Order of The Companions of Honour

Saturday, 24 November 2012

BIGSTAR Fanclub Name is...? 빅스타의 팬클럽명은..?

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Australia
Australia ( /əˈstreɪljə/), officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent as well as the island of Tasmania and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area. Neighbouring countries include Indonesia, East Timor and Papua New Guinea to the north; the Solomon Islands, Vanuatu and New Caledonia to the north-east; and New Zealand to the south-east.
For at least 40,000 years before European settlement in the late 18th century, Australia was inhabited by indigenous Australians, who belonged to one or more of roughly 250 language groups. After discovery by Dutch explorers in 1606, Australia's eastern half was claimed by Great Britain in 1770 and settled through penal transportation to the colony of New South Wales from 26 January 1788. The population grew steadily in subsequent decades; the continent was explored and an additional five self-governing Crown Colonies were established.

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Canada
Canada ( /ˈkænədə/) is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean. Spanning over 9,900,000 km² (3,800,000 sq mi), Canada is the world's second-largest country by total area, and its common border with the United States is the longest land border in the world.
The land that is now Canada has been inhabited for millennia by various Aboriginal peoples. Beginning in the late 15th century, British and French colonial expeditions explored, and later settled, along the region's Atlantic coast. France ceded nearly all of its colonies in North America in 1763 after the Seven Years' War. In 1867, with the union of three British North American colonies through Confederation, Canada was formed as a federal dominion of four provinces. This began an accretion of provinces and territories and a process of increasing autonomy; the Canada Act 1982 severed the vestiges of legal dependence on Britain.

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India
India (ⁱ/ˈɪndiə/), officially the Republic of India (Bhārat Gaṇarājya), is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world. Bounded by the Indian Ocean on the south, the Arabian Sea on the south-west, and the Bay of Bengal on the south-east, it shares land borders with Pakistan to the west;China, Nepal, and Bhutan to the north-east; and Burma and Bangladesh to the east. In the Indian Ocean, India is in the vicinity of Sri Lanka and the Maldives; in addition, India's Andaman and Nicobar Islands share a maritime border with Thailand and Indonesia.
Home to the ancient Indus Valley Civilisation and a region of historic trade routes and vast empires, the Indian subcontinent was identified with its commercial and cultural wealth for much of its long history. Four of the world's major religions—Hinduism, Buddhism, Jainism, and Sikhism—originated here, whereas Zoroastrianism, Christianity, and Islam arrived in the 1st millennium CE and also helped shape the region's diverse culture. Gradually annexed by and brought under the administration of the British East India Company from the early 18th century and administered directly by the United Kingdom from the mid-19th century, India became an independent nation in 1947 after a struggle for independence that was marked by non-violent resistance led by Mahatma Gandhi.

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Netherlands
The Netherlands (ⁱ/ˈnɛðərləndz/; Dutch: Nederland [ˈneːdə(r)lɑnt] or [ˈneɪ̯də(r)lɑnt] ( listen); West Frisian: Nederlân; Papiamento: Hulanda) is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with some islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders with Belgium, Germany and the United Kingdom. It is a parliamentary democracy organised as a unitary state. The country capital is Amsterdam and the seat of government is The Hague. The Netherlands in its entirety is often referred to as Holland, although North and South Holland are actually only two of its twelve provinces (a case of pars pro toto; see terminology of "the Netherlands").

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New Zealand
New Zealand (Māori: Aotearoa) is an island country located in the southwestern Pacific Ocean. The country geographically comprises two main landmasses ‒ that of the North and South Islands ‒ as well as numerous smaller islands. New Zealand is situated some 1,500 kilometres (900 mi) east of Australia across the Tasman Sea and roughly 1,000 kilometres (600 mi) south of the Pacific island nations of New Caledonia, Fiji, and Tonga. Because of its remoteness, it was one of the last lands to be settled by humans.
During its long isolation, New Zealand developed a distinctive biodiversity of both animal and plant life. Most notable are the large number of unique bird species, many of which became extinct after the arrival of humans and introduced mammals. With a mild maritime climate, the land was mostly covered in forest. The country's varied topography and its sharp mountain peaks owe much to the tectonic uplift of land and volcanic eruptions caused by the Pacific and Indo-Australian Plates clashing beneath the earth's surface.

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Papua New Guinea
Papua New Guinea (PNG; Tok Pisin: Papua Niugini), officially the Independent State of Papua New Guinea, is a country in Oceania, occupying the eastern half of the island of New Guinea and numerous offshore islands (the western portion of the island is a part of the Indonesian provinces of Papua and West Papua). It is located in the southwestern Pacific Ocean, in a region defined since the early 19th century as Melanesia. The capital is Port Moresby.
Papua New Guinea is one of the most culturally diverse countries on Earth. According to recent data, 841 different languages are listed for the country, although 11 of these have no known living speakers. . (A detailed series of language maps of Papua New Guinea may be found at Ethnologue) There may be at least as many traditional societies,out of a population of about 6.2 million. It is also one of the most rural, as only 18% of its people live in urban centres. The country is one of the world's least explored, culturally and geographically, and many undiscovered species of plants and animals are thought to exist in the interior of Papua New Guinea.

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Singapore
Singapore (ⁱ/ˈsɪŋəpɔər/ SI-ngə-pohr), officially the Republic of Singapore, is a Southeast Asian city-state off the southern tip of the Malay Peninsula, 137 kilometres (85 mi) north of the equator. An island country made up of 63 islands, it is separated from Malaysia by the Straits of Johor to its north and from Indonesia's Riau Islands by the Singapore Strait to its south. Singapore is highly urbanised but almost half of the country is covered by greenery. More land is being created for development through land reclamation.

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South Africa
Coordinates: 30°S 25°E / 30°S 25°E / -30; 25
South Africa, officially the Republic of South Africa, is a country located at the southern tip of Africa. It is divided into nine provinces and has 2,798 kilometres (1,739 mi) of coastline. To the north of the country lie the neighbouring territories of Namibia, Botswana and Zimbabwe; to the east are Mozambique and Swaziland; while Lesotho is an enclave surrounded by South African territory.
South Africa is multi-ethnic and has diverse cultures and languages. Eleven official languages are recognised in the constitution. Two of these languages are of European origin: South African English and Afrikaans, a language which originated mainly from Dutch that is spoken by the majority of white and Coloured South Africans. Though English is commonly used in public and commercial life, it is only the fifth most-spoken home language. All ethnic and language groups have political representation in the country's constitutional democracy comprising a parliamentary republic; unlike most parliamentary republics, the positions of head of state and head of government are merged in a parliament-dependent President.

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United Kingdom
The United Kingdom of Great Britain and Northern Ireland (commonly known as the United Kingdom, the UK, or Britain) is a sovereign state located off the north-western coast of continental Europe. The country includes the island of Great Britain, the north-eastern part of the island of Ireland and many smaller islands. Northern Ireland is the only part of the UK that shares a land border with another sovereign state—the Republic of Ireland. Apart from this land border the UK is surrounded by the Atlantic Ocean, the North Sea, the English Channel and the Irish Sea.
The United Kingdom is a unitary state governed under a constitutional monarchy and a parliamentary system, with its seat of government in the capital city of London. It is a country in its own right and consists of four countries: England, Northern Ireland, Scotland and Wales. There are three devolved administrations, each with varying powers, based in Belfast, Edinburgh and Cardiff, the capitals of Northern Ireland, Scotland, and Wales. Associated with the UK, but not constitutionally part of it, are three Crown Dependencies. The United Kingdom has fourteen overseas territories. These are remnants of the British Empire which, at its height in 1922, encompassed almost a quarter of the world's land surface and was the largest empire in history. British influence can still be observed in the language, culture and legal systems of many of its former territories.

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Westminster Abbey
The Collegiate Church of St Peter at Westminster, popularly known as Westminster Abbey, is a large, mainly Gothic church, in the City of Westminster, London, United Kingdom, located just to the west of the Palace of Westminster. It is the traditional place of coronation and burial site for English, later British and later still (and currently) monarchs of the Commonwealth realms. The abbey is a Royal Peculiar and briefly held the status of a cathedral from 1540 to 1550.
Westminster Abbey is a collegiate church governed by the Dean and Chapter of Westminster, as established by Royal charter of Queen Elizabeth I in 1560, which created it as the Collegiate Church of St Peter Westminster and a Royal Peculiar under the personal jurisdiction of the Sovereign. The members of the Chapter are the Dean and four residentiary canons, assisted by the Receiver General and Chapter Clerk. One of the canons is also Rector of St Margaret's Church, Westminster, and often holds also the post of Chaplain to the Speaker of the House of Commons. In addition to the Dean and canons, there are at present two full-time minor canons, one precentor, the other succentor. The office of Priest Vicar was created in the 1970s for those who assist the minor canons. Together with the clergy and Receiver General and Chapter Clerk, various lay officers constitute the college, including the Organist and Master of the Choristers, the Registrar, the Auditor, the Legal Secretary, the Surveyor of the Fabric, the Head Master of the Choir School, the Keeper of the Muniments and the Clerk of the Works, as well as 12 lay vicars, 10 choristers and the High Steward and High Bailiff. There are also 40 Queen's Scholars who are pupils at Westminster School (the School has its own Governing Body). Those who are most directly concerned with liturgical and ceremonial matters are the two minor canons and the organist and Master of the Choristers.

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Alexander Pope
Alexander Pope (21 May 1688 – 30 May 1744) was an 18th-century English poet, best known for his satirical verse and for his translation of Homer. Famous for his use of the heroic couplet, he is the third-most frequently quoted writer in The Oxford Dictionary of Quotations, after Shakespeare and Tennyson.
Pope was born to Alexander Pope Senior (1646–1717), a linen merchant of Plough Court, Lombard Street, London, and his wife Edith (née Turner) (1643–1733), who were both Catholics. Edith's sister Christiana was the wife of the famous miniature painter Samuel Cooper. Pope's education was affected by the recently enacted Test Acts, which upheld the status of the established Church of England and banned Catholics from teaching, attending a university, voting, or holding public office on pain of perpetual imprisonment. Pope was taught to read by his aunt, and went to Twyford School in about 1698/99. He then went to two Catholic schools in London. Such schools, while illegal, were tolerated in some areas.

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Amartya Sen
Amartya Sen, CH (Bengali: অমর্ত্য সেন, translit. Ômorto Shen; born 3 November 1933) is an Indian economist who was awarded the 1998 Nobel Memorial Prize in Economic Sciences for his contributions to welfare economics and social choice theory, and for his interest in the problems of society's poorest members. Sen is best known for his work on the causes of famine, which led to the development of practical solutions for preventing or limiting the effects of real or perceived shortages of food. He helped to create the United Nations Human Development Index. In 2012, he became the first non-American recipient of the National Humanities Medal.
He is currently the Thomas W. Lamont University Professor and Professor of Economics and Philosophy at Harvard University. He is also a senior fellow at the Harvard Society of Fellows, distinguished fellow of All Souls College, Oxford and a Fellow of Trinity College, Cambridge, where he previously served as Master from 1998 to 2004. He is the first Indian and the first Asian academic to head an Oxbridge college.

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Anthony Pawson
Anthony 'Tony' James Pawson, OC OOnt CH FRS FRSC (born October 18, 1952), British-born Canadian scientist whose research has revolutionized the understanding of signal transduction, the molecular mechanisms by which cells respond to external cues, and how they communicate with each other. He identified the phosphotyrosine-binding Src homology 2 (SH2 domain) as the prototypic non-catalytic interaction module. SH2 domains serve as a model for a large family of protein modules that act together to control many aspects of cellular signaling. Since the discovery of SH2 domains, hundreds of different modules have been identified in many proteins.
Born in Maidstone, England, he received a MA in biochemistry from the University of Cambridge. Dr. Pawson received his Ph.D. from King's College London in 1976. From 1976 to 1980 he pursued postdoctoral work at the University of California, Berkeley. From 1981 to 1985, he was Assistant Professor at the University of British Columbia.
Dr. Pawson is a Distinguished Investigator and former Director of Research at the Samuel Lunenfeld Research Institute of Mount Sinai Hospital and Professor in the Department of Molecular Genetics at the University of Toronto both of which he joined in 1985.

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Bernard Haitink
Bernard Johan Herman Haitink, CH, KBE (Dutch pronunciation: [ˈbɛrnɑrd ˈhaːi̯tɪŋk]) (born 4 March 1929) is a Dutch conductor and violinist.
Haitink was born in Amsterdam, the son of Willem Haitink and Anna Haitink. He studied music at the conservatoire in Amsterdam. He played the violin in orchestras before taking courses in conducting under Ferdinand Leitner in 1954 and 1955.
Haitink became second conductor of the Netherlands Radio Union Orchestra in 1955. He took the post of chief conductor of the Netherlands Radio Philharmonic in 1957. His conducting debut with the Concertgebouw Orchestra was on 7 November 1956, substituting for Carlo Maria Giulini. After the sudden death of Eduard van Beinum, Haitink was named first conductor of the Concertgebouw Orchestra on 1 September 1959. He became principal conductor of the Concertgebouw Orchestra in 1961, and shared that position jointly with Eugen Jochum until 1963, when Haitink became sole principal conductor. With the Concertgebouw Orchestra, Haitink made many recordings for the Philips label, and later Decca and EMI Classics, and toured widely with the orchestra.

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Chris Patten
Christopher Francis Patten, Baron Patten of Barnes, CH, PC (born 12 May 1944) is a British Conservative politician and administrator. He is the current chairman of the BBC Trust, and was the last Governor of British Hong Kong.
Patten was Member of Parliament for Bath, eventually rising to a cabinet minister and party chairman. In the latter capacity, he orchestrated the Conservatives' unexpected fourth consecutive electoral victory in 1992, but lost his own seat in the House of Commons. He then accepted the post of Governor and Commander-in-Chief of Hong Kong, and oversaw its handover to the People's Republic of China in July 1997. As Governor and Commander-in-Chief, Patten presided over a steady rise in the living standards of ordinary Hong Kongers while encouraging a significant expansion of Hong Kong's social welfare system.
From 2000 to 2004 he served as one of the United Kingdom's two members of the European Commission. After leaving that post, he returned to the UK and became the Chancellor of the University of Oxford in 2003, and he was made a Life Peer in 2005.

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Colin Davis
Sir Colin Rex Davis, CH, CBE (born 25 September 1927) is an English conductor. His repertoire is broad, but among the composers with whom he is particularly associated are Mozart, Berlioz, Elgar, Sibelius, Stravinsky and Tippett.
He studied as a clarinettist, but was intent on becoming a conductor. After struggles as a freelance conductor from 1949 to 1957, he gained a series of appointments with orchestras including the BBC Scottish Orchestra, the BBC Symphony Orchestra and the Bavarian Radio Symphony Orchestra. He has been associated with the London Symphony Orchestra for over 50 years, including over ten years as its principal conductor. He has also held the musical directorships of Sadler's Wells Opera and the Royal Opera House, where he was principal conductor for over fifteen years. His guest conductorships include the Boston Symphony Orchestra, the New York Philharmonic Orchestra, and the Dresden Staatskapelle, among many others.
As a teacher, Davis holds posts at the Royal Academy of Music, London, and the Carl Maria von Weber High School of Music in Dresden. He made his first gramophone recordings in 1958, and his discography built up in the succeeding five decades is extensive, with a large number of studio recordings for Philips Records and a growing catalogue of live recordings for the London Symphony Orchestra's own label.

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David Attenborough
Sir David Frederick Attenborough ( /ˈætənbərə/) OM, CH, CVO, CBE, FRS, FZS, FSA (born 8 May 1926) is a British broadcaster and naturalist.
His career as the face and voice of natural history programmes has endured for more than 50 years. He is best known for writing and presenting the nine Life series, in conjunction with the BBC Natural History Unit, which collectively form a comprehensive survey of all life on the planet. He is also a former senior manager at the BBC, having served as controller of BBC Two and director of programming for BBC Television in the 1960s and 1970s.
Attenborough is widely considered a national treasure in Britain, although he himself does not care for the term. He is a younger brother of director, producer and actor Richard Attenborough.
Attenborough was born in Isleworth, west London, but grew up in College House on the campus of the University College, Leicester, where his father, Frederick, was principal. He is the middle of three sons (his elder brother, Richard, became an actor and his younger brother, John, an executive at Italian car manufacturer Alfa Romeo). During World War II his parents also adopted two Jewish refugee girls from Europe.

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David Hockney
David Hockney, OM, CH, RA, (born 9 July 1937) is an English painter, draughtsman, printmaker, stage designer and photographer, who is based in Bridlington, Yorkshire and Kensington, London.
An important contributor to the Pop art movement of the 1960s, he is considered one of the most influential British artists of the twentieth century.
Hockney was born in Bradford, England on 9 July 1937 to Laura and Kenneth Hockney and was educated first at Wellington Primary School, then Bradford Grammar School, Bradford College of Art and the Royal College of Art in London, where he met R. B. Kitaj. While he was there Hockney said he felt at home, he took pride and success in his work here. While a student at the Royal College of Art, Hockney was featured in the exhibition Young Contemporaries – alongside Peter Blake – that announced the arrival of British Pop art. He was associated with the movement, but his early works also display expressionist elements, not dissimilar to certain works by Francis Bacon. Sometimes, as in We Two Boys Together Clinging (1961), named after a poem by Walt Whitman, these works make reference to his love for men. From 1963, Hockney was represented by the art dealer John Kasmin. In 1963 Hockney visited New York, making contact with Andy Warhol. A subsequent visit to California, where he lived for many years, inspired Hockney to make a series of paintings of swimming pools in Los Angeles, using the comparatively new acrylic medium and rendered in a highly realistic style using vibrant colours. In 1967, his painting, Peter Getting Out Of Nick's Pool, won the John Moores Painting Prize at the Walker Art Gallery in Liverpool. He made prints, portraits of friends, and stage designs for the Royal Court Theatre, Glyndebourne, La Scala and the Metropolitan Opera in New York City.

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David Owen
David Anthony Llewellyn Owen, Baron Owen CH PC FRCP (born 2 July 1938) is a British politician.
Owen served as British Foreign Secretary from 1977 to 1979, the youngest person in over forty years to hold the post; he co-authored the failed Vance-Owen and Owen-Stoltenberg peace plans offered during the Bosnian War. In 1981, Owen was one of the "Gang of Four" who left the Labour Party to found the Social Democratic Party (SDP). Owen led the SDP from 1983 to 1987, and the continuing SDP from 1988 to 1990. He sits in the House of Lords as a crossbencher.
In the course of his career, Owen has held, and resigned from, a number of senior posts. He first quit as Labour's spokesman on defence in 1972 in protest at the Labour leader Harold Wilson's attitude to the EEC; he left the Labour Shadow cabinet over the same issue later; and over unilateral disarmament in November 1980 when Michael Foot became Labour leader. He resigned from the Labour Party when it rejected one member, one vote in February 1981 and later as Leader of the Social Democratic Party, which he had helped to found, after the party's rank-and-file membership voted to merge with the Liberal Party.

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Doris Lessing
Doris May Lessing CH (née Tayler; born 22 October 1919) is a Zimbabwean-British novelist, poet, playwright, librettist, biographer and short story writer. Her novels include The Grass Is Singing, The Golden Notebook, The Good Terrorist, and five novels collectively known as Canopus in Argos.
Lessing was awarded the 2007 Nobel Prize in Literature. In doing so the Swedish Academy described her as "that epicist of the female experience, who with scepticism, fire and visionary power has subjected a divided civilisation to scrutiny". Lessing was the eleventh woman and the oldest ever person to receive the Nobel Prize in Literature.
In 2001, Lessing was awarded the David Cohen Prize for a lifetime's achievement in British Literature. In 2008, The Times ranked her fifth on a list of "The 50 greatest British writers since 1945".
Lessing was born in Iran, then known as Persia, on 22 October 1919, to Captain Alfred Tayler and Emily Maude Tayler (née McVeagh), who were both English and of British nationality. Her father, who had lost a leg during his service in World War I, met his future wife, a nurse, at the Royal Free Hospital where he was recovering from his amputation. Alfred Tayler and his wife moved to Kermanshah, Iran, in order to take up a job as a clerk for the Imperial Bank of Persia and it was here that Doris was born in 1919. The family then moved to the British colony of Southern Rhodesia (now Zimbabwe) in 1925 to farm maize, among other plants, when her father purchased around one thousand acres of bush. Lessing's mother attempted to lead an Edwardian lifestyle amidst the rough environment, which would have been easy had the family been wealthy; in reality, such a lifestyle was not feasible. The farm failed to deliver any monetary value in return .

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Doug Anthony
John Douglas Anthony, AC, CH (born 31 December 1929), is a former Australian politician. He was leader of the National Party from 1971 to 1984, and Deputy Prime Minister from 1971 to 1972 and again from 1975 to 1983.
Anthony was born in Murwillumbah in northern New South Wales. He was the son of Hubert Lawrence "Larry" Anthony, a well-known Country Party politician. Doug Anthony was educated at The King's School in Sydney and at an agricultural college in Queensland. After graduating he took up dairy-farming near Murwillumbah. In 1957 he married Margot Budd, with whom he had three children: Dougald, Jane and Larry.
In 1957 Larry Anthony sr., who was Postmaster-General in the Liberal-Country Party coalition government led by Robert Menzies, died suddenly, and Doug was elected to succeed his father in the Division of Richmond in the House of Representatives, aged 27. He was appointed Minister for the Interior in 1964. In 1967 he became Minister for Primary Industry. It was obvious that the Country Party leader, John McEwen, was grooming Anthony to succeed him.

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Douglas Hurd
Douglas Richard Hurd, Baron Hurd of Westwell, CH, CBE, PC (born 8 March 1930), is a British Conservative politician and novelist, who served in the governments of Margaret Thatcher and John Major between 1979 and his retirement in 1995.
Born in Marlborough, Wiltshire, Hurd first entered parliament in February 1974, as MP for the Mid Oxfordshire constituency (Witney from 1983). His first government post was as Minister for Europe, from 1979-83 (being the post's inaugural holder), and served in several cabinet posts from 1984 onwards, including Secretary of State for Northern Ireland (1984–85), Home Secretary (1985–89) and Foreign Secretary (1989–95). He stood unsuccessfully for the Conservative Party leadership in 1990, but retired from frontline politics during a cabinet re-shuffle in 1995.
In 1997, Hurd entered the House of Lords. Viewed as one of the Conservative Party's senior elder statesmen, he is a patron of the Tory Reform Group, and remains an active figure in public life.
Douglas Hurd was born in Marlborough, Wiltshire, England, in 1930. His father Anthony Hurd (later Baron Hurd) and grandfather Sir Percy Hurd had both been Members of Parliament. Hurd attended Twyford School and Eton College. He then went on to Cambridge University, where he earned a first class degree in History at Trinity College and served as the President of the Cambridge Union Society.

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Elizabeth II
Elizabeth II (Elizabeth Alexandra Mary; born 21 April 1926) is the constitutional monarch of 16 sovereign states known as the Commonwealth realms, and head of the 54-member Commonwealth of Nations. She is also head of state of the Crown Dependencies.
Elizabeth was born in London, and educated privately at home. Her father acceded to the throne as George VI in 1936 on the abdication of his brother Edward VIII. She began to undertake public duties during the Second World War, in which she served in the Auxiliary Territorial Service. On the death of her father in 1952, she became Head of the Commonwealth and queen regnant of seven independent Commonwealth countries: the United Kingdom, Canada, Australia, New Zealand, South Africa, Pakistan, and Ceylon. Her coronation service in 1953 was the first to be televised. Between 1956 and 1992, the number of her realms varied as territories gained independence and some realms became republics. Today, in addition to the first four aforementioned countries, Elizabeth is Queen of Jamaica, Barbados, the Bahamas, Grenada, Papua New Guinea, the Solomon Islands, Tuvalu, Saint Lucia, Saint Vincent and the Grenadines, Belize, Antigua and Barbuda, and Saint Kitts and Nevis.

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Eric Hobsbawm
Eric John Ernest Hobsbawm, CH, FBA, FRSL (born 9 June 1917) is a British Marxist historian, public intellectual, and author. His best known works include the trilogy about the long 19th century: The Age of Revolution: Europe 1789-1848, The Age of Capital: 1848-1875 and The Age of Empire: 1875-1914; and an edited volume which introduced the influential idea of 'invented traditions'.
Hobsbawm was born in 1917 in Alexandria, Egypt, to Leopold Percy Obstbaum and Nelly Grün, both Jewish, and he grew up in Vienna, Austria and Berlin, Germany. A clerical error at birth altered his surname from Obstbaum to Hobsbawm. Although the family lived in German-speaking countries, his parents spoke to him and his younger sister Nancy in English.
His father died in 1929, and he started working as an au pair and English tutor. He became an orphan at age 14 upon the death of his mother. Subsequently, he and Nancy were adopted by his maternal aunt, Gretl, and paternal uncle, Sidney, who married and had a son named Peter. Hobsbawm was a schoolboy in Berlin when Hitler came to power in 1933; that year the family moved to London where he attended St Marylebone Grammar School.

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Frederick Sanger
Frederick Sanger, OM, CH, CBE, FRS (born 13 August 1918) is an English biochemist and a two-time Nobel laureate in chemistry, the only person to have been so. In 1958 he was awarded a Nobel prize in chemistry "for his work on the structure of proteins, especially that of insulin". In 1980, Walter Gilbert and Sanger shared half of the chemistry prize "for their contributions concerning the determination of base sequences in nucleic acids". The other half was awarded to Paul Berg "for his fundamental studies of the biochemistry of nucleic acids, with particular regard to recombinant-DNA".
He is the fourth (and only living) person to have been awarded two Nobel Prizes, either individually or in tandem with others.
Frederick Sanger was born on 13 August 1918 in Rendcomb, a small village in Gloucestershire, the second son of Frederick Sanger, a general practitioner, and his wife, Cicely Sanger née Crewdson. He was one of three children. His brother, Theodore was only a year older while his sister May (Mary) was five years younger. His father, Frederick Sanger senior, had worked as an Anglican medical missionary in China but returned to England because of ill health. He was 40 in 1916 when he married Cicely who was 4 years younger. Sanger’s father converted to Quakerism soon after his two sons were born and brought up the children as Quakers. Sanger’s mother was the daughter of a wealthy cotton manufacturer and had a Quaker background but Cicely herself was not a Quaker.

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Geoffrey Howe
Richard Edward Geoffrey Howe, Baron Howe of Aberavon, CH, QC, PC (born 20 December 1926) is a former British Conservative politician. He was Margaret Thatcher's longest-serving Cabinet minister, successively holding the posts of Chancellor of the Exchequer, Foreign Secretary, and finally Leader of the House of Commons and Deputy Prime Minister. His resignation on 1 November 1990 is widely considered to have precipitated Thatcher's own downfall three weeks later.
Geoffrey Howe was born in 1926 at Port Talbot in Wales. A pupil of the Bridgend Preparatory School, Bryntirion, he then attended Abberley Hall School, Worcestershire and Winchester College. He then did National Service as a Lieutenant with the Royal Corps of Signals in East Africa, by his own account giving political lectures in Swahili about how Africans should avoid communism and remain loyal to "Bwana Kingy George". Having declined an offer to remain in the army as a captain, he went up to Trinity Hall, Cambridge, where he read Law and was chairman of the Cambridge University Conservative Association, and on the committee of the Cambridge Union Society. He was called to the Bar in 1952 and was made a QC in 1965. He stood as the Conservative Party candidate in Aberavon at the 1955 and 1959 general elections, losing in a very safe Labour Party seat. He became chairman of the Bow Group, an internal Conservative think tank of 'young modernisers' in the 1960s, and edited its magazine Crossbow.

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Howard Hodgkin
Sir Gordon Howard Eliot Hodgkin CH, CBE (born 6 August 1932) is a British painter and printmaker. His work is most often associated with abstraction.
Hodgkin, a cousin of the English still life painter Eliot Hodgkin (1905–87), was educated at Bryanston School in Dorset. He then studied at the Camberwell Art School and later at the Bath Academy of Art in Corsham, where Edward Piper studied drawing under him.
Hodgkin's first solo show was in London in 1962. His early paintings tend to be made up of hard-edged curved forms in a limited number of colours.
Around the beginning of the 1970s, Hodgkin's style became more spontaneous, with vaguely recognisable shapes presented in bright colours and bold forms. His works may then be called "semi-abstract", and are often compared to the paintings of Henri Matisse.^[who?]
In 1984, Hodgkin represented Britain at the Venice Biennale, in 1985 he won the Turner Prize, and in 1992 he was knighted.
In 1995, Hodgkin printed the Venetian Views series, which depict the same view of Venice at four different times of day. Venice, Afternoon - one of the four prints - uses sixteen sheets, or fragments, in a hugely complex printing process which creates a colourful, painterly effect. This piece was given to the Yale Centre of British Art in June 2006 by the Israel family to complement their already-impressive collection of Hodgkins.

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Ian McKellen
Sir Ian Murray McKellen, CH CBE (born 25 May 1939) is an English actor. He has won multiple Laurence Olivier Awards, a Tony Award, two Academy Award nominations, and five Emmy Award nominations. His work has spanned genres from Shakespearean and modern theatre to popular fantasy and science fiction. He is known for film roles such as Gandalf in the Lord of the Rings film trilogy, Magneto in the X-Men films, and as Sir Leigh Teabing in The Da Vinci Code.
McKellen was made a Commander of the Order of the British Empire in 1979, was knighted in 1991 for services to the performing arts, and was made a Companion of Honour for services to drama and to equality, in the 2008 New Year Honours.
McKellen was born in Burnley, Lancashire, England, though he spent most of his early life in Wigan. Born shortly before the outbreak of World War II, the experience had some lasting impact on him. In response to an interview question when an interviewer remarked that he seemed quite calm in the aftermath of the 11 September attacks, he said: "Well, darling, you forget — I slept under a steel plate until I was four years old."

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James Lovelock
James Ephraim Lovelock, CH, CBE, FRS, Ph.D (born 26 July 1919) is an independent scientist, environmentalist and futurologist who lives in Devon, England. He is best known for proposing the Gaia hypothesis, which postulates that the biosphere is a self-regulating entity with the capacity to keep our planet healthy by controlling the chemical and physical environment.

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Janet Baker
Dame Janet Abbott Baker, CH, DBE, FRSA (born 21 August 1933) is an English mezzo-soprano best known as an opera, concert, and lieder singer.
She was particularly closely associated with baroque and early Italian opera and the works of Benjamin Britten. During her career, which spanned the 1950s to the 1980s, she was considered an outstanding singing actress and widely admired for her dramatic intensity, perhaps best represented in her famous portrayal as Dido, the tragic heroine of Berlioz's magnum opus Les Troyens. As a concert performer, Baker was noted for her interpretations of the music of Gustav Mahler and Edward Elgar. David Gutman, writing in the Gramophone, described her performance of Mahler's Kindertotenlieder as "intimate, almost self-communing."
Baker was born in Hatfield, South Yorkshire, northern England. Her father was an engineer who sang in a male voice choir. Members of her family worked at Bentley pit, in Doncaster, South Yorkshire.^{[citation needed]} She attended York College for Girls and then Wintringham Girls' Grammar School in Grimsby. The death of her elder brother, Peter, when she was 10 years old, from a heart condition, was a formative moment that made her take responsibility for the rest of her life, she revealed in a BBC Radio 3 Lebrecht Interview in September 2011.

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John de Chastelain
Alfred John Gardyne Drummond de Chastelain OC CMM CD CH (born July 30, 1937) is a retired Canadian soldier and diplomat.
De Chastelain was born in Romania and educated in England and in Scotland before his family immigrated to Canada in 1954. There, de Chastelain enrolled in military college and went on to pursue a career in the Canadian Army, rising through the ranks to eventually be twice appointed Chief of the Defence Staff, with a year serving as Canada's ambassador to the United States in between. In 1995, de Chastelain became involved in the Northern Ireland peace process and eventually focused on the disarmament of paramilitary groups in Northern Ireland.
De Chastelain was born in 1937 in Bucharest, Romania, to Alfred Gardyne de Chastelain, a Scottish oil engineer then working in Bucharest for British Petroleum, (and later an agent of SOE) and Marion Elizabeth de Chastelain, an American. He was educated in England and later at Fettes College, Edinburgh. The de Chastelain family emigrated to Canada in 1954 and a year later, after finishing his education at Fettes College, De Chastelain joined his family and attended Mount Royal College (now Mount Royal University) in Calgary.

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John Major
Sir John Major, KG, CH, PC, ACIB (born 29 March 1943) is a British Conservative politician, who served as Prime Minister of the United Kingdom and Leader of the Conservative Party from 1990 to 1997. He held the posts of Foreign Secretary and Chancellor of the Exchequer in the Cabinet of Margaret Thatcher and was the Member of Parliament (MP) for Huntingdon from 1979 to 2001.
Despite Thatcher's "notorious" assertion that "she expected to continue in control as a backseat driver," Major's mild and consensual style was seen as complete contrast to Thatcher's forceful and confrontational manner. Early in his term, he presided over British participation in the First Gulf War (March 1991) and negotiated "Game, Set and Match for Britain" at the Maastricht Treaty (December 1991). Despite the British economy then being in recession he led the Conservatives to a fourth consecutive election victory, winning the most votes in British electoral history in the 1992 general election, albeit with a much reduced majority in the House of Commons. He is to date, the last Conservative leader to win an outright majority in a general election.

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Judi Dench
Dame Judith Olivia “Judi” Dench, CH, DBE, FRSA (born 9 December 1934) is an English film, stage and television actress.
Dench made her professional debut in 1957 with the Old Vic Company. Over the following few years she played in several of William Shakespeare's plays in such roles as Ophelia in Hamlet, Juliet in Romeo and Juliet and Lady Macbeth in Macbeth. She branched into film work, and won a BAFTA Award as Most Promising Newcomer; however, most of her work during this period was in theatre. Not generally known as a singer, she drew strong reviews for her leading role in the musical Cabaret in 1968.
During the next two decades, she established herself as one of the most significant British theatre performers, working for the National Theatre Company and the Royal Shakespeare Company. In television, she achieved success during this period, in the series A Fine Romance from 1981 until 1984 and in 1992 began a continuing role in the television romantic comedy series As Time Goes By.
Her film appearances were infrequent until she was cast as M in GoldenEye (1995), a role she has played in each James Bond film since. She received several notable film awards for her role as Queen Victoria in Mrs. Brown (1997), and has since been acclaimed for her work in such films as Shakespeare in Love (1998), Chocolat (2000), Iris (2001), Mrs Henderson Presents (2005) and Notes on a Scandal (2006), and the television production The Last of the Blonde Bombshells (2001).

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Lee Kuan Yew
Lee Kuan Yew, GCMG CH (born 16 September 1923) is a Singaporean statesman. He was the first Prime Minister of the Republic of Singapore, governing for three decades. By the time he stepped down, he had become the world's longest-serving Prime Minister.
As the co-founder and first secretary-general of the People's Action Party (PAP), he led the party to eight victories from 1959 to 1990, and oversaw the separation of Singapore from Malaysia in 1965 and its subsequent transformation from a relatively underdeveloped colonial outpost with no natural resources into a "First World" Asian Tiger. He is one of the most influential political figures in South-East Asia.
Singapore's second prime minister, Goh Chok Tong, appointed him as Senior Minister in 1990. He held the advisory post of Minister Mentor, created by his son, Lee Hsien Loong, when the latter became the nation's third prime minister in August 2004. With his successive ministerial positions spanning over 50 years, Lee is also one of history's longest-serving ministers. On 14 May 2011, Lee and Senior Minister Goh Chok Tong announced their retirement from the cabinet after the 2011 General Election.

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Malcolm Fraser
John Malcolm Fraser AC, CH, GCL, PC ( /ˈfreɪzə/; born 21 May 1930) is a former Australian Liberal Party politician who was the 22nd Prime Minister of Australia. He came to power in 1975 through the dismissal of the Whitlam Labor government, in which he played a key role. After three election victories, he was defeated by Bob Hawke in the 1983 election and ended his career alienated from his own party.
Malcolm Fraser was born in Toorak to a family with a history of involvement in politics and the pastoral/grazing industry. His grandfather, Simon Fraser, emigrated from Nova Scotia in 1853, becoming a successful pastoralist and speculator, as well as a member of the Victorian Parliament, the Federation Conventions of 1897–98 and the Australian Senate. Malcolm Fraser's father, John Neville Fraser, was a pastoralist at Deniliquin in the Riverina region of New South Wales and later at a property called "Nareen", near Hamilton in the Western District of Victoria. Malcolm Fraser's mother, Una Woolf, was of Jewish descent on her own father's side.

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Michael Heseltine
Michael Ray Dibdin Heseltine, Baron Heseltine, CH, PC (born 21 March 1933) is a British businessman, Conservative politician and patron of the Tory Reform Group. He was a Member of Parliament from 1966 to 2001 and was a prominent figure in the governments of Margaret Thatcher and John Major. In 1990, he stood for leadership of the Conservative Party against Margaret Thatcher, and whilst he was unsuccessful, this triggered Thatcher's eventual resignation.
A self-made millionaire, Heseltine entered parliament in 1966, entered the Cabinet in 1979 as Secretary of State for the Environment, and was Secretary of State for Defence from 1983 to 1986. In the latter role, he was instrumental in the political battle against the Campaign for Nuclear Disarmament. Heseltine was widely considered an adept media performer and charismatic Minister, although frequently at odds with Prime Minister Margaret Thatcher. He resigned from the Cabinet in 1986 over the Westland Affair and returned to the backbenches. Following Geoffrey Howe's resignation speech in November 1990, Heseltine challenged the incumbent Margaret Thatcher for the leadership of the Conservative Party, polling well enough to deny her an outright victory on the first ballot. Following her resignation in a dramatic Tory "palace coup", Heseltine returned to the Cabinet under her successor John Major.

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Michael Somare
The Rt. Hon. Sir Michael Thomas Somare, GCL, GCMG, CH, CF, KStJ, PC, MP (born 9 April 1936), was Prime Minister of Papua New Guinea from 2002 to 2011; he had previously been Prime Minister from independence in 1975 until 1980 and again from 1982 until 1985. Somare's first two terms were as a member of the Pangu Party, but he then became leader of the National Alliance Party. He was re-elected as Prime Minister in the 2007 election, and his family announced his resignation due to ill health in June 2011.
On 12 December 2011, the Supreme Court of Papua New Guinea ordered that Somare be reinstated as Prime Minister, ruling that his successor, Peter O'Neill, was not lawfully appointed.
Somare is the son of Ludwig Somare Sana and Kambe Somare. Ludwig Somare was a policeman from 1922 to 1947, rising to the rank of sergeant. Having taught himself to read and write, he was subsequently active in encouraging formation of small businesses and co-operatives, founding the Angoram Co-operative Society which he chaired from 1961 until 1967, and remaining active in trading until his death in 1972. In all, Ludwig Somare Sana had four wives and six children, of whom Somare was the eldest.

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Norman Tebbit
Norman Beresford Tebbit, Baron Tebbit, CH, PC (born 29 March 1931), is a British politician. A member of the Conservative Party, he served in the Cabinet from 1981 to 1987 as Secretary of State for Employment (1981–83), Secretary of State for Trade and Industry (1983–85), Chancellor of the Duchy of Lancaster (1985–87) and Chairman of the Conservative Party (1985–87). He was a Member of Parliament (MP) from 1970 to 1992, representing the constituencies of Epping (1970–74) and Chingford (1974–92). In 1984, he was injured in the Provisional Irish Republican Army's bombing of the Conservative Party Conference in Brighton, and his wife Margaret was left permanently disabled.
Born into a working class family in Ponders End, Middlesex, Tebbit went to Edmonton County School, an academically selective state school in north London. He was then a journalist for the Financial Times before serving with the Royal Air Force, during which he flew Meteor and Vampire jets and once had to break open the cockpit canopy of a burning Mosquito aircraft to escape from it. On leaving the RAF he joined BOAC in 1953 as a pilot, during which time he was an official in the British Air Line Pilots' Association. He was elected MP for Epping in 1970 and then for Chingford in Feb 1974. He is recorded as an MP member of the Conservative Monday Club in 1970.

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Peter Brook
Peter Stephen Paul Brook CH, CBE (born 21 March 1925) is an English theatre and film director and innovator, who has been based in France since the early 1970s.
Brook was born in London in March 1925, the son of Simon Brook and his wife Ida (Jansen), two Jewish immigrants. He was educated at Gresham's School and Magdalen College, Oxford.
He directed Dr Faustus, his first production, in 1943 at the Torch Theatre in London, followed at the Chanticleer Theatre in 1945 with a revival of The Infernal Machine. In 1947, he went to Stratford-upon-Avon as assistant director on Romeo and Juliet and Love's Labour's Lost. From 1947 to 1950, he was Director of Productions at the Royal Opera House, Covent Garden. His work there included a highly controversial staging of Strauss’ Salome with sets by Salvador Dalí and also an effective re-staging of Puccini’s La Boheme using sets dating from 1899. A proliferation of stage and screen work as producer and director followed.
In 1951, Brook married the actress Natasha Parry; the couple have a daughter.

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Peter Brooke
Peter Leonard Brooke, Baron Brooke of Sutton Mandeville, CH, PC (born 3 March 1934), is a British politician. A member of the Conservative Party, he served in the Cabinet under Prime Ministers Margaret Thatcher and John Major, and was a Member of Parliament (MP) representing the Cities of London and Westminster from 1977-2001.
He is the son of Henry Brooke, Baron Brooke of Cumnor, a former Home Secretary, and Barbara Brooke, Baroness Brooke of Ystradfellte. (His parents were one of the few married couples where both partners held noble titles in their own right.) He was educated at Marlborough College and Balliol College, Oxford, (where he was President of the Oxford Union) before going on to the Harvard Business School in the United States. After leaving university he worked as a headhunter and was Chairman of Spencer Stuart.
He was elected as MP for the Cities of London and Westminster in 1977. He was sworn of the Privy Council in 1988. He was made Chairman of the Conservative Party in 1987, and then Secretary of State for Northern Ireland in 1989. He remained in this position until the reshuffle after the 1992 general election. After the 1992 general election, Brooke left the Cabinet and stood unsuccessfully for the position of Speaker of the House of Commons. (The House instead elected the Labour MP Betty Boothroyd to the role, with several Conservative MPs voting against Brooke on the grounds that he had too recently been in the Cabinet and was thus not sufficiently close to the backbenches.) Brooke then remained on the backbenches for a short time before being brought back into the Cabinet later in the year as Secretary of State for National Heritage, a role he held until 1994. During his time as Heritage Secretary, he oversaw the restoration of Windsor Castle following the fire that had struck the state apartments in 1992.

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Richard Rogers
Richard George Rogers, Baron Rogers of Riverside CH Kt FRIBA FCSD (born 23 July 1933) is a British architect noted for his modernist and functionalist designs.
Rogers is perhaps best known for his work on the Pompidou Centre in Paris, the Lloyd's building and Millennium Dome both in London, and the European Court of Human Rights building in Strasbourg. He is a winner of the RIBA Gold Medal, the Thomas Jefferson Medal, the RIBA Stirling Prize, the Minerva Medal and Pritzker Prize.
Rogers was born in Florence in 1933 and attended the Architectural Association School of Architecture in London, before graduating with a master's degree from the Yale School of Architecture in 1962. While studying at Yale, Rogers met fellow architecture student Norman Foster and planning student Su Brumwell. On returning to England he, Foster and Brumwell set up architectural practice as Team 4 with Wendy Cheeseman (Brumwell later married Rogers, Cheeseman married Foster). Rogers and Foster earned a reputation for what was later termed by the media high-tech architecture.

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Stephen Hawking
Stephen William Hawking, CH, CBE, FRS, FRSA (born 8 January 1942) is a British theoretical physicist, cosmologist, and author. His key scientific works to date have included providing, with Roger Penrose, theorems regarding gravitational singularities in the framework of general relativity, and the theoretical prediction that black holes should emit radiation, which is today known as Hawking radiation (or sometimes as Bekenstein–Hawking radiation).
He is an Honorary Fellow of the Royal Society of Arts, a lifetime member of the Pontifical Academy of Sciences, and in 2009 was awarded the Presidential Medal of Freedom, the highest civilian award in the United States. Hawking was the Lucasian Professor of Mathematics at the University of Cambridge between 1979 and 2009. Subsequently, he became research director at the university's Centre for Theoretical Cosmology.
Hawking has a motor neurone disease related to amyotrophic lateral sclerosis, a condition that has progressed over the years. He is now almost completely paralysed and communicates through a speech generating device. He has been married twice and has three children. Hawking has achieved success with works of popular science in which he discusses his own theories and cosmology in general; these include A Brief History of Time, which stayed on the British Sunday Times best-sellers list for a record-breaking 237 weeks.

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Sydney Brenner
Sydney Brenner, CH FRS (born 13 January 1927) is a South African biologist and a 2002 Nobel prize in Physiology or Medicine laureate, shared with H. Robert Horvitz and John Sulston.
Brenner made significant contributions to work on the genetic code, and other areas of molecular biology.
He established the roundworm Caenorhabditis elegans as a model organism for the investigation of developmental biology, and founded the Molecular Sciences Institute in Berkeley, California, U.S..
Brenner was born in the small town of Germiston, South Africa. His parents were Jewish immigrants. His father came to South Africa from Lithuania in 1910, and his mother, from Riga, Latvia, in 1922. Educated at Germiston High School and the University of the Witwatersrand, he went on to complete a D.Phil. from Exeter College, Oxford.
He was married to Dr. May Brenner (née Covitz, subsequently Balkind) from December 1952 until her death in January 2010; their children include Belinda, Carla, Stefan, and his stepson Jonathan Balkind from his wife's first marriage. He lives in Ely, Cambridgeshire.

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Order of the Companions of Honour

Riband and badge of the "Companions of Honour"
Awarded by the sovereign of the Commonwealth realms
Type	Order
Motto	IN ACTION FAITHFUL AND IN HONOUR CLEAR
Awarded for	Nationally important service
Status	Currently constituted
Sovereign	Queen Elizabeth II
Grades (w/ post-nominals)	Companion (CH)
Established	1917

Ribbon bar of the order

Lord Tweedsmuir, as Governor General of Canada, wearing the Companion of Honour badge around his neck as part of his ceremonial Windsor uniform

The Order of the Companions of Honour is an order of the Commonwealth realms. It was founded by King George V in June 1917, as a reward for outstanding achievements in the arts, literature, music, science, politics, industry or religion. It is sometimes regarded as the junior order to the Order of Merit.^[1]

The order consists of the sovereign, plus no more than 65 Companions of Honour with, originally, a quota of 45 members for the United Kingdom, 7 for Australia, 2 for New Zealand and 11 for other Commonwealth realms. It has not been awarded in Australia since 1992 (the last recipient being in 1981), where it has been displaced by the Order of Australia. Additionally, foreigners from outside the realm may be added as "honorary members". Membership of the order confers no title, or precedence, but recipients of this one-class order are entitled to use the post-nominal letters "CH".

The insignia of the Order consists of an oval medallion with an oak tree, a shield with the royal arms hanging from one branch, and on the left a mounted knight in armour. The badge's clear blue border bears the motto IN ACTION FAITHFUL AND IN HONOUR CLEAR (Alexander Pope's description (in iambic pentameter) in his Epistle to Mr Addison of James Craggs, later used on Craggs' monument in Westminster Abbey) in gold letters, and the oval is surmounted by an imperial crown. Men wear the badge on a ribbon (red with golden border threads) around their necks, and women on a bow at the left shoulder.

Current members [link]

Sovereign
- Elizabeth II
Members
1. Malcolm Fraser AC CH (1976)
2. Sir Michael Somare GCL GCMG CH CF (1978)
3. The Lord Healey CH MBE PC (1979)
4. Sir Brian Talboys AC CH KCB (1981)
5. Doug Anthony AC CH (1981)
6. Frederick Sanger OM CH CBE FRS (1981)
7. The Lord Carrington KG GCMG CH MC PC DL (1983)
8. The Lord Tebbit CH PC (1987)
9. Stephen Hawking CH CBE FRS FRSA (1989)
10. The Lord Baker of Dorking CH PC (1992)
11. The Lord Brooke of Sutton Mandeville CH PC (1992)
12. The Lord King of Bridgwater CH PC (1992)
13. Dame Janet Baker CH DBE (1993)
14. The Lord Owen CH PC (1994)
15. The Lord Hurd of Westwell CH CBE PC (1995)
16. Sir David Attenborough OM CH CVO CBE FRS FZS (1996)
17. The Lord Howe of Aberavon CH PC QC (1996)
18. The Lord Heseltine CH PC (1997)
19. David Hockney OM CH RA (1997)
20. Peter Brook CH CBE (1998)
21. Eric Hobsbawm CH FBA (1998)
22. Sir John Major KG CH PC (1998)
23. The Lord Patten of Barnes CH PC (1998)
24. Bridget Riley CH CBE (1998)
25. John de Chastelain CH OC CMM CD (1999)
26. Doris Lessing CH OBE (2000)
27. Sir Harrison Birtwistle CH (2001)
28. Sir Colin Davis CH CBE (2001)
29. Sir George Christie CH (2001)
30. Sir Michael Howard OM CH CBE MC FBA (2002)
31. The Lord Hannay of Chiswick GCMG CH (2003)
32. Sir Howard Hodgkin CH CBE (2003)
33. James Lovelock CH CBE FRS (2003)
34. Dan McKenzie CH FRS (2003)
35. Dame Judi Dench CH DBE FRSA (2005)
36. Anthony Pawson CH OC OOnt FRS FRSC (2006)
37. Sir Ian McKellen CH CBE (2008)
38. The Lord Rogers of Riverside CH (2008)
39. The Lord Howard of Lympne CH PC QC (2011)
Honorary Members
1. Lee Kuan Yew GCMG CH (1970)
2. Sydney Brenner CH FRS (1987)
3. Amartya Sen CH (2000)
4. Bernard Haitink CH KBE (2002)

References [link]

^ "The Monarchy Today > Queen and public > Honours > Companions of Honour". Royal.gov.uk. http://www.royal.gov.uk/MonarchUK/Honours/CompanionsofHonour.aspx. Retrieved 2011-12-03.

v t e Orders and Honours of the United Kingdom

Current orders	Garter Thistle Bath Merit St Michael and St George Royal Victorian Distinguished Service British Empire Crown of India x10px Imperial Service Companions of Honour King George V King George VI Queen Elizabeth II St. John

Noncurrent orders	St. Patrick Royal Guelphic Star of India Indian Empire Indian Merit British India Burma King George IV Victoria and Albert King Edward VII

Other honours	Royal Victorian Chain Hereditary peerage Life peerage Privy Counsellor x10px Baronet Knight Bachelor Hereditary knighthood ADC more

Revocations Orders, decorations and medals Campaign medals

v t e Orders, decorations and medals of Canada

Awards of valour	Victoria Cross Cross of Valour

National orders	Order of Merit Order of Canada Order of Military Merit Order of Merit of the Police Forces Royal Victorian Order Most Venerable Order of the Hospital of St. John of Jerusalem

Provincial orders	Orders, decorations, and medals of the Canadian provinces

Other decorations and medals	Decorations War and operational service medals Special service medals United Nations medals NATO medals International mission medals Commemorative medals Long service and good conduct medals Exemplary service medals Special Medals

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This page contains text from Wikipedia, the Free Encyclopedia - http://en.wikipedia.org/wiki/Order_of_the_Companions_of_Honour

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Example of Big O notation: f(x) ∈ O(g(x)) as there exists c > 0 (e.g. c = 1) and x₀ (e.g. x₀ = 5) such that f(x) < cg(x) whenever x > x₀.

In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann–Landau notation (after Edmund Landau and Paul Bachmann), or asymptotic notation. In computer science, big O notation is used to classify algorithms by how they respond (e.g., in their processing time or working space requirements) to changes in input size.

Big O notation characterizes functions according to their growth rates: different functions with the same growth rate may be represented using the same O notation. A description of a function in terms of big O notation usually only provides an upper bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates.

Big O notation is also used in many other fields to provide similar estimates.

Formal definition [link]

Let f(x) and g(x) be two functions defined on some subset of the real numbers. One writes

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=O(g(x))\text{ as }x\to\infty\,

if and only if there is a positive constant M such that for all sufficiently large values of x, f(x) is at most M multiplied by g(x) in absolute value. That is, f(x) = O(g(x)) if and only if there exists a positive real number M and a real number x₀ such that

Failed to parse (Missing texvc executable; please see math/README to configure.): |f(x)| \le \; M |g(x)|\text{ for all }x>x_0.

In many contexts, the assumption that we are interested in the growth rate as the variable x goes to infinity is left unstated, and one writes more simply that f(x) = O(g(x)). The notation can also be used to describe the behavior of f near some real number a (often, a = 0): we say

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=O(g(x))\text{ as }x\to a\,

if and only if there exist positive numbers δ and M such that

Failed to parse (Missing texvc executable; please see math/README to configure.): |f(x)| \le \; M |g(x)|\text{ for }|x - a| < \delta.

If g(x) is non-zero for values of x sufficiently close to a, both of these definitions can be unified using the limit superior:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=O(g(x))\text{ as }x \to a\,

if and only if

Failed to parse (Missing texvc executable; please see math/README to configure.): \limsup_{x\to a} \left|\frac{f(x)}{g(x)}\right| < \infty.

Example [link]

In typical usage, the formal definition of O notation is not used directly; rather, the O notation for a function f(x) is derived by the following simplification rules:

If f(x) is a sum of several terms, the one with the largest growth rate is kept, and all others omitted.
If f(x) is a product of several factors, any constants (terms in the product that do not depend on x) are omitted.

For example, let Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = 6x^4 - 2x^3 +5 , and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms: 6x⁴, −2x³, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x⁴. Now one may apply the second rule: 6x⁴ is a product of 6 and x⁴ in which the first factor does not depend on x. Omitting this factor results in the simplified form x⁴. Thus, we say that f(x) is a big-oh of (x⁴) or mathematically we can write f(x) = O(x⁴). One may confirm this calculation using the formal definition: let f(x) = 6x⁴ − 2x³ + 5 and g(x) = x⁴. Applying the formal definition from above, the statement that f(x) = O(x⁴) is equivalent to its expansion,

Failed to parse (Missing texvc executable; please see math/README to configure.): |f(x)| \le \; M |g(x)|

for some suitable choice of x₀ and M and for all x > x₀. To prove this, let x₀ = 1 and M = 13. Then, for all x > x₀:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align}|6x^4 - 2x^3 + 5| &\le 6x^4 + |2x^3| + 5\\ &\le 6x^4 + 2x^4 + 5x^4\\ &\le 13x^4\\ &\le 13|x^4|\end{align}

Failed to parse (Missing texvc executable; please see math/README to configure.): |6x^4 - 2x^3 + 5| \le 13 \,|x^4 |.

Usage [link]

Big O notation has two main areas of application. In mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion. In computer science, it is useful in the analysis of algorithms. In both applications, the function g(x) appearing within the O(...) is typically chosen to be as simple as possible, omitting constant factors and lower order terms. There are two formally close, but noticeably different, usages of this notation: infinite asymptotics and infinitesimal asymptotics. This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument.

Infinite asymptotics [link]

Big O notation is useful when analyzing algorithms for efficiency. For example, the time (or the number of steps) it takes to complete a problem of size n might be found to be T(n) = 4n² − 2n + 2. As n grows large, the n² term will come to dominate, so that all other terms can be neglected — for instance when n = 500, the term 4n² is 1000 times as large as the 2n term. Ignoring the latter would have negligible effect on the expression's value for most purposes. Further, the coefficients become irrelevant if we compare to any other order of expression, such as an expression containing a term n³ or n⁴. Even if T(n) = 1,000,000n², if U(n) = n³, the latter will always exceed the former once n grows larger than 1,000,000 (T(1,000,000) = 1,000,000³= U(1,000,000)). Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm. So the big O notation captures what remains: we write either

Failed to parse (Missing texvc executable; please see math/README to configure.): \ T(n)= O(n^2) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): T(n)\in O(n^2) \,

and say that the algorithm has order of n² time complexity. Note that "=" is not meant to express "is equal to" in its normal mathematical sense, but rather a more colloquial "is", so the second expression is technically accurate (see the "Equals sign" discussion below) while the first is a common abuse of notation.^[1]

Infinitesimal asymptotics [link]

Big O can also be used to describe the error term in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term. For example,

Failed to parse (Missing texvc executable; please see math/README to configure.): e^x=1+x+\frac{x^2}{2}+O(x^3)\qquad\text{as } x\to 0

expresses the fact that the error, the difference Failed to parse (Missing texvc executable; please see math/README to configure.): \ e^x - (1 + x + x^2/2) , is smaller in absolute value than some constant times Failed to parse (Missing texvc executable; please see math/README to configure.): |x^3|

when Failed to parse (Missing texvc executable; please see math/README to configure.): x
is close enough to 0.

Properties [link]

If a function f(n) can be written as a finite sum of other functions, then the fastest growing one determines the order of f(n). For example

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) = 9 \log n + 5 (\log n)^3 + 3n^2 + 2n^3 = O(n^3)\,\!.

In particular, if a function may be bounded by a polynomial in n, then as n tends to infinity, one may disregard lower-order terms of the polynomial. O(n^c) and O(cⁿ) are very different. If c is greater than one, then the later grows much faster. A function that grows faster than n^c for any c is called superpolynomial. One that grows more slowly than any exponential function of the form Failed to parse (Missing texvc executable; please see math/README to configure.): c^n

is called subexponential. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest known algorithms for integer factorization.

O(log n) is exactly the same as O(log(n^c)). The logarithms differ only by a constant factor (since Failed to parse (Missing texvc executable; please see math/README to configure.): \log(n^c)=c \log n ) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. Exponentials with different bases, on the other hand, are not of the same order. For example, Failed to parse (Missing texvc executable; please see math/README to configure.): 2^n

and Failed to parse (Missing texvc executable; please see math/README to configure.): 3^n
are not of the same order.

Changing units may or may not affect the order of the resulting algorithm. Changing units is equivalent to multiplying the appropriate variable by a constant wherever it appears. For example, if an algorithm runs in the order of n², replacing n by cn means the algorithm runs in the order of Failed to parse (Missing texvc executable; please see math/README to configure.): c^2n^2 , and the big O notation ignores the constant Failed to parse (Missing texvc executable; please see math/README to configure.): c^2 . This can be written as Failed to parse (Missing texvc executable; please see math/README to configure.): c^2n^2 \in O(n^2) . If, however, an algorithm runs in the order of Failed to parse (Missing texvc executable; please see math/README to configure.): 2^n , replacing n with cn gives Failed to parse (Missing texvc executable; please see math/README to configure.): 2^{cn} = (2^c)^n . This is not equivalent to Failed to parse (Missing texvc executable; please see math/README to configure.): 2^n

in general.

Changing of variable may affect the order of the resulting algorithm. For example, if an algorithm's running time is O(n) when measured in terms of the number n of digits of an input number x, then its running time is O(log x) when measured as a function of the input number x itself, because n = Θ(log x).

Product [link]

Failed to parse (Missing texvc executable; please see math/README to configure.): f_1 \in O(g_1) \text{ and } f_2\in O(g_2)\, \Rightarrow f_1 f_2\in O(g_1 g_2)\,

Failed to parse (Missing texvc executable; please see math/README to configure.): f\cdot O(g) \subset O(f g)

Sum [link]

Failed to parse (Missing texvc executable; please see math/README to configure.): f_1 \in O(g_1) \text{ and } f_2\in O(g_2)\, \Rightarrow f_1 + f_2\in O(|g_1| + |g_2|)\,

This implies Failed to parse (Missing texvc executable; please see math/README to configure.): f_1 \in O(g) \text{ and } f_2 \in O(g) \Rightarrow f_1+f_2 \in O(g)

, which means that Failed to parse (Missing texvc executable; please see math/README to configure.): O(g)

is a convex cone.

If f and g are positive functions, Failed to parse (Missing texvc executable; please see math/README to configure.): f + O(g) \in O(f + g)

Multiplication by a constant [link]

Let k be a constant. Then:

Failed to parse (Missing texvc executable; please see math/README to configure.): \ O(k g) = O(g)

if k is nonzero.

Failed to parse (Missing texvc executable; please see math/README to configure.): f\in O(g) \Rightarrow kf\in O(g).

Multiple variables [link]

Big O (and little o, and Ω...) can also be used with multiple variables. To define Big O formally for multiple variables, suppose Failed to parse (Missing texvc executable; please see math/README to configure.): f(\vec{x})

and Failed to parse (Missing texvc executable; please see math/README to configure.): g(\vec{x})
are two functions defined on some subset of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^n

. We say

Failed to parse (Missing texvc executable; please see math/README to configure.): f(\vec{x})\text{ is }O(g(\vec{x}))\text{ as }\vec{x}\to\infty

if and only if

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists C\,\exists M>0\text{ such that } |f(\vec{x})| \le C |g(\vec{x})|\text{ for all }\vec{x} \text{ with } x_i>M \text{ for all }i.

For example, the statement

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n,m) = n^2 + m^3 + O(n+m) \text{ as } n,m\to\infty\,

asserts that there exist constants C and M such that

Failed to parse (Missing texvc executable; please see math/README to configure.): \forall n, m>M\colon |g(n,m)| \le C(n+m),

where g(n,m) is defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n,m) = n^2 + m^3 + g(n,m).\,

Note that this definition allows all of the coordinates of Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{x}

to increase to infinity. In particular, the statement

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n,m) = O(n^m) \text{ as } n,m\to\infty\,

(i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): \exists C\,\exists M\,\forall n\,\forall m\dots ) is quite different from

Failed to parse (Missing texvc executable; please see math/README to configure.): \forall m\colon f(n,m) = O(n^m) \text{ as } n\to\infty

(i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): \forall m\,\exists C\,\exists M\,\forall n\dots ).

Matters of notation [link]

Equals sign [link]

The statement "f(x) is O(g(x))" as defined above is usually written as f(x) = O(g(x)). Some consider this to be an abuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As de Bruijn says, O(x) = O(x²) is true but O(x²) = O(x) is not.^[2] Knuth describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like n = n² from the identities n = O(n²) and n² = O(n²)."^[3] For these reasons, it would be more precise to use set notation and write f(x) ∈ O(g(x)), thinking of O(g(x)) as the class of all functions h(x) such that |h(x)| ≤ C|g(x)| for some constant C.^[3] However, the use of the equals sign is customary. Knuth pointed out that "mathematicians customarily use the = sign as they use the word 'is' in English: Aristotle is a man, but a man isn't necessarily Aristotle."^[4]

Other arithmetic operators [link]

Big O notation can also be used in conjunction with other arithmetic operators in more complicated equations. For example, h(x) + O(f(x)) denotes the collection of functions having the growth of h(x) plus a part whose growth is limited to that of f(x). Thus,

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) = h(x) + O(f(x))\,

expresses the same as

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) - h(x) \in O(f(x))\,.

[edit] Example

Suppose an algorithm is being developed to operate on a set of n elements. Its developers are interested in finding a function T(n) that will express how long the algorithm will take to run (in some arbitrary measurement of time) in terms of the number of elements in the input set. The algorithm works by first calling a subroutine to sort the elements in the set and then perform its own operations. The sort has a known time complexity of O(n²), and after the subroutine runs the algorithm must take an additional Failed to parse (Missing texvc executable; please see math/README to configure.): 55n^3+2n+10

time before it terminates.  Thus the overall time complexity of the algorithm can be expressed as

Failed to parse (Missing texvc executable; please see math/README to configure.): T(n)=O(n^2)+55n^3+2n+10.\

This can perhaps be most easily read by replacing O(n²) with "some function that grows asymptotically no faster than n² ". Again, this usage disregards some of the formal meaning of the "=" and "+" symbols, but it does allow one to use the big O notation as a kind of convenient placeholder.

Declaration of variables [link]

Another feature of the notation, although less exceptional, is that function arguments may need to be inferred from the context when several variables are involved. The following two right-hand side big O notations have dramatically different meanings:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(m) = O(m^n)\,,

Failed to parse (Missing texvc executable; please see math/README to configure.): g(n)\,\, = O(m^n)\,.

The first case states that f(m) exhibits polynomial growth, while the second, assuming m > 1, states that g(n) exhibits exponential growth. To avoid confusion, some authors use the notation

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) \in O(f(x))\,.

rather than the less explicit

Failed to parse (Missing texvc executable; please see math/README to configure.): g \in O(f)\,,

Multiple usages [link]

In more complicated usage, O(...) can appear in different places in an equation, even several times on each side. For example, the following are true for Failed to parse (Missing texvc executable; please see math/README to configure.): n\to\infty

Failed to parse (Missing texvc executable; please see math/README to configure.): (n+1)^2 = n^2 + O(n)\

Failed to parse (Missing texvc executable; please see math/README to configure.): (n+O(n^{1/2}))(n + O(\log n))^2 = n^3 + O(n^{5/2})\

Failed to parse (Missing texvc executable; please see math/README to configure.): n^{O(1)} = O(e^n).\

The meaning of such statements is as follows: for any functions which satisfy each O(...) on the left side, there are some functions satisfying each O(...) on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)=O(1)\, , there is some function Failed to parse (Missing texvc executable; please see math/README to configure.): g(n)=O(e^n)\,

such that Failed to parse (Missing texvc executable; please see math/README to configure.): n^{f(n)}=g(n)

." In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side. In this use the "=" is a formal symbol that unlike the usual use of "=" is not a symmetric relation. Thus for example Failed to parse (Missing texvc executable; please see math/README to configure.): n^{O(1)} = O(e^n)\,

does not imply the false statement Failed to parse (Missing texvc executable; please see math/README to configure.): O(e^n) = n^{O(1)}\,

Orders of common functions [link]

Further information: Time complexity#Table of common time complexities

Here is a list of classes of functions that are commonly encountered when analyzing the running time of an algorithm. In each case, c is a constant and n increases without bound. The slower-growing functions are generally listed first.

Notation	Name	Example
Failed to parse (Missing texvc executable; please see math/README to configure.): O(1)\,	constant	Determining if a number is even or odd; using a constant-size lookup table or hash table
Failed to parse (Missing texvc executable; please see math/README to configure.): O(\log \log n)\,	double logarithmic	Finding an item using interpolation search in a sorted array of uniformly distributed values.
Failed to parse (Missing texvc executable; please see math/README to configure.): O(\log n)\,	logarithmic	Finding an item in a sorted array with a binary search or a balanced search tree as well as all operations in a Binomial heap.
Failed to parse (Missing texvc executable; please see math/README to configure.): O(n^c),\;0<c<1	fractional power	Searching in a kd-tree
Failed to parse (Missing texvc executable; please see math/README to configure.): O(n)\,	linear	Finding an item in an unsorted list or a malformed tree (worst case) or in an unsorted array; Adding two n-bit integers by ripple carry.
Failed to parse (Missing texvc executable; please see math/README to configure.): O(n\log n)=O(\log n!)\,	linearithmic, loglinear, or quasilinear	Performing a Fast Fourier transform; heapsort, quicksort (best and average case), or merge sort
Failed to parse (Missing texvc executable; please see math/README to configure.): O(n^2)\,	quadratic	Multiplying two n-digit numbers by a simple algorithm; bubble sort (worst case or naive implementation), Shell sort, quicksort (worst case), selection sort or insertion sort
Failed to parse (Missing texvc executable; please see math/README to configure.): O(n^c),\;c>1	polynomial or algebraic	Tree-adjoining grammar parsing; maximum matching for bipartite graphs
Failed to parse (Missing texvc executable; please see math/README to configure.): L_n[\alpha,c],\;0 < \alpha < 1=\, Failed to parse (Missing texvc executable; please see math/README to configure.): e^{(c+o(1)) (\ln n)^\alpha (\ln \ln n)^{1-\alpha}}	L-notation or sub-exponential	Factoring a number using the quadratic sieve or number field sieve
Failed to parse (Missing texvc executable; please see math/README to configure.): O(c^n),\;c>1	exponential	Finding the (exact) solution to the travelling salesman problem using dynamic programming; determining if two logical statements are equivalent using brute-force search
Failed to parse (Missing texvc executable; please see math/README to configure.): O(n!)\,	factorial	Solving the traveling salesman problem via brute-force search; generating all unrestricted permutations of a poset; finding the determinant with expansion by minors.

The statement Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)=O(n!)\,

is sometimes weakened to Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)=O\left(n^n\right)
to derive simpler formulas for asymptotic complexity.

For any Failed to parse (Missing texvc executable; please see math/README to configure.): k>0

and Failed to parse (Missing texvc executable; please see math/README to configure.): c>0

, Failed to parse (Missing texvc executable; please see math/README to configure.): O(n^c(\log n)^k)

is a subset of Failed to parse (Missing texvc executable; please see math/README to configure.): O(n^{c+\varepsilon })
for any Failed to parse (Missing texvc executable; please see math/README to configure.):  \varepsilon >0

, so may be considered as a polynomial with some bigger order.

Related asymptotic notations [link]

Big O is the most commonly used asymptotic notation for comparing functions, although in many cases Big O may be replaced with Big Theta Θ for asymptotically tighter bounds. Here, we define some related notations in terms of Big O, progressing up to the family of Bachmann–Landau notations to which Big O notation belongs.

Little-o notation [link]

The relation Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) \in o(g(x))

is read as "Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)
is little-o of Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)

". Intuitively, it means that Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)

grows much faster than Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

, or similarly, the growth of Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

is nothing compared to that of Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)

. It assumes that f and g are both functions of one variable. Formally, f(n) = o(g(n)) as $n \to \infty$ means that for every positive constant ε there exists a constant N such that

Failed to parse (Missing texvc executable; please see math/README to configure.): |f(n)|\leq\epsilon|g(n)|\qquad\text{for all }n\geq N~.

^[3]

Note the difference between the earlier formal definition for the big-O notation, and the present definition of little-o: while the former has to be true for at least one constant M the latter must hold for every positive constant ε, however small.^[1] In this way little-o notation makes a stronger statement than the corresponding big-O notation: every function that is little-o of g is also big-O of g, but not every function that is big-O g is also little-o of g (for instance g itself is not, unless it is identically zero near ∞).

If g(x) is nonzero, or at least becomes nonzero beyond a certain point, the relation f(x) = o(g(x)) is equivalent to

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{x \to \infty}\frac{f(x)}{g(x)}=0.

For example,

Failed to parse (Missing texvc executable; please see math/README to configure.): 2x \in o(x^2) \,\!

Failed to parse (Missing texvc executable; please see math/README to configure.): 2x^2 \not \in o(x^2)

Failed to parse (Missing texvc executable; please see math/README to configure.): 1/x \in o(1)

Little-o notation is common in mathematics but rarer in computer science. In computer science the variable (and function value) is most often a natural number. In mathematics, the variable and function values are often real numbers. The following properties can be useful:

Failed to parse (Missing texvc executable; please see math/README to configure.): o(f) + o(f) \subseteq o(f)

Failed to parse (Missing texvc executable; please see math/README to configure.): o(f)o(g) \subseteq o(fg)

Failed to parse (Missing texvc executable; please see math/README to configure.): o(o(f)) \subseteq o(f)

Failed to parse (Missing texvc executable; please see math/README to configure.): o(f) \subset O(f)

(and thus the above properties apply with most combinations of o and O).

As with big O notation, the statement "Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

is Failed to parse (Missing texvc executable; please see math/README to configure.): o(g(x))

" is usually written as Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = o(g(x)) , which is a slight abuse of notation.

Family of Bachmann–Landau notations [link]

Notation	Name	Intuition	Informal definition: for sufficiently large Failed to parse (Missing texvc executable; please see math/README to configure.): n ...	Formal Definition	Notes
Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in O(g(n))	Big Omicron; Big O; Big Oh	Failed to parse (Missing texvc executable; please see math/README to configure.): f is bounded above by Failed to parse (Missing texvc executable; please see math/README to configure.): g (up to constant factor) asymptotically	Failed to parse (Missing texvc executable; please see math/README to configure.): \|f(n)\| \leq g(n)\cdot k for some k	Failed to parse (Missing texvc executable; please see math/README to configure.): \exists k>0 \; \exists n_0 \; \forall n>n_0 \; \|f(n)\| \leq \|g(n)\cdot k\| or Failed to parse (Missing texvc executable; please see math/README to configure.): \exists k>0 \; \exists n_0 \; \forall n>n_0 \; f(n) \leq g(n)\cdot k
Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \Omega(g(n))	Big Omega	Failed to parse (Missing texvc executable; please see math/README to configure.): f is bounded below by Failed to parse (Missing texvc executable; please see math/README to configure.): g (up to constant factor) asymptotically	Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \geq g(n)\cdot k for some positive k	Failed to parse (Missing texvc executable; please see math/README to configure.): \exists k>0 \; \exists n_0 \; \forall n>n_0 \; g(n)\cdot k \leq f(n)	Since the beginning of the 20th century, papers in number theory have been increasingly and widely using this notation in the weaker sense that f = o(g) is false.
Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \Theta(g(n))	Big Theta	Failed to parse (Missing texvc executable; please see math/README to configure.): f is bounded both above and below by Failed to parse (Missing texvc executable; please see math/README to configure.): g asymptotically	Failed to parse (Missing texvc executable; please see math/README to configure.): g(n)\cdot k_1 \leq f(n) \leq g(n)\cdot k_2 for some positive k₁, k₂	Failed to parse (Missing texvc executable; please see math/README to configure.): \exists k_1>0 \; \exists k_2>0 \; \exists n_0 \; \forall n>n_0 Failed to parse (Missing texvc executable; please see math/README to configure.): g(n) \cdot k_1 \leq f(n) \leq g(n) \cdot k_2
Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in o(g(n))	Small Omicron; Small O; Small Oh	Failed to parse (Missing texvc executable; please see math/README to configure.): f is dominated by Failed to parse (Missing texvc executable; please see math/README to configure.): g asymptotically	Failed to parse (Missing texvc executable; please see math/README to configure.): \|f(n)\| \le \|g(n)\|\cdot \varepsilon for every Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon	Failed to parse (Missing texvc executable; please see math/README to configure.): \forall \varepsilon>0 \; \exists n_0 \; \forall n>n_0 \; \|f(n)\| \le \|g(n)\cdot \varepsilon\|
Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \omega(g(n))	Small Omega	Failed to parse (Missing texvc executable; please see math/README to configure.): f dominates Failed to parse (Missing texvc executable; please see math/README to configure.): g asymptotically	Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \ge g(n)\cdot k for every k	Failed to parse (Missing texvc executable; please see math/README to configure.): \forall k>0 \; \exists n_0 \; \forall n>n_0 \; g(n)\cdot k < f(n)
Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)\sim g(n)\!	On the order of	Failed to parse (Missing texvc executable; please see math/README to configure.): f is equal to Failed to parse (Missing texvc executable; please see math/README to configure.): g asymptotically	Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)/g(n) \to 1	Failed to parse (Missing texvc executable; please see math/README to configure.): \forall \varepsilon>0\;\exists n_0\;\forall n>n_0\;\left\|{f(n) \over g(n)}-1\right\|<\varepsilon

Bachmann–Landau notation was designed around several mnemonics, as shown in the As Failed to parse (Missing texvc executable; please see math/README to configure.): n \to \infty , eventually... column above and in the bullets below. To conceptually access these mnemonics, "omicron" can be read "o-micron" and "omega" can be read "o-mega". Also, the lower-case versus capitalization of the Greek letters in Bachmann–Landau notation is mnemonic.

The o-micron mnemonic: The o-micron reading of Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in O(g(n))

and of Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in o(g(n))
 can be thought of as "O-smaller than" and "o-smaller than", respectively.  This micro/smaller mnemonic refers to: for sufficiently large input parameter(s), Failed to parse (Missing texvc executable; please see math/README to configure.): f
grows at a rate that may henceforth be less than Failed to parse (Missing texvc executable; please see math/README to configure.): cg
regarding Failed to parse (Missing texvc executable; please see math/README to configure.): g \in O(f)
or Failed to parse (Missing texvc executable; please see math/README to configure.): g \in o(f)

The o-mega mnemonic: The o-mega reading of Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \Omega(g(n))

and of Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \omega(g(n))
can be thought of as "O-larger than".  This mega/larger mnemonic refers to: for sufficiently large input parameter(s), Failed to parse (Missing texvc executable; please see math/README to configure.): f
grows at a rate that may henceforth be greater than Failed to parse (Missing texvc executable; please see math/README to configure.): cg
regarding Failed to parse (Missing texvc executable; please see math/README to configure.): g \in \Omega(f)
or Failed to parse (Missing texvc executable; please see math/README to configure.): g \in \omega(f)

The upper-case mnemonic: This mnemonic reminds us when to use the upper-case Greek letters in Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in O(g(n))

and Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \Omega(g(n))

for sufficiently large input parameter(s), Failed to parse (Missing texvc executable; please see math/README to configure.): f

grows at a rate that may henceforth be equal to Failed to parse (Missing texvc executable; please see math/README to configure.): cg
regarding Failed to parse (Missing texvc executable; please see math/README to configure.): g \in O(f)

The lower-case mnemonic: This mnemonic reminds us when to use the lower-case Greek letters in Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in o(g(n))

and Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \omega(g(n))

for sufficiently large input parameter(s), Failed to parse (Missing texvc executable; please see math/README to configure.): f

grows at a rate that is henceforth inequal to Failed to parse (Missing texvc executable; please see math/README to configure.): cg
regarding Failed to parse (Missing texvc executable; please see math/README to configure.): g \in O(f)

. Aside from Big O notation, the Big Theta Θ and Big Omega Ω notations are the two most often used in computer science; the Small Omega ω notation is rarely used in computer science.

Use in computer science [link]

For more details on this topic, see Analysis of algorithms.

Informally, especially in computer science, the Big O notation often is permitted to be somewhat abused to describe an asymptotic tight bound where using Big Theta Θ notation might be more factually appropriate in a given context. For example, when considering a function Failed to parse (Missing texvc executable; please see math/README to configure.): T(n) = 73n^3 + 22n^2 + 58 , all of the following are generally acceptable, but tightnesses of bound (i.e., bullets 2 and 3 below) are usually strongly preferred over laxness of bound (i.e., number 1 below).

T(n) = O(n¹⁰⁰), which is identical to T(n) ∈ O(n¹⁰⁰)
T(n) = O(n³), which is identical to T(n) ∈ O(n³)
T(n) = Θ(n³), which is identical to T(n) ∈ Θ(n³).

The equivalent English statements are respectively:

T(n) grows asymptotically no faster than n¹⁰⁰
T(n) grows asymptotically no faster than n³
T(n) grows asymptotically as fast as n³.

So while all three statements are true, progressively more information is contained in each. In some fields, however, the Big O notation (number 2 in the lists above) would be used more commonly than the Big Theta notation (bullets number 3 in the lists above) because functions that grow more slowly are more desirable. For example, if Failed to parse (Missing texvc executable; please see math/README to configure.): T(n)

represents the running time of a newly developed algorithm for input size Failed to parse (Missing texvc executable; please see math/README to configure.): n

, the inventors and users of the algorithm might be more inclined to put an upper asymptotic bound on how long it will take to run without making an explicit statement about the lower asymptotic bound.

Extensions to the Bachmann–Landau notations [link]

Another notation sometimes used in computer science is Õ (read soft-O): f(n) = Õ(g(n)) is shorthand for f(n) = O(g(n) log^k g(n)) for some k. Essentially, it is Big O notation, ignoring logarithmic factors because the growth-rate effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s). This notation is often used to obviate the "nitpicking" within growth-rates that are stated as too tightly bounded for the matters at hand (since log^k n is always o(n^ε) for any constant k and any ε > 0). The L notation, defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): L_n[\alpha,c]=O\left(e^{(c+o(1))(\ln n)^\alpha(\ln\ln n)^{1-\alpha}}\right),

is convenient for functions that are between polynomial and exponential.

Generalizations and related usages [link]

The generalization to functions taking values in any normed vector space is straightforward (replacing absolute values by norms), where f and g need not take their values in the same space. A generalization to functions g taking values in any topological group is also possible. The "limiting process" x→x_o can also be generalized by introducing an arbitrary filter base, i.e. to directed nets f and g. The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions,

Failed to parse (Missing texvc executable; please see math/README to configure.): f\sim g \iff (f-g) \in o(g)

which is an equivalence relation and a more restrictive notion than the relationship "f is Θ(g)" from above. (It reduces to Failed to parse (Missing texvc executable; please see math/README to configure.): \lim f/g = 1

if f and g are positive real valued functions.)  For example, 2x is Θ(x), but 2x − x is not o(x).

Graph theory [link]

It is often useful to bound the running time of graph algorithms. Unlike most other computational problems, for a graph G = (V, E) there are two relevant parameters describing the size of the input: the number |V| of vertices in the graph and the number |E| of edges in the graph. Inside asymptotic notation (and only there), it is common to use the symbols V and E, when someone really means |V| and |E|. This convention simplifies asymptotic functions and make them easily readable. The symbols V and E are never used inside asymptotic notation with their literal meaning, since the number of vertices and edges must be non-negative, so this abuse of notation does not risk ambiguity. For example Failed to parse (Missing texvc executable; please see math/README to configure.): O(E + V \log V)\,

means Failed to parse (Missing texvc executable; please see math/README to configure.): O((V,E) \mapsto |E| + |V|\cdot\log|V|)\,
for a suitable metric of graphs.  Another common convention—referring to the values |V| and |E| by the names n and m, respectively—sidesteps this ambiguity.

History [link]

The notation was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book Analytische Zahlentheorie ("analytic number theory"), the first volume of which (not yet containing big O notation) was published in 1892.^[5] The notation was popularized in the work of number theorist Edmund Landau; hence it is sometimes called a Landau symbol. It was popularized in computer science by Donald Knuth, who re-introduced the related Omega and Theta notations.^[6] He also noted that the (then obscure) Omega notation had been introduced by Hardy and Littlewood^[7] under a slightly different meaning, and proposed the current definition. Hardy's symbols were (in terms of the modern O notation)

Failed to parse (Missing texvc executable; please see math/README to configure.): f\lesssim g \iff f \in O(g)

  and   Failed to parse (Missing texvc executable; please see math/README to configure.):  f\ll g \iff f\in o(g);

other similar symbols were sometimes used, such as Failed to parse (Missing texvc executable; please see math/README to configure.): \preceq

and Failed to parse (Missing texvc executable; please see math/README to configure.): \prec\!\!\!\!\!\!\!\!\prec

. The big-O, standing for "order of", was originally a capital omicron; today the identical-looking Latin capital letter O is used, but never the digit zero.

References [link]

^ ^a ^b Thomas H. Cormen et al., 2001, Introduction to Algorithms, Second Edition
^ N. G. de Bruijn (1958). Asymptotic Methods in Analysis. Amsterdam: North-Holland. pp. 5–7. ISBN 978-0-486-64221-5. http://books.google.com/?id=_tnwmvHmVwMC&pg=PA5&vq=%22The+trouble+is%22.
^ ^a ^b ^c Ronald Graham, Donald Knuth, and Oren Patashnik (1994). 0-201-55802-5 Concrete Mathematics (2 ed.). Reading, Massachusetts: Addison–Wesley. p. 446. ISBN 978-0-201-55802-9. http://books.google.com/?id=pntQAAAAMAAJ&dq=editions:ISBN 0-201-55802-5.
^ Donald Knuth (June/July 1998). "Teach Calculus with Big O". Notices of the American Mathematical Society 45 (6): 687. http://www.ams.org/notices/199806/commentary.pdf. (Unabridged version)
^ Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, p. 25
^ Donald Knuth. Big Omicron and big Omega and big Theta, ACM SIGACT News, Volume 8, Issue 2, 1976.
^ G. H. Hardy and J. E. Littlewood, Some problems of Diophantine approximation, Acta Mathematica 37 (1914), p. 225

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