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Amphitheatre

An amphitheatre (British English) or amphitheater (American English; both ) is an open-air venue used for entertainment, performances, and sports. The term derives from the ancient Greek ἀμφιθέατρον ('), from ἀμφί ('), meaning "on both sides" or "around" and θέατρον (), meaning "place for viewing".

Ancient Greek theatres were typically built on hillsides and semi-circular in design. The first amphitheatre may have been built at Pompeii around 70 BC. "The amphitheater (amphitheatrum) is so called because it is composed of two theaters, for an amphitheater is round, whereas a theater, having a semicircular shape, is half an ampitheater."

- Isidore of Seville, Etymologies, XV.ii.26–ii.40 307 Ancient Roman amphitheatres were oval or circular in plan, with seating tiers that surrounded the central performance area, like a modern open-air stadium. In contrast, both ancient Greek and ancient Roman theatres were built in a semicircle, with tiered seating rising on one side of the performance area.

Modern parlance uses "amphitheatre" for any structure with sloping seating, including theatre-style stages with spectator seating on only one side, theatres in the round, and stadia. They can be indoor or outdoor.

Natural landforms and rock formations of similar shape are sometimes known as natural amphitheatres.

Roman amphitheatres
About 230 Roman amphitheatres have been found across the area of the Roman Empire. Their typical shape, functions and name distinguish them from Roman theatres, which are more or less semicircular in shape; from the circuses (similar to hippodromes) whose much longer circuits were designed mainly for horse or chariot racing events; and from the smaller stadia, which were primarily designed for athletics and footraces.

Roman amphitheatres were circular or oval in plan, with a central arena surrounded by perimeter seating tiers. The seating tiers were pierced by entrance-ways controlling access to the arena floor, and isolating it from the audience. Temporary wooden structures functioning as amphitheaters would have been erected for the funeral games held in honour of deceased Roman magnates by their heirs, featuring fights to the death by gladiators, usually armed prisoners of war, at the funeral pyre or tomb of the deceased. These games are described in Roman histories as munera, gifts, entertainments or duties to honour deceased individuals, Rome's gods and the Roman community.

Some Roman writers interpret the earliest attempts to provide permanent amphitheaters and seating for the lower classes as populist political graft, rightly blocked by the Senate as morally objectionable; too-frequent, excessively "luxurious" munera would corrode traditional Roman morals. The provision of permanent seating was thought a particularly objectionable luxury.

The earliest permanent, stone and timber Roman amphitheatre with perimeter seating was built in the Campus Martius in 29 BCE. Most were built under Imperial rule, from the Augustan period (27 BCE–14 CE) onwards. Imperial amphitheatres were built throughout the Roman Empire, especial in provincial capitals and major colonies, as an essential aspect of Romanitas. There was no standard size; the largest could accommodate 40,000–60,000 spectators. The most elaborate featured multi-storeyed, arcaded façades and were decorated with marble, stucco and statuary. The best-known and largest Roman amphitheatre is the Colosseum in Rome, also known as the Flavian Amphitheatre (Amphitheatrum Flavium), after the Flavian dynasty who had it built. After the ending of gladiatorial games in the 5th century and of staged animal hunts in the 6th, most amphitheatres fell into disrepair. Their materials were mined or recycled. Some were razed, and others were converted into fortifications. A few continued as convenient open meeting places; in some of these, churches were sited.

Modern amphitheatres

In modern usage, an amphitheatre is a circular, semicircular or curved, performance space, particularly one located outdoors. Contemporary amphitheatres often include standing structures, called bandshells, sometimes curved or bowl-shaped, both behind the stage and behind the audience, creating an area which echoes or amplifies sound, making the amphitheatre ideal for musical or theatrical performances. Small-scale amphitheatres can serve to host outdoor local community performances.

Notable modern amphitheatres include the Shoreline Amphitheatre, the Hollywood Bowl and the Aula Magna at Stockholm University. The term "amphitheatre" is also used for some indoor venues, such as the (by now demolished) Gibson Amphitheatre and Chicago International Ampitheatre.

Natural amphitheatres

A natural amphitheatre is a performance space located in a spot where a steep mountain or a particular rock formation naturally amplifies or echoes sound, making it ideal for musical and theatrical performances. An amphitheatre can be naturally occurring formations which would be ideal for this purpose, even if no theatre has been constructed there.

Notable natural amphitheatres include the Drakensberg Amphitheatre in South Africa, Slane Castle in Ireland, the Supernatural Amphitheatre in Australia, and the Red Rocks and the Gorge Amphitheatres in the western United States.

There is evidence that the Anasazi people used natural amphitheatres for the public performance of music in Pre-Columbian times including a large constructed performance space in Chaco Canyon, New Mexico.

title: Amphitheatre


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Southwark

Southwark is a district of Central London situated on the south bank of the River Thames, forming the north-western part of the wider modern London Borough of Southwark. The district, which is the oldest part of South London, developed due to its position at the southern end of the early versions of London Bridge, for centuries the only dry crossing on the river. Around 43 AD, engineers of the Roman Empire found the geographic features of the south bank here suitable for the placement and construction of the first bridge.

London's historic core, the City of London, lay north of the bridge and for centuries the area of Southwark just south of the bridge was partially governed by the City, while other areas of the district were more loosely governed. The section known as Liberty of the Clink became a place of entertainment. By the 12th century Southwark had been incorporated as an ancient borough, and this historic status is reflected in the alternative name of the area, as Borough. The ancient borough of Southwark's river frontage extended from the modern borough boundary, just to the west of the Oxo Tower, to St Saviour's Dock (originally the mouth of the River Neckinger) in the east. In the 16th century, parts of Southwark near London Bridge became a formal City ward, Bridge Without.

The urban area expanded over the years and Southwark was completely separated administratively from the City in 1900. Local points of interest include Southwark Cathedral, Borough Market, Shakespeare's Globe theatre, The Shard, Tower Bridge, Butler's Wharf and the Tate Modern museum.

Toponymy
The name Suthriganaweorc or Suthringa geweorche is recorded for the area in the 10th-century Anglo-Saxon document known as the Burghal Hidage and means "fort of the men of Surrey" or "the defensive work of the men of Surrey". Southwark is recorded in the 1086 Domesday Book as Sudweca. The name means "southern defensive work" and is formed from the Old English sūþ (south) and weorc (work). The southern location is in reference to the City of London to the north, Southwark being at the southern end of London Bridge. In Old English, Surrey means "southern district (or the men of the southern district)", so the change from "southern district work" to the latter "southern work" may be an evolution based on the elision of the single syllable ge element, meaning district.

Rome
Recent excavation has revealed pre-Roman activity including evidence of early ploughing, burial mounds and ritual activity.

The natural geography of Southwark (now much altered by human activity), was the principal determining factor for the location of London Bridge, and therefore London itself.

Natural setting
Until relatively recent times, the Thames in central London was much wider and shallower at high tide. The natural shoreline of the City Of London was a short distance further back than it is now, and the high tide shoreline on the Southwark side was much further back, except for the area around London Bridge.

Southwark was mostly made up of a series of often marshy tidal islands in the Thames, with some of the waterways between these island formed by branches of the River Neckinger, a tributary of the Thames. A narrow strip of higher firmer ground ran on a N-S alignment and, even at high tide, provided a much narrower stretch of water, enabling the Romans to bridge the river.

As the lowest bridging point of the Thames in Roman Britain, it determined the position of Londinium; without London Bridge there is unlikely to have been a settlement of any importance in the area; previously the main crossing had been a ford near Vauxhall Bridge. Because of the bridge and the establishment of London, the Romans routed two Roman roads into Southwark: Stane Street and Watling Street which met in what in what is now Borough High Street.

For centuries London Bridge was the only Thames bridge in the area, until a bridge was built upstream more than 10 miles to the west.

Archaeological finds
In February 2022, archaeologists from the Museum of London Archaeology (MOLA) announced the discovery of a well-preserved massive Roman mosaic which is believed to date from A.D. 175–225. The dining room (triclinium) mosaic was patterned with knot patterns known as the Solomon's knot and dark red and blue floral and geometric shapes known as guilloche.

Archaeological work at Tabard Street in 2004 discovered a plaque with the earliest reference to 'Londoners' from the Roman period on it.

End of Roman Southwark
Londinium was abandoned at the end of the Roman occupation in the early 5th century and both the city and its bridge collapsed in decay. The settlement at Southwark, like the main settlement of London to the north of the bridge, had been more or less abandoned, a little earlier, by the end of the fourth century.

King Alfred the Great
Southwark appears to recover only during the time of King Alfred and his successors. Sometime about 886, the burh of Southwark was created and the Roman city area reoccupied. It was probably fortified to defend the bridge and hence the reemerging City of London to the north.

St Olaf
This defensive role is highlighted by the role of the bridge in the 1014-1016 war between King Ethelred the Unready and his ally Olaf II Haraldsson (later King of Norway, and afterwards known as St Olaf, or St Olave) on one side, and Sweyn Forkbeard and his son Cnut (later King Cnut), on the other.

London submitted to Swein in 1014, but on Swein's death, Ethelred returned, with Olaf in support. Swein had fortified London and the bridge, but according to Snorri Sturleson's saga, Edgar and Olaf tied ropes from the bridge's supporting posts and pulled it into the river, together with the Danish army, allowing Ethelred to recapture London. This may be the origin of the nursery rhyme "London Bridge Is Falling Down".

There was a church, St Olave's Church, dedicated to St Olaf before the Norman Conquest and this survived until the 1920's. St Olaf House (part of London Bridge Hospital), named after the church and its saint, stands on the spot. Tooley Street, being a corruption of St Olave's Street, also takes its name from the former church.

King Canute
Cnut returned in 1016, but capturing the city was a great challenge. To cut London off from upstream riverborne supplies, Cnut dug a trench around Southwark, so that he could sail or drag his ships around Southwark and get upstream in a way that allowed his boats to avoid the heavily defended London Bridge. In so doing he hoped to cut London off from river borne resupply from upstream. The Dane's efforts to recapture London were in vain, until he defeated Ethelred at the Battle of Assandun in Essex later that year, and became King of England. It is thought that the section of the Kent Road, at Lock Bridge, was Canute's Trench. In May, 1016, In 1173, a channel following a similar course was used to drain the Thames to allowing building work on London Bridge.

Later medieval period
Southwark and in particular the Bridge, proved a formidable obstacle against William the Conqueror in 1066. He failed to force the bridge during the Norman conquest of England, but Southwark was devastated.

At Domesday, the area's assets were: Bishop Odo of Bayeux held the monastery (the site of modern Southwark Cathedral) and the tideway, which still exists as St Mary Overie dock; the King owned the church (probably St Olave's) and its tidal stream (St Olave's Dock); the dues of the waterway or mooring place were shared between King William I and Earl Godwin; the King also had the toll of the strand; and "men of Southwark" had the right to "a haw and its toll". Southwark's value to the King was £16. Much of Southwark was originally owned by the church – the greatest reminder of monastic London is Southwark Cathedral, originally the priory of St Mary Overie.

During the early Middle Ages, Southwark developed and was one of the four Surrey towns which returned Members of Parliament for the first commons assembly in 1295. An important market occupied the High Street from some time in the 13th century, which was controlled by the city's officers—it was later removed in order to improve traffic to the Bridge, under a separate Trust by Act of Parliament of 1756 as the Borough Market on the present site. The area was renowned for its inns, especially The Tabard, from which Geoffrey Chaucer's pilgrims set off on their journey in The Canterbury Tales. The continuing defensive importance of London Bridge was demonstrated by its important role in thwarting Jack Cade's Rebellion in 1450; Cade's army tried to force its way across the bridge to enter the City, but was foiled in a battle which cost 200 lives. The bridge was also closed during the Siege of London in 1471, helping to foil attempts by the Bastard of Fauconberg to cross and capture the City.

Post-medieval
Just west of the Bridge was the Liberty of the Clink manor, which was never controlled by the city, but was held under the Bishopric of Winchester's nominal authority. This lack of oversight helped the area become the entertainment district for London, with a concentration of sometimes disreputable attractions such as bull and bear-baiting, taverns, theatre and brothels.

In the 1580s, Reasonable Blackman worked as a silk weaver in Southwark, as one of the first people of African heritage to work as independent business owners in London in that era. In 1587, Southwark's first playhouse theatre, The Rose, opened. The Rose was set up by Philip Henslowe, and soon became a popular place of entertainment for all classes of Londoners. Both Christopher Marlowe and William Shakespeare, two of the finest writers of the Elizabethan age, worked at the Rose.

In 1599 the Globe Theatre, in which Shakespeare was a shareholder, was erected on the Bankside in the Liberty of the Clink. It burned down in 1613, and was rebuilt in 1614, only to be closed by the Puritans in 1642 and subsequently pulled down not long thereafter. A modern replica, called Shakespeare's Globe, has been built near the original site. The impresario in the later Elizabethan period for these entertainments was Shakespeare's colleague Edward Alleyn, who left many local charitable endowments, most notably Dulwich College.

During the Second English Civil War, a force of Kentish Royalist Rebels approached London, hoping the lightly defended city might fall to them, or that the citizens would rise in their favour, however their hopes were quashed when Philip Skippon, in charge of the defence swiftly fortified the bridge making it all but impregnable to the modest Royalist force.

On 26 May 1676, ten years after the Great Fire of London, a great fire broke out, which continued for 17 hours before houses were blown up to create fire breaks. King Charles II and his brother, James, Duke of York, oversaw the effort.

There was also a famous fair in Southwark which took place near the Church of St George the Martyr. William Hogarth depicted this fair in his engraving of Southwark Fair (1733).

Southwark was also the location of several prisons, including those of the Crown or Prerogative Courts, the Marshalsea and King's Bench prisons, those of the local manors' courts, e.g., Borough Compter, The Clink and the Surrey county gaol originally housed at the White Lion Inn (also informally called the Borough Gaol) and eventually at Horsemonger Lane Gaol.

One other local family is of note, the Harvards. John Harvard went to the local parish free school of St Saviour's and on to Cambridge University. He migrated to the Massachusetts Colony and left his library and the residue of his will to the new college there, named after him as its first benefactor. Harvard University maintains a link, having paid for a memorial chapel within Southwark Cathedral (his family's parish church), and where its UK-based alumni hold services. John Harvard's mother's house is in Stratford-upon-Avon.

Urbanisation
In 1836 the first railway in the London area was created, the London and Greenwich Railway, originally terminating at Spa Road and later extended west to London Bridge. In 1861, another great fire in Southwark destroyed a large number of buildings between Tooley Street and the Thames, including those around Hays Wharf (later replaced by Hays Galleria) and blocks to the west almost as far as St Olave's Church.

The first deep-level underground tube line in London was the City and South London Railway, now the Bank branch of the Northern line, opened in 1890, running from King William Street south through Borough to Stockwell. Southwark, since 1999, is also now served by Southwark, Bermondsey and London Bridge stations on the Jubilee line.

Administrative history
Southwark is thought to have become a burh in 886. The area appears in the Domesday Book of 1086 within the hundred of Brixton as held by several Surrey manors.

The ancient borough of Southwark, enfranchised in 1295, initially consisted of the pre-existing Surrey parishes of St George the Martyr, St Olave, St Margaret and St Mary.

St Margaret and St Mary were abolished in 1541 and their former area combined to create Southwark St Saviour. Around 1555 Southwark St Thomas was split off from St Olave, and in 1733 Southwark St John Horsleydown was also split off.

In 1855 the parishes came into the area of responsibility of the Metropolitan Board of Works. The large St George the Martyr parish was governed by its own administrative vestry, but the smaller St John Horsleydown, St Olave and St Thomas parishes were grouped together to form the St Olave District. St Saviour was combined with Southwark Christchurch (the former liberty of Paris Garden) to form the St Saviour's District. In 1889 the area became part of the new County of London. St Olave and St Thomas were combined as a single parish in 1896.

The ancient borough of Southwark, was traditionally known simply as The Borough – or Borough – to distinguish it from 'The City', and this name has persisted as an alternative name for the area. The medieval heart of Southwark was also, simultaneously, referred to as the ward of Bridge Without when administered by the city (from 1550 to 1900) and as an aldermanry until 1978.

The local government arrangements were reorganised in 1900 with the creation of the Metropolitan Borough of Southwark. It comprised the parishes of Southwark Christchurch, Southwark St Saviours, Southwark St George the Martyr and Newington. The Metropolitan Borough of Southwark was based at the former Newington Vestry Hall, now known as Walworth Town Hall. The eastern parishes that had formed the St Olave District instead became part of the Metropolitan Borough of Bermondsey. In 1965 the two boroughs were combined with the Metropolitan Borough of Camberwell to form the current London Borough of Southwark.

A new Diocese of Southwark was established in 1905 from parts of the Diocese of Rochester; the diocese serves large parts of south London and Surrey.

Relationship with the City of London
Southwark was outside of the control of the City of London and was a haven for criminals and free traders, who would sell goods and conduct trades outside the regulation of the city's Livery Companies. In 1327 the City obtained control from King Edward III of the manor next to the south side of London Bridge known as the Town of Southwark (called latterly the Guildable Manor—i.e., the place of taxes and tolls). The Livery Companies also ensured that they had jurisdiction over the area.

From the Norman period manorial organisation obtained through major lay and ecclesiastic magnates. Southwark still has vestiges of this because of the connection with the City of London. In 1327 the city acquired from Edward III the original vill of Southwark and this was also described as "the borough". In 1536 Henry VIII acquired the Bermondsey Priory properties and in 1538 that of the Archbishop. In 1550 these were sold to the city.

After many decades of petitioning, in 1550 Southwark was incorporated into the City of London as the ward of Bridge Without. However, the Alderman was appointed by the Court of Aldermen and no Common Councilmen were ever elected. This ward was constituted of the original Guildable Manor and the properties previously held by the church, under a charter of Edward VI, latterly called the King's Manor or Great Liberty. These manors are still constituted by the City under a Bailiff and Steward with their Courts Leet and View of Frankpledge Juries and Officers which still meet—their annual assembly being held in November under the present High Steward (the Recorder of London). The Ward and Aldermanry were effectively abolished in 1978, by merging it with the Ward of Bridge Within. These manorial courts were preserved under the Administration of Justice Act 1977. Therefore, between 1750 and 1978 Southwark had two persons (the Alderman and the Recorder) who were members of the city's Court of Aldermen and Common Council who were elected neither by the City freemen or by the Southwark electorate but appointed by the Court of Aldermen.

Contemporary governance and representation
The Borough and Bankside Community Council corresponds to the Southwark electoral wards of Cathedrals and Chaucer. They are part of the Bermondsey and Old Southwark Parliament constituency whose Member of Parliament is Neil Coyle. It is within the Lambeth and Southwark London Assembly constituency. Until 2022 Southwark was the location of City Hall, the administrative headquarters of the Greater London Authority and the meeting place of the London Assembly and Mayor of London. Since 2009, Southwark London Borough Council has its main offices at 160 Tooley Street, having moved administrative staff from the Camberwell Town Hall.

Geography and attractions

In common with much of the south bank of the Thames, the Borough has seen extensive regeneration in the last decade. Declining wharfage trade, light industry and factories have given way to residential development, shops, restaurants, galleries, bars and most notably major office developments housing international headquarters of accountancy, legal and other professional services consultancies, most notably along London Bridge City and More London between Tooley Street and the riverside. The area is in easy walking distance of the City and the West End. As such it has become a major business centre with many national and international corporations, professional practices and publishers locating to the area. London's tallest skyscraper, the Shard, is next to London Bridge Station.

To the north is the River Thames, London Bridge station and Southwark Cathedral. Borough Market is a well-developed visitor attraction and has grown in size. The adjacent units have been converted and form a gastronomic focus for London. Borough High Street runs roughly north to south from London Bridge towards Elephant and Castle. The Borough runs further to the south than realised; both St George's Cathedral and the Imperial War Museum are within the ancient boundaries, which border nearby Lambeth.

Its entertainment district, in its heyday at the time of Shakespeare's Globe Theatre (which stood 1599–1642) has revived in the form of the post-1997 reinvention of the original theatre, Shakespeare's Globe, incorporating other smaller theatre spaces, an exhibition about Shakespeare's life and work and which neighbours Vinopolis and the London Dungeon. The Southbank area, primarily in Lambeth but shared with Southwark also hosts many artistic venues.

At its heart is the area known as Borough, which has an eclectic covered and semi-covered market and numerous food and drink venues as well as the skyscraper The Shard. The Borough is generally an area of mixed development, with council estates, major office developments, social housing and high value residential gated communities side by side with each other.

Another landmark is Southwark Cathedral, a priory then parish church, created a cathedral in 1905, noted for its Merbecke Choir. The area at an advanced stage of regeneration and has the City Hall offices of the Greater London Authority.

Transport
The area has three main tube stations: Borough, Southwark and one close to the river, which is combined with a railway terminus; London Bridge.

Notable people

 * Michael Caine (born 1933), actor
 * Bradley Gamble (born 1975), former footballer
 * King Krule (born 1994), singer, songwriter, musician, rapper, and record producer
 * Lucy Worsley (born 1973), historian, author, curator, and television presenter
 * Sir Keir Starmer (born 1962), Prime Minister of the United Kingdom

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Southwark


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Inductive logic programming

Inductive logic programming (ILP) is a subfield of symbolic artificial intelligence which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. The term "inductive" here refers to philosophical (i.e. suggesting a theory to explain observed facts) rather than mathematical (i.e. proving a property for all members of a well-ordered set) induction. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples.

 * Schema: positive examples + negative examples + background knowledge ⇒ hypothesis.

Inductive logic programming is particularly useful in bioinformatics and natural language processing.

History
Building on earlier work on Inductive inference, Gordon Plotkin was the first to formalise induction in a clausal setting around 1970, adopting an approach of generalising from examples. In 1981, Ehud Shapiro introduced several ideas that would shape the field in his new approach of model inference, an algorithm employing refinement and backtracing to search for a complete axiomatisation of given examples. His first implementation was the Model Inference System in 1981: a Prolog program that inductively inferred Horn clause logic programs from positive and negative examples. The term Inductive Logic Programming was first introduced in a paper by Stephen Muggleton in 1990, defined as the intersection of machine learning and logic programming. Muggleton and Wray Buntine introduced predicate invention and inverse resolution in 1988.

Several inductive logic programming systems that proved influential appeared in the early 1990s. FOIL, introduced by Ross Quinlan in 1990 was based on upgrading propositional learning algorithms AQ and ID3. Golem, introduced by Muggleton and Feng in 1990, went back to a restricted form of Plotkin's least generalisation algorithm. The Progol system, introduced by Muggleton in 1995, first implemented inverse entailment, and inspired many later systems. Aleph, a descendant of Progol introduced by Ashwin Srinivasan in 2001, is still one of the most widely used systems.

At around the same time, the first practical applications emerged, particularly in bioinformatics, where by 2000 inductive logic programming had been successfully applied to drug design, carcinogenicity and mutagenicity prediction, and elucidation of the structure and function of proteins. Unlike the focus on automatic programming inherent in the early work, these fields used inductive logic programming techniques from a viewpoint of relational data mining. The success of those initial applications and the lack of progress in recovering larger traditional logic programs shaped the focus of the field.

Recently, classical tasks from automated programming have moved back into focus, as the introduction of meta-interpretative learning makes predicate invention and learning recursive programs more feasible. This technique was pioneered with the Metagol system introduced by Muggleton, Dianhuan Lin, Niels Pahlavi and Alireza Tamaddoni-Nezhad in 2014. This allows ILP systems to work with fewer examples, and brought successes in learning string transformation programs, answer set grammars and general algorithms.

Setting
Inductive logic programming has adopted several different learning settings, the most common of which are learning from entailment and learning from interpretations. In both cases, the input is provided in the form of background knowledge $B$, a logical theory (commonly in the form of clauses used in logic programming), as well as positive and negative examples, denoted $E^+$ and $E^{-}$  respectively. The output is given as a hypothesis $H$, itself a logical theory that typically consists of one or more clauses.

The two settings differ in the format of examples presented.

Learning from entailment
, learning from entailment is by far the most popular setting for inductive logic programming. In this setting, the positive and negative examples are given as finite sets $E^+$ and $E^{-}$  of positive and negated ground literals, respectively. A correct hypothesis $H$ is a set of clauses satisfying the following requirements, where the turnstile symbol $$\models$$ stands for logical entailment: $$\begin{array}{llll} \text{Completeness:} & B \cup H & \models & E^+ \\ \text{Consistency: } & B \cup H \cup E^- & \not\models & \textit{false} \end{array}$$ Completeness requires any generated hypothesis $h$ to explain all positive examples $E^+$, and consistency forbids generation of any hypothesis $h$ that is inconsistent with the negative examples $E^{-}$ , both given the background knowledge $B$.

In Muggleton's setting of concept learning, "completeness" is referred to as "sufficiency", and "consistency" as "strong consistency". Two further conditions are added: "Necessity", which postulates that $B$ does not entail $E^+$,  does not impose a restriction on $h$, but forbids any generation of a hypothesis as long as the positive facts are explainable without it. . "Weak consistency", which states that no contradiction can be derived from $B\land H$, forbids generation of any hypothesis $h$ that contradicts the background knowledge $B$. Weak consistency is implied by strong consistency; if no negative examples are given, both requirements coincide. Weak consistency is particularly important in the case of noisy data, where completeness and strong consistency cannot be guaranteed.

Learning from interpretations
In learning from interpretations, the positive and negative examples are given as a set of complete or partial Herbrand structures, each of which are themselves a finite set of ground literals. Such a structure $e$ is said to be a model of the set of clauses $B \cup H$ if for any substitution $\theta$  and any clause $\mathrm{head} \leftarrow \mathrm{body}$  in $B \cup H$  such that $\mathrm{body}\theta \subseteq e$, $$\mathrm{head}\theta \subseteq e$$ also holds. The goal is then to output a hypothesis that is complete, meaning every positive example is a model of $B \cup H$, and consistent, meaning that no negative example is a model of $B \cup H$.

Approaches to ILP
An inductive logic programming system is a program that takes as an input logic theories $$B, E^+, E^-$$ and outputs a correct hypothesis $H$ with respect to theories $$B, E^+, E^-$$. A system is complete if and only if for any input logic theories $$B, E^+, E^-$$ any correct hypothesis $H$ with respect to these input theories can be found with its hypothesis search procedure. Inductive logic programming systems can be roughly divided into two classes, search-based and meta-interpretative systems.

Search-based systems exploit that the space of possible clauses forms a complete lattice under the subsumption relation, where one clause $C_1$  subsumes another clause $C_2$  if there is a substitution $\theta$  such that $C_1\theta$, the result of applying $\theta$  to $C_1$ , is a subset of $C_2$. This lattice can be traversed either bottom-up or top-down.

Bottom-up search
Bottom-up methods to search the subsumption lattice have been investigated since Plotkin's first work on formalising induction in clausal logic in 1970. Techniques used include least general generalisation, based on anti-unification, and inverse resolution, based on inverting the resolution inference rule.

Least general generalisation
A least general generalisation algorithm takes as input two clauses $C_1$ and $C_2$  and outputs the least general generalisation of  $C_1$  and $C_2$, that is, a clause  $C$  that subsumes  $C_1$  and $C_2$ , and that is subsumed by every other clause that subsumes  $C_1$  and $C_2$. The least general generalisation can be computed by first computing all selections from $C$ and $D$, which are pairs of literals  $$(L,M) \in (C_1, C_2)$$ sharing the same predicate symbol and negated/unnegated status. Then, the least general generalisation is obtained as the disjunction of the least general generalisations of the individual selections, which can be obtained by first-order syntactical anti-unification.

To account for background knowledge, inductive logic programming systems employ relative least general generalisations, which are defined in terms of subsumption relative to a background theory. In general, such relative least general generalisations are not guaranteed to exist; however, if the background theory $B$ is a finite set of ground literals, then the negation of $B$ is itself a clause. In this case, a relative least general generalisation can be computed by disjoining the negation of $B$ with both $C_1$ and $C_2$  and then computing their least general generalisation as before.

Relative least general generalisations are the foundation of the bottom-up system Golem.

Inverse resolution
Inverse resolution is an inductive reasoning technique that involves inverting the resolution operator.

Inverse resolution takes information about the resolvent of a resolution step to compute possible resolving clauses. Two types of inverse resolution operator are in use in inductive logic programming: V-operators and W-operators. A V-operator takes clauses $R$ and $C_1$ as input and returns a clause $C_2$  such that $R$  is the resolvent of $C_1$  and  $C_2$. A W-operator takes two clauses $R_1$ and $R_2$  and returns thre clauses $C_1$, $C_2$  and $C_3$  such that $R_1$  is the resolvent of $C_1$  and $C_2$  and $R_2$  is the resolvent of  $C_2$  and $C_3$.

Inverse resolution was first introduced by Stephen Muggleton and Wray Buntine in 1988 for use in the inductive logic programming system Cigol. By 1993, this spawned a surge of research into inverse resolution operators and their properties.

Top-down search
The ILP systems Progol, Hail and Imparo find a hypothesis $H$ using the principle of the inverse entailment for theories $B$, $E$, $H$: $$B \land H \models E \iff B \land \neg E \models \neg H$$. First they construct an intermediate theory $F$ called a bridge theory satisfying the conditions $$B \land \neg E \models F$$ and $$F \models \neg H$$. Then as $$H \models \neg F$$, they generalize the negation of the bridge theory $F$ with anti-entailment. However, the operation of anti-entailment is computationally more expensive since it is highly nondeterministic. Therefore, an alternative hypothesis search can be conducted using the inverse subsumption (anti-subsumption) operation instead, which is less non-deterministic than anti-entailment.

Questions of completeness of a hypothesis search procedure of specific inductive logic programming system arise. For example, the Progol hypothesis search procedure based on the inverse entailment inference rule is not complete by Yamamoto's example. On the other hand, Imparo is complete by both anti-entailment procedure and its extended inverse subsumption procedure.

Metainterpretive learning
Rather than explicitly searching the hypothesis graph, metainterpretive or meta-level systems encode the inductive logic programming program as a meta-level logic program which is then solved to obtain an optimal hypothesis. Formalisms used to express the problem specification include Prolog and answer set programming, with existing Prolog systems and answer set solvers used for solving the constraints.

And example of a Prolog-based system is Metagol, which is based on a meta-interpreter in Prolog, while ASPAL and ILASP are based on an encoding of the inductive logic programming problem in answer set programming.

List of implementations

 * 1BC and 1BC2: first-order naive Bayesian classifiers:
 * ACE (A Combined Engine)
 * Aleph
 * Atom
 * Claudien
 * DL-Learner
 * DMax
 * FastLAS (Fast Learning from Answer Sets)
 * FOIL (First Order Inductive Learner)
 * Golem
 * ILASP (Inductive Learning of Answer Set Programs)
 * Imparo
 * Inthelex (INcremental THEory Learner from EXamples)
 * Lime
 * Metagol
 * Mio
 * MIS (Model Inference System) by Ehud Shapiro
 * Popper
 * PROGOL
 * RSD
 * Warmr (now included in ACE)
 * ProGolem

Probabilistic inductive logic programming
Probabilistic inductive logic programming adapts the setting of inductive logic programming to learning probabilistic logic programs. It can be considered as a form of statistical relational learning within the formalism of probabilistic logic programming.

Given

 * 1) background knowledge as a probabilistic logic program $B$, and
 * 2) a set of positive and negative examples $E^{+}$  and $E^{-}$

the goal of probabilistic inductive logic programming is to find a probabilistic logic program $H$ such that the probability of positive examples according to ${H \cup B}$  is maximized and the probability of negative examples is minimized.

This problem has two variants: parameter learning and structure learning. In the former, one is given the structure (the clauses) of $H$ and the goal is to infer the probabilities annotations of the given clauses, while in the latter the goal is to infer both the structure and the probability parameters of $H$. Just as in classical inductive logic programming, the examples can be given as examples or as (partial) interpretations.

Parameter Learning
Parameter learning for languages following the distribution semantics has been performed by using an expectation-maximisation algorithm or by gradient descent. An expectation-maximisation algorithm consists of a cycle in which the steps of expectation and maximization are repeatedly performed. In the expectation step, the distribution of the hidden variables is computed according to the current values of the probability parameters, while in the maximisation step, the new values of the parameters are computed. Gradient descent methods compute the gradient of the target function and iteratively modify the parameters moving in the direction of the gradient.

Structure Learning
Structure learning was pioneered by Daphne Koller and Avi Pfeffer in 1997, where the authors learn the structure of first-order rules with associated probabilistic uncertainty parameters. Their approach involves generating the underlying graphical model in a preliminary step and then applying expectation-maximisation.

In 2008, De Raedt et al. presented an algorithm for performing theory compression on ProbLog programs, where theory compression refers to a process of removing as many clauses as possible from the theory in order to maximize the probability of a given set of positive and negative examples. No new clause can be added to the theory.

In the same year, Meert, W. et al. introduced a method for learning parameters and structure of ground probabilistic logic programs by considering the Bayesian networks equivalent to them and applying techniques for learning Bayesian networks.

ProbFOIL, introduced by De Raedt and Ingo Thon in 2010, combined the inductive logic programming system FOIL with ProbLog. Logical rules are learned from probabilistic data in the sense that both the examples themselves and their classifications can be probabilistic. The set of rules has to allow one to predict the probability of the examples from their description. In this setting, the parameters (the probability values) are fixed and the structure has to be learned.

In 2011, Elena Bellodi and Fabrizio Riguzzi introduced SLIPCASE, which performs a beam search among probabilistic logic programs by iteratively refining probabilistic theories and optimizing the parameters of each theory using expectation-maximisation. Its extension SLIPCOVER, proposed in 2014, uses bottom clauses generated as in Progol to guide the refinement process, thus reducing the number of revisions and exploring the search space more effectively. Moreover, SLIPCOVER separates the search for promising clauses from that of the theory: the space of clauses is explored with a beam search, while the space of theories is searched greedily.

title: Inductive logic programming


text:
Circus (disambiguation)

A circus is a traveling company of performers that may include acrobats, clowns, trained animals, and other novelty acts.

Circus or The Circus may also refer to:

Art and architecture

 * Roman circus, an open-air public space in the Roman Empire
 * The Circus, Bath, a structure in Bath, Somerset, England
 * The Circus (Seurat), an 1891 painting by Georges Seurat

Literature

 * Circus (magazine), an American rock music magazine 1966–2006
 * Circus (novel), a 1975 novel by Alistair MacLean

Film

 * The Circus (1928 film), an American film directed by and starring Charlie Chaplin
 * Circus (1936 film), a Soviet comedy musical directed by Grigori Aleksandrov
 * Circus (1939 film), a Swedish-Danish drama film directed by George Schnéevoigt
 * The Circus (1943 film), a Mexican comedy starring Cantinflas
 * Circus (2000 film), a British/American neo-noir crime film directed by Rob Walker
 * Circus (2009 film), an Indian Kannada-language film directed by Dayal Padmanabhan

Television

 * Circus (Canadian TV series), a 1978–1985 Canadian television variety show
 * Circus (Indian TV series), a 1989–1990 Indian drama series
 * Circus, a 2010 PBS documentary series on the Big Apple Circus
 * "Circus" (Aqua Teen Hunger Force), a 2002 episode
 * "Circus" (Glenn Martin, DDS), a 2009 episode
 * "The Circus", the debut episode of the second season of Helluva Boss
 * The Circus: Inside the Greatest Political Show on Earth, a 2016 documentary series

Performers

 * Circus (American band), a 1970–1975 power pop band
 * Circus (French band), a pop group formed in 2012
 * Circus (rapper) (21st century), American rapper
 * Circus, a 1967–1969 British jazz group that included Mel Collins
 * Circus, a 1972–1990 Swiss progressive rock band

Albums

 * Circus (Argent album) or the title song, 1975
 * Circus (Britney Spears album) or the title song (see below), 2008
 * The Circus Starring Britney Spears, a 2009 concert tour
 * Circus (Chiaki Kuriyama album), 2011
 * Circus (Eraserheads album), 1994
 * Circus (FictionJunction Yuuka album), 2007
 * Circus (Lenny Kravitz album) or the title song (see below), 1995
 * Circus (EP), by Stray Kids, or the title song (see below), 2022
 * Circus, by Flairck, 1981
 * Circus, by Mary Black, 1995
 * Circus, by Teflon Brothers, 2017
 * The Circus (Erasure album) or the title song (see below), 1987
 * The Circus (Take That album) or the title song, 2008
 * The Circus (EP), by Mick Jenkins, 2020
 * The Circus, by the Venetia Fair, 2009

Songs

 * "Circus" (Britney Spears song), 2008
 * "Circus" (Lenny Kravitz song), 1995
 * "Circus" (Stray Kids song), 2022
 * "Circus", by Brotherhood of Man later reissued on Images, 1977
 * "Circus", by Circle from Kollekt, 1998
 * "Circus", by Friend & Lover, 1969
 * "Circus", by John Cale from Walking on Locusts, 1996
 * "Circus", by Kim Taeyeon from Something New, 2018
 * "Circus", by John Denver from Rhymes & Reasons, 1969
 * "Circus", by MC Mong from Show's Just Begun, 2008
 * "Circus", by Sonny & Cher, 1969
 * "Circus", by String Driven Thing, 1972
 * "Circus", by Summer Walker from Still Over It, 2021
 * "Circus", by Tomoyasu Hotei, 1996
 * "Circus", by Uriah Heep from Sweet Freedom, 1973
 * "Circus", written by Louis Alter, 1949
 * "Circus (What I Am)", by Michael Quatro, 1972
 * "The Circus" (song), by Erasure, 1987
 * "The Circus", by the Clentele from Music for the Age of Miracles, 2017
 * "The Circus", by Toby Fox from Deltarune Chapter 1 OST, 2018

Other uses

 * Circus (bird), the largest genus of harriers
 * Circus (company), a Japanese visual novel studio
 * Circus (video game), a Breakout clone released by Exidy in 1977
 * Circus offensive, British Royal Air Force bombing operations during World War II
 * Secret Intelligence Service (MI6), nicknamed "The Circus", the UK foreign intelligence service
 * Circus, a United Kingdom term for a circular road junction

title: Circus (disambiguation)


text:
Talk:Division (mathematics)

Page title
Given Addition, Multiplication and Subtraction are all named without appending (mathematics) at the end... and the likely thing people are trying to find when they go to a page called Division will be the mathematical version... can't we just have Division go straight to this page with a thing at the top for disambiguation? --93.97.23.110 (talk) 02:22, 4 January 2011 (UTC)

 * Yes: your proposal would be more consistent with Wikipedia's style. --Hroðulf (or Hrothulf) (Talk) 10:30, 6 January 2011 (UTC)

 * While I'm generally in favour of making this move, one counterargument is that for addition, subtraction, and multiplication, all other uses of the word stem from or are to some extent indistinguishable from the mathematical sense of the word, while division as a mathematical concept stems from the more general concept of dividing something into smaller things. A set of 10 can be divided into 2 new sets, one with 7 and one with 3. This is very different from the mathematical sense, and some may consider this a compelling reason to keep Division as a disambiguation page, with the mathematical sense explicitly distinguished as such, while keeping the other operations where they are. That being said, the asymmetry does bother be, and in my opinion the move should be made. -Ramzuiv (talk) 02:34, 25 September 2019 (UTC)
 * Such a move has no chance to reach a WP:consensus, as the word has many meanings that have nothing to do with mathematics (14 meanings in the Wiktionary entry), and the mathematical meaning is certainly not the WP:primary topic. The primary topic, if any, is probably "The act or process of splitting anything". D.Lazard (talk) 08:06, 25 September 2019 (UTC)

Divided by - what is a succinct synonym?
20 ÷ 5 isn't dialectical English. When you write something, if possible write the wordish for of it.

twenty divided by five or ? — Preceding unsigned comment added by 2A02:2149:8464:BA00:BCE5:B794:8895:1FC4 (talk) 10:06, 15 February 2018 (UTC)
 * Good point. Fixed. D.Lazard (talk) 10:37, 15 February 2018 (UTC)

Division is anticommutative
Division is anti-commutative. Given that $$1$$ is the right identity for division

 * $$x \div 1 = x$$

then

 * $$x \div y = 1 \div (y \div x)$$

or in fraction form

 * $$\frac{x}{y} = \frac{1}{\frac{y}{x}}$$

Matt (talk) 11:41, 9 March 2018 (UTC)
 * It is not useful to duplicate your posts (here and in my talk page)
 * The assertion that division is anti-commutative is wrong, as anticommutativity is a well defined concept in mathematics. Division would be anti-commutative if one would have
 * $$\frac{y}{x} = -\frac{x}{y}$$
 * for all $x$ and $y$.
 * This is a true property that one gets the multiplicative inverse when commuting the operands of a division. This could be added to the article, but this should be written in a less confusing way (less jargon, more text, less formulas and less complicated formulas). D.Lazard (talk) 12:23, 9 March 2018 (UTC)

 * Limiting anticommutativity to just the additive inverse makes no sense, it is the simplest and most common but not the only case. Both are forms of the more general case; with operator $$\cdot$$ and right identity under operator $$\cdot$$ (a.k.a. $$I_{(\cdot)}$$) anything that in the general case conforms to this equation
 * $$x \cdot y = I_{(\cdot)} \cdot (y \cdot x)$$
 * can be considered to be anticommutative.
 * There is a general pattern here and if multiplication can be considered commutative (just as addition is commutative) it doesn't make sense to say that subtraction is anticommutative and division is not. Matt (talk) 18:33, 9 March 2018 (UTC)

 * While there is a pattern here, it is not the one you have settled on. The negative in the anti-commutative law that you are interpreting as additive inverse is not really that, as can be seen when you examine the more general n-ary definition. It is actually the sign of a permutation and can be positive or negative. It just happens to be negative in the common two variable case. Your attempt to redefine the meaning of anti-commutative will fall upon deaf ears and certainly has no place here on Wikipedia. --Bill Cherowitzo (talk) 19:07, 9 March 2018 (UTC)

 * I do not think that Matt refers to "additive inverse", but rather to "multiplicative inverse". So he arrives at "anticommutativity" of division, with respect to "multiplicative inverses", which is trivial in the context of division being the "inverse operation" of multiplication ($$\cdot$$). Certainly, "multiplying" something with its "multiplicative inverse" yields the "multiplicative unity". The pertinent WP-article Anticommutativity is not very explicit about a canonical or generally accepted setting. Personally, I prefer the notion "antisymmetry" for the setting described there (cf. "totally antisymmetric symbol"). My question at the ref-desk also did not return ultimate answers. Purgy (talk) 16:26, 10 March 2018 (UTC)

 * Aside from what Purgy has said above (I almost entirely agree with his description of his understanding of what I was trying to say) I consider the general n-ary definition as a moot point for two reasons a) the four basic operators I have mentioned are all binary, and b) if you consider either lambda calculus or Turing / von Neumann machines (where almost all the world's research effort on computability has gone) n-ary computations can always be reduced / decomposed into a sequence of binary operations. In the case of basic arithmetic and algebraic operations they can be reduced even further to the Peano axioms and the Successor function (itself often used for Church encoding in pure lambda calculus systems). Matt (talk) 16:24, 12 March 2018 (UTC)
 * Most of above discussion is WP:OR. In particular operator arity and computability have nothing to do with anti-commutativity. The point is: are there reliable sources that define anti-commutativity in terms of a multiplicative inverse? I do not know any, and I am quite sure the if there are some, they do not belong to the main stream of mathematics. Thus, unless somebody provide reliable sources, the extension of anti-commutativity to multiplicative inverses is WP:OR and does not belong to Wikipedia. D.Lazard (talk) 17:18, 12 March 2018 (UTC)

 * No, it's not "anti-commutative". The relevant properties are those of the inverse operation to multiplication in a group or - with application to arithmetic in mind - a commutative (a.k.a. Abelian) group. For general non-Abelian groups, the right quotient may be completely characterized by the properties: a/e = a, a/a = e and (a/c)/(b/c) = a/b. This can be further refined to a set of properties that eliminate explicit reference to the group identity, e, while yet implying the uniqueness of a group identity (and its existence, if the group is non-empty, as is normally always assumed). For Abelian groups, the relevant set of properties are a/(a/b) = b and a/(b/c) = c/(b/a), with the identity being (uniquely) definable as e = a/a. To ensure the group be non-empty, one can also add in the axiom e = e/e. In both cases, the defining properties of a group can be recovered from these by defining inverse by a⁻¹ = (a/a)/a and product by a b = a/(b⁻¹). The property you're referring to as "anti-commutativity" is actually the one I just cited: a/(b/c) = c/(b/a).

Adding an introduction
I have added an introduction section to the article because as it is, the beginning space was too complicated for something to work correctly as a start. As it was, it did not adequately describe division in a way that was approachable; trying to adequately describe the entire process of division without becoming to long was not something that it could do. Because division is more complex than the other elementary operations, it is too difficult to shove into the beginning. I have also merged the properties section into the intro to show how division works (the "properties" section was so far only one property, and not sufficient anyway). I would greatly appreciate any extra help in making the article more approachable without losing information. Thank you! IntegralPython (talk) 21:02, 23 October 2018 (UTC)

 * This article is already on my mind for a long time, but I am not sufficiently versed in the appropriate literature to make any changes on my behalf. I would like to revert the recent edit "making the text less cluttered", because I think its content should be discussed beforehand: there is no "remainder" in other basic operations at all, "keeping" the remainder may be usual lingo, but is imho un-mathy, creating the rationals therefrom is strange to me, "division by zero is only defined..." leads astray (similar to "summing" divergent series to -1/12), ...
 * I am not certain about, to which extent this article should, e.g.,
 * report the didactic stories invented to engage the beginners (see also Quotition and partition),
 * extend on the "operation" division (I am unsure to which extent there is mathematically RELEVANT literature on this) as a "basic" algebraic structure with e.g. "right-distributivity" (sic!), besides the important Euclidean "divides" with "remainder" in the integer environment,
 * refer to a function implementing multiplicative inverses, or simply demand their existence, with mentioning the final target of getting rid of "division" in all formal treatments on operations on rational/real/complex/quaternion-numbers,
 * delve into other settings (polynomials, algebra, ..., (wheels?))
 * I certainly would like to help. Purgy (talk) 08:25, 24 October 2018 (UTC)
 * The present version, has two major issues: the second paragraph is about teaching or learning division and does belongs to the lead (and, maybe, to this article, as consisting mainly of original research about the psychology of students). The third paragraph begin by the undefined concept of "mathematics of fields". If this means field theory, this is too technical for appearing here. If this means something else, the formulation is clearly not acceptable.
 * I'll try to improve this lead. D.Lazard (talk) 09:24, 24 October 2018 (UTC)

Symbols
I ended up here because I was idly curious about the background or history of the ÷ symbol and whether there was any truth to my perception that it's been largely disappearing in recent decades. Of course, I didn't find much about that, but I did find a line (under "Of integers") reading "Names and symbols used for integer division include div, /, \, and %". If that % wasn't a typo of ÷, it's in need of explanation. --01:00, 29 December 2018 (UTC)71.234.116.22 (talk)
 * The % appears only in the section on Euclidean division of integer, and more specifically in the paragraphs on division on computers. It is true that some computer languages use % for Euclidean division. As far as I know, this notation is not used outside computer programming. Nevertheless, this paragraph could be made clearer.
 * The symbol ÷ is called obelus, and there is an article about it, and it's history. Your perception of its disappearition is true, and is clearly stated in the article: "This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used." In fact, mathematicians do not use ÷ since a long time. I do not know whether it becomes less used in elementary arithmetic. If it is the case, a probable reason is that / is simpler to type on a keyboard than ÷. D.Lazard (talk) 09:48, 29 December 2018 (UTC)

Wikiprojects B class checklist
I'm going through the criteria for this page to be considered class B, and I'll just put some notes here for my thoughts as I go through the checklist. For Verifiability, I have started working my way through this page, placing markers on statements that may be worth verifying. We need much more verification before this article is in a good place.

As regards structure, I think the structure of this article is perfectly suitable, and have marked it as such, but it annoys me that the structure of this page and that of multiplication are quite different, so I would like to propose taking the effort to align this page's structure with that of multiplication, as well as subtraction and addition.

As far as accuracy, I find no reason to complain

I would like to see more content on this page. Off the top of my head, it seems worthwhile to mention the formal definition of arithmetic, division as it regards dimensional analysis, and probably also quotient sets. If you can think of anything else worth adding, please let me know.

Grammar and Style is good

This article seems quite lacking in support material. Some topics that stand to be illustrated include euclidean vs. rational division, illustrating both quotition and partition, and perhaps other interpretations of division, why commutativity and association don't hold, why it is left-distributative but not right-distributative, manual methods of division, and division of complex numbers and matrices. All of these topics have some way they can be visualized, and such illustrations will greatly aid understanding.

The standard I'm using for accessibility for this article is "can a 5 year old understand this". Maybe 5 years is a little extreme, but at the very least a 7-year old should be able to follow a good chunk of this article, which covers what many people would consider an elementary subject. Of course, some things (such as polynomial rings in one indeterminate) will be mentioned that I wouldn't expect a kid to understand - for more technical things such as these, I ask "would my father (that is, a knowledgeable person with a tangential interest in math) understand this?". I've gone through the introduction, channeling my inner 5-year-old to ask as many stupid questions as I could, taking notes, and I edited the article based on most of the notes I made. Considering how much I did edit, I don't feel that this article is at the level of accessibility that it should be at.

While I did pause at the lack of definition of what arithmetic is, I didn't really define it in my recent edit. Perhaps it should be defined, but I felt comfortable leaving it as it was. I feel the explanations for quotition and partition can be improved. I suspect this part: "In a ring the elements by which division is always possible are called the units (for example, 1 and –1 in the ring of integers)" can be improved.

- Ramzuiv (talk) 03:43, 6 September 2019 (UTC)

Can someone please rewrite the intro
A little pressed for time here but can someone please rewrite the intro? Current intro as of the time of this writing exemplifies the worst of Wikipedia intros: at some point I assume someone once wrote a couple basic starter sentences, and then some later editor began adding all this extra info (perhaps to show off their knowledge) about ISO 80000-2 standards and usage of symbols in anglophone countries and info about deprecating the division sign in favor of the solidus character blah blah blah. None of that stuff belongs in an intro lead paragraph about a general mathematical topic. The intro paragraph should be like one, at most two, concise, succinct sentences about division, and then all the ISO 80000-2 stuff and discussion about anglophone countries and other ridiculousness can be moved way further down in the article. Can’t express how much I hate intro paragraphs like this. —PowerPCG5 (talk) 03:23, 29 July 2021 (UTC)

a = c / b means a × b = c
Illustration is a pie divided by 4 to make 4 slices. slice = pie/4 letting a = slice, c = pie, b = 4 gives a = c / b $$4 \times slice = slice + slice + slice + slice = pie$$ b x a = c a = c / b means b × a = c 194.207.86.26 (talk) 11:16, 19 December 2021 (UTC)
 * This is in the 5th paragraph of the article. D.Lazard (talk) 11:43, 19 December 2021 (UTC)
 * The article says `a = c / b means a × b = c` but it should say `a = c / b means b × a = c` 194.207.86.26 (talk) 12:11, 20 December 2021 (UTC)
 * This does not matter, since $$a\times b = b\times a.$$ D.Lazard (talk) 15:20, 20 December 2021 (UTC)
 * It may as well be corrected though. Since division and multiplication are not always commutative. 194.207.86.26 (talk) 08:20, 2 May 2022 (UTC)
 * In the non-commutative case, the equivalence is obtained by multiplying by $b$ on the right. So, in this case, this is "a = c / b means a × b = c" that is correct. In any case, the symbol × is used only for multiplication of numbers, which is commutative, and, in the non-commutative case, notation $$cb^{-1}$$ is generally preferred to $$c/b$$ for avoiding any ambiguity. For example, if $B$ and $C$ are matrices $$CB^{-1}$$ is often encountered, and $$C/B$$ is never defined. D.Lazard (talk) 08:54, 2 May 2022 (UTC)
 * a = c / b, which is the right-quotient, always means a = c b⁻¹ which is equivalent to a×b = c. To get the equivalent of b×a = c would require a notation for left-quotient: a = b \ c = b⁻¹ c. For commutative products c / b = b \ c, but the two should not be confused for one another, since such confusion does not scale up to other cases of the product and quotient where commutativity does not apply.

"Delen" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Delen and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 February 14 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. User:1234qwer1234qwer4 (talk) 06:07, 14 February 2022 (UTC)

missing basics
There are no illustrations of how division is done at the introductory (grammar school), with __ in the U.S.  or other 'tools" in other countries.     ) Kdammers (talk) 03:09, 20 March 2022 (UTC) Look at the draft to see what I'm trying to show -- a figure a little like the square root figure.  Kdammers (talk) 03:12, 20 March 2022 (UTC)

title: Talk:Division (mathematics)


text:
Perpetual motion

Perpetual motion is the motion of bodies that continues forever in an unperturbed system. A perpetual motion machine is a hypothetical machine that can do work infinitely without an external energy source. This kind of machine is impossible, since its existence would violate the first and/or second laws of thermodynamics.

These laws of thermodynamics apply regardless of the size of the system. For example, the motions and rotations of celestial bodies such as planets may appear perpetual, but are actually subject to many processes that slowly dissipate their kinetic energy, such as solar wind, interstellar medium resistance, gravitational radiation and thermal radiation, so they will not keep moving forever.

Thus, machines that extract energy from finite sources cannot operate indefinitely because they are driven by the energy stored in the source, which will eventually be exhausted. A common example is devices powered by ocean currents, whose energy is ultimately derived from the Sun, which itself will eventually burn out.

In 2016, new states of matter, time crystals, were discovered in which, on a microscopic scale, the component atoms are in continual repetitive motion, thus satisfying the literal definition of "perpetual motion". However, these do not constitute perpetual motion machines in the traditional sense, or violate thermodynamic laws, because they are in their quantum ground state, so no energy can be extracted from them; they exhibit motion without energy.

History
The history of perpetual motion machines dates back to the Middle Ages. For millennia, it was not clear whether perpetual motion devices were possible or not, until the development of modern theories of thermodynamics showed that they were impossible. Despite this, many attempts have been made to create such machines, continuing into modern times. Modern designers and proponents often use other terms, such as "over unity", to describe their inventions.

Basic principles
There is a scientific consensus that perpetual motion in an isolated system violates either the first law of thermodynamics, the second law of thermodynamics, or both. The first law of thermodynamics is a version of the law of conservation of energy. The second law can be phrased in several different ways, the most intuitive of which is that heat flows spontaneously from hotter to colder places; relevant here is that the law observes that in every macroscopic process, there is friction or something close to it; another statement is that no heat engine (an engine which produces work while moving heat from a high temperature to a low temperature) can be more efficient than a Carnot heat engine operating between the same two temperatures.

In other words:

 * 1) In any isolated system, one cannot create new energy (law of conservation of energy). As a result, the thermal efficiency—the produced work power divided by the input heating power—cannot be greater than one.
 * 2) The output work power of heat engines is always smaller than the input heating power. The rest of the heat energy supplied is wasted as heat to the ambient surroundings. The thermal efficiency therefore has a maximum, given by the Carnot efficiency, which is always less than one.
 * 3) The efficiency of real heat engines is even lower than the Carnot efficiency due to irreversibility arising from the speed of processes, including friction.

Statements 2 and 3 apply to heat engines. Other types of engines that convert e.g. mechanical into electromagnetic energy, cannot operate with 100% efficiency, because it is impossible to design any system that is free of energy dissipation.

Machines that comply with both laws of thermodynamics by accessing energy from unconventional sources are sometimes referred to as perpetual motion machines, although they do not meet the standard criteria for the name. By way of example, clocks and other low-power machines, such as Cox's timepiece, have been designed to run on the differences in barometric pressure or temperature between night and day. These machines have a source of energy, albeit one which is not readily apparent, so that they only seem to violate the laws of thermodynamics.

Even machines that extract energy from long-lived sources - such as ocean currents - will run down when their energy sources inevitably do. They are not perpetual motion machines because they are consuming energy from an external source and are not isolated systems.

Classification
One classification of perpetual motion machines refers to the particular law of thermodynamics the machines purport to violate:
 * A perpetual motion machine of the first kind produces work without the input of energy. It thus violates the first law of thermodynamics: the law of conservation of energy.
 * A perpetual motion machine of the second kind is a machine that spontaneously converts thermal energy into mechanical work. When the thermal energy is equivalent to the work done, this does not violate the law of conservation of energy. However, it does violate the more subtle second law of thermodynamics in a cyclic process (see also entropy). The signature of a perpetual motion machine of the second kind is that there is only one heat reservoir involved, which is being spontaneously cooled without involving a transfer of heat to a cooler reservoir. This conversion of heat into useful work, without any side effect, is impossible, according to the second law of thermodynamics.
 * A perpetual motion machine of the third kind is defined as one that completely eliminates friction and other dissipative forces, to maintain motion forever due to its mass inertia (third in this case refers solely to the position in the above classification scheme, not the third law of thermodynamics). It is impossible to make such a machine, as dissipation can never be completely eliminated in a mechanical system, no matter how close a system gets to this ideal (see examples at  below).

Impossibility

"Epistemic impossibility" describes things which absolutely cannot occur within our current formulation of the physical laws. This interpretation of the word "impossible" is what is intended in discussions of the impossibility of perpetual motion in a closed system.

The conservation laws are particularly robust from a mathematical perspective. Noether's theorem, which was proven mathematically in 1915, states that any conservation law can be derived from a corresponding continuous symmetry of the action of a physical system. The symmetry which is equivalent to conservation of energy is the time invariance of physical laws. Therefore, if the laws of physics do not change with time, then the conservation of energy follows. For energy conservation to be violated to allow perpetual motion would require that the foundations of physics would change.

Scientific investigations as to whether the laws of physics are invariant over time use telescopes to examine the universe in the distant past to discover, to the limits of our measurements, whether ancient stars were identical to stars today. Combining different measurements such as spectroscopy, direct measurement of the speed of light in the past and similar measurements demonstrates that physics has remained substantially the same, if not identical, for all of observable time spanning billions of years.

The principles of thermodynamics are so well established, both theoretically and experimentally, that proposals for perpetual motion machines are universally dismissed by physicists. Any proposed perpetual motion design offers a potentially instructive challenge to physicists: one is certain that it cannot work, so one must explain how it fails to work. The difficulty (and the value) of such an exercise depends on the subtlety of the proposal; the best ones tend to arise from physicists' own thought experiments and often shed light upon certain aspects of physics. So, for example, the thought experiment of a Brownian ratchet as a perpetual motion machine was first discussed by Gabriel Lippmann in 1900 but it was not until 1912 that Marian Smoluchowski gave an adequate explanation for why it cannot work. However, during that twelve-year period scientists did not believe that the machine was possible. They were merely unaware of the exact mechanism by which it would inevitably fail.

"The law that entropy always increases – the second law of thermodynamics – holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation."

- Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)

In the mid-19th-century Henry Dircks investigated the history of perpetual motion experiments, writing a vitriolic attack on those who continued to attempt what he believed to be impossible:

"There is something lamentable, degrading, and almost insane in pursuing the visionary schemes of past ages with dogged determination, in paths of learning which have been investigated by superior minds, and with which such adventurous persons are totally unacquainted. The history of Perpetual Motion is a history of the fool-hardiness of either half-learned, or totally ignorant persons."

- Henry Dircks, Perpetuum Mobile: Or, A History of the Search for Self-motive (1861)

Techniques
Some common ideas recur repeatedly in perpetual motion machine designs. Many ideas that continue to appear today were stated as early as 1670 by John Wilkins, Bishop of Chester and an official of the Royal Society. He outlined three potential sources of power for a perpetual motion machine, "Chymical [sic] Extractions", "Magnetical Virtues" and "the Natural Affection of Gravity".

The seemingly mysterious ability of magnets to influence motion at a distance without any apparent energy source has long appealed to inventors. One of the earliest examples of a magnetic motor was proposed by Wilkins and has been widely copied since: it consists of a ramp with a magnet at the top, which pulled a metal ball up the ramp. Near the magnet was a small hole that was supposed to allow the ball to drop under the ramp and return to the bottom, where a flap allowed it to return to the top again. However, if the magnet is to be strong enough to pull the ball up the ramp, it cannot then be weak enough to allow gravity to pull it through the hole. Faced with this problem, more modern versions typically use a series of ramps and magnets, positioned so the ball is to be handed off from one magnet to another as it moves. The problem remains the same.

Gravity also acts at a distance, without an apparent energy source, but to get energy out of a gravitational field (for instance, by dropping a heavy object, producing kinetic energy as it falls) one has to put energy in (for instance, by lifting the object up), and some energy is always dissipated in the process. A typical application of gravity in a perpetual motion machine is Bhaskara's wheel in the 12th century, whose key idea is itself a recurring theme, often called the overbalanced wheel: moving weights are attached to a wheel in such a way that they fall to a position further from the wheel's center for one half of the wheel's rotation, and closer to the center for the other half. Since weights further from the center apply a greater torque, it was thought that the wheel would rotate forever. However, since the side with weights further from the center has fewer weights than the other side, at that moment, the torque is balanced and perpetual movement is not achieved. The moving weights may be hammers on pivoted arms, or rolling balls, or mercury in tubes; the principle is the same.

Another theoretical machine involves a frictionless environment for motion. This involves the use of diamagnetic or electromagnetic levitation to float an object. This is done in a vacuum to eliminate air friction and friction from an axle. The levitated object is then free to rotate around its center of gravity without interference. However, this machine has no practical purpose because the rotated object cannot do any work as work requires the levitated object to cause motion in other objects, bringing friction into the problem. Furthermore, a perfect vacuum is an unattainable goal since both the container and the object itself would slowly vaporize, thereby degrading the vacuum.

To extract work from heat, thus producing a perpetual motion machine of the second kind, the most common approach (dating back at least to Maxwell's demon) is unidirectionality. Only molecules moving fast enough and in the right direction are allowed through the demon's trap door. In a Brownian ratchet, forces tending to turn the ratchet one way are able to do so while forces in the other direction are not. A diode in a heat bath allows through currents in one direction and not the other. These schemes typically fail in two ways: either maintaining the unidirectionality costs energy (requiring Maxwell's demon to perform more thermodynamic work to gauge the speed of the molecules than the amount of energy gained by the difference of temperature caused) or the unidirectionality is an illusion and occasional big violations make up for the frequent small non-violations (the Brownian ratchet will be subject to internal Brownian forces and therefore will sometimes turn the wrong way).

Buoyancy is another frequently misunderstood phenomenon. Some proposed perpetual-motion machines miss the fact that to push a volume of air down in a fluid takes the same work as to raise a corresponding volume of fluid up against gravity. These types of machines may involve two chambers with pistons, and a mechanism to squeeze the air out of the top chamber into the bottom one, which then becomes buoyant and floats to the top. The squeezing mechanism in these designs would not be able to do enough work to move the air down, or would leave no excess work available to be extracted.

Patents
Proposals for such inoperable machines have become so common that the United States Patent and Trademark Office (USPTO) has made an official policy of refusing to grant patents for perpetual motion machines without a working model. The USPTO Manual of Patent Examining Practice states: "With the exception of cases involving perpetual motion, a model is not ordinarily required by the Office to demonstrate the operability of a device. If operability of a device is questioned, the applicant must establish it to the satisfaction of the examiner, but he or she may choose his or her own way of so doing."

And, further, that: "A rejection [of a patent application] on the ground of lack of utility includes the more specific grounds of inoperativeness, involving perpetual motion. A rejection under 35 U.S.C. 101 for lack of utility should not be based on grounds that the invention is frivolous, fraudulent or against public policy."

The filing of a patent application is a clerical task, and the USPTO will not refuse filings for perpetual motion machines; the application will be filed and then most probably rejected by the patent examiner, after he has done a formal examination. Even if a patent is granted, it does not mean that the invention actually works, it just means that the examiner believes that it works, or was unable to figure out why it would not work.

The United Kingdom Patent Office has a specific practice on perpetual motion; Section 4.05 of the UKPO Manual of Patent Practice states: "Processes or articles alleged to operate in a manner which is clearly contrary to well-established physical laws, such as perpetual motion machines, are regarded as not having industrial application."

Examples of decisions by the UK Patent Office to refuse patent applications for perpetual motion machines include:
 * Decision BL O/044/06, John Frederick Willmott's application no. 0502841
 * Decision BL O/150/06, Ezra Shimshi's application no. 0417271

The European Patent Classification (ECLA) has classes including patent applications on perpetual motion systems: ECLA classes "F03B17/04: Alleged perpetua mobilia" and "F03B17/00B: [... machines or engines] (with closed loop circulation or similar : ... Installations wherein the liquid circulates in a closed loop; Alleged perpetua mobilia of this or similar kind".

Apparent perpetual motion machines
As a perpetual motion machine can only be defined in a finite isolated system with discrete parameters, and since true isolated systems do not exist (among other things, due to quantum uncertainty), "perpetual motion" in the context of this article is better defined as a "perpetual motion machine" because a machine is a "A device that directs and controls energy, often in the form of movement or electricity, to produce a certain effect" whereas "motion" is simply movement (such as Brownian motion). Distinctions aside, on the macro scale, there are concepts and technical drafts that propose "perpetual motion", but on closer analysis it is revealed that they actually "consume" some sort of natural resource or latent energy, such as the phase changes of water or other fluids or small natural temperature gradients, or simply cannot sustain indefinite operation. In general, extracting work from these devices is impossible.

Resource consuming

Some examples of such devices include:
 * The drinking bird toy functions using small ambient temperature gradients and evaporation. It runs until all water is evaporated.
 * A capillary action-based water pump functions using small ambient temperature gradients and vapour pressure differences. With the "capillary bowl", it was thought that the capillary action would keep the water flowing in the tube, but since the cohesion force that draws the liquid up the tube in the first place holds the droplet from releasing into the bowl, the flow is not perpetual.
 * A Crookes radiometer consists of a partial vacuum glass container with a lightweight propeller moved by (light-induced) temperature gradients.
 * Any device picking up minimal amounts of energy from the natural electromagnetic radiation around it, such as a solar-powered motor.
 * Any device powered by changes in air pressure, such as some clocks (Cox's timepiece, Beverly Clock). The motion leeches energy from moving air which in turn gained its energy from being acted on.
 * A heat pump, due to it having a COP above 1: the energy it consumes as work is less than the energy it moves as heat.
 * The Atmos clock uses changes in the vapor pressure of ethyl chloride with temperature to wind the clock spring.
 * A device powered by induced nuclear reactions or by radioactive decay from an isotope with a relatively long half-life; such a device could plausibly operate for hundreds or thousands of years.
 * The Oxford Electric Bell and the Karpen Pile are driven by dry pile batteries.

Low friction

 * In flywheel energy storage, "modern flywheels can have a zero-load rundown time measurable in years".
 * Once spun up, objects in the vacuum of space—stars, black holes, planets, moons, spin-stabilized satellites, etc.—dissipate energy very slowly, allowing them to spin for long periods. Tides on Earth are dissipating the gravitational energy of the Moon/Earth system at an average rate of about 3.75 terawatts.
 * In certain quantum-mechanical systems (such as superfluidity and superconductivity), very low friction movement is possible. However, the motion stops when the system reaches an equilibrium state (e.g. all the liquid helium arrives at the same level). Similarly, seemingly entropy-reversing effects like superfluids climbing the walls of containers operate by ordinary capillary action.

Thought experiments
In some cases a thought experiment appears to suggest that perpetual motion may be possible through accepted and understood physical processes. However, in all cases, a flaw has been found when all of the relevant physics is considered. Examples include:
 * Maxwell's demon: This was originally proposed to show that the second law of thermodynamics applied in the statistical sense only, by postulating a "demon" that could select energetic molecules and extract their energy. Subsequent analysis (and experiment) have shown there is no way to physically implement such a system that does not result in an overall increase in entropy.
 * Brownian ratchet: In this thought experiment, one imagines a paddle wheel connected to a ratchet. Brownian motion would cause surrounding gas molecules to strike the paddles, but the ratchet would only allow it to turn in one direction. A more thorough analysis showed that when a physical ratchet was considered at this molecular scale, Brownian motion would also affect the ratchet and cause it to randomly fail resulting in no net gain. Thus, the device would not violate the laws of thermodynamics.
 * Vacuum energy and zero-point energy: In order to explain effects such as virtual particles and the Casimir effect, many formulations of quantum physics include a background energy which pervades empty space, known as vacuum or zero-point energy. The ability to harness zero-point energy for useful work is considered pseudoscience by the scientific community at large. Inventors have proposed various methods for extracting useful work from zero-point energy, but none have been found to be viable, no claims for extraction of zero-point energy have ever been validated by the scientific community, and there is no evidence that zero-point energy can be used in violation of conservation of energy.
 * Ellipsoid paradox: This paradox considers a perfectly reflecting cavity with two black bodies at points A and B. The reflecting surface is composed of two elliptical sections E1 and E2 and a spherical section S, and the bodies at A and B are located at the joint foci of the two ellipses and B is at the center of S. This configuration is such that apparently black body at B heat up relative to A: the radiation originating from the blackbody at A will land on and be absorbed by the blackbody at B. Similarly, rays originating from point B that land on E1 and E2 will be reflected to A. However, a significant proportion of rays that start from B will land on S will be reflected back to B. This paradox is solved when the black bodies' finite sizes are considered instead of punctual black bodies.

Conspiracy theories
Despite being dismissed as pseudoscientific, perpetual motion machines have become the focus of conspiracy theories, alleging that they are being hidden from the public by corporations or governments, who would lose economic control if a power source capable of producing energy cheaply was made available.

title: Perpetual motion


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Robert Ludlum

Robert Ludlum (May 25, 1927 – March 12, 2001) was an American author of 27 thriller novels, best known as the creator of Jason Bourne from the original The Bourne Trilogy series. The number of copies of his books in print is estimated between 300 million and 500 million. They have been published in 33 languages and 40 countries. Ludlum also published books under the pseudonyms Jonathan Ryder and Michael Shepherd.

Early life and education
Ludlum was born in New York City, the son of Margaret (née Wadsworth) and George Hartford Ludlum. He was educated at the Rectory School then Cheshire Academy and Wesleyan University in Middletown, Connecticut, where he earned a B.A. in Drama in 1951.

Career
Prior to becoming an author, he had been a United States Marine, a theatrical actor and producer. In the 1950s, he produced shows at the Grant Lee theater in Fort Lee, New Jersey. From 1960 to 1970, he managed and produced shows at the Playhouse on the Mall at Bergen Mall in Paramus, New Jersey. His theatrical experience may have contributed to his understanding of the energy, escapism and action that the public wanted in a novel. He once remarked: "I equate suspense and good theater in a very similar way. I think it's all suspense and what-happens-next. From that point of view, yes, I guess, I am theatrical."

Many of Ludlum's novels have been made into films and mini-series, including The Osterman Weekend, The Holcroft Covenant, The Apocalypse Watch, The Bourne Identity, The Bourne Supremacy and The Bourne Ultimatum. Covert One: The Hades Factor, a book co-written with Gayle Lynds, was originally conceived as a mini-series; the book evolved from a short treatment Ludlum wrote for NBC. The Bourne movies, starring Matt Damon in the title role, have been commercially and critically successful (The Bourne Ultimatum won three Academy Awards in 2008), although the story lines depart significantly from the source material.

During the 1970s, Ludlum lived in Leonia, New Jersey, where he spent hours each day writing at his home.

Death
Ludlum died of a heart attack on March 12, 2001, at his home in Naples, Florida, while recovering from severe burns caused by a mysterious fire which occurred on February 10, 2001.

In 2005, the company which held all merchandising rights for Ludlum's works via his estate, Ludlum Entertainment, signed an agreement with Vivendi Universal Games to handle video game rights for ten years. The license expired in 2008. On November 20, 2008, the Ludlum estate signed a deal with Universal Pictures to develop feature films based on the works of Robert Ludlum. On February 2, 2009, his estate transferred video game rights to Ludlum’s work to Electronic Arts after Activision took over Vivendi Games' assets.

Writing analysis and criticism
Ludlum's novels typically feature one heroic man, or a small group of crusading individuals, in a struggle against powerful adversaries whose intentions and motivations are evil and who are capable of using political and economic mechanisms in frightening ways. The world in his writings is one where global corporations, shadowy military forces and government organizations all conspired to preserve (if it was evil) or undermine (if it was law-abiding) the status quo.

Ludlum's novels were often inspired by conspiracy theories, both historical and contemporary. He wrote that The Matarese Circle was inspired by rumors about the Trilateral Commission, and it was published only a few years after the commission was founded. His depictions of terrorism in books such as The Holcroft Covenant and The Matarese Circle reflected the theory that terrorists, rather than being merely isolated bands of ideologically or politically motivated extremists, are actually pawns of governments or private organizations who are using them to facilitate the establishment of authoritarian rule.

Filmography
Many of Ludlum's novels have been made into films and miniseries, although the storylines might depart significantly from the source material. In general, a miniseries is more faithful to the original novel on which it is based. Adaptations of Ludlum's works are published under the trademark Treadstone, which is held by the executor of the Robert Ludlum estate.
 * 1977 – The Rhinemann Exchange — miniseries — Stephen Collins as David Spaulding, Lauren Hutton as Leslie Jenner Hawkewood
 * 1983 – The Osterman Weekend — film — Rutger Hauer as John Tanner, Sam Peckinpah directed
 * 1985 – The Holcroft Covenant — film — Michael Caine as Noel Holcroft
 * 1988 – The Bourne Identity — miniseries — Richard Chamberlain as Jason Bourne, Jaclyn Smith as Marie St. Jacques
 * 1997 – The Apocalypse Watch — miniseries — Patrick Bergin as Drew Latham
 * 2002 – The Bourne Identity — film — Matt Damon as Jason Bourne and Franka Potente as Marie Helena Kreutz
 * 2004 – The Bourne Supremacy — film — Matt Damon as Jason Bourne
 * 2006 – Covert One: The Hades Factor — miniseries — Stephen Dorff as Jon Smith
 * 2007 – The Bourne Ultimatum — film — Matt Damon as Jason Bourne
 * 2012 – The Bourne Legacy — film — starring Jeremy Renner, Rachel Weisz and Edward Norton
 * 2016 – Jason Bourne — film — Matt Damon as Jason Bourne
 * TBA – The Chancellor Manuscript — film — Leonardo DiCaprio as Peter Chancellor1
 * TBA – The Janson Directive — film — John Cena as Paul Janson1

1 announced/in development

title: Robert Ludlum


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Gentianales

Gentianales is an order of flowering plant, included within the asterid clade of eudicots. It comprises more than 20,000 species in about 1,200 genera in 5 families. More than 80% of the species in this order belong to the family Rubiaceae.

Many of these flowering plants are used in traditional medicine. They have been used to treat pain, anxiety, cancers and neurological conditions.

Taxonomy
In the classification system of Dahlgren the Gentiales were in the superorder Gentianiflorae (also called Gentiananae). The following families are included according to the APG III system:
 * Family Apocynaceae (424 genera)
 * Family Gelsemiaceae (2 genera)
 * Family Gentianaceae (87 genera)
 * Family Loganiaceae (13 genera)
 * Family Rubiaceae (611 genera)

Phylogeny
The following phylogenetic tree is based on molecular phylogenetic studies of DNA sequences.

Etymology
It takes its name from the family Gentianaceae, which in turn is based on the name of the type genus, Gentiana. The genus name is a tribute to Gentius, an Illyrian king.

Characteristics
This large order has a variety of different plants, ranging from small herbaceous plants and saprophytes to shrubs and large trees. Species are, however, united by their simple and opposite leaves and typically have showy pentamerous flowers (flowers in which components occur in multiples of five) and show nuclear endosperm formation (in which cell division takes place without the cell wall forming between divisions). Many species have structures between the leaf petioles, such as ridges or stipules. Many species also have colleters; thick hair-like structures that secrete mucilage, a thick gluey substance.

Distribution
Species of this order are found in moist climates around the world. They are most common in tropical regions.

Uses
Many gentianales contain toxic compounds and species have a variety of uses. Some species are also grown ornamentally. Well-known members of Gentianales are coffee, frangipani, Gardenia, gentian, oleander, and periwinkle.

Certain species belonging to the order Gentianales have been used in traditional medicine in rural southeastern Asia countries. Gelsemium sempervirens has been used in North American folk medicine to treat conditions such as anxiety, migraines/headaches, and neuralgia, while Gelsemium elegans has been used in China to treat rheumatoid arthritis pain, neuropathic pain, skin ulcers, and even cancers.

The compounds found in some species are used in the synthesis of modern medicines. Cinchona trees, for example, are a source of quinine, which is used to treat malaria. Vinblastine, which has anti-tumor properties as it disrupts cell division, is used in chemotherapy. It is extracted from the Madagascar periwinkle.

title: Gentianales


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Maloideae

The Maloideae C.Weber was the apple subfamily, a grouping used by some taxonomists within the rose family, Rosaceae. Recent molecular phylogenetic evidence has shown that the traditional Spiraeoideae and Amygdaloideae form part of the same clade as the traditional Maloideae, and the correct name for this group is Amygdaloideae. Earlier circumscriptions of Maloideae are more-or-less equivalent to subtribe Malinae or to tribe Maleae. The group includes a number of plants bearing commercially important fruits, such as apples and pears, while others are cultivated as ornamentals.

In its traditional circumscription this subfamily consisted exclusively of shrubs and small trees characterised by a pome, a type of accessory fruit that does not occur in other Rosaceae, and by a basal haploid chromosome count of 17 (instead of 7, 8, or 9 as in the other Rosaceae), involving approximately 28 genera with approximately 1100 species worldwide, with most species occurring in the temperate Northern Hemisphere.

Taxonomy
The subfamily was given the name Pomoideae Juss. in 1789, but this name is no longer accepted under the nomenclature codes because it is not based on a genus name. It has also been separated into its own family the Malaceae Small (formerly Pomaceae Lindl.).

An earlier intermediate classification expanded Maloideae to include four genera with dry non-pome fruit. These are Kageneckia, Lindleya, and Vauquelinia, which have a haploid chromosome count of 15 or 17, and Gillenia, which is herbaceous and has a haploid chromosome count of 9.

A traditional circumscription of Maloideae includes the following genera:
 * Amelanchier – serviceberry, juneberry
 * Aria (see Sorbus)
 * Aronia – chokeberry
 * Chaenomeles – Japanese quince
 * Chamaemeles
 * Chamaemespilus (see Sorbus chamaemespilus)
 * Cormus (see Sorbus)
 * Cotoneaster – cotoneaster
 * Crataegus – hawthorn
 * Cydonia – quince
 * Dichotomanthes
 * Docynia
 * Docyniopsis
 * Eriobotrya – loquat
 * Eriolobus
 * Hesperomeles
 * Heteromeles – toyon
 * Malacomeles
 * Malus – apple, crabapple
 * Mespilus – medlar
 * Osteomeles
 * Peraphyllum
 * Photinia
 * Pseudocydonia – Chinese quince
 * Pyracantha – firethorn
 * Pyrus – pear
 * Rhaphiolepis – hawthorn
 * Sorbus – rowan, whitebeam, service tree
 * Stranvaesia = Photinia pro parte
 * Torminalis (see Sorbus torminalis)

Intergeneric hybrids:
 * × Amelasorbus
 * × Crataegosorbus
 * × Crataemespilus
 * × Malosorbus
 * × Sorbocotoneaster
 * × Sorbopyrus

Graft chimeras:
 * + Crataegomespilus
 * + Pyrocydonia (Pirocydonia)

title: Maloideae