title: Hershey–Chase experiment

text: Hershey and Chase, along with others who had done related experiments, confirmed that DNA was the biomolecule that carried genetic information. Before that, Oswald Avery, Colin MacLeod, and Maclyn McCarty had shown that DNA led to the transformation of one strain of "Streptococcus pneumoniae" to another. The results of these experiments provided evidence that DNA was the biomolecule that carried genetic information.


title: Hershey–Chase experiment

text: Hershey and Chase needed to be able to examine different parts of the phages they were studying separately, so they needed to distinguish the phage subsections. Viruses were known to be composed of a protein shell and DNA, so they chose to uniquely label each with a different elemental isotope. This allowed each to be observed and analyzed separately. Since phosphorus is contained in DNA but not amino acids, radioactive phosphorus-32 was used to label the DNA contained in the T2 phage. Radioactive sulfur-35 was used to label the protein sections of the T2 phage, because sulfur is contained in protein but not DNA.


title: Hershey–Chase experiment

text: Hershey and Chase inserted the radioactive elements in the bacteriophages by adding the isotopes to separate media within which bacteria were allowed to grow for 4 hours before bacteriophage introduction. When the bacteriophages infected the bacteria, the progeny contained the radioactive isotopes in their structures. This procedure was performed once for the sulfur-labeled phages and once for phosphorus-labeled phages.


title: Hershey–Chase experiment

text: The labeled progeny were then allowed to infect unlabeled bacteria. The phage coats remained on the outside of the bacteria, while genetic material entered. Disruption of phage from the bacteria by agitation in a blender followed by centrifugation allowed for the separation of the phage coats from the bacteria. These bacteria were lysed to release phage progeny. The progeny of the phages that were labeled with radioactive phosphorus remained labeled, whereas the progeny of the phages labeled with radioactive sulfur were unlabeled. Thus, the Hershey–Chase experiment helped to confirm that DNA, not protein, is the genetic material.


title: Hershey–Chase experiment

text: Hershey and Chase showed that the introduction of deoxyribonuclease (referred to as DNase), an enzyme that breaks down DNA, into a solution containing the labeled bacteriophages did not introduce any P into the solution. This demonstrated that the phage is resistant to the enzyme while intact. Additionally, they were able to plasmolyze the bacteriophages so that they went into osmotic shock, which effectively created a solution containing most of the P and a heavier solution containing structures called "ghosts" that contained the S and the protein coat of the virus. It was found that these "ghosts" could adsorb to bacteria that were susceptible to T2, although they contained no DNA and were simply the remains of the original viral capsule. They concluded that the protein protected the DNA from DNase, but that once the two were separated and the phage was inactivated, the DNase could hydrolyze the phage DNA.


title: Hershey–Chase experiment

text: Hershey and Chase were also able to prove that the DNA from the phage is inserted into the bacteria shortly after the virus attaches to its host. Using a high-speed blender they were able to force the bacteriophages from the bacterial cells after adsorption. The lack of P-labeled DNA remaining in the solution after the bacteriophages had been allowed to adsorb to the bacteria showed that the phage DNA was transferred into the bacterial cell. The presence of almost all the radioactive S in the solution showed that the protein coat that protects the DNA before adsorption stayed outside the cell.


title: Hershey–Chase experiment

text: Hershey and Chase concluded that DNA, not protein, was the genetic material. They determined that a protective protein coat was formed around the bacteriophage, but that the internal DNA is what conferred its ability to produce progeny inside a bacterium. They showed that, in growth, protein has no function, while DNA has some function. They determined this from the amount of radioactive material remaining outside of the cell. Only 20% of the P remained outside the cell, demonstrating that it was incorporated with DNA in the cell's genetic material. All of the S in the protein coats remained outside the cell, showing it was not incorporated into the cell, and that protein is not the genetic material.


title: Hershey–Chase experiment

text: Hershey and Chase's experiment concluded that little sulfur-containing material entered the bacterial cell. However no specific conclusions can be made regarding whether material that is sulfur-free enters the bacterial cell after phage adsorption. Further research was necessary to conclude that it was solely bacteriophages' DNA that entered the cell and not a combination of protein and DNA where the protein did not contain any sulfur.


title: Hershey–Chase experiment

text: Hershey and Chase concluded that protein was not likely to be the hereditary genetic material. However, they did not make any conclusions regarding the specific function of DNA as hereditary material, and only said that it must have some undefined role.


title: Hershey–Chase experiment

text: Confirmation and clarity came a year later in 1953, when James D. Watson and Francis Crick correctly hypothesized, in their journal article "", the double helix structure of DNA, and suggested the copying mechanism by which DNA functions as hereditary material. Furthermore, Watson and Crick suggested that DNA, the genetic material, is responsible for the synthesis of the thousands of proteins found in cells. They had made this proposal based on the structural similarity that exists between the two macromolecules: both protein and DNA are linear sequences of monomers (amino acids and nucleotides, respectively).


title: Hershey–Chase experiment

text: Once the Hershey–Chase experiment was published, the scientific community generally acknowledged that DNA was the genetic code material. This discovery led to a more detailed investigation of DNA to determine its composition as well as its 3D structure. Using X-ray crystallography, the structure of DNA was discovered by James Watson and Francis Crick with the help of previously documented experimental evidence by Maurice Wilkins and Rosalind Franklin.


title: Hershey–Chase experiment

text: Knowledge of the structure of DNA led scientists to examine the nature of genetic coding and, in turn, understand the process of protein synthesis. George Gamow proposed that the genetic code was composed of sequences of three DNA base pairs known as triplets or codons which represent one of the twenty amino acids. Genetic coding helped researchers to understand the mechanism of gene expression, the process by which information from a gene is used in protein synthesis. Since then, much research has been conducted to modulate steps in the gene expression process. These steps include transcription, RNA splicing, translation, and post-translational modification which are used to control the chemical and structural nature of proteins. Moreover, genetic engineering gives engineers the ability to directly manipulate the genetic materials of organisms using recombinant DNA techniques. The first recombinant DNA molecule was created by Paul Berg in 1972 when he combined DNA from the monkey virus SV40 with that of the lambda phage.


title: Hershey–Chase experiment

text: Experiments on hereditary material during the time of the Hershey–Chase experiment often used bacteriophages as a model organism. Bacteriophages lend themselves to experiments on hereditary material because they incorporate their genetic material into their host cell's genetic material (making them useful tools), they multiply quickly, and they are easily collected by researchers.


title: Hershey–Chase experiment

text: The Hershey–Chase experiment, its predecessors, such as the Avery–MacLeod–McCarty experiment, and successors served to unequivocally establish that hereditary information was carried by DNA. This finding has numerous applications in forensic science, crime investigation and genealogy. It provided the background knowledge for further applications in DNA forensics, where DNA fingerprinting uses data originating from DNA, not protein sources, to deduce genetic variation.


title: Battle of Shrewsbury

text: The Battle of Shrewsbury was a battle fought on 21 July 1403, waged between an army led by the Lancastrian King Henry IV and a rebel army led by Henry "Harry Hotspur" Percy from Northumberland. The battle, the first in which English archers fought each other on English soil, reaffirmed the effectiveness of the longbow and ended the Percy challenge to King Henry IV of England.


title: Battle of Shrewsbury

text: Part of the fighting is believed to have taken place at what is now Battlefield, Shropshire, England, three miles (5 km) north of the centre of Shrewsbury. It is marked today by Battlefield Church and Battlefield Heritage Park.


title: Battle of Shrewsbury

text: The Percys had previously supported Henry IV in a war against King Richard II of England, which ended when Henry IV took the throne in 1399. The Percys subsequently supported Henry IV in Wales, early in the rebellion of Owain Glyndŵr, and in Scotland, in both negotiations and conflict against the Scots.


title: Battle of Shrewsbury

text: King Henry IV had been supported by a number of wealthy landowners to whom he had promised land, money and royal favour in return for their continued support. When the war ended, lands in and around Cumberland promised to the Percys were instead given to a rival. The promised money never materialised, and so the Percys revolted. Henry Percy, 1st Earl of Northumberland, and Thomas Percy, 1st Earl of Worcester, publicly renounced their allegiance to King Henry IV. They charged him with perjury because he claimed the throne in addition to his old lands and titles, taxed the clergy despite his promise not to without the consent of Parliament, imprisoned and murdered King Richard II, did not allow a free Parliamentary election, and refused to pay a just ransom to Owain Glyndŵr, who was then holding Edmund Mortimer. The King also retained custody of the Scottish nobles captured at Homildon Hill as prisoners of war rather than permitting the Percys to release them for ransom.


title: Battle of Shrewsbury

text: Henry Percy, nicknamed "Hotspur," raised a group initially of about 200 retainers in early July 1403, and started the long march south to meet his uncle, Thomas Percy. Some nobles joined him, such as Lord Bardolf, but he recruited most of his army in Cheshire, an area hostile to Henry IV, which provided many experienced soldiers, notably its Cheshire archers, some of whom had served as Richard II's bodyguard. Henry Percy may have hoped for reinforcements from Wales under the Prince of Wales, Owain Glyndŵr, but he was disappointed. Glyndŵr, at the time fighting in Carmarthenshire, was unaware that Hotspur had acted. Some Welsh forces from the Cheshire borders may have joined him. The rebels then marched towards Shrewsbury, the heavily defended county town in Shropshire.


title: Battle of Shrewsbury

text: King Henry IV only became aware of the Percy forces on 12 July, apparently while he was marching an army north to assist the Percys against the Scots, receiving the news at Burton-on-Trent. He may have anticipated the Percys' change of heart, but instantly altered his plans to meet the immediate threat posed by the Percys. He changed direction and marched west towards Shrewsbury with his army, arriving before the Percys could capture the town.


title: Battle of Shrewsbury

text: Both forces arrived in the Shrewsbury area on 20 July and set up camp to the north and south of the Severn River, which loops around the town. Hotspur based himself initially at the house of William Betton, his army camping close to the town. The next day the King's forces crossed the River Severn at Uffington, about a mile to the east of Shrewsbury to cut off Percy's line of retreat to Chester. They failed and the armies took up position in a field that was variously named: "Haytleyfield", "Husefeld", "Berwykfeld", "Bolefeld". The battle commenced in the manor of Harlescott about a mile south west of where Battlefield Church now stands. (The owner of this manor, Richard Hussey, swore to this fact under oath in the escheator's court in January 1416). The battle took place in a large field of peas.


title: Battle of Shrewsbury

text: Estimates of the sizes of the two armies vary widely, and the medieval chronicles are subject to exaggeration. "Annales Henrici Quarti" states 14,000 Royal troops, far fewer than Waurin's estimate of 60,000. Although Henry's army is generally agreed to have been larger, John Capgrave writing in the "Chronicle of England" quotes Percy's army as 14,000.


title: Battle of Shrewsbury

text: For much of the morning of Saturday 21 July, the two forces parleyed. Thomas Prestbury, the Abbot of Shrewsbury and the Abbot of Haughmond presented the King's terms. Hotspur declined any terms and Thomas Percy spoke to the King, trading insults. Henry Percy was somewhat inclined toward accepting the King's position, while his uncle Thomas Percy was not. Negotiations ended near noon, and the two forces advanced closer for the fight. One rebel, later pardoned, went over to the royal army and the king knighted several of his followers.


title: Battle of Shrewsbury

text: About two hours before dusk, King Henry IV raised his sword. The battle opened with a massive archery barrage, arrows killing or wounding many men before they could meet hand to hand in the field. Percy's Cheshire bowmen proved generally superior. Thomas Walsingham recorded how the King's men "fell like leaves in Autumn, every one [arrow] struck a mortal man". According to the "Dieulacres Chronicle" the King's right wing under the command of the Earl of Stafford (who was killed) fled from the field. Far more than this wing may have fled as well, as there is evidence that some baggage was looted and after the battle the Cheshire rebels were "prosecuted" for taking some 7,000 horses with them. Prince Henry, Prince of Wales was hit in the face with an arrow during the fighting, sustaining a terrible wound. He later recovered due to the skilled treatment of the Physician General John Bradmore, who used honey, alcohol and a specially designed surgical instrument to extract the arrowhead. He was left with a permanent scar. Bradmore recorded treating Prince Henry in his Latin manuscript, "Philomena." An account of this can also be found in Thomas Morstede's 1446 surgical treaty.


title: Battle of Shrewsbury

text: Enough of the King's men remained on the field, particularly on the left wing, which was under the command of the Prince of Wales. Perhaps in desperation, Hotspur led a charge aimed at killing the King himself, during which the Royal Standard was overthrown and its bearer, Sir Walter Blount, was hacked down by Archibald Douglas, 4th Earl of Douglas. Hotspur was killed in the charge, reputedly shot in the face with an arrow when he opened his visor. His death was initially not realised, and some point soon afterwards the Northumbrian knights hailed the death of Henry IV, exclaiming "Henry Percy King!". Henry IV was not dead however, and retaliated by shouting "Henry Percy is dead"; the absence of a reply confirmed that Henry Percy was indeed dead. The battle ended soon after. It is recorded that many did not know who had won. The King's forces sustained greater losses than the rebels, and Henry IV very nearly lost both his life and his throne.


title: Battle of Shrewsbury

text: Henry Percy was initially buried by his maternal first cousin, Thomas Nevill, 5th Baron Furnivall at Whitchurch, Shropshire, with honours, but rumours soon spread that he was not really dead. In response the King had him disinterred. His body was salted, set up in Shrewsbury impaled on a spear between two millstones in the marketplace pillory, with an armed guard, and was later quartered and put on display in Chester, London, Bristol and Newcastle upon Tyne. His head was sent to York and impaled on the north gate, looking toward his own lands. In November his grisly remains were returned to his widow Elizabeth.


title: Battle of Shrewsbury

text: The Earl of Worcester was beheaded and Sir Richard Venables, Sir Richard Vernon and Sir Henry Boynton were publicly hanged, drawn and quartered in Shrewsbury on 23 July and their heads publicly displayed, Thomas Percy's on London Bridge.


title: Battle of Shrewsbury

text: Battlefield Church is said to have been erected over the site of the mass burial pit dug immediately after the battle. It was built initially as a memorial chapel, on the orders of King Henry IV and paid for by him, with prayers and masses being said continually for the dead on both sides. The chapel was replaced in 1460 by a church, which was further restored in 1862. A drain being dug in a corner of the churchyard may have inadvertently opened part of the burial pit. Workmen were surprised by the mass of bones which they thought showed the hurried nature of the burials. They may have merely unearthed a charnel pit containing bones of a variety of different ages.


title: Battle of Shrewsbury

text: In 2006, the BBC show "Two Men in a Trench" proved only that an area to the west of the church was not in fact a graveyard.


title: Battle of Shrewsbury

text: There were two other 15th century churches associated with battles. The one at Towton no longer exists but remains of it have been identified as lying beneath the present day Towton Hall by the archaeologist Tim Sutherland. The other for the battle of Wakefield still stands on Wakefield Bridge. Neither was near the battlefield.


title: Battle of Shrewsbury

text: The battle itself and many of the key people involved appear in Shakespeare's "Henry IV, Part 1", with the fictional participation of Sir John Falstaff and Hotspur being killed by Prince Henry. It is also depicted in the 1966 movie "Chimes at Midnight" (released in the UK as "Falstaff"), in the 2012 BBC Two adaptation of Shakespeare's play, and in the climax to a historical novel by Edith Pargeter, "A Bloody Field by Shrewsbury" (1972) (US title: "The Bloody Field").


title: Battle of Shrewsbury

text: The battle features in an early and highly inaccurate sequence in the 2019 David Michod film "The King", in which the battle does not take place because Prince Henry, wresting command of the royal forces from his brother Thomas, and Hotspur fight in single combat rather than having the battle fought. After a lengthy scene in which the duel begins with swords and ends up a wrestling match, Prince Henry kills Hotspur by stabbing him in the neck and the day is won. The motivations for the rebellion - the king's refusal to ransom Edmund Mortimer and the discontent of the Percys with Henry IV's regime - are left intentionally vague. Prince Henry, later ascending the throne as Henry V, begins his reign with a policy of conciliation, with one of his first stated acts being the ransoming of Mortimer.


title: Peach Pit (band)

text: Peach Pit is an indie pop band from Vancouver, British Columbia, Canada. The band consists of singer and rhythm guitarist Neil Smith, lead guitarist Christopher Vanderkooy, bassist Peter Wilton, and drummer Mikey Pascuzzi. Lately they have been joined by multi-instrumentalist Dougal Bain McLean on synth, guitar, and fiddle. They describe their own music as "chewed bubblegum pop", whereas critics have described their sound as sad pop, and surf rock and is characterized by a soft spoken vocal style and guitar-driven rock melodies. The band's music videos are produced by videographer Lester Lyons-Hookham. The band sported the same clothes for every live performance during their "Being So Normal" album phase after buying them for their first music video shoot.


title: Peach Pit (band)

text: The group formed when high school friends, Neil Smith and Chris Vanderkooy, decided to collaborate on a musical project in 2014. Peter Wilton and Mikey Pascuzzi joined them on bass and drums. In June 2016, the band released its debut EP, "Sweet FA", produced and recorded by Smith's former roommate Harley Small at The Space Studios in Vancouver BC. They quickly began work on their follow up release and debut LP, "Being So Normal". The album was again produced and recorded by Harley Small at The Space Studios and was released in September 2017 on the Vancouver indie label Kingfisher Bluez.'


title: Peach Pit (band)

text: Vocalist/rhythm guitarist Neil Smith was formerly part of the folk band Dogwood and Dahlia. Smith and bassist Peter Wilton formerly worked as Amazon delivery drivers, guitarist Christopher Vanderkooy worked at a local Vancouver brewery, and drummer Mikey Pascuzzi worked as a carpenter. The band quit their jobs to go on their first tour of North America, Europe and Asia from 2017–2018. They have been touring steadily ever since the release of their debut album.


title: Peach Pit (band)

text: Peach Pit performed at Bonnaroo, Shaky Knees Music Festival, CBC Music Festival, and Capitol Hill Block Party in 2019 and opened for Two Door Cinema Club in a tour of the U.S. and Canada in 2019. "You and Your Friends" is their second album and was released on April 3rd 2020. Their third studio album, "From 2 to 3", was released on March 4th 2022.


title: Circuit Zandvoort

text: Circuit Zandvoort (), known for sponsorship reasons as CM.com Circuit Zandvoort, and previously known as Circuit Park Zandvoort until 2017, is a motorsport race track located in the dunes north of Zandvoort, the Netherlands, near the North Sea coast line. It returned to the Formula One calendar in 2021 as the location of the revived Dutch Grand Prix.


title: Circuit Zandvoort

text: There were plans for races at Zandvoort before World War II: the first street race was held on 3 June 1939. However, a permanent race track was not constructed until after the war, using communications roads built by the occupying German army. Contrary to popular belief John Hugenholtz cannot be credited with the design of the Zandvoort track, although he was involved as the chairman of the Nederlandse Automobiel Ren Club (Dutch Auto Racing Club) before becoming the first track director in 1949. Instead, it was 1927 Le Mans winner, S. C. H. "Sammy" Davis who was brought in as a track design advisor in July 1946 although the layout was partly dictated by the existing roads.


title: Circuit Zandvoort

text: The first race on the circuit, the "Prijs van Zandvoort", took place on 7 August 1948. The race was renamed the "Grote Prijs van Zandvoort" (Zandvoort Grand Prix) in 1949, then the "Grote Prijs van Nederland" (Dutch Grand Prix) in 1950. The 1952 race was the first to be run as a round of the World Championship, albeit to Formula Two regulations rather than Formula One regulations like all the European rounds of the championship that year; a similar situation also applied to the 1953. There was no Dutch Grand Prix in 1954, 1956 or 1957, but 1955 saw the first true Formula One race as part of the Drivers' Championship. The Dutch Grand Prix returned in 1958 and remained a permanent fixture on the F1 calendar (with the exception of 1972) until , when it was held for the last time in the 20th century.


title: Circuit Zandvoort

text: To solve a number of problems that had made it impossible to develop and upgrade the track, the most important one being noise pollution for the inhabitants of the part of Zandvoort closest to the track, the track management adopted and developed a plan to move the most southern part of the track away from the housing estate and rebuild a more compact track in the remaining former 'infield'. In January 1987 this plan got the necessary 'green light' when it was formally approved by the Provincial Council of North Holland. However, only a couple of months later a new problem arose: the company that commercially ran the circuit (CENAV), called in the receiver and went out of business, marking the end of "Circuit van Zandvoort". Again the track, owned by the municipality of Zandvoort, was in danger of being permanently lost for motorsports. However, a new operating company, the Stichting Exploitatie Circuit Park, was formed and started work at the realization of the track's reconstruction plans. Circuit Park Zandvoort was born and in the summer of 1989 the track was remodeled to an interim Club Circuit of , while the disposed southern part of the track was used to build a Vendorado Bungalow Park and new premises for the local football and hockey clubs.


title: Circuit Zandvoort

text: In 1995, CPZ ("Circuit Park Zandvoort") got the "A Status" of the government of the Netherlands and began building an international Grand Prix Circuit. This project was finished in 2001 when, after the track was redesigned to a long circuit and a new pits building was realized (by HPG, the development company of John Hugenholtz Jr., son of the former director), a new grandstand was situated along the long straight. One of the major events that is held at the circuit, along with DTM and A1GP, is the RTL Masters of Formula 3, where Formula Three cars of several national racing series compete with each other (originally called Marlboro Masters, before tobacco advertising ban). A noise restriction order was responsible for this event moving to the Belgian Circuit Zolder for 2007 and 2008. However, the race returned to its historical home in 2009.


title: Circuit Zandvoort

text: Circuit Park Zandvoort played host to the first race in the 2006/07 season of A1 Grand Prix from 29 September–1 October 2006. On 21 August 2008, the official A1GP site reported that the 2008/09 season's first race has moved from the Mugello Circuit, Italy to Zandvoort on the 4–5 October 2008 due to the delay in the building the new chassis for the new race cars. The Dutch round moved to TT Circuit Assen in 2010. A1GP bankrupted before its fifth season and the Dutch round was replaced with Superleague Formula.


title: Circuit Zandvoort

text: In November 2018 reported that Formula One Management (FOM) had invited the owners of the Zandvoort race track to make a proposal to stage a Grand Prix race in 2020. In March 2019, it was confirmed that a letter of intent had been signed between Zandvoort and FOM to stage the Dutch Grand Prix, dependent on private funding being secured to cover the cost of hosting the race. A deadline of 31 March 2019 was set for a final decision to be made. On 14 May 2019 it was confirmed that Zandvoort would host the Dutch Grand Prix for 2020 and beyond for a duration of at least three years, with the option to host another two years beyond that. Several alterations were made to the track to bring it up to date with F1 standards, including adding banking to turn 14 (Arie Luyendijkbocht) and turn 3 (Hugenholtzbocht), but the layout as a whole remained the same. The municipality of Zandvoort invested four million euros into the infrastructure around the circuit to improve the accessibility to the track. On 29 August 2019, the 2020 Dutch Grand Prix at Zandvoort was placed 5th race on the draft schedule, listed on 3 May 2020, between China and Spain. The 2020 scheduled appearance was canceled due to the COVID-19 pandemic, however F1 racing did finally return to the circuit on 5 September 2021. On 17 September 2019, it was announced that Zandvoort would host the FIA Formula 2 Championship and FIA Formula 3 Championship, replacing the series' support races at Circuit Paul Ricard.


title: Circuit Zandvoort

text: The circuit gained popularity because of its fast, sweeping corners such as Scheivlak as well as the "Tarzanbocht" (Tarzan corner) hairpin at the end of the start/finish straight. Tarzanbocht is the most famous corner in the circuit. Since there is a camber in the corner, it provides excellent overtaking opportunities. It is possible to pass around the outside as well as the easier inside lane. This corner is reportedly named after a local character who had earned the nickname of Tarzan and only wanted to give up his vegetable garden in the dunes if the track's designers named a nearby corner after him. On the other hand, many different stories about Tarzan Corner are known.


title: Circuit Zandvoort

text: The corners are named as follows (the numbers correspond to the present map, starting at the start/finish line):


title: Circuit Zandvoort

text: Turns 3 and 13/14 are extremely cambered corners; turn 3 has a 19-degree bank while turns 13/14 have an 18-degree bank.


title: Circuit Zandvoort

text: The official lap record for the current circuit layout is 1:11.097, set by Lewis Hamilton driving for Mercedes in the 2021 Dutch Grand Prix. The all-time fastest official track record set during a race weekend for the current Grand Prix Circuit layout is 1:08.885, set by Max Verstappen during qualifying for the aforementioned Grand Prix. The official race lap records at the Circuit Zandvoort are listed as:


title: Circuit Zandvoort

text: Motor racer Willy Koppen was the first woman to participate in motor trials in the early fifties on the circuit. In August 1959 the UCI Road World Championships men's race was held at Zandvoort. André Darrigade of France won the race, Tom Simpson (Britain) was 4th.


title: Circuit Zandvoort

text: Since 2008, the course has been used as the venue for the Runner's World Zandvoort Circuit Run, a 5-kilometre road running competition. The 2010 edition of the race attracted Lornah Kiplagat, a multiple world champion, who won the ladies 5 km race.


title: Circuit Zandvoort

text: The Cycling Zandvoort 24h race was first held on 25–26 May 2013. It is open for public for soloists and teams up to 8 riders. A 6-hours was added to the event in 2016. On 13./14. June 2015 (12:00) the Cycling Zandvoort – 24 hour race over 4307-m-laps took place.


title: Chief marketing officer

text: A chief marketing officer (CMO), also called a global marketing officer or marketing director, or chief brand officer, is a corporate executive responsible for managing marketing activities in an organization. Whilst historically these titles may have signified a legal responsibility, for example at Companies House in the UK, the titles are less strict/formal in the 21st Century and allow companies to acknowledge the evolving and increasingly significant role that marketers can play in an organisation, not least because of the inherent character of successful marketers. The CMO leads brand management, marketing communications (including advertising, promotions and public relations), market research, product marketing, distribution channel management, pricing, customer success, and customer service.


title: Chief marketing officer

text: The CMO is a member of the C-suite and typically reports to the chief executive officer. A number of senior vice presidents, vice presidents, directors, and other senior marketing managers responsible for various parts of the marketing strategy may report directly to the CMO.


title: Chief marketing officer

text: A study from consulting firm Spencer Stuart in 2021 showed that women made up 47% of CMO positions in 2020, an increase from the 43% reported in 2019. 13% of CMOs had racially or ethnically diverse background in 2020, down from 14% in 2019.


title: Chief marketing officer

text: The day-to-day tasks and responsibilities are often categorically different from one another, due to the fluid nature of the CMO's skill set: language is needed to stitch together all aspects of the company. Thus, in a given day the CMO completes tasks that fall into many different categories:


title: Chief marketing officer

text: The CMO must quickly react to the changing market conditions and competitive dynamics and must reshape, as needed, the company's strategy and execution plans based on real-time market scenarios. Each of these products comes from a different department, so the CMO must be a nexus of information: it is a highly receptive role, with involvement in departments such as production, information technology, corporate communications, documentation, public relations, law, human resources, and finance.


title: Chief marketing officer

text: In the 21st century, digitalization and the rise of consumer-centric marketing has changed the role of the CMO. They are now typically finding themselves handling customer-facing technology implementations in addition to the above tasks. One analyst predicted that in the future CMOs will spend more on IT than their counterpart CIOs. According to another analyst firm, few senior-executive positions will be subject to as much change over the next few years as that of the chief marketing officer.


title: Chief marketing officer

text: Peers to the CMO include chief human resources officer, chief technology officer, chief financial officer, chief communications officer, chief procurement officer, chief information officer, and general counsel.


title: Chief marketing officer

text: The CMO is responsible for facilitating growth, sales and marketing strategy. They must work towards objectives such as revenue generation, cost reduction, or risk mitigation. The unpredictable effect of marketing efforts, coupled with the need to drive profits, often leads to a short tenure for most CMOs. "Forbes" reported that the average CMO tenure in 2008 was just over 28 months.


title: Chief marketing officer

text: In a CMO mapping study done by Raines, CMOs are losing their influence (73% of Fortune 250 companies have a Global CMO, but only 44% of Global CMOs sit in the C-suite). CMOs don't feel valued (23% of surveyed Fortune 500 CMOs were unsure their CEO understood their job). CMOs don't have clear alignment with their CEOs on key performance metrics (42% of surveyed Fortune 500 CMOs considered "Topline Revenue Growth" a top priority) whereas "topline growth" is typically one of the major priorities for CEOs at any stage of a business.


title: Chief marketing officer

text: CMOs see customer loyalty as their top priority in the digital era; their second priority is to design experiences for tablets and mobile apps.


title: Bernard Purdie

text: Bernard Lee "Pretty" Purdie (born June 11, 1939) is an American drummer, and an influential R&B, soul and funk musician. He is known for his precise musical time keeping and his signature use of triplets against a half-time backbeat: the "Purdie Shuffle." He was inducted into the "Modern Drummer" Hall of Fame in 2013.


title: Bernard Purdie

text: Purdie recorded "Soul Drums" (1968) as a band leader and although he went on to record "Alexander's Ragtime Band", the album remained unreleased until "Soul Drums" was reissued on CD in 2009 with the "Alexander's Ragtime Band" sessions. Other solo albums include "Purdie Good!" (1971), "Soul Is... Pretty Purdie" (1972) and the soundtrack for the blaxploitation film "Lialeh" (1973).


title: Bernard Purdie

text: Purdie was born on June 11, 1939 in Elkton, Maryland, US, the eleventh of fifteen children. At an early age he began hitting cans with sticks and learned the elements of drumming techniques from overhearing lessons being given by Leonard Heywood. He later took lessons from Heywood and played in Heywood's big band. Purdie's other influences at that time were Papa Jo Jones, Buddy Rich, Gene Krupa, Joe Marshall, Art Blakey, as well as Cozy Cole, Sticks Evans, Panama Francis, Louis Bellson, and Herbie Lovelle.


title: Bernard Purdie

text: In 1961, he moved from his home town of Elkton, Maryland, to New York City. There he played sessions with Mickey and Sylvia and regularly visited the Turf Club on 50th and Broadway, where musicians, agents, and promoters met and touted for business. It was during this period that he played for the saxophonist Buddy Lucas, who nicknamed him 'Mississippi Bigfoot'. Eventually Barney Richmond contracted him to play session work.


title: Bernard Purdie

text: Purdie was contracted by arranger Sammy Lowe to play a session with James Brown in 1965 and recording session records also show that Purdie played on "Ain't That A Groove" at the same session. Purdie is credited on James Brown's albums "Say It Loud – I'm Black and I'm Proud" (1969) and "Get on the Good Foot" (1972).


title: Bernard Purdie

text: Purdie started working with Aretha Franklin as musical director in 1970 and held that position for five years, as well as drumming for Franklin's opening act, saxophonist King Curtis and The King Pins. March 5–7, 1971, he performed with both bands at the Fillmore West; the resulting live recordings were released as "Aretha Live At The Fillmore West" (May 19, 1971) and King Curtis's "Live At Fillmore West" (1971). His best known track with Franklin was "Rock Steady", on which he played what he described as "a funky and low down beat". Of his time with Franklin he once commented that "backing her was like floating in seventh heaven".


title: Bernard Purdie

text: Purdie was credited on the soundtrack album for the film "Sgt. Pepper's Lonely Hearts Club Band" (1978) and later he was the drummer for the 2009 Broadway revival of "Hair" and appeared on the associated Broadway cast recording. In 2016, he was awarded an honorary doctorate in music by Five Towns College.


title: Bernard Purdie

text: Purdie is known as a groove drummer with immaculate timing who makes use of precision half note, backbeats, and grooves. Purdie's signature sixteenth note hi-hat lick "pish-ship, pish-ship, pish-ship" is distinct. He often employs a straight eight groove sometimes fusing several influences such as swing, blues and funk. He created the now well-known drum pattern "Purdie Half-Time Shuffle" that is a blues shuffle variation with the addition of syncopated ghost notes on the snare drum. Variations on this shuffle can be heard on songs such as Led Zeppelin's "Fool in the Rain", the Police's "Walking on the Moon", and Toto's "Rosanna" (Rosanna shuffle). Purdie plays the shuffle on Steely Dan's "Babylon Sisters" and "Home At Last".


title: Jun Matsumoto

text: , often called by the portmanteau nickname , is a Japanese singer, actor, radio host, concert director, dancer, and model. He is a member of the boy band Arashi, and produces Arashi's concerts. He is best known to Japanese television drama audiences for his portrayal as Dōmyōuji Tsukasa in the "Hana Yori Dango" series, in which he won GQ Japan's Man of the Year Award under the singer/actor category for his work in the drama.


title: Jun Matsumoto

text: Matsumoto began his career in the entertainment industry when he joined the Japanese talent agency Johnny & Associates in 1996 at the age of . Prior to his debut as a singer with Arashi in 1999, Matsumoto started an acting career when he was cast as Teddy Duchamp for the stage play "Stand by Me", which was based on the film of the same name. Since then, he has gone on to appear in numerous dramas and movies, receiving a number of awards and nominations for his roles.


title: Jun Matsumoto

text: Matsumoto was born in Toshima, Tokyo, as the youngest child in his family. He has an older sister whose support of KinKi Kids influenced his decision to join Johnny & Associates in 1996. Thinking it might bode good luck, he sent his application to the agency on his elementary school graduation day and received a phone call weeks later from president Johnny Kitagawa himself, inviting him to attend a rehearsal instead of being auditioned. Due to this, Matsumoto is frequently referred to as one of the "elite" within the agency.


title: Jun Matsumoto

text: Matsumoto graduated from Horikoshi Gakuen, a renowned high school known for its many performing arts alumnae such as Kyoko Fukada and Ai Kato, in March 2002 at the age of .


title: Jun Matsumoto

text: Although the majority of Matsumoto's solos for albums and concerts are written by Arashi's staff, he has contributed lyrics for some of his solos: "La Familia" for the "2004 Arashi! Iza, Now Tour!!", "Naked" in 2008 for the album "Dream "A" Live" under the pen name "Jun", and "Stay Gold" in 2014 for the album "The Digitalian". He also supervised the song "DRIVE" in 2016 for the album "Are You Happy?" and took part in writing lyrics with other Arashi members as Arashi for their songs "Fight Song", "Energy Song - Zekkōchō Chō!!!!", "5×10", and "5×20".


title: Jun Matsumoto

text: Matsumoto began to be involved in concert production in 2000. He is officially credited as the director in the closing credits for the concerts Arashi Anniversary Tour 5×20 and Arafes 2020, and on the official website of "Johnny's Festival Thank you 2021 Hello 2022" for the concert of the same name. Although he primarily focuses on Arashi concerts, he also gave advice on concert production to younger groups of Johnny & Associates, such as for Hey! Say! JUMP in 2014 and King & Prince in 2018, and in 2021, he is the director for the "Johnny's Festival Thank you 2021 Hello 2022" concert, featuring many different groups from Johnny & Associates.


title: Jun Matsumoto

text: As a concert director, Matsumoto had also been conceptualizing new ideas for concert production. In 2005, he along with the concert staff team implemented the usage of the moving stage, which has been subsequently used by several Asian musical acts. 3 years prior to Arashi Live Tour 2014 The Digitalian, Matsumoto had been putting efforts into utilizing the idea of light color changing via bluetooth onto penlights, and the concept was successfully implemented in the previously mentioned concert.


title: Jun Matsumoto

text: In 1997, Matsumoto was cast in his first stage play, which was based on the American coming of age film "Stand by Me" with future bandmates Masaki Aiba and Kazunari Ninomiya. He did not return to do any major stage productions for nearly seven years after "Stand by Me", instead focusing on dramas and movies. However, in 2004, Matsumoto appeared in the stage play "West Side Story" with bandmates Satoshi Ohno and Sho Sakurai. In 2005 and 2006, Matsumoto was given his first lead stage play roles in and respectively.


title: Jun Matsumoto

text: It was announced on July 21, 2011, that Matsumoto will star in Yukio Ninagawa's production play, , which was his first stage play in five years.


title: Jun Matsumoto

text: Like bandmate Sakurai, Matsumoto made his acting debut as a television actor in April 1997 in the TBS drama special . A few months later, he co-starred with the members of KinKi Kids and future bandmate Masaki Aiba in the mystery-thriller series . In 1999, all the members of Arashi co-starred together for the first time in the volleyball-centered short drama .


title: Jun Matsumoto

text: Matsumoto made his breakthrough in 2001 when he was cast as Hajime Kinda'ichi in the third season of "Kinda'ichi Shōnen no Jikenbo", taking over the role from Tsuyoshi Domoto and co-starring opposite Anne Suzuki.


title: Jun Matsumoto

text: Matsumoto gained further popularity as an actor in 2002, when he starred in the first season of "Gokusen" with Yukie Nakama, Shun Oguri, Tomohiro Waki, Hiroki Narimiya and Yuma Ishigaki. His portrayal of the troubled but highly intelligent student, Shin Sawada, drew acclaim and won him Best Supporting Actor at the 33rd Television Drama Academy Awards. He later returned with most of the original cast to star in the special epilogue episode in 2003. Soon after, his character also made a cameo in Sakurai's comedy series .


title: Jun Matsumoto

text: In 2003, Matsumoto took another high-profile role in the live-action adaptation of manga series "Kimi wa Pet" as Takeshi "Momo" Goda, starring opposite Koyuki.


title: Jun Matsumoto

text: In 2005, Matsumoto took the most prominent role of his career to date, when he was cast as Domyōuji Tsukasa in the live-action adaptation of shōjo manga "Hana Yori Dango". Co-starring opposite Oguri again, and with Mao Inoue, Shota Matsuda and Tsuyoshi Abe, the series was a success with an average viewership rating of 19.6%. Matsumoto's portrayal as the air-headed and arrogant leader of four rich heirs won him Best Supporting Actor again at the 47th Television Drama Academy Awards. In 2007, due to the success of "Hana Yori Dango", it spawned a second season, which was an even bigger hit with television audiences as it had a peak rating of 27.6% on the final episode and an overall rating of 21.57%. Matsumoto won Best Supporting Actor at the 10th Nikkan Sports Drama Grand Prix for his role. The series ended with a movie in 2008. In the new series, "Hana Nochi Hare ~ Hanadan Next Season", that aired in 2018, he returned as Domyōuji (along with Oguri and Matsuda, two more of the original F4 members) for a one-episode guest appearance.


title: Jun Matsumoto

text: In 2007, Matsumoto starred in "Bambino!", which won him his first Best Actor award at the 53rd Television Drama Academy Awards.


title: Jun Matsumoto

text: In 2008, he re-united with "Bambino!" co-star Karina for the drama special , which aired as part of the "24-hour Television" telethon in 2008. He portrayed a man diagnosed with CIDP struggling to recuperate and return to normal life with his wife and young daughter. The drama special received a viewership rating of 25.6%.


title: Jun Matsumoto

text: In 2009, Matsumoto starred in his first drama series in nearly two years. He portrayed , a half-Japanese, half-Filipino young man who always faces life with a smile in the drama "Smile", co-starring Oguri once more. Matsumoto subsequently won Best Actor for his role in the 13th Nikkan Sports Drama Grand Prix. Fuji TV announced on September 10, 2009, that Matsumoto would star in a three-part drama special called scheduled to air for three consecutive days in the spring of 2010.


title: Jun Matsumoto

text: In January 2010, Matsumoto co-starred with the other members of Arashi in their first drama in nearly ten years in the human suspense drama special . Matsumoto portrayed , a 27-year-old motorcycle courier rider who is caught up in a building hijack. Matsumoto co-starred with Yūko Takeuchi in his first Getsuku drama titled . He also made a guest appearance on the final episode of bandmate Ohno's drama , which is based on the Fujiko Fujio's "Kaibutsu-kun" manga and anime series.


title: Jun Matsumoto

text: In 2012, co-starring with Eita Nagayama who played as Teru Nitta, Matsumoto played the role of Shuntaro Tokita in Lucky Seven, a series surrounding a group of quirky detectives.


title: Jun Matsumoto

text: In 2014, co-starring with Satomi Ishihara who played as Saeko Takahashi, Matsumoto played as Koyurugi Sota in the live-adaptation of romance Josei manga series Shitsuren Chocolatier.


title: Jun Matsumoto

text: In 2016, Matsumoto starred as Hiroto Miyama in . The series is about a lawyer always eager to find the 0.1% truth in the midst of 99.9% cases in Japan, where once one is prosecuted, they are presumably guilty. Later in 2018, Matsumoto returned for the sequel titled . The sequel also experienced success like the prequel, with an average viewership of 17.6% in the Kanto region, being one of the most praised drama series in 2018. The series continued in 2021, with a special on TV and a film , that were released on December 29 and 30, respectively.


title: Jun Matsumoto

text: In 2019, Matsumoto played lead role as Matsuura Takeshirō in the NHK drama special, which was made to commemorate the 150th anniversary of the naming of Hokkaido.


title: Jun Matsumoto

text: By the end of 2021, Matsumoto's next drama was announced, his first co-starring role with long time friend Aya Ueto, called , airing weekly from January 20, 2022. He portrayed Chikara Nakagoshi, a stay-at-home family man, who's also a struggling writer, that neglects other duties while being concerned with the wellbeing of people around him.


title: Jun Matsumoto

text: In 2002, Arashi co-starred in their first movie together called . They came together again for its sequel in 2004. He also starred in the film "Tokyo Tower" as a womanizer with a preference for older women with Junichi Okada the same year.


title: Jun Matsumoto

text: In 2007, Arashi starred in their third movie together "Kiiroi Namida" while Matsumoto was cast in the independent film "Boku wa Imōto ni Koi o Suru" with Nana Eikura as his co-star. He portrayed the lead character , a high school student who falls in love and develops a romantic relationship with his younger twin sister.


title: Jun Matsumoto

text: In 2008, Matsumoto took on his first jidaigeki role in a re-make of Akira Kurosawa's "", with award-winning actors Hiroshi Abe, Kippei Shiina and Masami Nagasawa. Soon after, TBS decided to end the "Hana Yori Dango" series through a film. "Hana Yori Dango Final", which hit the big screen on June 28, 2008, in Japan, became a box office hit.


title: Jun Matsumoto

text: In 2013, Matsumoto Jun played the role of Okuda Kosuke in the movie 'Hidamari no Kanojo'(Her Sunny Side) along with actress Ueno Juri in the latest movie of director Miki Takahiro ('Solanin', 'Boku ga Ita').'Hidamari no Kanojo' was filmed in January and to hit cinemas in October 2013.


title: Jun Matsumoto

text: In 2017, Matsumoto Jun played the role of Takashi Hayama in the Movie 'Narratage' along with actress Kasumi Arimura.


title: Jun Matsumoto

text: It is announced that in late 2021, Matsumoto Jun will be playing Hiroto Miyama in the , a sequel to the 99.9 Criminal Lawyer drama series.


title: Jun Matsumoto

text: From 2005 to 2007, Matsumoto along with Akiko Wada, Aya Matsuura, KAT-TUN, Tomochika, and Hinoi Team hosted Minna no Terebi/Utawara Hot Hit 10.


title: Scott Putesky

text: Scott Mitchell Putesky (April 28, 1968 – October 22, 2017), also known as Daisy Berkowitz, was an American musician, songwriter, composer, visual artist, and record producer. He was the co-founder of the rock band Marilyn Manson & the Spooky Kids (later shortened to just Marilyn Manson), and was the guitarist for the band until 1996. After leaving Marilyn Manson, Putesky was involved in a number of other projects such as Three ton Gate, The Linda Blairs, Jack Off Jill, Stuck on Evil (previously called Rednecks on Drugs), Kill Miss Pretty, and The Daisy Kids.


title: Scott Putesky

text: Scott Putesky was born in Los Angeles, California, and was adopted and raised in New Jersey. His family is Jewish, but he described himself as atheist. Putesky's first instruments were the flute and the snare drum, and in 6th grade he joined the school's chorus, for which he apparently had a talent. He started playing the guitar when he was 15. His early loves were drawing, movies and "Star Wars".


title: Scott Putesky

text: Putesky and Marilyn Manson met at a Fort Lauderdale club called The Reunion Room and later at a local after-party in December 1989. The two created the concept of Marilyn Manson & the Spooky Kids poking fun at American media hypocrisy and its obsessions with serial killers and beautiful women. For this act, he took the stage name of Daisy Berkowitz, devised by mixing the names of "The Dukes Of Hazzard" character Daisy Duke and serial killer David Berkowitz.


title: Scott Putesky

text: In 1996, creative differences with Manson caused Putesky to leave the studio before "Antichrist Superstar" was completed. Manson had begun to work more with the other members of the band, and the dark environment of "Antichrist Superstar"s production led to Berkowitz being muscled out of the group. He is, however, credited for writing the music for four songs on the record, among them being the album's second single, "Tourniquet". Due to unpaid royalties, Berkowitz filed a $15 million lawsuit against Manson, which was later settled under confidential terms.


title: Scott Putesky

text: In late 1998, Putesky joined up with longtime Marilyn Manson collaborators Jack Off Jill, replacing departing member Ho Ho Spade and playing live guitar on their 1999 West Coast tour which lasted for only a handful of gigs. His first recorded work with the band was the 1998 EP "Covetous Creature", to which he lent guitar and some production. By early 1999, SMP was no longer a member of Jack Off Jill. Putesky himself was vague, but he related to MTV News that the parting was amicable. He returned once again to Ft. Lauderdale to resume work on his own music, primarily seeking a live band to perform his Three Ton Gate material. In the meantime, he successfully sued Marilyn Manson for what he claimed were unpaid royalties for his contributions to "Antichrist Superstar". In the same lawsuit, Putesky was also awarded the rights to 21 unreleased recordings by Marilyn Manson & the Spooky Kids.


title: Scott Putesky

text: In July 2010, Putesky officially joined electro-punk band Kill Miss Pretty after collaborating with them on their single "Judy Garland". Following this the band began working on new material including a cover of the Spooky Kids' song "She's Not My Girlfriend".


title: Scott Putesky

text: Putesky revived Three Ton Gate shortly afterward to play live under that name in a 2011 series of dates across America starting in New York City on October 15.


title: Jacqui Smith

text: Jacqueline Jill Smith (born 3 November 1962) is a British broadcaster, political commentator and former Labour Party politician. She was Member of Parliament (MP) for Redditch from 1997 to 2010. She served as Home Secretary from 2007 to 2009 and was the first woman to hold the position.


title: Jacqui Smith

text: Smith was born and raised in Malvern, Worcestershire. She attended Hertford College, Oxford, before training to become a teacher at Worcester College of Higher Education and having a career as an economics and business studies teacher. She was elected for Redditch at the 1997 general election. She joined the government in 1999 and served in a series of ministerial positions under Prime Minister Tony Blair. In the 2006 cabinet reshuffle she was promoted to Chief Whip.


title: Jacqui Smith

text: Following Gordon Brown's appointment as Prime Minister, Smith became the first female Home Secretary. She resigned as Home Secretary in June 2009 following her involvement in the United Kingdom parliamentary expenses scandal in which she had falsely claimed that a room in her sister's house was her main home. Smith, one of the highest profile figures involved in the scandal, then lost her seat as MP for Redditch in the 2010 general election. After leaving the House of Commons, she remained in public life as a political pundit and took up roles in various other sectors, such as health and media.


title: Jacqui Smith

text: Smith was born in Malvern, Worcestershire. She attended Dyson Perrins High School in Malvern. Her parents were teachers, and both Labour councillors, although her mother briefly joined the Social Democratic Party. Her local MP, Conservative backbencher Michael Spicer, recalled in Parliament in 2003 how he had first met her when he addressed the sixth form at The Chase School, where her mother was a teacher; he joked: "So great was my eloquence that she immediately rushed off and joined the Labour Party." Smith obtained a place to study philosophy, politics and economics at Hertford College, Oxford.


title: Jacqui Smith

text: Deciding she wanted a career outside politics, Smith moved out of London and gained a Postgraduate Certificate in Education from Worcester College of Higher Education. Working as a school teacher, she taught economics at Arrow Vale High School in Redditch from 1986 to 1988 and at Worcester Sixth Form College, before becoming head of economics and General National Vocational Qualification co-ordinator at Haybridge High School, Hagley in 1990. During this time Smith held positions in the local Labour party and campaigned on behalf of the party.


title: Jacqui Smith

text: Smith worked as secretary of the National Organisation of Labour Students and describes herself as having a "feminist background". She served on Redditch Borough Council from 1991 to 1996, where she chaired the development committee. Smith was unsuccessful in an attempt to be elected as (Labour) MP for the safe Conservative seat of Mid Worcestershire in the 1992 general election, despite achieving a 4.9% swing. In early 1997 she was identified by "The Independent" as a potential future cabinet member.


title: Jacqui Smith

text: Smith was selected through an all-women shortlist as the Labour candidate for Redditch, a new constituency created after a boundary review. She won the seat in the 1997 general election, as part of a (then) record number of female MPs elected to the House of Commons.


title: Jacqui Smith

text: Smith entered the Government in July 1999, as a Parliamentary Under-Secretary of State at the Department for Education and Employment, working with the Minister for School Standards Estelle Morris. She then became a Minister of State at the Department of Health after the 2001 general election. She was appointed as deputy Minister for Women in 2003, working alongside Secretary of State Patricia Hewitt. In this role she published the government's proposals for same-sex civil partnerships, a system designed to offer same-sex couples an opportunity to gain legal recognition for their relationships with an associated set of rights and responsibilities.


title: Jacqui Smith

text: In the 2005 general election Smith had a majority of just 2,716 (6.7% of the vote), owing to boundary changes.


title: Jacqui Smith

text: Following the 2005 general election, Smith was appointed Minister of State for Schools at the Department for Education and Skills, replacing Stephen Twigg who had lost his seat. Teacher trade union sources stated that Smith "talked to us on our level".


title: Jacqui Smith

text: In the 2006 cabinet reshuffle Smith was appointed as the government's Chief Whip. In a period when supporters of Gordon Brown were pushing Prime Minister Tony Blair to resign, she was successfully able to calm the situation down. The BBC's political editor Nick Robinson described her as being effective at "making peace between the warring Blair and Brown factions".


title: Jacqui Smith

text: Smith was regarded as a loyal Blairite during Tony Blair's premiership, a position reflected in her voting record, and she was brought to tears by Blair's farewell appearance in the House of Commons.


title: Jacqui Smith

text: In Gordon Brown's first cabinet reshuffle on 28 June 2007, Smith expressed interest in becoming Secretary of State for Education, but was appointed Home Secretary. She became the first woman to hold the position and the third woman to hold one of the Great Offices of State, after Margaret Thatcher (Prime Minister) and Margaret Beckett (Foreign Secretary). Just one day into her new job bombs were found in London, and a terrorist attack took place in Glasgow the following day.


title: Jacqui Smith

text: On 24 January 2008, she announced new powers for the police, including the proposal to permit law enforcement services to hold terrorist suspects or those linked to terrorism for up to 42 days without charging them. In the same month Smith said that she would not feel safe on the streets of London at night. Critics suggested her statements were an admission that the government had failed to tackle crime effectively. Smith also introduced legislation to toughen the prostitution laws of England and Wales, making it a criminal offence to pay for sex with a prostitute controlled by a pimp, with the possibility that anyone caught paying for sex with an illegally trafficked woman could face criminal charges.


title: Jacqui Smith

text: Smith introduced a crime mapping scheme to allow citizens of England and Wales to access local crime information and how to combat crime. As Home Secretary, she was able to announce that minor crime dropped year-on-year under the Labour government, and continued to do so in 2008.


title: Jacqui Smith

text: Smith managed to pass the 42-day detention law plans in the House of Commons, despite heavy opposition. The House of Lords voted overwhelmingly against the law, with some of the Lords reportedly characterising it as "fatally flawed, ill-thought-through and unnecessary", stating that "it seeks to further erode fundamental legal and civil rights". In March 2009, Smith published the first ever public Counter Terror Strategy.


title: Jacqui Smith

text: When Conservative MP Damian Green was arrested in his Commons office, Smith stated that she was not informed of the impending arrest. The Metropolitan Police said that Green was "arrested on suspicion of conspiring to commit misconduct in a public office and aiding and abetting, counselling or procuring misconduct in a public office". A junior Home Office official, Christopher Galley, was later arrested regarding the same alleged offences as Green, and was released on bail. He was not charged, but he was suspended from his Home Office job while the investigation continued. He was later dismissed from his position for gross misconduct. Green did not dispute having dealings with the Home Office official.


title: Jacqui Smith

text: In March 2009, at the height of the expenses furore, a leaked poll of Labour Party members revealed that Smith was considered to be the worst performing member of the Cabinet, with only 56% of her party believing she was doing a good job.


title: Jacqui Smith

text: In October 2018 Chief Prosecutor in North West England, Nazir Afzal, alleged on BBC Radio 4 that the Home Office had issued a memo to all police forces in 2008, informing that the child victims of Pakistani grooming gangs had made an "informed choice" and "it's not for you police officers to get involved in". Smith ran the Home Office at the time of this alleged instruction.


title: Jacqui Smith

text: In May 2009, Smith announced that the cost of introducing the National Identity Card project (a scheme abandoned by the incoming Liberal-Conservative coalition government in May 2010), had risen to an estimated £5.3 billion, and that it would first become compulsory for foreign students and airport staff. It was planned that the cards would be made available from high-street shops at an estimated cost of £60. Smith defended her decision to use high-street shops, and stated that the hope was to make enrolment in the scheme a less intimidating experience and to make the cards easier to access. She claimed, despite evidence to the contrary, that the majority of the population was in favour of the scheme. In another privacy-related issue, Smith said she was disappointed at the European Court of Human Rights' decision to strike down a law allowing the government to store the DNA and fingerprints of people with no criminal record; in December 2008 an estimated 850,000 such DNA samples were being held in England and Wales. Her compromise was to scale down the length of time that data could be kept, with a maximum limit of 12 years. This went against the spirit of the Court's decision.


title: Jacqui Smith

text: On 19 July 2007, Smith admitted to smoking cannabis a few times in Oxford in the 1980s. "I did break the law... I was wrong... drugs are wrong", she said. Asked why students today should listen when she urged them not to try the drug, she said that the dangers of cannabis use had become clearer, including mental health issues and the increasing strength of the drug over the past 25 years. Smith's admission was made public the day after Gordon Brown appointed her head of a new government review of UK drugs strategy.


title: Jacqui Smith

text: In May 2008, against the recommendations of her own scientific advisers, Smith reversed the government's 2004 decision to downgrade cannabis to a class C drug, returning it to the status of class B, with the law change taking effect on 26 January 2009. According to her most senior expert drugs adviser Professor David Nutt, the following exchange took place between Smith and himself:


title: Jacqui Smith

text: In February 2009, Smith was accused by Nutt of making a political decision in rejecting the scientific advice to downgrade ecstasy from a class A drug. The advisory council on the misuse of drugs (ACMD) report on ecstasy, based on a 12-month study of 4,000 academic papers, concluded that it is nowhere near as dangerous as other class A drugs such as heroin and crack cocaine, and should be downgraded to class B alongside amphetamines and cannabis. The advice was not followed; the government saying that it was "not prepared to send a message to young people that we take ecstasy less seriously". Smith was also widely criticised by the scientific community for bullying Professor Nutt into apologising for his factual comments that, in the course of a normal year, more people died from falling off horses than died from taking ecstasy.


title: Jacqui Smith

text: On 5 May 2009, Smith named 16 "undesirable individuals", including convicted murderers and advocates of violence, who were to be banned from entering the United Kingdom over their alleged threat to public order. Controversially, the exclusion list included outspoken American talk radio host Michael Savage, who instructed London lawyers to sue Smith for "serious and damaging defamatory allegations".


title: Jacqui Smith

text: A Home Office spokeswoman said: "The home secretary has made it clear that if such a case was brought that any legal proceedings would be robustly defended."


title: Jacqui Smith

text: Smith defended the choice of individuals by declaring, "If you can't live by the rules that we live by, the standards and the values that we live by, we should exclude you from this country and, what's more, now we will make public those people that we have excluded." "The Guardian" criticised Smith's actions.


title: Jacqui Smith

text: Smith was investigated by the Parliamentary Commissioner for Standards over accusations that she had inappropriately designated her sister's home in London as her main residence. The arrangement had allowed Smith to claim over £116,000 on her family's Redditch home since becoming an MP. Smith said that she had followed advice from parliamentary authorities.


title: Jacqui Smith

text: On 8 February 2009, it was reported in the media that Smith had designated a house in London owned by her sister as her main residence in order to claim a parliamentary allowance for her house in Redditch as a secondary home, despite explicitly stating on her website that she "lives in Redditch". When asked whether it was fair that she made claims believed to have been made for items such as a flat-screen TV and scatter cushions, she said that analyses of her receipts had been very particular. In response to criticisms over her housing allowances, she said it was the "nature of the job" that MPs had to furnish and run two properties.


title: Jacqui Smith

text: It was also reported that Smith had claimed expenses for a telecoms bill that contained two pornographic films and two other pay-per-view films. Smith said it was a mistake, and she would repay the amount. The reports made clear that the films had been viewed in the family home at a time when Smith was not present, and that she had given her husband, Richard Timney, a "real ear-bashing" over the incident. This and other cases prompted calls for reform of the additional costs allowance and a new system of payments to be introduced. Gordon Brown supported her and said she had done nothing wrong. However, Sir Alistair Graham, the former chairman of the Committee on Standards in Public Life, was critical of her actions, stating that naming her sister's spare bedroom as her main home was "near fraudulent". Smith was one of the highest profile politicians involved in the expenses scandal and citing the impact on her family life she later resigned.


title: Jacqui Smith

text: In October 2009, it was reported that the Standards Commissioner, John Lyon, had looked into complaints over her expense claims. He concluded that, although her London home was a genuine home and she had spent more nights there than in her Redditch home, her constituency home was in fact her main home, and that she was in breach of Commons rules, despite "significant mitigating circumstances". The claims for pay-per-view films were also found to be in breach. Ms Smith was not asked to repay any money, but was told to "apologise to the House by means of a personal statement." Smith reacted by saying that she was "disappointed that this process has not led to a fairer set of conclusions, based on objective and consistent application of the rules as they were at the time."


title: Jacqui Smith

text: In an interview with "Radio Times" published in February 2011, Smith claimed that her expenses had been scrutinised because she was a woman, saying: "[I] know that it was my expenses people looked at first because I was a woman and should have been at home looking after my husband and children." Smith said that she had felt "frozen rather than angry" on learning that her husband had entered a parliamentary expenses claim for two pornographic films.


title: Jacqui Smith

text: On 2 June 2009, Smith confirmed that she would leave the Cabinet in the next reshuffle, expected after the local and European elections. She left office on 5 June and returned to the backbenches. She was replaced by Alan Johnson. In a subsequent interview with "Total Politics" magazine regarding her time as Home Secretary, Smith described how she felt under-qualified for her ministerial roles, adding "when I became Home Secretary, I'd never run a major organisation. I hope I did a good job. But if I did, it was more by luck than by any kind of development of skills. I think we should have been better trained. I think there should have been more induction." Smith's major achievements as Home Secretary were introduction of tougher prostitution laws, a reduction in crime rates and promotion of Police Community Support Officers. Journalist Andrew Pierce echoed Smith's comments about her unsuitability for the position of home secretary, going further by stating: "Smith, beset by gaffes and errors, was hopelessly out of her depth in one of the most demanding jobs in politics".


title: Jacqui Smith

text: In the general election on 6 May 2010, Smith lost her seat as MP for Redditch to Karen Lumley of the Conservative Party, who won the seat with a majority of 5,821 votes. Smith said that she had been "immensely honoured" to serve Redditch. Smith wrote an open letter to the new Conservative home secretary Theresa May, advising her that the post was often seen as a "poisoned chalice".


title: Jacqui Smith

text: In 2010, she began working as a consultant for KPMG and as an adviser to Sarina Russo Job Access. She applied to be vice-chairman of the BBC Trust. Smith presented a documentary on pornography, for BBC Radio 5 Live, called "Porn Again" which was broadcast on 3 March 2011. It was followed by a special edition of the Tony Livesey show, discussing pornography.


title: Jacqui Smith

text: She has regularly been on "This Week" and "Question Time" and was also a regular weekly commentator on Sky News' "Press Preview". She also contributed to "The Purple Book" in 2011, putting forward new ideas on crime and policing.


title: Jacqui Smith

text: On 24 August 2011 it emerged that Smith had arranged for two prisoners on day-release to paint a room in her house, when they were supposed to be undertaking work to benefit the community. The Ministry of Justice launched an internal investigation into the matter, and Smith made a donation to the charity overseeing the scheme. The episode was condemned by Matthew Elliot of the TaxPayers' Alliance, who stated: "It's a disgrace that a former home secretary has used prisoners as her personal handymen".


title: Jacqui Smith

text: Smith previously co-hosted a weekly show on talk radio station LBC alongside former Conservative cabinet minister David Mellor – she took over from Ken Livingstone after he left to contest the London mayoral race of 2012. She became chair of University Hospitals Birmingham NHS Foundation Trust in December 2013, though in 2020 she stepped down from this role temporarily to perform in "Strictly Come Dancing".


title: Jacqui Smith

text: Smith publicly supported the campaign for the UK to remain in the European Union in the 2016 EU referendum, and continued to argue for a second referendum on the issue until the 2019 general election.


title: Jacqui Smith

text: Since 2017, Smith has co-hosted a weekly political and current affairs podcast, entitled For The Many, alongside LBC broadcaster Iain Dale. She has appeared on "Good Morning Britain" on ITV. She is also the chair of both the Jo Cox Foundation and the Sandwell Children's Trust.


title: Jacqui Smith

text: In September 2020, it was announced that Smith would compete in the eighteenth series of "Strictly Come Dancing". She was partnered with Anton du Beke, and became the first celebrity to be voted off the show.


title: Jacqui Smith

text: In 2021 she was appointed as chair of both Barts Health NHS Trust and Barking, Havering and Redbridge University Hospitals NHS Trust.


title: Jacqui Smith

text: Smith married Richard Timney in October 1987 and they have two sons. Smith employed her husband as her parliamentary aide on a salary of £40,000. In January 2020, she and Timney announced they had ended their marriage.


title: Covariant derivative

text: In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component.


title: Covariant derivative

text: The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation.


title: Covariant derivative

text: This article presents an introduction to the covariant derivative of a vector field with respect to a vector field, both in a coordinate free language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension of the same concept. The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection.


title: Covariant derivative

text: Historically, at the turn of the 20th century, the covariant derivative was introduced by Gregorio Ricci-Curbastro and Tullio Levi-Civita in the theory of Riemannian and pseudo-Riemannian geometry. Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) observed that the Christoffel symbols used to define the curvature could also provide a notion of differentiation which generalized the classical directional derivative of vector fields on a manifold. This new derivative – the Levi-Civita connection – was "covariant" in the sense that it satisfied Riemann's requirement that objects in geometry should be independent of their description in a particular coordinate system.


title: Covariant derivative

text: It was soon noted by other mathematicians, prominent among these being Hermann Weyl, Jan Arnoldus Schouten, and Élie Cartan, that a covariant derivative could be defined abstractly without the presence of a metric. The crucial feature was not a particular dependence on the metric, but that the Christoffel symbols satisfied a certain precise second order transformation law. This transformation law could serve as a starting point for defining the derivative in a covariant manner. Thus the theory of covariant differentiation forked off from the strictly Riemannian context to include a wider range of possible geometries.


title: Covariant derivative

text: In the 1940s, practitioners of differential geometry began introducing other notions of covariant differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold. By and large, these generalized covariant derivatives had to be specified "ad hoc" by some version of the connection concept. In 1950, Jean-Louis Koszul unified these new ideas of covariant differentiation in a vector bundle by means of what is known today as a Koszul connection or a connection on a vector bundle. Using ideas from Lie algebra cohomology, Koszul successfully converted many of the analytic features of covariant differentiation into algebraic ones. In particular, Koszul connections eliminated the need for awkward manipulations of Christoffel symbols (and other analogous non-tensorial objects) in differential geometry. Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject.


title: Covariant derivative

text: The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule, formula_1, which takes as its inputs: (1) a vector, u, defined at a point "P", and (2) a vector field v defined in a neighborhood of "P". The output is the vector formula_2, also at the point "P". The primary difference from the usual directional derivative is that formula_1 must, in a certain precise sense, be "independent" of the manner in which it is expressed in a coordinate system.


title: Covariant derivative

text: A vector may be "described" as a list of numbers in terms of a basis, but as a geometrical object the vector retains its identity regardless of how it is described. For a geometric vector written in components with respect to one basis, when the basis is changed the components transform according to a change of basis formula, with the coordinates undergoing a covariant transformation. The covariant derivative is required to transform, under a change in coordinates, by a covariant transformation in the same way as a basis does (hence the name).


title: Covariant derivative

text: In the case of Euclidean space, one usually defines the directional derivative of a vector field in terms of the difference between two vectors at two nearby points.


title: Covariant derivative

text: In such a system one translates one of the vectors to the origin of the other, keeping it parallel, then taking their difference within the same vector space. With a Cartesian (fixed orthonormal) coordinate system "keeping it parallel" amounts to keeping the components constant. This ordinary directional derivative on Euclidean space is the first example of a covariant derivative .


title: Covariant derivative

text: Next, one must take into account changes of the coordinate system. For example, if the Euclidean plane is described by polar coordinates, "keeping it parallel" does "not" amount to keeping the polar components constant under translation, since the coordinate grid itself "rotates". Thus, the same covariant derivative written in polar coordinates contains extra terms that describe how the coordinate grid itself rotates, or how in more general coordinates the grid expands, contracts, twists, interweaves, etc.


title: Covariant derivative

text: Consider the example of a particle moving along a curve in the Euclidean plane. In polar coordinates, may be written in terms of its radial and angular coordinates by . A vector at a particular time (for instance, a constant acceleration of the particle) is expressed in terms of formula_4, where formula_5 and formula_6 are unit tangent vectors for the polar coordinates, serving as a basis to decompose a vector in terms of radial and tangential components. At a slightly later time, the new basis in polar coordinates appears slightly rotated with respect to the first set. The covariant derivative of the basis vectors (the Christoffel symbols) serve to express this change.


title: Covariant derivative

text: In a curved space, such as the surface of the Earth (regarded as a sphere), the translation of tangent vectors between different points is not well defined, and its analog, parallel transport, depends on the path along which the vector is translated. A vector on a globe on the equator at point Q is directed to the north. Suppose we transport the vector (keeping it parallel) first along the equator to the point P, then drag it along a meridian to the N pole, and finally transport it along another meridian back to Q. Then we notice that the parallel-transported vector along a closed circuit does not return as the same vector; instead, it has another orientation. This would not happen in Euclidean space and is caused by the "curvature" of the surface of the globe. The same effect occurs if we drag the vector along an infinitesimally small closed surface subsequently along two directions and then back. This infinitesimal change of the vector is a measure of the curvature, and can be defined in terms of the covariant derivative.


title: Covariant derivative

text: Suppose an open subset formula_8 of a formula_9-dimensional Riemannian manifold formula_10 is embedded into Euclidean space formula_11 via a twice continuously-differentiable (C) mapping formula_12 such that the tangent space at formula_13 is spanned by the vectors


title: Covariant derivative

text: The last term is not tangential to "M", but can be expressed as a linear combination of the tangent space base vectors using the Christoffel symbols as linear factors plus a vector orthogonal to the tangent space:


title: Covariant derivative

text: In the case of the Levi-Civita connection, the covariant derivative formula_20, also written is defined as the orthogonal projection of the usual derivative onto tangent space:


title: Covariant derivative

text: To obtain the relation between Christoffel symbols for the Levi-Civita connection and the metric, first we must note that, since formula_22 in previous equation is orthogonal to tangent space:


title: Covariant derivative

text: implies for a basis using the symmetry of the scalar product and swapping the order of partial differentiation:


title: Covariant derivative

text: For a very simple example that captures the essence of the description above, draw a circle on a flat sheet of paper. Travel around the circle at a constant speed. The derivative of your velocity, your acceleration vector, always points radially inward. Roll this sheet of paper into a cylinder. Now the (Euclidean) derivative of your velocity has a component that sometimes points inward toward the axis of the cylinder depending on whether you're near a solstice or an equinox. (At the point of the circle when you are moving parallel to the axis, there is no inward acceleration. Conversely, at a point (1/4 of a circle later) when the velocity is along the cylinder's bend, the inward acceleration is maximum.) This is the (Euclidean) normal component. The covariant derivative component is the component parallel to the cylinder's surface, and is the same as that before you rolled the sheet into a cylinder.


title: Covariant derivative

text: A covariant derivative is a (Koszul) connection on the tangent bundle and other tensor bundles: it differentiates vector fields in a way analogous to the usual differential on functions. The definition extends to a differentiation on the dual of vector fields (i.e. covector fields) and to arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction).


title: Covariant derivative

text: Given a point formula_28 of the manifold formula_10, a real function formula_30 on the manifold and a tangent vector formula_31, the covariant derivative of at along is the scalar at , denoted formula_32, that represents the principal part of the change in the value of "f" when the argument of is changed by the infinitesimal displacement vector . (This is the differential of evaluated against the vector .) Formally, there is a differentiable curve formula_33 such that formula_34 and formula_35, and the covariant derivative of "f" at "p" is defined by


title: Covariant derivative

text: When formula_37 is a vector field on formula_10, the covariant derivative formula_39 is the function that associates with each point "p" in the common domain of "f" and v the scalar formula_40.


title: Covariant derivative

text: For a scalar function "f" and vector field v, the covariant derivative formula_41 coincides with the Lie derivative formula_42, and with the exterior derivative formula_43.


title: Covariant derivative

text: Given a point formula_44 of the manifold formula_10, a vector field formula_46 defined in a neighborhood of "p" and a tangent vector formula_31, the covariant derivative of u at "p" along v is the tangent vector at "p", denoted formula_48, such that the following properties hold (for any tangent vectors v, x and y at "p", vector fields u and w defined in a neighborhood of "p", scalar values "g" and "h" at "p", and scalar function "f" defined in a neighborhood of "p"):


title: Covariant derivative

text: Note that formula_49 depends not only on the value of u at "p" but also on values of u in an infinitesimal neighborhood of "p" because of the last property, the product rule.


title: Covariant derivative

text: If and are both vector fields defined over a common domain, then formula_59 denotes the vector field whose value at each point "p" of the domain is the tangent vector formula_60.


title: Covariant derivative

text: Given a field of covectors (or one-form) formula_61 defined in a neighborhood of "p", its covariant derivative formula_62 is defined in a way to make the resulting operation compatible with tensor contraction and the product rule. That is, formula_62 is defined as the unique one-form at "p" such that the following identity is satisfied for all vector fields u in a neighborhood of "p"


title: Covariant derivative

text: Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary tensor fields by imposing the following identities for every pair of tensor fields formula_65 and formula_66 in a neighborhood of the point "p":


title: Covariant derivative

text: The covariant derivative of a tensor field along a vector field v is again a tensor field of the same type.


title: Covariant derivative

text: Explicitly, let "T" be a tensor field of type . Consider "T" to be a differentiable multilinear map of smooth sections "α", "α", …, "α" of the cotangent bundle "T""M" and of sections "X", "X", …, "X" of the tangent bundle "TM", written "T"("α", "α", …, "X", "X", …) into R. The covariant derivative of "T" along "Y" is given by the formula


title: Covariant derivative

text: Given coordinate functions formula_72 any tangent vector can be described by its components in the basis formula_73


title: Covariant derivative

text: The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination formula_74.


title: Covariant derivative

text: To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field formula_75 along formula_76.


title: Covariant derivative

text: the coefficients formula_78 are the components of the connection with respect to a system of local coordinates. In the theory of Riemannian and pseudo-Riemannian manifolds, the components of the Levi-Civita connection with respect to a system of local coordinates are called Christoffel symbols.


title: Covariant derivative

text: Then using the rules in the definition, we find that for general vector fields formula_79 and formula_80 we get


title: Covariant derivative

text: The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field "u". In particular


title: Covariant derivative

text: In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change.


title: Covariant derivative

text: Or, in words: take the partial derivative of the tensor and add: formula_88 for every upper index formula_89, and formula_90 for every lower index formula_91.


title: Covariant derivative

text: If instead of a tensor, one is trying to differentiate a "tensor density" (of weight +1), then one also adds a term


title: Covariant derivative

text: where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. Incidentally, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is always zero.


title: Covariant derivative

text: In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its components in this equation.


title: Covariant derivative

text: Often a notation is used in which the covariant derivative is given with a semicolon, while a normal partial derivative is indicated by a comma. In this notation we write the same as:


title: Covariant derivative

text: In case two or more indexes appear after the semicolon, all of them must be understood as covariant derivatives:


title: Covariant derivative

text: In some older texts (notably Adler, Bazin & Schiffer, "Introduction to General Relativity"), the covariant derivative is denoted by a double pipe and the partial derivative by single pipe:


title: Covariant derivative

text: In general, covariant derivatives do not commute. By example, the covariant derivatives of vector field formula_111. Riemann tensor formula_112 is defined such that:


title: Covariant derivative

text: Since the covariant derivative formula_117 of a tensor field formula_118 at a point formula_44 depends only on the value of the vector field formula_120 at formula_44 one can define the covariant derivative along a smooth curve formula_122 in a manifold:


title: Covariant derivative

text: Note that the tensor field formula_118 only needs to be defined on the curve formula_122 for this definition to make sense.


title: Covariant derivative

text: In particular, formula_126 is a vector field along the curve formula_127 itself. If formula_128 vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a positive-definite metric then the geodesics for the connection are precisely the geodesics of the metric that are parametrized by arc length.


title: Covariant derivative

text: A covariant derivative introduces an extra geometric structure on a manifold that allows vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system.


title: Covariant derivative

text: There is however another generalization of directional derivatives which "is" canonical: the Lie derivative, which evaluates the change of one vector field along the flow of another vector field. Thus, one must know both vector fields in an open neighborhood, not merely at a single point. The covariant derivative on the other hand introduces its own change for vectors in a given direction, and it only depends on the vector direction at a single point, rather than a vector field in an open neighborhood of a point. In other words, the covariant derivative is linear (over "C"("M")) in the direction argument, while the Lie derivative is linear in neither argument.


title: Covariant derivative

text: Note that the antisymmetrized covariant derivative , and the Lie derivative differ by the torsion of the connection, so that if a connection is torsion free, then its antisymmetrization "is" the Lie derivative.


title: Affine space

text: In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.


title: Affine space

text: In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead "displacement vectors", also called "translation" vectors or simply "translations", between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector.


title: Affine space

text: Any vector space may be viewed as an affine space; this amounts to forgetting the special role played by the zero vector. In this case, elements of the vector space may be viewed either as "points" of the affine space or as "displacement vectors" or "translations". When considered as a point, the zero vector is called the "origin". Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an "affine subspace". One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear space). In finite dimensions, such an "affine subspace" is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding "homogeneous" linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.


title: Affine space

text: The "dimension" of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension in an affine space or a vector space of dimension is an affine hyperplane.


title: Affine space

text: The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after one has forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps"). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point—call it —is the origin. Two vectors, and , are to be added. Bob draws an arrow from point to point and another arrow from point to point , and completes the parallelogram to find what Bob thinks is , but Alice knows that he has actually computed


title: Affine space

text: Similarly, Alice and Bob may evaluate any linear combination of and , or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a linear combination is 1, then Alice and Bob will arrive at the same answer.


title: Affine space

text: Under this condition, for all coefficients , Alice and Bob describe the same point with the same linear combination, despite using different origins.


title: Affine space

text: While only Alice knows the "linear structure", both Alice and Bob know the "affine structure"—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. A set with an affine structure is an affine space.


title: Affine space

text: An "affine space" is a set together with a vector space formula_1, and a transitive and free action of the additive group of formula_1 on the set . The elements of the affine space are called "points". The vector space formula_1 is said to be "associated" to the affine space, and its elements are called "vectors", "translations", or sometimes "free vectors".


title: Affine space

text: Explicitly, the definition above means that the action is a mapping, generally denoted as an addition,


title: Affine space

text: The first two properties are simply defining properties of a (right) group action. The third property characterizes free and transitive actions, the onto character coming from transitivity, and then the injective character follows from the action being free. There is a fourth property that follows from 1, 2 above:


title: Affine space

text: Another way to express the definition is that an affine space is a principal homogeneous space for the action of the additive group of a vector space. Homogeneous spaces are by definition endowed with a transitive group action, and for a principal homogeneous space such a transitive action is by definition free.


title: Affine space

text: The properties of the group action allows for the definition of subtraction for any given ordered pair of points in , producing a vector of formula_1. This vector, denoted formula_16 or formula_17, is defined to be the unique vector in formula_1 such that


title: Affine space

text: Existence follows from the transitivity of the action, and uniqueness follows because the action is free.


title: Affine space

text: Affine spaces can be equivalently defined as a point set , together with a vector space formula_1, and a subtraction satisfying Weyl's axioms. In this case, the addition of a vector to a point is defined from the first of Weyl's axioms.


title: Affine space

text: An "affine subspace" (also called, in some contexts, a "linear variety", a flat, or, over the real numbers, a "linear manifold") of an affine space is a subset of such that, given a point formula_25, the set of vectors formula_26 is a linear subspace of formula_1. This property, which does not depend on the choice of , implies that is an affine space, which has formula_28 as its associated vector space.


title: Affine space

text: The linear subspace associated with an affine subspace is often called its "", and two subspaces that share the same direction are said to be "parallel".


title: Affine space

text: This implies the following generalization of Playfair's axiom: Given a direction , for any point of there is one and only one affine subspace of direction , which passes through , namely the subspace .


title: Affine space

text: The term "parallel" is also used for two affine subspaces such that the direction of one is included in the direction of the other.