title: Nikki Grahame

text: Grahame refused to willingly accept treatment and commented; "I was on a mission to starve myself to death ... I was obsessed". This culminated in several suicide bids. At the age of 12, she stole a packet of paracetamol tablets from the hospital's supplies and swallowed some. Shortly after, her weight drastically dropped and she fell into a coma; at one stage, her doctors predicted she was minutes from death. After this incident, Grahame was force-fed through a tube that was stitched inside her stomach but when she regained consciousness, she continuously tried to remove the feeding apparatus.


title: Nikki Grahame

text: Grahame was eventually admitted to Rhodes Farm, an eating disorders clinic in Mill Hill, North London, and after six months, she began showing signs of improvement. She commented; "I woke up one morning and thought, 'What are you doing to yourself?'" At the age of 16, her weight had increased and she was released from the clinic. Grahame relapsed when she was 18, and spent time in an adult eating disorder ward. In the process of overcoming her eating disorder, she developed obsessive-compulsive disorder involving a chronic fear of germs.


title: Nikki Grahame

text: Before Grahame appeared on "Big Brother", she appeared as an extra in the BBC One soap opera "EastEnders" and played a footballer's wife in Sky One's "Dream Team". In 2003, she was a contestant on the ITV dating show "Blind Date". Grahame, who had earned an NVQ in beauty therapy, took part in the 2004 Miss Hertfordshire pageant, where she came fourth.


title: Nikki Grahame

text: In May 2006, Grahame registered as a contestant on the seventh series of the Channel 4 reality television series "Big Brother", in which she became known for her behaviour such as temper tantrums, Diary Room theatrics, and becoming romantically linked to Pete Bennett, the winner of the series. During her "Big Brother" tenure, tabloid newspapers reported Grahame had suffered from anorexia nervosa for most of her adolescence and several times had attempted suicide by taking an overdose of paracetamol. Media criticised the programme's decision to include a contestant with prior mental health issues; experts said she should not have been allowed to take part. However, Grahame herself admitted in her autobiography that she intentionally withheld her diagnoses and psychiatric history from the Big Brother team, and only admitted the truth when she was already in the House and struggling.


title: Nikki Grahame

text: During her first stay in the "Big Brother" house, Grahame was nominated for eviction four times and was eventually evicted on Day 58 after gaining 37.2% of the public vote. Less than four weeks after her exit, she and three other ex-housemates were voted into the "House Next Door" by 63% of the voting public. The remaining contestants chose Grahame to return to the main house, although she had already been evicted. Grahame was reinstated as an eligible housemate and was again given the opportunity to receive the show's £100,000 prize; prompting 2,700 viewers to complain to the premium-rate telephone watchdog ICSTIS.


title: Nikki Grahame

text: Grahame finished the show in fifth place with 6.5% of the final vote. The crowd response was less positive than it had been at her prior eviction, when she departed to unanimous chanting and cheers. Host Davina McCall attempted to interview Grahame but she was reluctant to speak, saying she was scared and shocked by the booing from the crowd. Grahame's live interview was cut short and she was led off the stage.


title: Nikki Grahame

text: After leaving the "Big Brother" House, Grahame presented the "Celebrity Spotting" section on one episode of "The Friday Night Project" that was normally presented by Debra Stephenson. Grahame has also co-presented several episodes of "Celebrity Soup", alongside Iain Lee on the satellite channel E! Entertainment Television. After her first eviction, a reality television series similar to "Simple Life" in which Grahame would star was announced. The programme followed her attempts to hold an everyday job and was inspired by her unsuccessful stint as a personal assistant in a "Big Brother" task. "Princess Nikki" premiered on E4 on 6 September and ran for six weeks.


title: Nikki Grahame

text: Grahame made personal appearances, which included participation in various television shows such as "Celebrity Scissorhands" and "8 out of 10 Cats", and appearances at student nights and clubs. On 24 March 2007, Grahame appeared on a special "Goodies and Baddies" edition of the game show "The Weakest Link". During round three, she was the strongest link of the team (answered the most questions correctly and earned the most money), but was eliminated in round five after then becoming the weakest link of the team. Grahame took part in the Soccer Six charity tournament at Wrexham's Racecourse Ground in May 2007.


title: Nikki Grahame

text: In June 2007, Grahame returned to the "Big Brother" franchise as a reporter for the spin-off show "Big Brother's Little Brother". Known as "The B-Team", Grahame and several other former housemates were sent to interview people about their opinions of "Big Brother 8". She also wrote a "Big Brother" column in the magazine "OK!". In September 2007, Grahame was given a regular column in "OK!'s" "Hot Stars" section.


title: Nikki Grahame

text: On 24 August 2010, Grahame competed in "Ultimate Big Brother", in which she was the only housemate to represent "Big Brother 7". Grahame was nominated for a double eviction in the show's second week of the show's run alongside five of her housemate. Grahame survived the public vote after polling fewer votes than at least two of the other nominees. On 10 September, Grahame became the show's runner-up, receiving a positive reaction from the audience after losing to "Big Brother 2" winner Brian Dowling in the final vote. Grahame finished the show three places higher than her placing in "Big Brother 7".


title: Nikki Grahame

text: During Grahame's time in "Ultimate Big Brother", she appeared to be more comfortable discussing her anorexia; in particular, she conversed with Vanessa Feltz, who has also had problems with her weight.


title: Nikki Grahame

text: In 2015, Grahame participated in the sixteenth series of "Big Brother" as a "time warp housemate". In September 2015, she collaborated with JYY London to design a range of fashion garments for petite women. In an interview with "Reveal" magazine, Grahame stated:


title: Nikki Grahame

text: I've been on buying trips and been designing a lot of things myself. From the very start I said I wanted to be really hands on, from picking fabrics to designing labels and I feel like I've succeeded. I'm excited that it's a petite collection because sizing is always something I struggle with.


title: Nikki Grahame

text: On 23 February 2016, Grahame was announced as one of four 'international wildcards' representing "Big Brother UK" who could be voted into the "Big Brother Canada" house via an online vote. Following the series' first eviction on 3 March 2016, it was revealed Grahame, along with Tim Dormer, the winner of the tenth series of "Big Brother Australia", had been voted into the house. They both became immune from eviction for that week. On 14 April, Grahame was voted in the top two Canadian choices to become the Head of Household (HoH) for the week but she lost a competition to fellow house guests Nick & Phil Paquette. On 24 April, HoH Cassandra nominated Grahame for eviction as a "pawn" to evict a bigger target. On 28 April, she survived this eviction attempt when Jared Kesler was evicted in a 3:1 eviction vote. In a double-eviction twist, however, Grahame was again nominated for eviction alongside Dormer by the HoHs Nick and Phil. Dormer was vetoed off the block and was replaced by Joel Lefevre. Grahame then became the tenth house guest to be evicted in a 3:0 eviction vote and received a standing ovation from the studio audience. Grahame was evicted on Day 63, her 34th birthday. As a result of her eviction, Grahame was the fifth member of the nine-person jury who decided the season's winner on Finale Night. She, along with six other jurors, voted for Nick and Phil to be the winners of the series.


title: Nikki Grahame

text: In August 2017, Grahame appeared on the Channel 5 documentary "In Therapy", which aired on 3 August 2017. The programme features Grahame in intensive, on-camera therapy sessions with psychotherapist Mandy Saligari, during which Grahame spoke about her depression and anorexia.


title: Nikki Grahame

text: Grahame often appeared on the panel of "Big Brother's Bit on the Side." In 2017, Grahame was a special guest at the "Big Brother Canada" awards. In 2018, Grahame entered the "Big Brother" house for the final time. Speaking about it being the last series, Grahame said:There is something about this house where nothing else matters. "Big Brother" has played such a huge part of my life, it changed my life for the better. I don’t have one regret, not one. It will always have a place in my heart. Thank you "Big Brother" for everything. Literally, everything.


title: Nikki Grahame

text: At the age of 18, Grahame was given breast implants on the National Health Service because her anorexia had left her flat-chested. In April 2007, Grahame said she had undergone corrective plastic surgery on her breast implants to fix "ripples" left by the first operation.


title: Nikki Grahame

text: In October 2011, Grahame confirmed she had relapsed and was again suffering from anorexia, stating she had checked into her 11th institution. In 2012, Grahame was sectioned following an overdose.


title: Nikki Grahame

text: In March 2021, Grahame's family announced she had again relapsed, that she "constantly feels weak and is struggling on a day to day basis", and that she wanted to recover but felt it was "impossible". They opened a GoFundMe fundraiser to pay for private treatment because the NHS treatments had "failed". The family said they had "exhausted every avenue possible" and raised over £60,000 for Grahame's treatment. On 24 March, following a fall Grahame entered Dorset County Hospital, where she remained until she was discharged on 8 April. The next day, she was found dead in her Stanmore flat; she had died from complications of anorexia nervosa at the age of 38. She had been due to enter a private facility in Devon to receive specialist treatment the following week. At the time of her death, Grahame weighed less than .


title: Nikki Grahame

text: Among those who paid tributes were former "Big Brother UK" presenter Davina McCall, "Big Brother Canada" host Arisa Cox, television presenter Paddy McGuinness, and other "Big Brother" alumni including Rylan Clark-Neal, Aisleyne Horgan-Wallace, Pete Bennett, Katie Price, Imogen Thomas, and Richard Newman. A tribute to Grahame was aired at the start of episode 18 of "Big Brother Canada 9". The week following Grahame's death, her GoFundMe page was re-opened to raise money for her funeral and for eating disorder charities.


title: AK-103

text: It is an AK-100 derivative of the AK-74M that is chambered for the 7.62x39mm M43 cartridge, similar to the AKM. The AK-103 can be fitted with a variety of sights, including night vision and telescopic sights, plus a knife-bayonet or a grenade launcher like the GP-34. Newer versions can fit Picatinny rails, allowing more accessories to be mounted. It uses plastic components where possible instead of wood or metal, with such components being the pistol grip, handguards, folding stock and depending on the type, the magazine.


title: AK-103

text: Protective coatings for corrosion resistance of metal parts. Forearm, magazine, butt stock and pistol grip are made of high strength shatterproof plastic.


title: AK-103

text: The AK-104 is a compact version of the AK-103. It has a muzzle brake derived from the older AKS-74U combined with a shorter barrel. It is chambered for 7.62×39mm ammunition.


title: AK-103

text: The current issue steel-reinforced matte true black nonreflective surface finished 7.62×39mm 30-round magazines, fabricated from ABS plastic weigh empty. Early steel AK-47 magazines are long, and the later ribbed steel AKM and newer plastic 7.62×39mm magazines are about shorter.


title: AK-103

text: The transition from steel to mainly plastic magazines yielded a significant weight reduction and allow a soldier to carry more rounds for the same weight.


title: AK-103

text: The semi-automatic only variant of the AK-103 is designated the AK-103-1, and the three round burst is designated the AK-103-2.


title: AK-103

text: A Vietnamese version known as the STL-1A is made by Factory Z111 and is used by the People's Army of Vietnam. Another modernized version called the STL-1B is currently planned. It first appeared in the 2018 Indo Defence Expo & Forum.


title: West Nile virus

text: West Nile virus (WNV) is a single-stranded RNA virus that causes West Nile fever. It is a member of the family "Flaviviridae", from the genus "Flavivirus", which also contains the Zika virus, dengue virus, and yellow fever virus. The virus is primarily transmitted by mosquitoes, mostly species of "Culex". The primary hosts of WNV are birds, so that the virus remains within a "bird–mosquito–bird" transmission cycle. The virus is genetically related to the Japanese encephalitis family of viruses.


title: West Nile virus

text: Humans and horses both exhibit disease symptoms from the virus, and symptoms rarely occur in other animals. Identification of the human disease was first made in 1937 in Uganda and in the latter half of the 20th century spread to many other parts of the world.


title: West Nile virus

text: Like most other flaviviruses, WNV is an enveloped virus with icosahedral symmetry. Electron microscope studies reveal a 45–50 nm virion covered with a relatively smooth protein shell; this structure is similar to the dengue fever virus, another "Flavivirus". The protein shell is made of two structural proteins: the glycoprotein E and the small membrane protein M. Protein E has numerous functions including receptor binding, viral attachment, and entry into the cell through membrane fusion.


title: West Nile virus

text: The outer protein shell is covered by a host-derived lipid membrane, the viral envelope. The flavivirus lipid membrane has been found to contain cholesterol and phosphatidylserine, but other elements of the membrane have yet to be identified. The lipid membrane has many roles in viral infection, including acting as signaling molecules and enhancing entry into the cell. Cholesterol, in particular, plays an integral part in WNV entering a host cell. The two viral envelope proteins, E and M, are inserted into the membrane.


title: West Nile virus

text: The RNA genome is bound to capsid (C) proteins, which are 105 amino-acid residues long, to form the nucleocapsid. The capsid proteins are one of the first proteins created in an infected cell; the capsid protein is a structural protein whose main purpose is to package RNA into the developing viruses. The capsid has been found to prevent apoptosis by affecting the Akt pathway.


title: West Nile virus

text: WNV is a positive-sense, single-stranded RNA virus. Its genome is approximately 11,000 nucleotides long and is flanked by 5′ and 3′ non-coding stem loop structures. The coding region of the genome codes for three structural proteins and seven nonstructural (NS) proteins, proteins that are not incorporated into the structure of new viruses. The WNV genome is first translated into a polyprotein and later cleaved by virus and host proteases into separate proteins (i.e. NS1, C, E).


title: West Nile virus

text: Structural proteins (C, prM/M, E) are capsid, precursor membrane proteins, and envelope proteins, respectively. The structural proteins are located at the 5′ end of the genome and are cleaved into mature proteins by both host and viral proteases.


title: West Nile virus

text: Nonstructural proteins consist of NS1, NS2A, NS2B, NS3, NS4A, NS4B, and NS5. These proteins mainly assist with viral replication or act as proteases. The nonstructural proteins are located near the 3′ end of the genome.


title: West Nile virus

text: Once WNV has successfully entered the bloodstream of a host animal, the envelope protein, E, binds to attachment factors called glycosaminoglycans on the host cell. These attachment factors aid entry into the cell, however, binding to primary receptors is also necessary. Primary receptors include DC-SIGN, DC-SIGN-R, and the integrin αβ. By binding to these primary receptors, WNV enters the cell through clathrin-mediated endocytosis. As a result of endocytosis, WNV enters the cell within an endosome.


title: West Nile virus

text: The acidity of the endosome catalyzes the fusion of the endosomal and viral membranes, allowing the genome to be released into the cytoplasm. Translation of the positive-sense single-stranded RNA occurs at the endoplasmic reticulum; the RNA is translated into a polyprotein which is then cleaved by both host and viral proteases NS2B-NS3 to produce mature proteins.


title: West Nile virus

text: In order to replicate its genome, NS5, a RNA polymerase, forms a replication complex with other nonstructural proteins to produce an intermediary negative-sense single-stranded RNA; the negative-sense strand serves as a template for synthesis of the final positive-sense RNA. Once the positive-sense RNA has been synthesized, the capsid protein, C, encloses the RNA strands into immature virions. The rest of the virus is assembled along the endoplasmic reticulum and through the Golgi apparatus, and results in non-infectious immature virions. The E protein is then glycosylated and prM is cleaved by furin, a host cell protease, into the M protein, thereby producing an infectious mature virion. The mature viruses are then secreted out of the cell.


title: West Nile virus

text: WNV is one of the Japanese encephalitis antigenic serocomplex of viruses, together with Japanese encephalitis virus, Murray Valley encephalitis virus, Saint Louis encephalitis virus and some other flaviviruses. Studies of phylogenetic lineages have determined that WNV emerged as a distinct virus around 1000 years ago. This initial virus developed into two distinct lineages. Lineage 1 and its multiple profiles is the source of the epidemic transmission in Africa and throughout the world. Lineage 2 was considered an African zoonosis. However, in 2008, lineage 2, previously only seen in horses in sub-Saharan Africa and Madagascar, began to appear in horses in Europe, where the first known outbreak affected 18 animals in Hungary. Lineage 1 West Nile virus was detected in South Africa in 2010 in a mare and her aborted fetus; previously, only lineage 2 West Nile virus had been detected in horses and humans in South Africa. Kunjin virus is a subtype of West Nile virus endemic to Oceania. A 2007 fatal case in a killer whale in Texas broadened the known host range of West Nile virus to include cetaceans.


title: West Nile virus

text: Since the first North American cases in 1999, the virus has been reported throughout the United States, Canada, Mexico, the Caribbean, and Central America. There have been human cases and equine cases, and many birds are infected. The Barbary macaque, "Macaca sylvanus", was the first nonhuman primate to contract WNV. Both the American and Israeli strains are marked by high mortality rates in infected avian populations; the presence of dead birds—especially Corvidae—can be an early indicator of the arrival of the virus.


title: West Nile virus

text: The natural hosts for WNV are birds and mosquitoes. Over 300 different species of bird have been shown to be infected with the virus. Some birds, including the American crow ("Corvus brachyrhynchos"), blue jay ("Cyanocitta cristata") and greater sage-grouse ("Centrocercus urophasianus"), are killed by the infection, but others survive. The American robin ("Turdus migratorius") and house sparrow ("Passer domesticus") are thought to be among the most important reservoir species in N. American and European cities. Brown thrashers ("Toxostoma rufum"), gray catbirds ("Dumetella carolinensis"), northern cardinals ("Cardinalis cardinalis"), northern mockingbirds ("Mimus polyglottos"), wood thrushes ("Hylocichla mustelina") and the dove family are among the other common N. American birds in which high levels of antibodies against WNV have been found.


title: West Nile virus

text: WNV has been demonstrated in a large number of mosquito species, but the most significant for viral transmission are "Culex" species that feed on birds, including "Culex pipiens", "C. restuans", "C. salinarius", "C. quinquefasciatus", "C. nigripalpus", "C. erraticus" and "C. tarsalis". Experimental infection has also been demonstrated with soft tick vectors, but is unlikely to be important in natural transmission.


title: West Nile virus

text: WNV has a broad host range, and is also known to be able to infect at least 30 mammalian species, including humans, some non-human primates, horses, dogs and cats. Some infected humans and horses experience disease but dogs and cats rarely show symptoms. Reptiles and amphibians can also be infected, including some species of crocodiles, alligators, snakes, lizards and frogs. Mammals are considered incidental or dead-end hosts for the virus: they do not usually develop a high enough level of virus in the blood (viremia) to infect another mosquito feeding on them and carry on the transmission cycle; some birds are also dead-end hosts.


title: West Nile virus

text: In the normal rural or enzootic transmission cycle, the virus alternates between the bird reservoir and the mosquito vector. It can also be transmitted between birds via direct contact, by eating an infected bird carcass or by drinking infected water. Vertical transmission between female and offspring is possible in mosquitoes, and might potentially be important in overwintering. In the urban or spillover cycle, infected mosquitoes that have fed on infected birds transmit the virus to humans. This requires mosquito species that bite both birds and humans, which are termed bridge vectors. The virus can also rarely be spread through blood transfusions, organ transplants, or from mother to baby during pregnancy, delivery, or breastfeeding. Unlike in birds, it does not otherwise spread directly between people.


title: West Nile virus

text: Severe disease may also occur in horses. Several vaccines for these animals are now available. Before the availability of veterinary vaccines, around 40% of horses infected in North America died.


title: West Nile virus

text: According to the Center for Disease Control, infection with West Nile Virus is seasonal in temperate zones. Climates that are temperate, such as those in the United States and Europe, see peak season from July to October. Peak season changes depending on geographic region and warmer and humid climates can see longer peak seasons. All ages are equally likely to be infected but there is a higher amount of death and neuroinvasive West Nile Virus in people 60–89 years old. People of older age are more likely to have adverse effects.


title: West Nile virus

text: There are several modes of transmission, but the most common cause of infection in humans is by being bitten by an infected mosquito. Other modes of transmission include blood transfusion, organ transplantation, breast-feeding, transplacental transmission, and laboratory acquisition. These alternative modes of transmission are extremely rare.


title: West Nile virus

text: Prevention efforts against WNV mainly focus on preventing human contact with and being bitten by infected mosquitoes. This is twofold, first by personal protective actions and second by mosquito-control actions. When a person is in an area that has WNV, it is important to avoid outdoor activity, and if they go outside they should use a mosquito repellent with DEET. A person can also wear clothing that covers more skin, such as long sleeves and pants. Mosquito control can be done at the community level and include surveillance programs and control programs including pesticides and reducing mosquito habitats. This includes draining standing water. Surveillance systems in birds is particularly useful. If dead birds are found in a neighborhood, the event should be reported to local authorities. This may help health departments do surveillance and determine if the birds are infected with West Nile Virus.


title: West Nile virus

text: Despite the commercial availability of four veterinary vaccines for horses, no human vaccine has progressed beyond phase II clinical trials. Efforts have been made to produce a vaccine for human use and several candidates have been produced but none are licensed to use. The best method to reduce the risk of infections is avoiding mosquito bites. This may be done by eliminating standing pools of water, such as in old tires, buckets, gutters, and swimming pools. Mosquito repellent, window screens, mosquito nets, and avoiding areas where mosquitoes occur may also be useful.


title: West Nile virus

text: Like other tropical diseases which are expected to have increased spread due to climate change, there is concern that changing weather conditions will increase West Nile Virus spread. Climate change will affect disease rates, ranges, seasonality and affects the distribution of West Nile Virus.


title: West Nile virus

text: Projected changes in flood frequency and severity can bring new challenges in flood risk management, allowing for increased mosquito populations in urban areas. Weather conditions affected by climate change including temperature, precipitation and wind may affect the survival and reproduction rates of mosquitoes, suitable habitats, distribution, and abundance. Ambient temperatures drive mosquito replication rates and transmission of WNV by affecting the peak season of mosquitoes and geographic variations. For example, increased temperatures can affect the rate of virus replication, speed up the virus evolution rate, and viral transmission efficiency. Furthermore, higher winter temperatures and warmer spring may lead to larger summer mosquito populations, increasing the risk for WNV. Similarly, rainfall may also drive mosquito replication rates and affect the seasonality and geographic variations of the virus. Studies show an association between heavy precipitation and higher incidence of reported WNV. Likewise, wind is another environmental factor that serves as a dispersal mechanism for mosquitoes.


title: West Nile virus

text: Mosquitoes have extremely wide environmental tolerances and a nearly ubiquitous geographical distribution, being present on all major land masses except Antarctica and Iceland. Nevertheless, changes in climate and land use on ecological timescales can variously expand or fragment their distribution patterns, raising consequent concerns for human health.


title: Marla Gibbs

text: Marla Gibbs (born Margaret Theresa Bradley; June 14, 1931) is an American actress, singer, comedian, writer and television producer, whose career spans six decades. Gibbs is known for her role as George Jefferson's maid, Florence Johnston, in the CBS sitcom, "The Jeffersons" (1975–1985), for which she received five nominations for Primetime Emmy Award for Outstanding Supporting Actress in a Comedy Series.


title: Marla Gibbs

text: Gibbs also starred in the show's spin-off "Checking In" (1981) and the NBC sitcom, "227" (1985–1990); Gibbs co-produced the latter series, played the lead role of Mary Jenkins, and sang the theme song. Gibbs has won seven NAACP Image Awards. In later years, Gibbs played supporting roles in films "The Meteor Man" (1993), "Lost & Found" (1999), "The Visit" (2000), "The Brothers" (2001), "Madea's Witness Protection" (2012), "Grantham & Rose" (2015), and "Lemon" (2017) and the Shondaland produced TV shows "Station 19" (2018) and "Grey's Anatomy" (2022). In August 2021, Gibbs began a run as Olivia Price on the NBC daytime drama, "Days of Our Lives".


title: Marla Gibbs

text: Marla Gibbs was born Margaret Theresa Bradley in Chicago, Illinois at Cook County Hospital. The middle of three sisters, Gibbs was raised in the Bronzeville neighborhood on the city's south side. Her father, Douglas Bradley, a native of Waterproof, Louisiana (d. 1947) was a self-taught mechanic who owned an ice company. Her mother, Ophelia Birdie (née Kemp; d. 1967) was a businesswoman who occasionally ran numbers in the gambling trade. In 1936, Gibbs's parents divorced, with her mother remarrying Charles Cady and relocating to Detroit, Michigan. Gibbs and her siblings remained in Chicago with their father and grandmother Hattie Sims. Gibbs attended Corpus Christi Elementary School, graduating in 1945.


title: Marla Gibbs

text: After graduating from elementary school, Gibbs attended St. Elizabeth High School, where she completed her freshman year and first semester of her sophomore year. During the middle of her sophomore year, Gibbs transferred to Wendell Phillips Academy High School. When she was 16, her father died, and Gibbs was sent to live with her mother and step-father in Detroit, Michigan where she attended Northern High School for a semester during her junior year. After months in Detroit, Gibbs returned to Chicago, re-enrolling at Phillips where she graduated in June 1949. Several years later, Gibbs returned to Detroit, Michigan, where she attended Peters Business School.


title: Marla Gibbs

text: In 1963, Gibbs relocated to Los Angeles to recuperate from an ulcer, and began working as a reservations agent for United Airlines. She first acted in local Los Angeles black theater before she got her first acting job in the early 1970s, in the blaxploitation films "Sweet Jesus", "Preacher Man" and "Black Belt Jones". In 1975, she was cast as Florence Johnston, the family's maid, in the CBS comedy series "The Jeffersons". For the first two years of her acting job with CBS, Gibbs continued working at her job as a United Airlines reservations agent. She would work at the CBS television studio during the day and then go to United Airlines, where she worked the night shift. When the series became an established success, CBS requested that she take a leave of absence from her job at United Airlines.


title: Marla Gibbs

text: Gibbs's performance on the series garnered her five nominations for a Primetime Emmy Award for Outstanding Supporting Actress in a Comedy Series, and one nomination for a Golden Globe Award for Best Supporting Actress – Series, Miniseries or Television Film. In 1981, she starred in the short-lived spin-off of "The Jeffersons", titled "Checking In". Gibbs responded in a 2015 interview on Broadway Showbiz, when asked if she'd based any of her characters on real-life people:


title: Marla Gibbs

text: "Yes, Florence was like my aunt and grandmother so I lived it. She came easy to me so I'm like Florence in giving smart answers, but I was also shy so I wouldn't have dared to say some of the things Florence said. I prefer to do whatever I can do at the moment. Whoever's hiring me at the moment...that's what I'm supposed to do. My favorite is drama. I'm doing that now (on "Scandal"), but also still doing comedy on "Hot in Cleveland"."


title: Marla Gibbs

text: In 1985, when "The Jeffersons" was cancelled after 11 seasons, Gibbs was the lead actress in the NBC sitcom, "227". "227" was adapted from a play directed by Cambridge Players' then-president Ed Cambridge and was presented to NBC by Cambridge, at Gibbs's Crossroads Theater in L.A. He served as artistic director. The series aired until 1990, producing 116 episodes. Two decades later, Gibbs teamed again with former "227" co-star Jackée Harry in "The First Family", where Gibbs had a recurring role as Harry's on-screen mother Grandma Eddy. Then, she again worked with Harry in the independent film "Forbidden Woman".


title: Marla Gibbs

text: Gibbs had a number of supporting film roles, and guest starred on "The Fresh Prince of Bel-Air", "Martin", "Touched by an Angel" (with Della Reese), "Judging Amy", "ER", and "Southland". From 1998 to 2002 she had a recurring role on "The Hughleys". In 2012, Gibbs appeared in the Tyler Perry film "Madea's Witness Protection", and in 2014 she starred in the independent film, "Grantham & Rose".


title: Marla Gibbs

text: In 2015, Gibbs made two appearances in the Shonda Rhimes' drama series, "Scandal". She later guest starred on "Hot in Cleveland", "" and "This Is Us". In 2018, she was cast in a recurring role on the ABC drama series "Station 19". Gibbs also has had starring roles in two television pilots: "Old Soul" alongside Ellen Burstyn and Rita Moreno for NBC in 2014, and ABC's "Jalen Vs. Everybody" in 2017. In film, she co-starred in "Lemon" and "Please Stand By".


title: Marla Gibbs

text: In 2021, Gibbs began appearing as Olivia Price in a recurring role on "Days of Our Lives". That same year, for her contributions to the television industry, she was awarded a star on the Hollywood Walk of Fame.


title: Marla Gibbs

text: Gibbs owned a jazz club in South Central L.A. called Marla's Memory Lane Jazz and Supper Club from 1981 to 1999. As a singer, she released a number of albums. In 1990, she moved her Crossroads Arts Academy and Theatre into the former Leimert Theatre in Los Angeles. Plans included the construction of a second stage, but the project ended in debt in June 1997.


title: Marla Gibbs

text: Gibbs has been married once and had three children. Gibbs was married to her high school sweetheart Jordan Gibbs from 1955 to 1973; the couple had three children: Angela, Dorian and Joseph. Her older sister is the late actress Susie Garrett, who played Cherie's grandmother Betty Johnson on the NBC series "Punky Brewster". Her daughter Angela, also an actress, appeared on the TV show "Sanford and Son" and films such as "Together Brothers", "Drumline" and "Think Like A Man Too". On July 31, 2014, Gibbs attended the Leimert Park Village Book Fair in Los Angeles, California, to pay tribute to Maya Angelou, who died on May 28, and Ruby Dee, who died on June 11. On January 11, 2016, Gibbs and former "227" co-star, Jackée Harry, attended the funeral of Grammy Award-winning singer Natalie Cole at the West Angeles Church of God in Christ in Los Angeles, California. Gibbs co-starred opposite Cole in the television film "Lily in Winter".


title: Sikorsky CH-53K King Stallion

text: The Sikorsky CH-53K King Stallion (Sikorsky S-95) is a heavy-lift cargo helicopter designed and produced by Sikorsky Aircraft. The King Stallion is an evolution of the long running CH-53 series of helicopters which have been in continuous service since 1966, and features three uprated engines, new composite rotor blades, and a wider aircraft cabin than its predecessors. It is the largest and heaviest helicopter in the U.S. military.


title: Sikorsky CH-53K King Stallion

text: The United States Marine Corps plans to receive 200 helicopters at a total cost of $25 billion. Ground Test Vehicle (GTV) testing started in April 2014; flight testing began with the maiden flight on 27 October 2015. In May 2018, the first CH-53K was delivered to the Marine Corps. On 22 April 2022, it was declared to have passed initial operational capability. Israel has also reportedly ordered the type; other potential export customers include Japan and Germany.


title: Sikorsky CH-53K King Stallion

text: The Sikorsky CH-53 Sea Stallion came out of the US Marine Corps' (USMC) "Heavy Helicopter Experimental" (HH(X)) competition begun in 1962. Sikorsky's S-65 was selected over Boeing Vertol's modified CH-47 Chinook version. The prototype YCH-53A first flew on 14 October 1964. It was designated "CH-53A Sea Stallion", delivery of production helicopters began in 1966. The CH-53A was equipped with two T64-GE-6 turboshaft engines, and had a maximum gross weight of .


title: Sikorsky CH-53K King Stallion

text: Variants of the original CH-53A Sea Stallion include the RH-53A/D, HH-53B/C, CH-53D, CH-53G, and MH-53H /J/M. The RH-53A and RH-53D were used by the United States Navy for minesweeping. The CH-53D had a more powerful version of the General Electric T64 engine, used in all H-53 variants, and external fuel tanks. The US Air Force's "HH-53B/C Super Jolly Green Giant" were for special operations and combat rescue. The Air Force's "MH-53H/J/M Pave Low" were the final twin-engined H-53s, and had extensive avionics upgrades for all-weather operation.


title: Sikorsky CH-53K King Stallion

text: In October 1967, the USMC issued a requirement for a helicopter with a lifting capacity 1.8 times that of the CH-53D, that could fit on amphibious assault ships. Before this, Sikorsky had been working on an enhancement to the CH-53D, under the company designation "S-80", featuring a third turboshaft engine and a more powerful rotor system. Sikorsky proposed the S-80 design to the Marines in 1968. The Marines considered this a good, quick solution, and funded development of a testbed helicopter. Changes on the CH-53E also included a stronger transmission and a fuselage stretched . The main rotor blades' material was changed to a titanium-fiberglass composite. A new automatic flight control system was added. The vertical tail was also enlarged, with the tail rotor tilted upwards slightly to provide some lift in hover.


title: Sikorsky CH-53K King Stallion

text: The initial YCH-53E first flew in 1974. Following successful testing, the initial production contract was awarded in 1978, and service introduction followed in February 1981. The US Navy acquired the CH-53E in small numbers for shipboard resupply. The Marines and Navy acquired a total of 177. For the airborne mine countermeasures role, the Navy obtained a CH-53E variant, designated "MH-53E Sea Dragon", with enlarged sponsons and fuel tanks for greater fuel storage, in the 1980s. The Navy obtained 46 Sea Dragons.


title: Sikorsky CH-53K King Stallion

text: The USMC had planned upgrades to retain most CH-53Es, but this effort stalled. Sikorsky proposed a new model, originally designated "CH-53X"; in April 2006, the USMC signed a $18.8 billion contract for 156 "CH-53K" helicopters, with deliveries to be completed by 2021. The USMC was to begin retiring CH-53Es in 2009, and needed replacements as rotorcraft reached their structural life limits in 2011–12. CH-53K flight testing was expected to begin in 2011. In August 2007, the USMC increased the order from 156 to 227. By that time, the first flight was planned for November 2011 with initial operating capability (IOC) by 2015. The CH-53K will be the USMC's heavy lift helicopter with the MV-22 (medium lift) and UH-1Y (light lift). A 2007 RAND report on seabasing found that a higher ratio of CH-53Ks to MV-22s would reduce ship-based deployment times.


title: Sikorsky CH-53K King Stallion

text: In 2008, design work was well underway, along with weight reduction efforts to meet operational requirements; increased engine performance and rotor blade improvements are options to help meet requirements if needed. The rotor mast tilt was decreased and components shifted to ensure the center of gravity does not shift too far rearward as fuel is burned. Design requirements were frozen in 2009–10. On 22 January 2010, Sikorsky opened a $20 million Precision Components Technology Center in Stratford, Connecticut, for producing CH-53K parts, such as the rotating and stationary swashplates, main and tail rotor hubs, and main rotor sleeves. On 3 August 2010, the CH-53K passed its Critical Design Review, reading it for test production. However, the IOC fielding date was deferred to 2018. Sikorsky proposed building four pre-production aircraft for evaluation.


title: Sikorsky CH-53K King Stallion

text: On 4 December 2012, Sikorsky delivered the first CH-53K, a Ground Test Vehicle (GTV) airframe. Early tests included fuel system calibration and attaching test sensors across the airframe to record temperature, aerodynamic load, pressure, and vibration. Two additional static GTVs underwent structural testing at the firm's Stratford manufacturing plant. In January 2013, the program had an estimated cost of US$23.17 billion after procurement of the planned 200 CH-53Ks. In April 2013, the U.S. Navy program manager stated that work had gone well and it may become operational ahead of schedule. On 31 May 2013, the Navy awarded Sikorsky a $435 million contract to deliver four prototype CH-53Ks for evaluation and mission testing; The first two prototypes focused on structural flight loads while the third and fourth validated general performance, propulsion and avionics.


title: Sikorsky CH-53K King Stallion

text: Major subcontractors include Aurora Flight Sciences (main rotor pylon), Exelis Aerostructures (tail rotor pylon and sponsons), GKN Aerospace (aft transition), Onboard Systems International (external cargo hook), Rockwell Collins (avionics management system), Sanmina-SCI Corporation (communications), and Spirit AeroSystems (cockpit and cabin). In October 2013, Sikorsky gave Kratos Defense & Security Solutions a $8.5 million contract for CH-53K maintenance training aids, such as the Maintenance Training Device Suite (MTDS) and Helicopter Emulation Maintenance Trainer (HEMT). The MTDS is a realistic training and evaluation environment for various avionics, electrical, and hydraulic subsystems. The HEMT is a 3D simulation of multiple scenarios, such as functional test, troubleshoot, fault isolation, removal and installation of 27 subsystems.


title: Sikorsky CH-53K King Stallion

text: On 24 January 2014, the CH-53K GTV ignited its engines, spinning the rotor head without rotors attached. Low-rate production is planned to proceed from 2015 to 2017. Initial operating capability (IOC) was set to occur in 2019, with full-rate production commencing between then and 2022. The USMC intends to have eight active squadrons, one training squadron, and one reserve squadron. In April 2014, testing with blades attached began, system integration followed. Flight testing was set to start in late 2014, each test aircraft flying approximately 500 hours over three years. The maiden flight was delayed, due to issues with the titanium quill shafts in the transmission and gear box.


title: Sikorsky CH-53K King Stallion

text: On 5 May 2014, General James F. Amos announced during the official rollout that it will be called the "King Stallion". On 27 October 2015, the CH-53K took its first flight. On 7 March 2018, one lifted a payload of , the maximum weight on the single center point cargo hook. The first CH-53K was delivered to the USMC on 16 May 2018; at the time, 18 additional helicopters were in production, and the second was planned for delivery in early 2019.


title: Sikorsky CH-53K King Stallion

text: In December 2018, the CH-53K was projected to not be combat ready as expected in late 2019, due to delivery delays caused by technical flaws found in testing, which resulted in a major program restructuring. Flaws included the engine re-ingesting exhaust gas, limited service life for the rotor gear boxes, late deliveries of redesigned parts, and deficiencies with the tail rotor and driveshaft. It is estimated that the delay will push back delivery of combat-ready CH-53Ks until May 2020.


title: Sikorsky CH-53K King Stallion

text: The CH-53K King Stallion is a heavy lift helicopter, being a general redesign of the preceding CH-53E, the main improvements being the new engines and cockpit layout. It has over twice the lift capacity and radius of action of the CH-53E, and a wider cargo hold to allow it to carry a Humvee internally. A new composite rotor blade system is also used, featuring technology similar to that of the UH-60 Black Hawk helicopter. The CH-53K is powered by the General Electric GE38-1B engine, which was selected over the Pratt and Whitney Canada PW150 and a variant of the Rolls-Royce AE 1107C-Liberty used on the V-22 Osprey. Each of the three T408 engines is rated at , and gives the CH-53K the ability to fly faster than its CH-53E predecessor.


title: Sikorsky CH-53K King Stallion

text: The CH-53K features a new digital glass cockpit with fly-by-wire controls and haptic feedback, HUMS, a new elastomeric hub system, and composite rotor blades to improve "hot and high" performance. The split torque gearbox with quill shafts started development around 2007. The gearbox assembly including rotor hub and rotating control system weighs around . The split torque gearbox weighs . For comparison, the twin-engine Mil Mi-26's split torque gearbox weighs .


title: Sikorsky CH-53K King Stallion

text: The CH-53K has an improved external cargo handling system, survivability enhancements, and improvements to extend service life. The cabin will be long by wide by tall. Its cabin will be wider and 15% larger, but will have new shorter composite sponsons. The CH-53K can carry two 463L master pallets, eliminating the need to break apart pallets between airlifter and helicopter.


title: Sikorsky CH-53K King Stallion

text: The CH-53K is to surpass the capability of its predecessor by carrying nearly 30% more than the CH-53E's external payload of over the same radius of . The CH-53K's payload reaches a maximum of . The CH-53K's maximum gross weight will be , which is increased over the CH-53E's . The CH-53K will keep approximately the same footprint as the CH-53E. To this end, it has redesigned composite sponsons that cut overall width for a narrower footprint, which is better for shipboard service.


title: Sikorsky CH-53K King Stallion

text: The U.S. Marine Corps received its first CH-53K simulator at Marine Corps Air Station New River in Jacksonville, North Carolina on 1 May 2020. It is a Containerized Flight Training Device (CFTD) built by Lockheed Martin, Sikorsky's parent company.


title: Sikorsky CH-53K King Stallion

text: On 22 April 2022, Lt. General Mark R. Wise, Deputy Commandant for Aviation, declared initial operational capability for the CH-53K.


title: Sikorsky CH-53K King Stallion

text: In 2009, the Israeli Air Force (IAF) said it would evaluate the new variant after it flies. In August 2015, it formalized a requirement for the CH-53K, listing the type as a "very high priority" item to enable the service to perform missions only the platform is capable of. Israel's current CH-53 "Yasur" fleet is to remain operational until 2025.


title: Sikorsky CH-53K King Stallion

text: The CH-53K competed with the Boeing CH-47F Chinook for an order of approximately 20 helicopters to replace the CH-53 Yasur. In February 2021, the Israeli Ministry of Defense announced the CH-53K's selection. On 31 December 2021, it was announced that Israel had signed a deal to buy 12 CH-53Ks.


title: Sikorsky CH-53K King Stallion

text: In February 2018, Sikorsky signed an agreement valued at around 4 billion euro with Rheinmetall to team up for the German Air Force's CH-53G heavy lift helicopter replacement program, in which the CH-53K competed against the Boeing CH-47 Chinook. The German Federal Ministry of Defence was expected to issue an official request for information in late 2018, to award a contract in 2020, and for deliveries to begin in 2023 for an expected order of around 40 helicopters.


title: Sikorsky CH-53K King Stallion

text: On 29 September 2020, the German Ministry of Defense cancelled the "Schwerer Transporthubschrauber" (STH) heavy-lifting helicopter program, it having been judged to be too expensive; instead, the CH-53Gs are still to be replaced after reexamining the project. In 2022, Germany decided to procure the CH-47F Chinook instead, citing interoperability advantages with other European NATO countries - especially the Netherlands - as well as the lower unit cost compared to the CH-53K which would allow the purchase of more helicopters for the same budget.


title: Cluster analysis

text: Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). It is a main task of exploratory data analysis, and a common technique for statistical data analysis, used in many fields, including pattern recognition, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning.


title: Cluster analysis

text: Cluster analysis itself is not one specific algorithm, but the general task to be solved. It can be achieved by various algorithms that differ significantly in their understanding of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with small distances between cluster members, dense areas of the data space, intervals or particular statistical distributions. Clustering can therefore be formulated as a multi-objective optimization problem. The appropriate clustering algorithm and parameter settings (including parameters such as the distance function to use, a density threshold or the number of expected clusters) depend on the individual data set and intended use of the results. Cluster analysis as such is not an automatic task, but an iterative process of knowledge discovery or interactive multi-objective optimization that involves trial and failure. It is often necessary to modify data preprocessing and model parameters until the result achieves the desired properties.


title: Cluster analysis

text: Besides the term "clustering", there is a number of terms with similar meanings, including "automatic classification", "numerical taxonomy", "botryology" (from Greek βότρυς "grape"), "typological analysis", and "community detection". The subtle differences are often in the use of the results: while in data mining, the resulting groups are the matter of interest, in automatic classification the resulting discriminative power is of interest.


title: Cluster analysis

text: Cluster analysis was originated in anthropology by Driver and Kroeber in 1932 and introduced to psychology by Joseph Zubin in 1938 and Robert Tryon in 1939 and famously used by Cattell beginning in 1943 for trait theory classification in personality psychology.


title: Cluster analysis

text: The notion of a "cluster" cannot be precisely defined, which is one of the reasons why there are so many clustering algorithms. There is a common denominator: a group of data objects. However, different researchers employ different cluster models, and for each of these cluster models again different algorithms can be given. The notion of a cluster, as found by different algorithms, varies significantly in its properties. Understanding these "cluster models" is key to understanding the differences between the various algorithms. Typical cluster models include:


title: Cluster analysis

text: A "clustering" is essentially a set of such clusters, usually containing all objects in the data set. Additionally, it may specify the relationship of the clusters to each other, for example, a hierarchy of clusters embedded in each other. Clusterings can be roughly distinguished as:


title: Cluster analysis

text: As listed above, clustering algorithms can be categorized based on their cluster model. The following overview will only list the most prominent examples of clustering algorithms, as there are possibly over 100 published clustering algorithms. Not all provide models for their clusters and can thus not easily be categorized. An overview of algorithms explained in Wikipedia can be found in the list of statistics algorithms.


title: Cluster analysis

text: There is no objectively "correct" clustering algorithm, but as it was noted, "clustering is in the eye of the beholder." The most appropriate clustering algorithm for a particular problem often needs to be chosen experimentally, unless there is a mathematical reason to prefer one cluster model over another. An algorithm that is designed for one kind of model will generally fail on a data set that contains a radically different kind of model. For example, k-means cannot find non-convex clusters.


title: Cluster analysis

text: Connectivity-based clustering, also known as "hierarchical clustering", is based on the core idea of objects being more related to nearby objects than to objects farther away. These algorithms connect "objects" to form "clusters" based on their distance. A cluster can be described largely by the maximum distance needed to connect parts of the cluster. At different distances, different clusters will form, which can be represented using a dendrogram, which explains where the common name "hierarchical clustering" comes from: these algorithms do not provide a single partitioning of the data set, but instead provide an extensive hierarchy of clusters that merge with each other at certain distances. In a dendrogram, the y-axis marks the distance at which the clusters merge, while the objects are placed along the x-axis such that the clusters don't mix.


title: Cluster analysis

text: Connectivity-based clustering is a whole family of methods that differ by the way distances are computed. Apart from the usual choice of distance functions, the user also needs to decide on the linkage criterion (since a cluster consists of multiple objects, there are multiple candidates to compute the distance) to use. Popular choices are known as single-linkage clustering (the minimum of object distances), complete linkage clustering (the maximum of object distances), and UPGMA or WPGMA ("Unweighted or Weighted Pair Group Method with Arithmetic Mean", also known as average linkage clustering). Furthermore, hierarchical clustering can be agglomerative (starting with single elements and aggregating them into clusters) or divisive (starting with the complete data set and dividing it into partitions).


title: Cluster analysis

text: These methods will not produce a unique partitioning of the data set, but a hierarchy from which the user still needs to choose appropriate clusters. They are not very robust towards outliers, which will either show up as additional clusters or even cause other clusters to merge (known as "chaining phenomenon", in particular with single-linkage clustering). In the general case, the complexity is formula_1 for agglomerative clustering and formula_2 for divisive clustering, which makes them too slow for large data sets. For some special cases, optimal efficient methods (of complexity formula_3) are known: SLINK for single-linkage and CLINK for complete-linkage clustering.


title: Cluster analysis

text: In centroid-based clustering, each cluster is represented by a central vector, which is not necessarily a member of the data set. When the number of clusters is fixed to "k", "k"-means clustering gives a formal definition as an optimization problem: find the "k" cluster centers and assign the objects to the nearest cluster center, such that the squared distances from the cluster are minimized.


title: Cluster analysis

text: The optimization problem itself is known to be NP-hard, and thus the common approach is to search only for approximate solutions. A particularly well known approximate method is Lloyd's algorithm, often just referred to as ""k-means algorithm"" (although another algorithm introduced this name). It does however only find a local optimum, and is commonly run multiple times with different random initializations. Variations of "k"-means often include such optimizations as choosing the best of multiple runs, but also restricting the centroids to members of the data set ("k"-medoids), choosing medians ("k"-medians clustering), choosing the initial centers less randomly ("k"-means++) or allowing a fuzzy cluster assignment (fuzzy c-means).


title: Cluster analysis

text: Most "k"-means-type algorithms require the number of clusters – "k" – to be specified in advance, which is considered to be one of the biggest drawbacks of these algorithms. Furthermore, the algorithms prefer clusters of approximately similar size, as they will always assign an object to the nearest centroid. This often leads to incorrectly cut borders of clusters (which is not surprising since the algorithm optimizes cluster centers, not cluster borders).


title: Cluster analysis

text: K-means has a number of interesting theoretical properties. First, it partitions the data space into a structure known as a Voronoi diagram. Second, it is conceptually close to nearest neighbor classification, and as such is popular in machine learning. Third, it can be seen as a variation of model based clustering, and Lloyd's algorithm as a variation of the Expectation-maximization algorithm for this model discussed below.


title: Cluster analysis

text: Centroid-based clustering problems such as "k"-means and "k"-medoids are special cases of the uncapacitated, metric facility location problem, a canonical problem in the operations research and computational geometry communities. In a basic facility location problem (of which there are numerous variants that model more elaborate settings), the task is to find the best warehouse locations to optimally service a given set of consumers. One may view "warehouses" as cluster centroids and "consumer locations" as the data to be clustered. This makes it possible to apply the well-developed algorithmic solutions from the facility location literature to the presently considered centroid-based clustering problem.


title: Cluster analysis

text: The clustering model most closely related to statistics is based on distribution models. Clusters can then easily be defined as objects belonging most likely to the same distribution. A convenient property of this approach is that this closely resembles the way artificial data sets are generated: by sampling random objects from a distribution.


title: Cluster analysis

text: While the theoretical foundation of these methods is excellent, they suffer from one key problem known as overfitting, unless constraints are put on the model complexity. A more complex model will usually be able to explain the data better, which makes choosing the appropriate model complexity inherently difficult.


title: Cluster analysis

text: One prominent method is known as Gaussian mixture models (using the expectation-maximization algorithm). Here, the data set is usually modeled with a fixed (to avoid overfitting) number of Gaussian distributions that are initialized randomly and whose parameters are iteratively optimized to better fit the data set. This will converge to a local optimum, so multiple runs may produce different results. In order to obtain a hard clustering, objects are often then assigned to the Gaussian distribution they most likely belong to; for soft clusterings, this is not necessary.


title: Cluster analysis

text: Distribution-based clustering produces complex models for clusters that can capture correlation and dependence between attributes. However, these algorithms put an extra burden on the user: for many real data sets, there may be no concisely defined mathematical model (e.g. assuming Gaussian distributions is a rather strong assumption on the data).


title: Cluster analysis

text: In density-based clustering, clusters are defined as areas of higher density than the remainder of the data set. Objects in sparse areas – that are required to separate clusters – are usually considered to be noise and border points.


title: Cluster analysis

text: The most popular density based clustering method is DBSCAN. In contrast to many newer methods, it features a well-defined cluster model called "density-reachability". Similar to linkage based clustering, it is based on connecting points within certain distance thresholds. However, it only connects points that satisfy a density criterion, in the original variant defined as a minimum number of other objects within this radius. A cluster consists of all density-connected objects (which can form a cluster of an arbitrary shape, in contrast to many other methods) plus all objects that are within these objects' range. Another interesting property of DBSCAN is that its complexity is fairly low – it requires a linear number of range queries on the database – and that it will discover essentially the same results (it is deterministic for core and noise points, but not for border points) in each run, therefore there is no need to run it multiple times. OPTICS is a generalization of DBSCAN that removes the need to choose an appropriate value for the range parameter formula_4, and produces a hierarchical result related to that of linkage clustering. DeLi-Clu, Density-Link-Clustering combines ideas from single-linkage clustering and OPTICS, eliminating the formula_4 parameter entirely and offering performance improvements over OPTICS by using an R-tree index.


title: Cluster analysis

text: The key drawback of DBSCAN and OPTICS is that they expect some kind of density drop to detect cluster borders. On data sets with, for example, overlapping Gaussian distributions – a common use case in artificial data – the cluster borders produced by these algorithms will often look arbitrary, because the cluster density decreases continuously. On a data set consisting of mixtures of Gaussians, these algorithms are nearly always outperformed by methods such as EM clustering that are able to precisely model this kind of data.


title: Cluster analysis

text: Mean-shift is a clustering approach where each object is moved to the densest area in its vicinity, based on kernel density estimation. Eventually, objects converge to local maxima of density. Similar to k-means clustering, these "density attractors" can serve as representatives for the data set, but mean-shift can detect arbitrary-shaped clusters similar to DBSCAN. Due to the expensive iterative procedure and density estimation, mean-shift is usually slower than DBSCAN or k-Means. Besides that, the applicability of the mean-shift algorithm to multidimensional data is hindered by the unsmooth behaviour of the kernel density estimate, which results in over-fragmentation of cluster tails.


title: Cluster analysis

text: The grid-based technique is used for a multi-dimensional data set. In this technique, we create a grid structure, and the comparison is performed on grids (also known as cells). The grid-based technique is fast and has low computational complexity. There are two types of grid-based clustering methods: STING and CLIQUE. Steps involved in grid-based clustering algorithm are:


title: Cluster analysis

text: In recent years, considerable effort has been put into improving the performance of existing algorithms. Among them are "CLARANS", and "BIRCH". With the recent need to process larger and larger data sets (also known as big data), the willingness to trade semantic meaning of the generated clusters for performance has been increasing. This led to the development of pre-clustering methods such as canopy clustering, which can process huge data sets efficiently, but the resulting "clusters" are merely a rough pre-partitioning of the data set to then analyze the partitions with existing slower methods such as k-means clustering.


title: Cluster analysis

text: For high-dimensional data, many of the existing methods fail due to the curse of dimensionality, which renders particular distance functions problematic in high-dimensional spaces. This led to new clustering algorithms for high-dimensional data that focus on subspace clustering (where only some attributes are used, and cluster models include the relevant attributes for the cluster) and correlation clustering that also looks for arbitrary rotated ("correlated") subspace clusters that can be modeled by giving a correlation of their attributes. Examples for such clustering algorithms are CLIQUE and SUBCLU.


title: Cluster analysis

text: Ideas from density-based clustering methods (in particular the DBSCAN/OPTICS family of algorithms) have been adapted to subspace clustering (HiSC, hierarchical subspace clustering and DiSH) and correlation clustering (HiCO, hierarchical correlation clustering, 4C using "correlation connectivity" and ERiC exploring hierarchical density-based correlation clusters).


title: Cluster analysis

text: Several different clustering systems based on mutual information have been proposed. One is Marina Meilă's "variation of information" metric; another provides hierarchical clustering. Using genetic algorithms, a wide range of different fit-functions can be optimized, including mutual information. Also belief propagation, a recent development in computer science and statistical physics, has led to the creation of new types of clustering algorithms.


title: Cluster analysis

text: Evaluation (or "validation") of clustering results is as difficult as the clustering itself. Popular approaches involve ""internal"" evaluation, where the clustering is summarized to a single quality score, ""external"" evaluation, where the clustering is compared to an existing "ground truth" classification, ""manual"" evaluation by a human expert, and ""indirect"" evaluation by evaluating the utility of the clustering in its intended application.


title: Cluster analysis

text: Internal evaluation measures suffer from the problem that they represent functions that themselves can be seen as a clustering objective. For example, one could cluster the data set by the Silhouette coefficient; except that there is no known efficient algorithm for this. By using such an internal measure for evaluation, one rather compares the similarity of the optimization problems, and not necessarily how useful the clustering is.


title: Cluster analysis

text: External evaluation has similar problems: if we have such "ground truth" labels, then we would not need to cluster; and in practical applications we usually do not have such labels. On the other hand, the labels only reflect one possible partitioning of the data set, which does not imply that there does not exist a different, and maybe even better, clustering.


title: Cluster analysis

text: Neither of these approaches can therefore ultimately judge the actual quality of a clustering, but this needs human evaluation, which is highly subjective. Nevertheless, such statistics can be quite informative in identifying bad clusterings, but one should not dismiss subjective human evaluation.


title: Cluster analysis

text: When a clustering result is evaluated based on the data that was clustered itself, this is called internal evaluation. These methods usually assign the best score to the algorithm that produces clusters with high similarity within a cluster and low similarity between clusters. One drawback of using internal criteria in cluster evaluation is that high scores on an internal measure do not necessarily result in effective information retrieval applications. Additionally, this evaluation is biased towards algorithms that use the same cluster model. For example, k-means clustering naturally optimizes object distances, and a distance-based internal criterion will likely overrate the resulting clustering.


title: Cluster analysis

text: Therefore, the internal evaluation measures are best suited to get some insight into situations where one algorithm performs better than another, but this shall not imply that one algorithm produces more valid results than another. Validity as measured by such an index depends on the claim that this kind of structure exists in the data set. An algorithm designed for some kind of models has no chance if the data set contains a radically different set of models, or if the evaluation measures a radically different criterion. For example, k-means clustering can only find convex clusters, and many evaluation indexes assume convex clusters. On a data set with non-convex clusters neither the use of "k"-means, nor of an evaluation criterion that assumes convexity, is sound.


title: Cluster analysis

text: More than a dozen of internal evaluation measures exist, usually based on the intuition that items in the same cluster should be more similar than items in different clusters. For example, the following methods can be used to assess the quality of clustering algorithms based on internal criterion:


title: Cluster analysis

text: In external evaluation, clustering results are evaluated based on data that was not used for clustering, such as known class labels and external benchmarks. Such benchmarks consist of a set of pre-classified items, and these sets are often created by (expert) humans. Thus, the benchmark sets can be thought of as a gold standard for evaluation. These types of evaluation methods measure how close the clustering is to the predetermined benchmark classes. However, it has recently been discussed whether this is adequate for real data, or only on synthetic data sets with a factual ground truth, since classes can contain internal structure, the attributes present may not allow separation of clusters or the classes may contain anomalies. Additionally, from a knowledge discovery point of view, the reproduction of known knowledge may not necessarily be the intended result. In the special scenario of constrained clustering, where meta information (such as class labels) is used already in the clustering process, the hold-out of information for evaluation purposes is non-trivial.


title: Cluster analysis

text: A number of measures are adapted from variants used to evaluate classification tasks. In place of counting the number of times a class was correctly assigned to a single data point (known as true positives), such "pair counting" metrics assess whether each pair of data points that is truly in the same cluster is predicted to be in the same cluster.


title: Cluster analysis

text: One issue with the Rand index is that false positives and false negatives are equally weighted. This may be an undesirable characteristic for some clustering applications. The F-measure addresses this concern, as does the chance-corrected adjusted Rand index.


title: Cluster analysis

text: To measure cluster tendency is to measure to what degree clusters exist in the data to be clustered, and may be performed as an initial test, before attempting clustering. One way to do this is to compare the data against random data. On average, random data should not have clusters.


title: Hugh Grant filmography

text: Hugh Grant is a British actor. His career spans four decades. He has been recognised as an international film star since 1994, and has received a Golden Globe Award and a British Academy Film Award. His films have collectively grossed more than $2.4 billion worldwide. The following are his film and television credits.


title: 2022–23 Formula E World Championship

text: The 2022–23 ABB FIA Formula E World Championship is due to be the ninth season of the FIA Formula E championship.


title: 2022–23 Formula E World Championship

text: While it was originally planned to return to the format of starting the season at the end of a year and running until the European summer, the championship remained with a start early in the year for the third season running.


title: 2022–23 Formula E World Championship

text: The championship will begin its third generation of technical regulations. This will see the introduction of a completely new car, with the new chassis again built by Spark Racing Technology. The cars will now be powered by two powertrains, with a second one added to the rear axle, increasing the maximum power output from 250kW to 600kW. This new car will be the smallest and lightest car ever used in the championship, with its weight being reduced by 60kg. Increased power and reduced weight will see the maximum speeds climb to over 320km/h. The addition of a second powertrain will increase regenerative ability from 25% to 40%, enough that the new cars will have no hydraulic rear brakes. The tyre supplier will change from Michelin to Hankook.


title: 2022–23 Formula E World Championship

text: In addition to the new car, a number of race format changes were also made. The format will return to races run until a set number of laps instead of a fixed time, with safety cars and full course cautions increasing the laps needed to finish the race. Fanboost was also announced to be removed for this season.


title: 2022–23 Formula E World Championship

text: Fast recharging will be tested in select races under a so-called "attack charge" format. During an attack charge race, each driver is required to pit for a recharge during the course of the race, but they receive two additional attack mode activation periods for doing so.


title: Moussa Dembélé (French footballer)

text: Moussa Dembélé (born 12 July 1996) is a French professional footballer who plays as a striker for Ligue 1 club Lyon.


title: Moussa Dembélé (French footballer)

text: Developed at Paris Saint-Germain, Dembélé made his professional debut for Fulham in the Premier League in November 2013. He scored 19 goals in 64 matches for Fulham before joining Celtic in 2016. Dembélé helped Celtic win consecutive Scottish domestic trebles, before moving to Lyon in August 2018.


title: Moussa Dembélé (French footballer)

text: Dembélé has earned over 50 caps for France at youth level, and in October 2016, was called-up to the under-21 team.


title: Moussa Dembélé (French footballer)

text: In 2012, Dembélé signed for Fulham at the age of 16 from Paris Saint-Germain. He subsequently became a regular in the Fulham under-18 team and won the Premier Academy League in his first season with the club. He signed his first contract with Fulham in July 2013, lasting until the summer of 2015.


title: Moussa Dembélé (French footballer)

text: Dembélé was an unused substitute for Fulham's Premier League match against Swansea City on 23 November 2013. He made his competitive debut one week later against West Ham United, coming on as a substitute in the 83rd minute for Kieran Richardson in a 3–0 loss at the Boleyn Ground, but finished the match with an injury. On 30 March 2014, he started for the first time in the Premier League as Fulham lost 3–1 at home to Everton. The club ended the season with relegation to the Championship. He was still eligible for the academy teams, and scored in both legs of the 2014 FA Youth Cup Final.


title: Moussa Dembélé (French footballer)

text: On 28 October 2014, in the fourth round of the League Cup, Dembélé scored his first professional goals, a brace which put Fulham 2–0 up at home against Derby County. However, the match ended in a 5–2 defeat.


title: Moussa Dembélé (French footballer)

text: Dembélé scored his first league goal for Fulham in a 2–1 win against Blackburn Rovers on 13 September 2015, and scored a brace against Reading on 24 October as Fulham overcame a 2–0 deficit to win 4–2. Dembélé scored twice again in the following match a week later as Fulham beat Bristol City 4–1 at Ashton Gate. By the end of the calendar year, he had totalled ten goals for the season.


title: Moussa Dembélé (French footballer)

text: On 28 June 2016, Scottish Premiership club Celtic signed Dembélé on a four-year contract. He made his debut on 12 July, his 20th birthday, partnering Leigh Griffiths up front in a 1–0 loss at Gibraltar's Lincoln Red Imps in the first leg of the second qualifying round of the 2016–17 UEFA Champions League.


title: Moussa Dembélé (French footballer)

text: On 3 August, Dembélé scored his first goal for the club with a 92nd-minute penalty against Astana (after having won the penalty himself) to put Celtic into the Champions League play-offs. He followed this up by scoring twice against Motherwell in the Scottish League Cup last 16 on 10 August, as Celtic ran out 5–0 winners. He also scored the fourth goal in a 5–2 win over Hapoel Be'er Sheva the following week.


title: Moussa Dembélé (French footballer)

text: On 10 September, as his club defeated Rangers 5–1 at Parkhead, he scored Celtic's first Old Firm derby league hat-trick since Stevie Chalmers in 1966, and the first Celtic player to score a hat-trick in any game against Rangers since Harry Hood in 1973. It was also considered to be a "perfect hat-trick", given his goals were scored with his head, his right foot and his left foot. On 24 September, Dembélé scored twice in a 6–1 win over Kilmarnock, then four days later, he scored his first two Champions League group stage goals against Manchester City, as Celtic finished the match with a 3–3 draw. His goalscoring form at that time saw him named the Premiership Player of the Month for September.


title: Moussa Dembélé (French footballer)

text: Dembélé scored a late winner in a 1–0 victory over Rangers on 23 October, sending Celtic to the League Cup final, and scored again in the final against Aberdeen, scoring from the penalty spot as Celtic won 3–0 for his first senior honour.


title: Moussa Dembélé (French footballer)

text: Dembélé was linked with a move to Chelsea during the January 2017 transfer window. Coincidentally, he was sent to a hospital in Chelsea for medical treatment on 31 January, the final day of the transfer window. However, Chelsea manager Antonio Conte dismissed the rumours, stating that he had never spoke to his board about Dembélé. Later that day, Dembélé confirmed he was staying by tweeting a picture of himself celebrating a goal above an image of Leonardo DiCaprio's character Jordan Belfort from the film "The Wolf of Wall Street".


title: Moussa Dembélé (French footballer)

text: Dembélé scored back-to-back hat-tricks in February 2017, in 5–2 and 6–0 victories over St Johnstone in the Scottish Premiership and Inverness Caledonian Thistle in the Scottish Cup, respectively with his final goal in the 5–2 victory over St Johnstone awarding him Scotland's 'Goal of the Season' for 2016–17 as well as the goal being nominated for the prestigious FIFA Puskas Award. The goal had 25 passes, a Rabona and a back heel, with every player getting a touch on the ball and a clinical finish at the end which completed Dembélé's hat-trick.


title: Moussa Dembélé (French footballer)

text: Throughout the 2017–18 Scottish Premiership season, Dembélé suffered three injuries keeping him out of action for a total of 82 days. Due to the injuries, Dembélé struggled to find the form of his 2016–17 Scottish Premiership season, but still managed 16 goals in 39 appearances, contributing greatly to Celtic, thus completing their second consecutive domestic treble.


title: Moussa Dembélé (French footballer)

text: On 31 August 2018, Dembélé signed for Ligue 1 club Olympique Lyonnais on a five-year contract. The transfer fee was reported as €22 million (£19.7 million), payable to Celtic over 5 years. He made his competitive debut for the club on 15 September in the 2–2 Ligue 1 draw at Caen, playing the full 90 minutes. On 26 September, Dembélé scored his first competitive goals for the club when he scored two first-half goals in the 3–0 Ligue 1 away win over Dijon. On 3 February 2019, he headed in the equaliser and later won the penalty which Nabil Fekir converted in the 2–1 Ligue 1 home win over Paris Saint-Germain and handed the leaders their first 2018–19 Ligue 1 defeat.


title: Moussa Dembélé (French footballer)

text: On 15 August 2020, Dembélé came on as a substitute on the 75th minute, scoring two goals against Manchester City in the 2019–20 UEFA Champions League quarter-finals which Lyon won 3–1 and progressed to the semi-finals.


title: Moussa Dembélé (French footballer)

text: On 13 January 2021, Dembélé moved from Lyon to Atlético Madrid, on a 'loan with option to buy' deal until the end of the season. According to "Marca", the transfer fee for the loan reportedly amounted to €1.5M and Atlético reserved a purchase option for €33.5M, plus a variable additional amount depending on the player fulfilling certain goals.


title: Moussa Dembélé (French footballer)

text: Dembélé was part of the France under-19 squad which reached the semi-finals of the 2015 European Championship, scoring in group stage victories over Ukraine and hosts Greece. He made his debut for the under-21 team in October 2016, coming on as a second-half substitute and scoring one of France's goals in a 5–1 win over Georgia. On 13 November 2017, Dembélé scored an 11-minute hat-trick for the France U-21 team in a 3–1 away win over Slovenia, which was the forward's 7th goal in 10 games for the international outfit at the time. The hat-trick was also the quickest treble by a player in any French age group for 15 years.


title: Mean value theorem

text: In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.


title: Mean value theorem

text: More precisely, the theorem states that if formula_1 is a continuous function on the closed interval formula_2 and differentiable on the open interval formula_3, then there exists a point formula_4 in formula_3 such that the tangent at formula_4 is parallel to the secant line through the endpoints formula_7 and formula_8, that is,


title: Mean value theorem

text: A special case of this theorem for inverse interpolation of the sine was first described by Parameshvara (1380–1460), from the Kerala School of Astronomy and Mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II. A restricted form of the theorem was proved by Michel Rolle in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern form was stated and proved by Augustin Louis Cauchy in 1823. Many variations of this theorem have been proved since then.


title: Mean value theorem

text: Let formula_10 be a continuous function on the closed interval and differentiable on the open interval where Then there exists some formula_4 in formula_3 such that


title: Mean value theorem

text: The mean value theorem is a generalization of Rolle's theorem, which assumes formula_14, so that the right-hand side above is zero.


title: Mean value theorem

text: The mean value theorem is still valid in a slightly more general setting. One only needs to assume that formula_10 is continuous on formula_16, and that for every formula_17 in formula_3 the limit


title: Mean value theorem

text: exists as a finite number or equals formula_20 or formula_21. If finite, that limit equals formula_22. An example where this version of the theorem applies is given by the real-valued cube root function mapping formula_23, whose derivative tends to infinity at the origin.


title: Mean value theorem

text: Note that the theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define formula_24 for all real Then


title: Mean value theorem

text: The expression formula_27 gives the slope of the line joining the points formula_28 and formula_29, which is a chord of the graph of formula_1, while formula_22 gives the slope of the tangent to the curve at the point formula_32. Thus the mean value theorem says that given any chord of a smooth curve, we can find a point on the curve lying between the end-points of the chord such that the tangent of the curve at that point is parallel to the chord. The following proof illustrates this idea.


title: Mean value theorem

text: Define formula_33, where formula_34 is a constant. Since formula_1 is continuous on formula_16 and differentiable on formula_3, the same is true for formula_38. We now want to choose formula_34 so that formula_38 satisfies the conditions of Rolle's theorem. Namely


title: Mean value theorem

text: By Rolle's theorem, since formula_38 is differentiable and formula_43, there is some formula_4 in formula_3 for which formula_46 , and it follows from the equality formula_33 that,


title: Mean value theorem

text: Theorem 1: Assume that "f" is a continuous, real-valued function, defined on an arbitrary interval "I" of the real line. If the derivative of "f" at every interior point of the interval "I" exists and is zero, then "f" is constant in the interior.


title: Mean value theorem

text: Proof: Assume the derivative of "f" at every interior point of the interval "I" exists and is zero. Let ("a", "b") be an arbitrary open interval in "I". By the mean value theorem, there exists a point "c" in ("a", "b") such that


title: Mean value theorem

text: This implies that . Thus, "f" is constant on the interior of "I" and thus is constant on "I" by continuity. (See below for a multivariable version of this result.)


title: Mean value theorem

text: Theorem 2: If "f' "("x") = "g' "("x") for all "x" in an interval ("a", "b") of the domain of these functions, then "f - g" is constant, i.e. "f = g + c" where "c" is a constant on ("a", "b").


title: Mean value theorem

text: Proof: Let "F = f − g", then "F' = f' − g' =" 0 on the interval ("a", "b"), so the above theorem 1 tells that "F = f − g" is a constant "c" or "f = g + c".


title: Mean value theorem

text: Theorem 3: If "F" is an antiderivative of "f" on an interval "I", then the most general antiderivative of "f" on "I" is "F(x) + c" where "c" is a constant.


title: Mean value theorem

text: Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem. It states: if the functions formula_1 and formula_38 are both continuous on the closed interval formula_16 and differentiable on the open interval formula_3, then there exists some formula_54, such that


title: Mean value theorem

text: which is parallel to the line defined by the points formula_60 and formula_61. However, Cauchy's theorem does not claim the existence of such a tangent in all cases where formula_60 and formula_61 are distinct points, since it might be satisfied only for some value formula_4 with formula_65, in other words a value for which the mentioned curve is stationary; in such points no tangent to the curve is likely to be defined at all. An example of this situation is the curve given by


title: Mean value theorem

text: which on the interval formula_67 goes from the point formula_68 to formula_69, yet never has a horizontal tangent; however it has a stationary point (in fact a cusp) at formula_70.


title: Mean value theorem

text: Cauchy's mean value theorem can be used to prove L'Hôpital's rule. The mean value theorem is the special case of Cauchy's mean value theorem when formula_71.


title: Mean value theorem

text: The proof of Cauchy's mean value theorem is based on the same idea as the proof of the mean value theorem.


title: Mean value theorem

text: Assume that formula_72 and formula_73 are differentiable functions on formula_3 that are continuous on formula_16. Define


title: Mean value theorem

text: and if we place formula_80, we get Cauchy's mean value theorem. If we place formula_80 and formula_82 we get Lagrange's mean value theorem.


title: Mean value theorem

text: The proof of the generalization is quite simple: each of formula_83 and formula_84 are determinants with two identical rows, hence formula_85. The Rolle's theorem implies that there exists formula_86 such that formula_87.


title: Mean value theorem

text: The mean value theorem generalizes to real functions of multiple variables. The trick is to use parametrization to create a real function of one variable, and then apply the one-variable theorem.


title: Mean value theorem

text: Let formula_88 be an open subset of formula_89, and let formula_90 be a differentiable function. Fix points formula_91 such that the line segment between formula_92 lies in formula_88, and define formula_94. Since formula_38 is a differentiable function in one variable, the mean value theorem gives:


title: Mean value theorem

text: for some formula_4 between 0 and 1. But since formula_98 and formula_99, computing formula_100 explicitly we have:


title: Mean value theorem

text: where formula_102 denotes a gradient and formula_103 a dot product. Note that this is an exact analog of the theorem in one variable (in the case formula_104 this "is" the theorem in one variable). By the Cauchy–Schwarz inequality, the equation gives the estimate:


title: Mean value theorem

text: In particular, when the partial derivatives of formula_1 are bounded, formula_1 is Lipschitz continuous (and therefore uniformly continuous).


title: Mean value theorem

text: As an application of the above, we prove that formula_1 is constant if the open subset formula_88 is connected and every partial derivative of formula_1 is 0. Pick some point formula_111, and let formula_112. We want to show formula_113 for every formula_114. For that, let formula_115. Then "E" is closed and nonempty. It is open too: for every formula_116 ,


title: Mean value theorem

text: for every formula_118 in some neighborhood of formula_17. (Here, it is crucial that formula_17 and formula_118 are sufficiently close to each other.) Since formula_88 is connected, we conclude formula_123.


title: Mean value theorem

text: The above arguments are made in a coordinate-free manner; hence, they generalize to the case when formula_88 is a subset of a Banach space.


title: Mean value theorem

text: There is no exact analog of the mean value theorem for vector-valued functions (see below). However, there is an inequality which can be applied to many of the same situations to which the mean value theorem is applicable in the one dimensional case:


title: Mean value theorem

text: The theorem follows from the mean value theorem. Indeed, take formula_125. Then formula_126 is real-valued and thus, by the mean value theorem,


title: Mean value theorem

text: for some formula_128. Now, formula_129 and formula_130 Hence, using the Cauchy–Schwarz inequality, from the above equation, we get:


title: Mean value theorem

text: If formula_132, the theorem is trivial (any "c" works). Otherwise, dividing both sides by formula_133 yields the theorem. formula_134


title: Mean value theorem

text: Jean Dieudonné in his classic treatise "Foundations of Modern Analysis" discards the mean value theorem and replaces it by mean inequality (which is given below) as the proof is not constructive and one cannot find the mean value and in applications one only needs mean inequality. Serge Lang in "Analysis I "uses the mean value theorem, in integral form, as an instant reflex but this use requires the continuity of the derivative. If one uses the Henstock–Kurzweil integral one can have the mean value theorem in integral form without the additional assumption that derivative should be continuous as every derivative is Henstock–Kurzweil integrable.


title: Mean value theorem

text: The reason why there is no analog of mean value equality is the following: If is a differentiable function (where is open) and if , is the line segment in question (lying inside ), then one can apply the above parametrization procedure to each of the component functions of "f" (in the above notation set ). In doing so one finds points on the line segment satisfying


title: Mean value theorem

text: Then formula_138, but formula_139 and formula_140 are never simultaneously zero as formula_17 ranges over formula_142.


title: Mean value theorem

text: In fact, the above statement suffices for many applications and can be proved directly as follows. (We shall write formula_1 for formula_144 for readability.) First assume formula_1 is differentiable at formula_146 too. If formula_147 is unbounded on formula_148, there is nothing to prove. Thus, assume formula_149. Let formula_150 be some real number. Let


title: Mean value theorem

text: We want to show formula_152. By continuity of formula_1, the set formula_154 is closed. It is also nonempty as formula_155 is in it. Hence, the set formula_154 has the largest element formula_157. If formula_158, then formula_152 and we are done. Thus suppose otherwise. For formula_160,


title: Mean value theorem

text: Let formula_162 be such that formula_163. By the differentiability of formula_1 at formula_165 (note formula_157 may be 0), if formula_167 is sufficiently close to formula_157, the first term is formula_169. The second term is formula_170. The third term is formula_171. Hence, summing the estimates up, we get: formula_172, a contradiction to the maximality of formula_157. Hence, formula_174 and that means:


title: Mean value theorem

text: Since formula_176 is arbitrary, this then implies the assertion. Finally, if formula_1 is not differentiable at formula_146, let formula_179 and apply the first case to formula_1 restricted on formula_181, giving us:


title: Mean value theorem

text: For some applications of mean value inequality to establish basic results in calculus, see also Calculus on Euclidean space#Basic notions.


title: Mean value theorem

text: A certain type of generalization of the mean value theorem to vector-valued functions is obtained as follows: Let be a continuously differentiable real-valued function defined on an open interval , and let as well as be points of . The mean value theorem in one variable tells us that there exists some between 0 and 1 such that


title: Mean value theorem

text: On the other hand, we have, by the fundamental theorem of calculus followed by a change of variables,


title: Mean value theorem

text: The mean value inequality can then be obtained as a corollary of the above proposition (though under the assumption the derivatives are continuous).


title: Mean value theorem

text: Where one of the above conditions is not satisfied, Mean Value Theorem is not valid in general, and so it cannot be applied.


title: Mean value theorem

text: The necessity of the first condition can be seen by the counterexample where the function formula_193 on [-1,1] is not differentiable.


title: Mean value theorem

text: The necessity of the second condition can be seen by the counterexample where the function formula_194


title: Mean value theorem

text: But not criteria 2 since formula_198 and formula_199 for all formula_200 so no such formula_201 exists


title: Mean value theorem

text: In general, if "f" : ["a", "b"] → R is continuous and "g" is an integrable function that does not change sign on ["a", "b"], then there exists "c" in ("a", "b") such that


title: Mean value theorem

text: Suppose "f" : ["a", "b"] → R is continuous and "g" is a nonnegative integrable function on ["a", "b"]. By the extreme value theorem, there exists "m" and "M" such that for each "x" in ["a", "b"], formula_205 and formula_206. Since "g" is nonnegative,


title: Mean value theorem

text: By the intermediate value theorem, "f" attains every value of the interval ["m", "M"], so for some "c" in ["a", "b"]


title: Mean value theorem

text: There are various slightly different theorems called the second mean value theorem for definite integrals. A commonly found version is as follows:


title: Mean value theorem

text: Here formula_218 stands for formula_219, the existence of which follows from the conditions. Note that it is essential that the interval ("a", "b"] contains "b". A variant not having this requirement is:


title: Mean value theorem

text: If the function formula_88 returns a multi-dimensional vector, then the MVT for integration is not true, even if the domain of formula_88 is also multi-dimensional.


title: Mean value theorem

text: For example, consider the following 2-dimensional function defined on an formula_223-dimensional cube:


title: Mean value theorem

text: Let "X" and "Y" be non-negative random variables such that E["X"] < E["Y"] < ∞ and formula_229 (i.e. "X" is smaller than "Y" in the usual stochastic order). Then there exists an absolutely continuous non-negative random variable "Z" having probability density function


title: Mean value theorem

text: Let "g" be a measurable and differentiable function such that E["g"("X")], E["g"("Y")] < ∞, and let its derivative "g′" be measurable and Riemann-integrable on the interval ["x", "y"] for all "y" ≥ "x" ≥ 0. Then, E["g′"("Z")] is finite and


title: Mean value theorem

text: As noted above, the theorem does not hold for differentiable complex-valued functions. Instead, a generalization of the theorem is stated such:


title: Mean value theorem

text: Let "f" : Ω → C be a holomorphic function on the open convex set Ω, and let "a" and "b" be distinct points in Ω. Then there exist points "u", "v" on the interior of the line segment from "a" to "b" such that


title: Philip Hamilton

text: Philip Hamilton (January 22, 1782 – November 24, 1801) was the eldest child of Alexander Hamilton, the first U.S. Secretary of the Treasury, and Elizabeth Schuyler Hamilton. He died at age 19, fatally shot in a duel with George Eacker.


title: Philip Hamilton

text: Philip Hamilton was born in Albany, New York, on January 22, 1782. His father, Alexander Hamilton, was the first U.S. Secretary of the Treasury and one of the Founding Fathers of the United States. His mother, Elizabeth Schuyler Hamilton, was the daughter of Philip Schuyler, for whom he was named.


title: Philip Hamilton

text: Alexander Hamilton wrote that Philip's birth was "attended with all the omens of future greatness," and continued to express high expectations and hopes for the future of his firstborn:


title: Philip Hamilton

text: Late in 1791, at the age of nine, Philip was sent to attend a boarding school in Trenton, New Jersey, studying with William Frazer, an Episcopal clergyman and rector of St. Michael's Church. In early December that year, his father wrote encouragingly from Philadelphia: By 1794, his younger brother Alexander Jr., then eight years old, joined him at the boarding school.


title: Philip Hamilton

text: Philip enrolled in Columbia College, where his knowledge and enthusiasm were compared to that of his father, already a renowned alumnus. Robert Troup, a family friend who had been Alexander Hamilton's college roommate, wrote that Philip "was very promising in genius and acquirements, and Hamilton formed high expectations of his future greatness!" Troup wrote privately, however, that despite Hamilton's certainty that Philip was destined for greatness, "alas Philip is a sad rake and I have serious doubts whether he would ever be an honour to his family or his country."


title: Philip Hamilton

text: Philip graduated with honors from Columbia College in 1800, and went on to study law. His father prescribed rigorous study routines, including waking for study at 6 o'clock every day from April through September, and not later than 7 o'clock for the rest of the year, after which, "From the time he is dressed in the morning til nine o'clock (the time for breakfast excluded) he is to read law."