups such as real reductive Lie groups (discussed below), remains an important open problem in representation theory. It has been solved for many particular groups, such as SL(2,R) and the Lorentz group.

Harmonic analysis.

The duality between the circle group "S" and the integers Z, or more generally, between a torus "T" and Z is well known in analysis as the theory of Fourier series, and the Fourier transform similarly expresses the fact that the space of characters on a real vector space is the dual vector space. Thus unitary representation theory and harmonic analysis are intimately related, and abstract harmonic analysis exploits this relationship, by developing the analysis of functions on locally compact topological groups and related spaces.  A major goal is to provide a general form of the Fourier transform and the Plancherel theorem. This is done by constructing a measure on the unitary dual and an isomorphism between the regular representation of "G" on the space L("G") of square integrable functions on "G" and its representation on the space of L functions on the unitary dual. Pontrjagin duality and the Peter–Weyl theorem achieve this for abelian and compact "G" respectively.  Another approach involves considering all unitary representations, not just the irreducible ones. These form a category, and Tannaka–Krein duality provides a way to recover a compact group from its category of unitary representations.  If the group is neither abelian nor compact, no general theory is known with an analogue of the Plancherel theorem or Fourier inversion, although Alexander Grothendieck extended Tannaka–Krein duality to a relationship between linear algebraic groups and tannakian categories.  Harmonic analysis has also been extended from the analysis of functions on a group "G" to functions on homogeneous spaces for "G". The theory is particularly well developed for symmetric spaces and provides a theory of automorphic forms (discussed below).

Lie groups.

A Lie group is a group that is also a smooth manifold. Many classical groups of matrices over the real or complex numbers are Lie groups. Many of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields.  The representation theory of Lie groups can be developed first by considering the compact groups, to which results of compact representation theory apply. This theory can be extended to finite-dimensional representations of semisimple Lie groups using Weyl's unitary trick: each semisimple real Lie group "G" has a complexification, which is a complex Lie group "G", and this complex Lie group has a maximal compact subgroup "K". The finite-dimensional representations of "G" closely correspond to those of "K".  A general Lie group is a semidirect product of a solvable Lie group and a semisimple Lie group (the Levi decomposition). The classification of representations of solvable Lie groups is intractable in general, but often easy in practical cases. Representations of semidirect products can then be analysed by means of general results called "Mackey theory", which is a generalization of the methods used in Wigner's classification of representations of the Poincaré group.

Lie algebras.

A Lie algebra over a field F is a vector space over F equipped with a skew-symmetric bilinear operation called the Lie bracket, which satisfies the Jacobi identity. Lie algebras arise in particular as tangent spaces to Lie groups at the identity element, leading to their interpretation as "infinitesimal symmetries". An important approach to the representation theory of Lie groups is to study the corresponding representation theory of Lie algebras, but representations of Lie algebras also have an intrinsic interest.  Lie algebras, like Lie groups, have a Levi decomposition into semisimple and solvable parts, with the representation theory of solvable Lie algebras being intractable in general. In contrast, the finite-dimensional representations of semisimple Lie algebras are completely understood, after work of Élie Cartan. A representation of a semisimple Lie algebra g is analysed by choosing a Cartan subalgebra, which is essentially a generic maximal subalgebra h of g on which the Lie bracket is zero ("abelian"). The representation of g can be decomposed into weight spaces that are eigenspaces for the action of h and the infinitesimal analogue of characters. The structure of semisimple Lie algebras then reduces the analysis of representations to easily understood combinatorics of the possible weights that can occur.

Infinite-dimensional Lie algebras.

There are many classes of infinite-dimensional Lie algebras whose representations have been studied. Among these, an important class are the Kac–Moody algebras. They are named after Victor Kac and Robert Moody, who independently discovered them. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and share many of their combinatorial properties. This means that they have a class of representations that can be understood in the same way as representations of semisimple Lie algebras.  Affine Lie algebras are a special case of Kac–Moody algebras, which have particular importance in mathematics and theoretical physics, especially conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, Macdonald identities, which is based on the representation theory of affine Kac–Moody algebras.

Lie superalgebras.

Lie superalgebras are generalizations of Lie algebras in which the underlying vector space has a Z-grading, and skew-symmetry and Jacobi identity properties of the Lie bracket are modified by signs. Their representation theory is similar to the representation theory of Lie algebras.

Linear algebraic groups.

Linear algebraic groups (or more generally, affine group schemes) are analogues in algebraic geometry of Lie groups, but over more general fields than just R or C. In particular, over finite fields, they give rise to finite groups of Lie type. Although linear algebraic groups have a classification that is very similar to that of Lie groups, their representation theory is rather different (and much less well understood) and requires different techniques, since the Zariski topology is relatively weak, and techniques from analysis are no longer available.

Invariant theory.

Invariant theory studies actions on algebraic varieties from the point of view of their effect on functions, which form representations of the group. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are "invariant", under the transformations from a given linear group. The modern approach analyses the decomposition of these representations into irreducibles.  Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theories of quadratic forms and determinants. Another subject with strong mutual influence is projective geometry, where invariant theory can be used to organize the subject, and during the 1960s, new life was breathed into the subject by David Mumford in the form of his geometric invariant theory.  The representation theory of semisimple Lie groups has its roots in invariant theory and the strong links between representation theory and algebraic geometry have many parallels in differential geometry, beginning with Felix Klein's Erlangen program and Élie Cartan's connections, which place groups and symmetry at the heart of geometry. Modern developments link representation theory and invariant theory to areas as diverse as holonomy, differential operators and the theory of several complex variables.

Automorphic forms and number theory.

Automorphic forms are a generalization of modular forms to more general analytic functions, perhaps of several complex variables, with similar transformation properties. The generalization involves replacing the modular group PSL (R) and a chosen congruence subgroup by a semisimple Lie group "G" and a discrete subgroup "Γ". Just as modular forms can be viewed as differential forms on a quotient of the upper half space H = PSL (R)/SO(2), automorphic forms can be viewed as differential forms (or similar objects) on "Γ"\"G"/"K", where "K" is (typically) a maximal compact subgroup of "G". Some care is required, however, as the quotient typically has singularities. The quotient of a semisimple Lie group by a compact subgroup is a symmetric space and so the theory of automorphic forms is intimately related to harmonic analysis on symmetric spaces.  Before the development of the general theory, many important special cases were worked out in detail, including the Hilbert modular forms and Siegel modular forms. Important results in the theory include the Selberg trace formula and the realization by Robert Langlands that the Riemann-Roch theorem could be applied to calculate the dimension of the space of automorphic forms. The subsequent notion of "automorphic representation" has proved of great technical value for dealing with the case that "G" is an algebraic group, treated as an adelic algebraic group. As a result, an entire philosophy, the Langlands program has developed around the relation between representation and number theoretic properties of automorphic forms.

Associative algebras.

In one sense, associative algebra representations generalize both representations of groups and Lie algebras. A representation of a group induces a representation of a corresponding group ring or group algebra, while representations of a Lie algebra correspond bijectively to representations of its universal enveloping algebra. However, the representation theory of general associative algebras does not have all of the nice properties of the representation theory of groups and Lie algebras.

Module theory.

When considering representations of an associative algebra, one can forget the underlying field, and simply regard the associative algebra as a ring, and its representations as modules. This approach is surprisingly fruitful: many results in representation theory can be interpreted as special cases of results about modules over a ring.

Hopf algebras and quantum groups.

Hopf algebras provide a way to improve the representation theory of associative algebras, while retaining the representation theory of groups and Lie algebras as special cases. In particular, the tensor product of two representations is a representation, as is the dual vector space.  The Hopf algebras associated to groups have a commutative algebra structure, and so general Hopf algebras are known as quantum groups, although this term is often restricted to certain Hopf algebras arising as deformations of groups or their universal enveloping algebras. The representation theory of quantum groups has added surprising insights to the representation theory of Lie groups and Lie algebras, for instance through the crystal basis of Kashiwara.

Generalizations.

Set-theoretic representations.

A "set-theoretic representation" (also known as a group action or "permutation representation") of a group "G" on a set "X" is given by a function "ρ" from "G" to "X", the set of functions from "X" to "X", such that for all "g", "g" in "G" and all "x" in "X":  This condition and the axioms for a group imply that "ρ"("g") is a bijection (or permutation) for all "g" in "G". Thus we may equivalently define a permutation representation to be a group homomorphism from G to the symmetric group S of "X".

Representations in other categories.

Every group "G" can be viewed as a category with a single object; morphisms in this category are just the elements of "G". Given an arbitrary category "C", a "representation" of "G" in "C" is a functor from "G" to "C". Such a functor selects an object "X" in "C" and a group homomorphism from "G" to Aut("X"), the automorphism group of "X".  In the case where "C" is Vect, the category of vector spaces over a field F, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of "G" in the category of sets.  For another example consider the category of topological spaces, Top. Representations in Top are homomorphisms from "G" to the homeomorphism group of a topological space "X".  Two types of representations closely related to linear representations are:

Representations of categories.

Since groups are categories, one can also consider representation of other categories. The simplest generalization is to monoids, which are categories with one object. Groups are monoids for which every morphism is invertible. General monoids have representations in any category. In the category of sets, these are monoid actions, but monoid representations on vector spaces and other objects can be studied.  More generally, one can relax the assumption that the category being represented has only one object. In full generality, this is simply the theory of functors between categories, and little can be said.  One special case has had a significant impact on representation theory, namely the representation theory of quivers. A quiver is simply a directed graph (with loops and multiple arrows allowed), but it can be made into a category (and also an algebra) by considering paths in the graph. Representations of such categories/algebras have illuminated several aspects of representation theory, for instance by allowing non-semisimple representation theory questions about a group to be reduced in some cases to semisimple representation theory questions about a quiver.

Representation theory is a branch of mathematics that studies abstract algebraic structures by "representing" their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.

表示论是数学中抽象代数的一支。旨在将抽象代数结构中的元素「表示」成向量空间上的线性变换,并研究这些代数结构上的模,借以研究结构的性质。略言之,表示论将一代数对象表作较具体的矩阵,并使得原结构中的代数运算对应到矩阵加法和矩阵乘法。此法可施于群、结合代数及李代数等多种代数结构;其中肇源最早,用途也最广的是群表示论。设formula_1为群,其在域formula_2(常取复数域formula_3)表示是一formula_2-矢量空间formula_5及映至一般线性群之群同态 假设formula_5有限维,则上述同态即是将formula_1的元素映成可逆矩阵,并使得群运算对应到矩阵乘法。