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free her. In Fresnes she was treated well, but when this ploy did not work, she was shipped to Ravensbrück concentration camp in Germany. She later died of the effects of malnutrition, only a few days after liberation by the Russians. During the occupation, Jean was arrested by both French gendarmerie and French Milice, including the Swiss border police. The French gendarmerie beat him up brutally, but they had to release him later due to lack of evidence. In another arrest by the Milice in Toulouse he was tortured, but he managed to escape before they could transfer him to

Johan Hendrik Weidner

the Gestapo. The Gestapo were never able to get a hold of him. After the war. In November 1944, after the Liberation of France Weidner was invited to London by Queen Wilhelmina, to come to tell her about the "Dutch-Paris" escape route, and the situation of Dutch civilians in France and Belgium. In the same year he was made a Captain in the Dutch Armed Forces, after which he could be in charge of the Dutch Security Service based in Paris. His service was in charge of vetting all the Dutch citizens in France and Belgium to look for

Johan Hendrik Weidner

any that collaborated with the Germans. The Bureau of National Security, the Department of Justice, and the Dutch Embassy in Paris all claimed authority for Netherlands Security Service. Therefore, it has never become entirely clear under whose direction he fell. In mid 1946, Jean was suddenly dismissed by the Dutch government, arguing that they needed a professional policeman on the post. After his work with the security he picked up the threads of normal life again, and returned to his import/export textile business. In 1955 he emigrated to the United States, eventually settling in California where from 1958 he

Johan Hendrik Weidner

and his wife Naomi operated a chain of health food stores. Recognition. For his War efforts, Weidner was awarded the United States Medal of Freedom with Gold Palm, made an Honorary Officer of the Order of the British Empire, an Officer in the Dutch Order of Orange-Nassau. The French Government honored him with the Croix de Guerre and Médaille de la Résistance and the Légion d'honneur. The Belgian Government made him an Officer of the Order of Leopold. At the 1993 opening of the United States Holocaust Memorial Museum in Washington, D.C. he was one of seven persons chosen

Johan Hendrik Weidner

to light candles recognizing the rescuers. The Israeli government honored Weidner as one of the gentiles designated as Righteous Among the Nations at Israel's national Holocaust Memorial, Yad Vashem where a grove of trees was planted in his name on the Hill of Remembrance along the Avenue of the Righteous. Johan Hendrik Weidner Johan Hendrik Weidner (October 22, 1912, Brussels, Belgium - May 21, 1994, Monterey Park, California, United States) was a highly decorated Dutch hero of World War II. Early life. Johan Hendrik Weidner Jr. was born in Brussels to Dutch parents. Although his birth name was Johan Hendrik

Johan Hendrik Weidner

Gold (British TV channel) Gold (British TV channel) Gold is a British pay television channel from the UKTV network that was launched on 1 November 1992 as UK Gold before it was rebranded UKTV Gold in 2004. In 2008, it was split into current flagship channel Gold and miscellaneous channel, W, with classic comedy based programming now airing on Gold, non-crime drama and entertainment programming airing on W, and quiz shows and more high-brow comedy airing on Dave. It shows repeats of classic programming from the BBC, ITV and other broadcasters. Every December, from 2015 until 2018, the channel was temporarily renamed

Gold (British TV channel)

Christmas Gold. This has since been discontinued, although the channel still continues to broadcast Christmas comedy. History. The channel was formed as a joint venture between the BBC, through commercial arm BBC Enterprises, American company Cox Enterprises and outgoing ITV London weekday franchisee Thames Television. The channel, named "UK Gold", was to show repeats of the 'classic' archive programming from the two broadcasters. The channel launched on 1 November 1992 at 7pm with "Just Good Friends". The rights to the BBC programmes previously were held by the BSB entertainment channel Galaxy, prior to the merger with Sky Television plc

Gold (British TV channel)

to form BSkyB in November 1990. The channel was initially broadcast on an analogue transponder from an SES satellite at 19.2°E which was less well suited for UK reception. As a result, the channel used to be notorious for being marred with interference, known as 'sparklies', in large parts of the UK. Another initial drawback was the cutting of programming down to fit commercial time slots, and the intensive use of commercial breaks. Reception improved however with the channel added to BSkyB's basic subscription package in 1993, and the launch of the channel on cable services. In

Gold (British TV channel)

1993, Flextech gained its first stake in the station after acquiring Tele-Communications' (TCI) TV interests in Europe. In 1996, it started discussions about increasing its stake, to gain full control. At that point, Flextech held 27% with Cox (38%), BBC (20%) and Pearson (15%). By the Autumn, Flextech held 80% of UK Gold. Flextech's main reason for increasing its stake in UK Gold was in participation of new talks with the BBC. UKTV. The channel's success led to the launch of the UKTV network on 1 November 1997, owned by BBC Worldwide and Flextech, and consisting of

Gold (British TV channel)

three other channels: UK Arena, UK Horizons and UK Style, focusing on the arts, factual and lifestyle programmes respectively. The UKTV network would expand to include numerous more channels as the years progressed. The UK Gold brand was expanded in October 1998 with the launch of the digital only channel UK Gold Classics, broadcasting some of the older comedy serials that were being lost from the UK Gold schedule, as the channel moved towards more modern programming. UK Gold Classics was not to last however, and was rebranded as UK Gold 2 on 2 April 1999, which acted as a

Gold (British TV channel)

time shift of the original channel, showing the daytime programmes from UK Gold in the evening on UK Gold 2. In 2003 however, UK Gold 2 was rebranded and repositioned as UK G, with some programming transferring to the new channel. On 8 March 2004, the channel was rebranded as "UKTV Gold" in line with the other channels in the UKTV network. At approximately the same time, Granada-run archive channel Granada Plus closed to make way for ITV3. In late 2004, to show films produced in Hollywood in a marathon, UKTV Gold temporarily changed its name to "USTV Gold

Gold (British TV channel)

. The channels had always been the main rivals to Gold due to the direct mix of archive BBC and ITV programming. ITV3 currently has a higher viewer share, often put down to the fact that the terrestrial platform Freeview shows ITV3 but not Gold. Gold began transmitting in widescreen on 31 January 2008, although some programmes made in format are screened in the compromise semi-letterboxed ratio for a short while, before the 16:9 format became standard later in the year. The channel has been criticised by some, particularly in recent years, for featuring many recent programmes as opposed to Gold (British TV channel) 7904576 'classics' as was the original concept, with some shows appearing on the channel mere months or weeks after their first television broadcast. 2008 rebrand. In 2008, UKTV began a process of rebranding and expanding its channels, removing UKTV" from their name, following the rebranding of UKTV G2 as Dave in October 2007. On 7 October 2008, UKTV Gold became "G.O.L.D.", exclusively showing comedy, both old and new. This is reflected by its new slogan, which now represents Gold as a backronym—"Go On Laugh Daily". Unlike the complete name change for Dave, the Gold name was retained as having a resonance

Gold (British TV channel)

with viewers. The same day also saw the rebranding of UKTV Drama as Alibi, and the launch of a new channel, Watch. In Spring 2010, the channel dropped the acronym and is now known simply as "Gold". In October 2011, Virgin Media, owner of half of Gold and the rest of the UKTV network, sold their share to Scripps Networks Interactive, with the remaining half still retained by the BBC's commercial arm, BBC Worldwide. 2012 programming investment. In February 2012, it was announced that UKTV was to invest millions into producing its own original shows. According to the trade

Gold (British TV channel)

magazine, Broadcast, Gold "has secured a budget running into "double-digit millions" to create a raft of new comedy shows over the next two years". The article went on to say "the channel is looking to develop a mix of panel shows, sketch shows, sitcoms and comedy dramas". This move meant that Gold would follow its sister, Dave, which has resurrected "Red Dwarf" and produced numerous different panel and entertainment series, and BSkyB, who have invested £600m into original comedy for Sky One and Sky Atlantic. The first set of new series for Gold were broadcast in the

Gold (British TV channel)

latter half of 2012. UKTV executive Jane Rogers stated to Broadcast that the commissions would be high-quality, as "they would need to sit confidently alongside classics such as "Only Fools And Horses" and "The Vicar of Dibley"". She also added: "Gold is well entrenched in the UK's psyche as the home of national treasure comedies, so we cannot afford to look cheap next to those programmes. It's important that anything we order continues that love and feel, but we don't want to look back; we want a contemporary stamp on the channel." Commissioning editor Sarah

Gold (British TV channel)

Fraser commented that "there's never been a better time to invest in homegrown comedy. Comedians are selling out arena tours, being cast in the West End and on the big screen here and in the US." The first programme announced as part of the investment was a reboot of the BBC sitcom, "Yes, Prime Minister", to be based on the 2010 stage production and written by original writers Sir Antony Jay and Jonathan Lynn. The reboot was the second classic BBC sitcom to be resurrected by a UKTV network, following the two Dave-commissioned series of "Red Dwarf". News

Gold (British TV channel)

about other new commissions for the channel were announced during the summer. Freeview. The channel was removed from Freeview in 2013, along with Home as part of the closure of Top Up TV and was replaced with Drama in July 2013. However, Home relaunched on the platform in March 2016. Gold and Home were not available to watch on Freeview boxes and televisions due to them being encrypted as subscription channels on Top Up TV. Subsidiary channels. Gold +1. Alongside the main channel, a time shift channel is also operated: Gold +1. The channel, previously named UK Gold +1 and

Gold (British TV channel)

UKTV Gold +1 when corresponding to the name on the main channel, shows all programming from the channel one hour later. The channel UK Gold 2 also used to be a time shift, showing the daytime schedule from UK Gold in the evening prime time slot. This service ended in 2003. Gold HD. A HD version of Gold was launched on 2 October 2017 exclusively to Sky, replacing Eden HD on the platform. Gold HD was added to Virgin Media on 25 September 2018, replacing Gold in standard definition. It was added to BT TV on 11 March 2019, along

Gold (British TV channel)

with Vice HD. On-air identity. For the first few years, idents on UK Gold featured an animated golden retriever mascot named "Goldie" posing with the UK Gold logo. Goldie was never name-checked as such on air, possibly owing to the death of the "Blue Peter" dog Goldie some weeks before launch, although the late-night music video slot "Dog House" was originally listed as "Goldie's Video Bites" in initial pre-launch listings. The Goldie idents were kept until 1993, when they were replaced with a form-up of the first logo against a blue background. A re

Gold (British TV channel)

-branding in 1994 saw UK Gold adopt idents based on the forging of gold bars, with the station's logo appearing to have been stamped into gold. Variations on this theme were used until 1997, when the channel received a revamp as part of the formation of the UKTV network. The new network's corporate identity saw all its channels logos simplified to a boxed "UK" followed by the name (e.g. "Gold") in the Gill Sans font, which had also been adopted by the BBC. The new network-wide ident theme would involve the splitting of the screen for different

Gold (British TV channel)

purposes. UK Gold's new idents depicted objects such as apples or leaves falling through the top half of the screen, with only the gold coloured ones reaching the bottom half of the screen. These idents were briefly adopted for UK Gold 2 when it launched in 1998. In 1999, the theme changed again, this time with idents featuring fireworks making shapes in the air. The fireworks theme was carried on in a new set of idents, alongside another network-wide re-branding of the logo in 2002, adopting a bolder font and merging UK into a single composite character

Gold (British TV channel)

. A range of live-action idents showing everyday activities from unusual perspectives appeared in 2002. The 2003 to 2007 idents showed channel hopping viewers with their TV set "off stage" being brought to a halt by the appearance of a golden light accompanied by the channel's ident jingle. This new identity also featured a series of shifting yellow, orange and red blocks which suggested a gold bar at the centre of the screen. On 4 April 2007, UKTV Gold unveiled a new on-air identity centred around a branded golden space hopper, playing to contemporary trends toward 1970s nostalgia

Gold (British TV channel)

, and emphasising the station's re-run content. Twelve new live action idents featured the branded space hopper, either with people on them bouncing around normally serious scenes, or let loose to bounce around the natural environment, aired from 5 April 2007 to 7 October 2008. On 7 October 2008, following the rebranding, Gold's 2008 presentation debuted, featuring cartoon objects making a giant chain, in Heath Robinson fashion, which triggers an event to herald the next programme. The Gold logo features in the centre, with the channel slogan usually appearing alongside in the sequence. Programmes are announced by sole

Gold (British TV channel)

channel continuity announcer David Flynn, who has had the position since June 2009. In 2012, the logo was changed slightly, with an updated ident package. These followed a similar style to the previous set, but instead of the previous backronym, the idents focussed on the slogan of ‘Stick Something Funny On’. In July 2014, Gold rebranded after two years in its second identity. The logo is a ribbon with the letters "GOLD". The new idents have funny and wacky stuff happening (such as dancing legs). At the end of the idents, the ribbon comes out saying "GOLD", while the activity

Gold (British TV channel)

behind it is still going. Programming. The output of the channel is mainly British comedy programmes including repeats of BBC shows and sometimes feature-length films. The following is a list of all the programmes that have been shown over the years, plus ones that are currently being shown: Previous. Programming of classic general entertainment and movies now unused or superseded on 7 October 2008. Some programmes that were shown on UKTV G2 moved to other channels sometime later when it was renamed Dave in October 2007. However, most of the classic comedy series includes re-runs of some BBC

Gold (British TV channel)

shows which are used only on this channel. Some of its classic general entertainment shows moved to other current UKTV channels such as Drama, Dave, W, and Alibi. Others are currently airing on Forces TV (e.g. Blake’s 7, Get Some In and Sykes) and ITV-based Freeview TV Channels (e.g. The Sweeney, Bless This House and Kojak). Other ventures. An analogue teletext service known as GoldText used to be available on the channel, but has since closed down. Gold (British TV channel) Gold is a British pay television channel from the UKTV network that was launched on 1 November

Gold (British TV channel)

Marinus of Neapolis Marinus of Neapolis Marinus (; born c. 440 AD) was a Neoplatonist philosopher, mathematician and rhetorician born in Flavia Neapolis (modern Nablus), Palestine. He was a student of Proclus in Athens. His surviving works are an introduction to Euclid's "Data"; a "Life of Proclus", and two astronomical texts. Most of what we know of his life comes from an epitome of a work by Damascius conserved in the Byzantine Suda encyclopaedia. Life. He was, according to his pupil Damascius, born a Samaritan, though some uncertainty remains about this attribution of his ethnicity. Damascius also adds that he had converted from

Marinus of Neapolis

Samaritanism. He came to Athens at a time when, with the exception of Proclus, there was a great dearth of eminent men in the Neoplatonist school. He was appointed as successor ("diadochos") to Proclus, sometime before the latter's death, during the period of the teacher's infirmity. Proclus dedicated to Marinus his commentary to the Plato's Myth of Er. Proclus himself, it is reported, worried that Marinus himself was of delicate constitution. During this period, the professors of the old Greek religion suffered persecution at the hands of the Christians and Marinus was compelled to seek refuge at

Marinus of Neapolis

Epidaurus, where he died, at a date unknown. Works. Only a remnant of his output survives. His chief surviving work was a biography of Proclus since it is the main source of information on Proclus' life. This was written in a combination of prose and epic hexameters, of which only the former survives. The publication of the biography is fixed by internal evidence to the year of Proclus's death; for he mentions an eclipse which will happen when the first year after that event is completed. It was first published with the works of Marcus Aurelius in 1559; it

Marinus of Neapolis

was republished separately by Fabricius at Hamburg in 1700, and re-edited in 1814 by Boissonade with emendations and notes. He is also the author of a commentary (or introduction) on the "Data" of Euclid, and a commentary on Theon's "Little Commentary". There is also a surviving astronomical text which discusses the Milky Way. His lost works included commentaries on Aristotle and on the "Philebus" of Plato. He destroyed his commentary on the Philebus on the advice of a pupil he was tutoring, Isidorus. According to a version of the story written by Damascius, when Marinus showed his student

Marinus of Neapolis

, to whom he taught Aristotelianism, this commentary, which he had just completed, Isidorus prevailed on him to destroy it, arguing that since the 'divine' Proclus had himself written a definitive commentary which was the final word on the topic. Current scholarship suspects that this advice arose from fears that Marinus's commentary would, despite his best efforts, betray traces of material that might undermine the reigning Neoplatonic paradigm. Marinus of Neapolis Marinus (; born c. 440 AD) was a Neoplatonist philosopher, mathematician and rhetorician born in Flavia Neapolis (modern Nablus), Palestine. He was a student of Proclus in Athens. His surviving

Marinus of Neapolis

Isidore of Alexandria Isidore of Alexandria Isidore of Alexandria (; also Isidorus ; ; ) was a Greek philosopher and one of the last of the Neoplatonists. He lived in Athens and Alexandria toward the end of the 5th century AD. He became head of the school in Athens in succession to Marinus, who followed Proclus. Life. Isidore was born in Alexandria. In Athens, he studied under Proclus, and learned the doctrine of Aristotle from Marinus. According to Damascius, "Isidore was awestruck at the sight of Proclus, venerable and marvelous to see; he thought he was seeing in him the very face of true philosophy." Proclus for

Isidore of Alexandria

his part used to "marvel at Isidore's appearance, as it was possessed by the divine and full of the philosophical life within." Damascius further tells us that "Isidore, besides simplicity, loved truthfulness especially, and undertook to be straight-talking beyond what was necessary, and had no pretence in himself whatsoever." The claim made in the "Suda" that Isidore was the husband of Hypatia, must be in error since Isidore was born long after Hypatia died. It is elsewhere related that Isidore had a wife called Domna, who died five days after the birth of their son whom they named

Isidore of Alexandria

Proclus. Isidore returned to Alexandria accompanied by Sallustius. In Alexandria he taught philosophy. He was in Athens when Proclus died (in 485), and later when Marinus took over as head ("scholarch") of the Neoplatonist school. Marinus persuaded him to be his successor as head of the school, but he left Athens not long after Marinus died, resigning his position to Hegias. Isidore is known principally as the teacher of Damascius, whose testimony in his "Life of Isidore" presents Isidore in a very favourable light as a man and a thinker. Damascius' "Life", which is dedicated to Theodora, a disciple of

Isidore of Alexandria

both Isidore and Damascius, is preserved in summary form by Photius in his "Bibliotheca", and in fragments in the "Suda". Philosophy. It is generally admitted that he was rather an enthusiast than a thinker; reasoning with him was subsidiary to inspiration, and he preferred the theories of Pythagoras and Plato to the unimaginative logic and the practical ethics of the Stoics and Aristotelians. He seems to have given loose rein to theosophical speculation and attached great importance to dreams and waking visions, on which he used to expatiate in his public discourses. Isidore of Alexandria Isidore of Alexandria (; also Isidorus

Isidore of Alexandria

George Hill George Hill George Hill may refer to: George Hill George Hill may refer to:

George Hill

Zenodotus Zenodotus Zenodotus () was a Greek grammarian, literary critic, Homeric scholar, and the first librarian of the Library of Alexandria. A native of Ephesus and a pupil of Philitas of Cos, he lived during the reigns of the first two Ptolemies, and was at the height of his reputation about 280 BC. Biography. Zenodotus was the first superintendent of the Library of Alexandria and the first critical editor ( "diorthōtes") of Homer. In 284 BC, the Ptolemaic court appointed Zenodotus as the first Director of the library and also the official tutor to the royal children. His colleagues in the librarianship were

Zenodotus

Alexander of Aetolia and Lycophron of Chalcis, to whom were allotted the tragic and comic writers respectively, Homer and other epic poets being assigned to Zenodotus. Work. Although he has been reproached with arbitrariness and insufficient knowledge of Greek, his recension undoubtedly laid a sound foundation for future criticism. Having collated the different manuscripts in the library, he expunged or obelized doubtful verses, transposed or altered lines, and introduced new readings. It is probable that he was responsible for the division of the Homeric poems into twenty-four books each, and possibly was the author of the calculation of the

Zenodotus

days of the "Iliad" in the "Tabula Iliaca". Homeric glosses. He does not appear to have written any regular commentary on Homer, but his Homeric ("glōssai", "lists of unusual words, glosses") probably formed the source of the explanations of Homer attributed by the grammarians to Zenodotus. He also lectured upon Hesiod, Anacreon and Pindar, if he did not publish editions of them. He is further called an epic poet by the "Suda," and three epigrams in the "Greek Anthology" are assigned to him. Library organisation. Classification. In addition to his other scholarly work, Zenodotus introduced an organization system on the

Zenodotus

materials in the Library of Alexandria whereby texts were assigned to different rooms based on their subject matter (verse or prose, literary or scientific), and the various sub-classifications within each. Ordering. Within their subjects, Zenodotus organized the works alphabetically by the first letter of the name of their author. The principle of the alphabetic organization was introduced by Zenodotus. Tagging. In addition, library staff attached a small tag to the end of each scroll. These tags gave authors' names as well as other identification and had been added during the accessions procedure but often without a title; many a

Zenodotus

roll contained more than one work, and many works, such as compilations of poetry, warranted more than a single title. When a title was lacking, Zenodotus had to unroll and pass an eye over the text. Such tags enabled the scrolls to be easily returned to the area in which they had been classified and also ensured that library users did not have to unroll each scroll in order to see what it contained. This was the first recorded use of metadata, a landmark in library history. Not until the second century A.D. does fuller alphabetization make an appearance. References

Zenodotus

. Attribution: Zenodotus Zenodotus () was a Greek grammarian, literary critic, Homeric scholar, and the first librarian of the Library of Alexandria. A native of Ephesus and a pupil of Philitas of Cos, he lived during the reigns of the first two Ptolemies, and was at the height of his reputation about 280 BC. Biography. Zenodotus was the first superintendent of the Library of Alexandria and the first critical editor ( "diorthōtes") of Homer. In 284 BC, the Ptolemaic court appointed Zenodotus as the first Director of the library and also the official tutor to the royal children. His colleagues in the librarianship

Zenodotus

Crates of Mallus Crates of Mallus Crates of Mallus (, "Krátēs ho Mallṓtēs"; century BC) was a Greek grammarian and Stoic philosopher, leader of the literary school and head of the library of Pergamum. He was described as the Crates from Mallus to distinguish him from other philosophers by the same name. His chief work was a critical and exegetical commentary on Homer. He is also famous for constructing the earliest known globe of the Earth. Life. He was born in Mallus in Cilicia, and was brought up at Tarsus, and then moved to Pergamon, and there lived under the patronage of Eumenes II

Crates of Mallus

, and Attalus II. He was the founder of the Pergamon school of grammar, and seems to have been at one time the head of the library of Pergamon. Among his followers were Hermias (Κρατήτειος Ἑρμείας mentioned in sch. Hom. "Il". 16.207a), Zenodotus of Mallus and Herodicus of Babylon. He visited Rome as ambassador of either Eumenes, in 168 BC, or Attalus in 159 BC. Having broken his leg after falling into an open sewer, he was compelled to stay in Rome for some time and delivered lectures which gave the first impulse to the study of grammar and criticism among

Crates of Mallus

the Romans. Works. Crates made a strong distinction between criticism and grammar, the latter of which he regarded as subordinate to the former. A critic, according to Crates, should investigate everything which could throw light upon literature; the grammarian was only to apply the rules of language to clear up the meaning of particular passages, and to settle the text, prosody, accentuation, etc. From this part of his system, Crates derived the surname of "Kritikos". Like Aristarchus of Samothrace, Crates gave the greatest attention to the works of Homer, from his labours upon which he was also surnamed "Homerikos". He

Crates of Mallus

wrote a commentary on the Iliad and Odyssey in nine books. Some fragments of this commentary are preserved by the scholiasts and other ancient writers. His principles were opposed to those of Aristarchus, who was the leader of the Alexandrian school. Crates was the chief representative of the allegorical theory of exegesis, and maintained that Homer intended to express scientific or philosophical truths in the form of poetry. Besides his work on Homer, Crates wrote commentaries on the "Theogony" of Hesiod, on Euripides, on Aristophanes, and probably on other ancient authors; a work on the Attic dialect; and works on

Crates of Mallus

geography, natural history, and agriculture, of which only a few fragments exist. The Globe of Crates. According to Strabo, Crates devised a globe representing the Earth, which is thus the earliest known globe representing the Earth: We have now traced on a spherical surface the area in which we say the inhabited world is situated; and the man who would most closely approximate the truth by constructed figures must necessarily take for the earth a globe like that of Crates, and lay off on it the quadrilateral, and within the quadrilateral put down the map of the inhabited world. But

Crates of Mallus

since the need of a large globe, so that the section in question (being a small fraction of the globe) may be large enough to receive distinctly the appropriate parts of the inhabited world and to present the proper appearance to observers, it is better for him to construct a globe of adequate size, if he can do so; and let it be no less than ten feet in diameter. Following the theory of five climatic zones, Crates considered that the torrid zone is occupied by the Ocean and that, by analogy, one can imagine people living beyond the torrid

Crates of Mallus

zone: For Crates, following the mere form of mathematical demonstration, says that the torrid zone is "occupied" by Oceanus and that on both sides of this zone are the temperate zones, the one being on our side, while the other is on the other side of it. Now, just as these Ethiopians on our side of Oceanus, who face the south throughout the whole length of the inhabited land, are called the most remote of the one group of peoples, since they dwell on the shores of Oceanus, so too, Crates thinks, we must conceive that on the other side

Crates of Mallus

of Oceanus also there are certain Ethiopians, the most remote of the other group of peoples in the temperate zone, since they dwell on the shores of this same Oceanus; and that they are in two groups and are "sundered in twain" by Oceanus. The classic drawing of the sphere displays the known world, or Oecumene (Europe, North Africa, and Asia), with three other continents, labeled the Perioeci, the Antipodes, and the Antioeci. Crates' Perioeci and Antipodes arguably do exist, corresponding roughly to North America and South America respectively, but the continent of the Antioeci, Terra Australis, does not, except

Crates of Mallus

in fragments (Australasia and southern Africa). And the earth does in fact have a ring of water around it, but at 60 degrees South latitude, not at the Equator. Honours. Crates Bay in Antarctica is named after Crates of Mallus. Crates of Mallus Crates of Mallus (, "Krátēs ho Mallṓtēs"; century BC) was a Greek grammarian and Stoic philosopher, leader of the literary school and head of the library of Pergamum. He was described as the Crates from Mallus to distinguish him from other philosophers by the same name. His chief work was a critical and exegetical commentary on Homer. He

Crates of Mallus

Musaeum Clausum Musaeum Clausum Musaeum Clausum (Latin for "Sealed Museum"), also known as Bibliotheca abscondita ("Secret Library" in Latin), is a tract written by Sir Thomas Browne which was first published posthumously in 1684. The tract contains short sentence descriptions of supposed, rumoured or lost books, pictures, and objects. The subtitle describes the tract as an inventory of "remarkable books, antiquities, pictures and rarities of several kinds, scarce or never seen by any man now living". Its date is unknown: however, an event from the year 1673 is cited. Like his "Pseudodoxia Epidemica", "Musaeum Clausum" is a catalogue of doubts and queries

Musaeum Clausum

, only this time, in a style which anticipates the 20th-century Argentinian short-story writer Jorge Luis Borges, who once declared: "To write vast books is a laborious nonsense; much better is to offer a summary as if those books actually existed." Browne however was not the first author to engage in such fantasy. The French author Rabelais, in his epic "Gargantua and Pantagruel", also penned a list of imaginary and often obscene book titles in his "Library of Pantagruel", an inventory which Browne himself alludes to in his "Religio Medici". As the 17th-century Scientific Revolution progressed the popularity

Musaeum Clausum

and growth of antiquarian collections, those claiming to house highly improbable items grew. Browne was an avid collector of antiquities and natural specimens, possessing a supposed unicorn's horn, presented to him by Arthur Dee. Browne's eldest son Edward visited the famous scholar Athanasius Kircher, founder of the "Museo Kircherano" at Rome in 1667, whose exhibits included an engine for attempting perpetual motion and a speaking head, which Kircher called his "Oraculum Delphinium". He wrote to his father of his visit to the Jesuit priest's "closet of rarities". The sheer volume of book-titles, pictures and objects listed

Musaeum Clausum

in "Musaeum Clausum" is testimony to Browne's fertile imagination. However, his major editors, Simon Wilkin in the nineteenth century (1834) and Sir Geoffrey Keynes in the twentieth (1924), summarily dismissed it. Keynes considered its humour too erudite and "not to everyone's taste". Browne's miscellaneous tract may also be read as a parody of the rising trend of private museum collections with their curios of doubtful origin, and perhaps also of publications such as the so-called "Museum Hermeticum" (1678), one of the last great anthologies of alchemical literature, with their divulging of near common-place alchemical concepts

Musaeum Clausum

and symbols. Musaeum Clausum Musaeum Clausum (Latin for "Sealed Museum"), also known as Bibliotheca abscondita ("Secret Library" in Latin), is a tract written by Sir Thomas Browne which was first published posthumously in 1684. The tract contains short sentence descriptions of supposed, rumoured or lost books, pictures, and objects. The subtitle describes the tract as an inventory of "remarkable books, antiquities, pictures and rarities of several kinds, scarce or never seen by any man now living". Its date is unknown: however, an event from the year 1673 is cited. Like his "Pseudodoxia Epidemica", "Musaeum Clausum" is a catalogue of doubts

Musaeum Clausum

Function of several complex variables Function of several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variables (and analytic space (Complex analytic space) ), that has become a common name for that whole field of study and Mathematics Subject Classification has, as a top-level heading. A function formula_1 is -tuples of complex numbers, classically studied on the complex coordinate space formula_2. As in complex analysis of functions of one variable, which is the case

Function of several complex variables

, the functions studied are "holomorphic" or "complex analytic" so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domain(formula_3), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy. For several complex variables, this is not the case; there exist domains (formula_4) that are not the domain of holomorphy of any function, and so is not always the domain of holomorphy, so

Function of several complex variables

the domain of holomorphy is one of the themes in this field. Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and complex projective varieties (formula_5) and has a different flavour to complex analytic geometry in formula_6 or on Stein manifolds. Historical perspective. Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series. Naturally also

Function of several complex variables

same function of one variable that depends on some complex parameter is a candidate. The theory, however, for many years didn't become a full-fledged field in mathematical analysis, since its characteristic phenomena weren't uncovered. The Weierstrass preparation theorem would now be classed as commutative algebra; it did justify the local picture, ramification, that addresses the generalization of the branch points of Riemann surface theory. With work of Friedrich Hartogs, and of Kiyoshi Oka in the 1930s, a general theory began to emerge; others working in the area at the time were Heinrich Behnke, Peter Thullen and Karl

Function of several complex variables

Stein. Hartogs proved some basic results, such as every isolated singularity is removable, for every analytic function formula_7 whenever . Naturally the analogues of contour integrals will be harder to handle; when an integral surrounding a point should be over a three-dimensional manifold (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a double integral over a two-dimensional surface. This means that the residue calculus will have to take a very different character. After 1945 important work in France, in the seminar of Henri Cartan, and Germany with

Function of several complex variables

Hans Grauert and Reinhold Remmert, quickly changed the picture of the theory. A number of issues were clarified, in particular that of analytic continuation. Here a major difference is evident from the one-variable theory; while for every open connected set "D" in formula_8 we can find a function that will nowhere continue analytically over the boundary, that cannot be said for . In fact the "D" of that kind are rather special in nature (especially in complex coordinate spaces formula_2 and Stein manifolds, satisfying a condition called "pseudoconvexity"). The natural domains of definition of functions, continued to the limit

Function of several complex variables

, are called "Stein manifolds" and their nature was to make sheaf cohomology groups vanish, also, the property that the sheaf cohomology group disappears is also found in other high-dimensional complex manifolds, indicating that the Hodge manifold is projective. In fact it was the need to put (in particular) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the formulation of the theory (with major repercussions for algebraic geometry, in particular from Grauert's work). From this point onwards there was a foundational theory, which could be applied to analytic geometry

Function of several complex variables

, automorphic forms of several variables, and partial differential equations. The deformation theory of complex structures and complex manifolds was described in general terms by Kunihiko Kodaira and D. C. Spencer. The celebrated paper "GAGA" of Serre pinned down the crossover point from "géometrie analytique" to "géometrie algébrique". C. L. Siegel was heard to complain that the new "theory of functions of several complex variables" had few "functions" in it, meaning that the special function side of the theory was subordinated to sheaves. The interest for number theory, certainly, is in specific generalizations of modular forms. The classical candidates are the

Function of several complex variables

Hilbert modular forms and Siegel modular forms. These days these are associated to algebraic groups (respectively the Weil restriction from a totally real number field of , and the symplectic group), for which it happens that automorphic representations can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions. Subsequent developments included the hyperfunction theory, and the edge-of-the-wedge theorem, both of which had some inspiration from quantum field theory. There are a number of other fields, such as Banach algebra theory, that draw on several complex variables

Function of several complex variables

. The complex coordinate space. The complex coordinate space formula_2 is the Cartesian product of copies of formula_8, and when formula_2 is a domain of holomorphy, formula_2 can be regarded as a Stein manifold, and more generalized Stein space. formula_2 is also considered to be a complex projective variety, a Kähler manifold, etc. It is also an -dimensional vector space over the complex numbers, which gives its dimension over formula_15. Hence, as a set and as a topological space, formula_2 may be identified to the real coordinate space formula_17 and its topological dimension is thus . In coordinate-free language, any vector

Function of several complex variables

space over complex numbers may be thought of as a real vector space of twice as many dimensions, where a complex structure is specified by a linear operator (such that ) which defines multiplication by the imaginary unit . Any such space, as a real space, is oriented. On the complex plane thought of as a Cartesian plane, multiplication by a complex number may be represented by the real matrix with determinant Likewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be composed from 2 × 2 blocks of the aforementioned form), then its determinant

Function of several complex variables

equals to the square of absolute value of the corresponding complex determinant. It is a non-negative number, which implies that the (real) orientation of the space is never reversed by a complex operator. The same applies to Jacobians of holomorphic functions from formula_2 to formula_2. Connected space. Every product of a family of an connected (resp. path-connected) spaces is connected (resp. path-connected). Compact. From Tychonoff's theorem, the space mapped by the cartesian product consisting of any combination of compact spaces is a compact space. Holomorphic functions. Definition. When a function "f" defined on the domain "D

Function of several complex variables

is complex-differentiable at each point on D", "f" is said to be holomorphic on "D". When the function "f" defined on the domain "D" satisfies the following conditions, it is complex-differentiable at the point formula_22 on "D"; Therefore, a function "f" defined on a domain formula_27 is called holomorphic if "f" satisfies the following two conditions. Cauchy–Riemann equations. For each index ν let and Then as expected, through, let formula_31 be the Kronecker delta, that is formula_32, and formula_33 if formula_34. When, formula_35 then, therefore, This satisfies the Cauchy–Riemann equation for one variable for each index

Function of several complex variables

ν, then "f" is a separate holomorphic. Cauchy's integral formula I (Polydisc version). "f" meets the conditions of being continuous and separately homorphic on domain "D". Each disk has a rectifiable curve formula_38, formula_39 is piecewise smoothness, class formula_40 Jordan closed curve. (formula_41) Let formula_42 be the domain surrounded by each formula_39. Cartesian product closure formula_44 is formula_45. Also, take the polydisc formula_46 so that it becomes formula_47. (formula_48 and let formula_49 be the center of each disk.) Using Cauchy's integral formula of one variable repeatedly, Because formula_51 is a rectifiable Jordanian closed curve and "f" is continuous

Function of several complex variables

, so the order of products and sums can be exchanged so the iterated integral can be calculated as a multiple integral. Therefore, Cauchy's evaluation formula. Because the order of products and sums is interchangeable, from () we get "f" is class formula_52-function. From (2), if "f" is holomorphic, on polydisc formula_53 and formula_54, the following evaluation equation is obtained. Therefore, Liouville's theorem hold. Power series expansion of holomorphic functions on polydisc. If function "f" is holomorphic, on polydisc formula_56, from Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series

Function of several complex variables

. formula_57 In addition, "f" that satisfies the following conditions is called an analytic function. For each point formula_58, formula_59 is expressed as a power series expansion that is convergent on "D" : We have already explained that holomorphic functions on polydisc are analytic. Also, from the theorem derived by Weierstrass , we can see that the analytic function on polydisc (convergent power series) is holomorphic. Radius of convergence of power series. It is possible to define a combination of positive real numbers formula_65 such that the power series formula_66 converges uniformly at formula_67 and does not converge uniformly at formula_68. In this

Function of several complex variables

way it is possible to have a similar, combination of radius of convergence for a one complex variable. This combination is generally not unique and there are an infinite number of combinations. Laurent series expansion. Let formula_69 be holomorphic in the annulus formula_70 and continuous on their circumference, then there exists the following expansion ; formula_71 The integral in the second term, of the right-hand side is performed so as to see the zero on the left in every plane, also this integrated series is uniformly convergent in the annulus formula_72, where formula_73 and formula_74, and so it is possible

Function of several complex variables

to integrate term. Bochner–Martinelli formula (Cauchy's integral formula II). The Cauchy integral formula holds only for polydiscs, and in the domain of several complex variables, polydiscs are only one of many domain, so we introduce Bochner–Martinelli formula. Suppose that "f" is a continuously differentiable function on the closure of a domain "D" on formula_2 with piecewise smooth boundary formula_51. Then the Bochner–Martinelli formula states that if "z" is in the domain "D" then, for formula_77, "z" in formula_2 the Bochner–Martinelli kernel formula_79 is a differential form in formula_77 of bidegree formula_81 defined by In particular

Function of several complex variables

if "f" is holomorphic the second term vanishes, so Identity theorem. When the function "f,g" is analytic in the domain "D", even for several complex variables, the identity theorem holds on the domain "D", because it has a power series expansion the neighbourhood of point of analytic. Therefore, the maximal principle hold. Also, the inverse function theorem and implicit function theorem hold. Biholomorphism. From the establishment of the inverse function theorem, the following mapping can be defined. For the domain U, V of the "n"-dimensional complex space formula_2, the bijective holomorphic function formula_86 and the inverse mapping formula_87 is

Function of several complex variables

also holomorphic. At this time, formula_88 is called a U, V biholomorphism also, we say that "U" and "V" are biholomorphically equivalent or that they are biholomorphic. The Riemann mapping theorem does not hold. When formula_89, open balls and open polydiscs are "not" biholomorphically equivalent, that is, there is no biholomorphic mapping between the two. This was proven by Poincaré in 1907 by showing that their automorphism groups have different dimensions as Lie groups. However, even in the case of several complex variables, there are some results similar to the results of the theory of uniformization in one complex variable

Function of several complex variables

. Analytic continuation. Let "U, V" be domain on formula_6, formula_91 and formula_92. Assume that formula_93 and formula_94 is a connected component of formula_95. If formula_96 then "f" is said to be connected to "V", and "g" is said to be analytic continuation of "f". From the identity theorem, if "g" exists, for each way of choosing "w" it is unique. Whether or not the definition of this analytic continuation is well-defined should be considered whether the domains "U,V" and "W" can be defined arbitrarily. Several complex variables have restrictions on this domain, and depending on the shape of

Function of several complex variables

the domain , all analytic functions "g" belonging to "V" are connected to formula_97, and there may be not exist function "g" with formula_98 as the natural boundary. In other words, "V" cannot be defined arbitrarily. There is called the Hartogs's phenomenon. Therefore, researching when domain boundaries become natural boundaries has become one of the main research themes of several complex variables. Also, in the general dimension, there may be multiple intersections between "U" and "V". That is, "f" is not connected as a single-valued holomorphic function, but as a multivalued analytic function. This means that "W" is

Function of several complex variables

not unique and has different properties in the neighborhood of the branch point than in the case of one variable. Reinhardt domain. In polydisks, Cauchy's integral formula holds and the power series expansion of holomorphic functions is defined, but the unique radius of convergence is not defined for each variable. Also, since the Riemann mapping theorem does not hold, polydisks and open unit balls are not biholomorphic mapping, and also, polydisks was possible to separation of variables, but it doesn't always hold for any domain. Therefore, in order to study of the domain of convergence of the power

Function of several complex variables

series, it was necessary to make additional restriction on the domain, this was the Reinhardt domain. Early Knowledge into the properties of field of study of several complex variables, such as Logarithmically-convex, Hartogs's extension theorem, etc. , were given in the Reinhardt domain. A domain "D" in the complex coordinate space formula_2, formula_100, with centre at a point formula_101, with the following property; Together with each point formula_102, the domain also contains the set A Reinhardt domain "D" with formula_104 is invariant under the transformations formula_105, formula_106, formula_107. The Reinhardt domains constitute a subclass of the Hartogs domains and

Function of several complex variables

a subclass of the circular domains, which are defined by the following condition; Together with all points of formula_108, the domain contains the set i.e. all points of the circle with center formula_110 and radius formula_111 that lie on the complex line through formula_110 and formula_113. A Reinhardt domain "D" is called a complete Reinhardt domain if together with all point formula_114 it also contains the polydisc A complete Reinhardt domain is star-like with respect to its centre "a".　Therefore, the complete Reinhardt domain is simply connected, also when the complete Reinhardt domain is the boundary line, there is

Function of several complex variables

a way to prove Cauchy's integral theorem without using the Jordan curve theorem. Logarithmically-convex. A Reinhardt domain "D" is called logarithmically convex if the image formula_116 of the set under the mapping is a convex set in the real coordinate space formula_119. Every such domain in formula_2 is the interior of the set of points of absolute convergence (i.e. the domain of convergence) of some power series in formula_66, and conversely; The domain of convergence of every power series in formula_122 is a logarithmically-convex Reinhardt domain with centre formula_104. Some results. Hartogs's extension theorem and Hartogs

Function of several complex variables

's phenomenon. When examining the domain of convergence on the Reinhardt domain, Hartogs found the Hartogs's phenomenon in which holomorphic functions in some domain on the formula_6 were all connected to larger domain. From Hartogs's extension theorem the domain of convergence extends from formula_129 to formula_130. Looking at this from the perspective of the Reinhardt domain, formula_129 is the Reinhardt domain containing the center z = 0, and the domain of convergence of formula_129 has been extended to the smallest complete Reinhardt domain formula_130 containing formula_129. Thullen's classic results. Thullen's classical result says that a 2

Function of several complex variables

-dimensional bounded Reinhard domain containing the origin is biholomorphic to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension: Sunada's results. Toshikazu Sunada (1978) established a generalization of Thullen's result: Natural domain of the holomorphic function (domain of holomorphy). When moving from the theory of one complex variable to the theory of several complex variables, depending on the range of the domain, it may not be possible to define a holomorphic function such that the boundary of the domain becomes a natural boundary. Considering the domain where the

Function of several complex variables

boundaries of the domain are natural boundaries (In the complex coordinate space formula_2 call the domain of holomorphy), the first result of the domain of holomorphy was the holomorphic convexity of "H". Cartan and Thullen. Levi's problem shows that the pseudoconvex domain was a domain of holomorphy. (First for formula_145, later extended to formula_2.) Kiyoshi Oka's notion of "idéal de domaines indéterminés" is interpreted theory of sheaf cohomology by "H". Cartan and more development Serre. In sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds. The notion of the domain

Function of several complex variables

of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization. Domain of holomorphy. When a function "f" is holomorpic on the domain formula_147 and cannot directly connect to the domain outside "D", including the point of the domain boundary formula_51, the domain "D" is called the domain of holomorphy of "f" and the boundary is called the natural boundary of "f". In other words, the domain of holomorphy "D" is the supremum of the domain where the holomorphic function "f" is holomorphic, and the domain "D", which is holomorphic, can

Function of several complex variables

not be extended any more. For several complex variables, i.e. domain formula_149, the boundaries may not be natural boundaries. Hartogs' extension theorem gives an example of a domain where boundaries are not natural boundaries. Formally, a domain "D" in the "n"-dimensional complex coordinate space formula_2 is called a "domain of holomorphy" if there do not exist non-empty domain formula_151 and formula_152, formula_153 and formula_154 such that for every holomorphic function "f" on "D" there exists a holomorphic function "g" on "V" with formula_155 on "U". For the formula_156 case, the every domain (formula_157) was the domain of holomorphy; we

Function of several complex variables

can define a holomorphic function with zeros accumulating everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal. Holomorphically convex hull. Let formula_166 be a domain , or alternatively for a more general definition, let formula_167 be an formula_168 dimensional complex analytic manifold. Further let formula_169 stand for the set of holomorphic functions on "G". For a compact set formula_170, the holomorphically convex hull of "K" is One obtains a narrower concept of polynomially convex hull by taking formula_172 instead to be the set of complex-valued polynomial functions

Function of several complex variables

on "G". The polynomially convex hull contains the holomorphically convex hull. The domain formula_167 is called holomorphically convex if for every compact subset formula_174 is also compact in "G". Sometimes this is just abbreviated as "holomorph-convex". When formula_156, every domain formula_167 is holomorphically convex since then formula_177 is the union of "K" with the relatively compact components of formula_178. When formula_179, if "f" satisfies the above holomorphic convexity on "D" it has the following properties. formula_180 for every compact subset "K" in "D", where formula_181 denotes the distance between K and formula_182. Also, at this time, D is a

Function of several complex variables

domain of holomorphy. Therefore, every convex domain formula_183 is domain of holomorphy. Pseudoconvex. Pseudoconvex Hartogs showed that formula_184 is subharmonic for the radius of convergence in the Hartogs series formula_185 when the Hartogs series is a one-variable formula_186. If such a relations holds in the domain of holomorphy of several complex variables, it looks like a more manageable condition than a holomorphically convex. The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex domain (Hartogs pseudoconvex). Pseudoconvex domain are important, as they allow for classification of domains of holomorphy. Definition

Function of several complex variables

of plurisubharmonic function. is called plurisubharmonic if it is upper semi-continuous, and for every complex line formula_195 the function is subharmonic, where formula_197 denotes the unit disk. In one-complex variable, necessary and sufficient condition that the real-valued function formula_198, that can be second-order differentiable with respect to "z" of one-variable complex function is subharmonic is formula_199. There fore, if formula_200 is of class formula_201, then formula_200 is plurisubharmonic if and only if the hermitian matrix formula_203 is positive semidefinite. Equivalently, a formula_201-function "u" is plurisubharmonic if and only if formula_205 is a positive (1,1)-form

Function of several complex variables

. Strictly plurisubharmonic function. When the hermitian matrix of "u" is positive-definite and class formula_201, we call "u" a strict plurisubharmonic function. Section::::(Weakly) pseudoconvex (p-pseudoconvex). Weak pseudoconvex is defined as : Let formula_207 be a domain. One says that "X" is "pseudoconvex" if there exists a continuous plurisubharmonic function formula_208 on "X" such that the set formula_209 is a relatively compact subset of "X" for all real numbers "x". i.e. there exists a smooth plurisubharmonic exhaustion function formula_210. Often, the definition of pseudoconvex is used here and is written as; Let "X" be a complex "n"-dimensional manifold. Then is said

Function of several complex variables

to be weeak pseudoconvex there exists a smooth plurisubharmonic exhaustion function formula_210. Strongly (Strictly) pseudoconvex. Let "X" be a complex "n"-dimensional manifold. Strongly pseudoconvex if there exists a smooth strictly plurisubharmonic exhaustion function formula_210,i.e. formula_213 is positive definite at every point. The strongly pseudoconvex domain is the pseudoconvex domain. The strong Levi pseudoconvex domain is simply called strong pseudoconvex and is often called strictly pseudoconvex to make it clear that it has a strictly plurisubharmonic exhaustion function in relation to the fact that it may not have a strictly plurisubharmonic exhaustion function. Section::::(Weakly) Levi(–Krzoska) pseudoconvexity. If formula_201 boundary , it can

Function of several complex variables

be shown that "D" has a defining function; i.e., that there exists formula_215 which is formula_201 so that formula_217, and formula_218. Now, "D" is pseudoconvex iff for every formula_219 and formula_220 in the complex tangent space at p, that is, For arbitrary complex manifold, Levi (–Krzoska) pseudoconvexity does not always have an plurisubharmonic exhaustion function, i.e. it does not necessarily have a (p-)pseudoconvex domain. If "D" does not have a formula_201 boundary, the following approximation result can be useful. Proposition 1 "If "D" is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains formula_224 with class formula_225-boundary which are

Function of several complex variables

relatively compact in "D", such that" This is because once we have a formula_208 as in the definition we can actually find a formula_225 exhaustion function. Strongly Levi (–Krzoska) pseudoconvexity (Strongly pseudoconvex). When the Levi (–Krzoska) form is positive-definite, it is called Strongly Levi (–Krzoska) pseudoconvex or often called simply Strongly pseudoconvex. Levi total pseudoconvex. If for every boundary point formula_229 of "D", there exists an analytic variety formula_230 passing formula_229 which lies entirely outside "D" in some neighborhood around formula_229, except the point formula_229 itself. Domain "D" that satisfies these conditions is called Levi total pseudoconvex. Oka pseudoconvex

Function of several complex variables

. Family of Oka's disk. Let "n"-functions formula_234 be continuous on formula_235, holomorphic in formula_236 when the parameter "t" is fixed in [0, 1], and assume that formula_237 are not all zero at any point on formula_238. Then the set formula_239 is called an analytic disc de-pending on a parameter "t", and formula_240 is called its shell. If formula_241 and formula_242, Q(t) is called Family of Oka's disk. Definition. When formula_243 holds on any family of Oka's disk, "D" is called Oka pseudoconvex. Oka's proof of Levi's problem was that when the unramified Riemann domain

Function of several complex variables

was a domain of holomorphy (holomorphically convex), it was proved that it was necessary and sufficient that each boundary point of the domain of holomorphy is an Oka pseudoconvex. Locally pseudoconvex (locally Stein, Cartan pseudoconvex, local Levi property). For every point formula_244 there exist a neighbourhood "U" of "x" and "f" holomorphic. ( i.e. formula_245 be holomorphically convex.) such that "f" cannot be extended to any neighbourhood of "x". i.e. A holomorphic map formula_246 will be said to be locally pseudoconvex if every point formula_247 has a neighborhood U such that formula_248 is Stein (weakly 1-complete). In this situation

Function of several complex variables

, we shall also say that X is locally pseudoconvex over Y. This was also called locally Stein and was classically called Cartan Pseudoconvex. In formula_2 the Clocally pseudoconvexdomain is itself a pseudoconvex domain and is a domain of holomorphy. Conditions equivalent to domain of holomorphy. For a domain formula_250 the following conditions are equivalent.: The implications formula_251, formula_252, and formula_253 are standard results. Proving formula_254, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the Levi problem (after E. E. Levi) and was first solved by Kiyoshi Oka, and

Function of several complex variables

then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of formula_255-problem). Sheaf. Idéal de domaines indéterminés (The predecessor of the notion of the coherent (sheaf)). Oka introduced the notion which he termed "idéal de domaines indéterminés" or "ideal of indeterminate domains". Specifically, it is a set formula_256 of pairs formula_257, formula_258 holomorphic on a non-empty open set formula_259, such that The origin of indeterminate domains comes from the fact that domains change depending on the pair formula_257. Cartan translated this notion into the notion of the coherent (Especially, coherent analytic sheaf) (sheaf

Function of several complex variables

) in sheaf cohomology. This name comes from H. Cartan. Also, Serre (1955) introduced the notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf. The notion of coherent (coherent sheaf cohomology) helped solve the problems in several complex variables. Coherent sheaf. Definition. The definition of the coherent sheaf is as follows. A coherent sheaf on a ringed space formula_261 is a sheaf formula_262 satisfying the following two properties: Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of formula_263-modules. Also, Jean-Pierre Serre (1955) proves that A quasi-coherent sheaf

Function of several complex variables