title: Paul Rusesabagina

text: In 2007, in an interview with American journalist Keith Harmon Snow, Rusesabagina blamed the 1994 genocide on the RPF, claiming that Interahamwe leaders, including Robert Kaguja, were working for Paul Kagame, who had ordered them to begin the genocide. He also claimed that the RPF shot down Juvénal Habyarimana's plane, a theory ruled out by a ballistics report, and that the killings committed by the RPF rebels during the conflict constituted genocide. The historian Gérard Prunier agrees that the RPF committed "horrendous crimes", but he rejects the notion of a "double genocide", which he argues "does not stand up to serious inquiry".


title: Paul Rusesabagina

text: In a 2012 speech, Rusesabagina expressed a disillusionment with the RPF, casting doubt on their ability to institute a democratic process and calling for general mobilization to remove them from power.


title: Paul Rusesabagina

text: In an open letter to President Bill Clinton in 2012, Rusesabagina warned against the Clinton Foundation's support for Kagame. In this letter, he made claims that Rwandan nurses were being ordered to kill by the government. He said that one nurse was "ordered to give tainted vaccines to prisoners, Hutus and other enemies of the Kagame administration" and to "control population growth among undesirable populations by causing birth and surgical complications". For example, during C-sections, nurses were to place metal objects such as scissors into the abdomen before closing the wound to cause infections and death".


title: Paul Rusesabagina

text: His comments have attracted strong criticism in Rwanda. On 6 April 2006, Kagame suggested, "[Rusesabagina] should try his talents elsewhere and not climb on the falsehood of being a hero, because it's totally false". Francois Xavier Ngarambe, the president of Ibuka, the umbrella body of survivors' associations for the genocide, said of Rusesabagina, "he has hijacked heroism. He is trading with the genocide. He should be charged." Terry George, the director of "Hotel Rwanda", characterized the comment as part of a smear campaign. The Rwandan government has accused Rusesabagina of genocide denial. Various media outlets in Rwanda, including The New Times, have accused him of genocide denial.


title: Paul Rusesabagina

text: Rusesabagina has consistently denied allegations put forward by the Rwandan government accusing Rusesabagina of helping the Democratic Forces for the Liberation of Rwanda (FDLR), a Rwandan Hutu Power rebel group, which has been condemned by the UN Security Council for "serious violations of international law involving the targeting of women and children". In an open letter to the UN, he equated the FDLR to the RPF, and cast them as "bona fide refugees" who had been "collectively demonized". In a 2010 interview with CNN, Rusesabagina said: "I have sent no money to terrorists... He [the prosecutor] is not only lying, but lying with bad logic... This is pure and simple fabrication from Kigali". Text messages intercepted by German intelligence, between Rusesabagina and the FDLR's former leader, Ignace Murwanashyaka, came to light during Murwanashyaka's trial in 2011. Rusesabagina's wife also denies his association with FDLR. She told the New York Times: "the rebel group “hates Paul because he protected Tutsis during the genocide.”


title: Paul Rusesabagina

text: The armed wing of Paul Rusesabagina's PDR-Ihumure and MRCD political parties, the FLN, was accused of terror attacks in South-West Rwanda in 2018. In an interview broadcast on Voice of America Kinyarwanda in 2018, when asked if FLN rebels were in the Nyungwe forest near where the attacks took place, Rusesabagina responded: "We are angry. We did not enter it to abandon it, we are there to demand our rights as Rwandan natives". After his arrest, he admitted that he had founded the FLN "as an armed wing, not as a terrorist group... I do not deny that the FLN committed crimes but my role was diplomacy". He has denied any wrongdoing.


title: Paul Rusesabagina

text: Rusesabagina has admitted to backing and "diplomatically" supporting the group, as evidenced in a widely disseminated video in which he pledges his "unreserved support" for the FLN and denies any wrongdoing. In the video, Rusesabagina's speech includes:


title: Paul Rusesabagina

text: "The time has come for us to use any means possible to bring about change in Rwanda, as all political means have been tried and failed. It is time to attempt our last resort. Hence, I plead my unreserved support that our youth, The National Liberation forces, NLF, launches against the Kagame army in order to free the Rwandan people. As Rwandans it is important to understand that this is the only way to bring about change in the whole country."


title: Paul Rusesabagina

text: Rusesabagina, a permanent resident of the United States who has not lived in Rwanda since an assassination attempt was made on him in 1996, had gone on a trip to Dubai shortly before being arrested. In a jailhouse interview with "The New York Times", Rusesabagina stated that in Dubai, he boarded a private plane that he thought was bound for Burundi, where he planned to speak at the invitation of a Christian pastor; instead, the plane took him to Kigali. The Minister of Justice, Johnston Busingye, admitted in February 2021 that the Rwandan government had paid for the private jet that brought Rusesabagina to Kigali in an interview with Marc Lamont Hill on Al Jazeera.


title: Paul Rusesabagina

text: On 31 August 2020, Rusesabagina was kidnapped and taken to Kigali where he was arrested on charges of terrorism, arson, kidnap and "murder perpetrated against unarmed, innocent Rwandan civilians on Rwandan territory". The charges refer to terrorist attacks that took place in the south of Rwanda, near the Burundi border, in 2018. At least 9 people were killed in these attacks, including two children.


title: Paul Rusesabagina

text: In October 2020, the Rwandan Prosecution Authority announced that it would try Rusesabagina along with 16 alleged rebels. His trial was initially scheduled for the 26 January 2021, but was postponed due to ongoing complications with the COVID-19 situation in Kigali. His trial, alongside 20 co-defendants, began on 17 February 2021. Rusesabagina told the court that he did not have Rwandan citizenship, so he could not face trial in Rwanda. Another defendant, FLN spokesperson Callixte Nsabimana, seemed to be shocked by these comments, testifying that Rusesabagina "had ambitions to become the president of Rwanda. Now how do you have such ambitions when you're not Rwandan? We waged war on Rwanda, and failed and were captured. It is embarrassing for him to now claim that he is not Rwandan."


title: Paul Rusesabagina

text: Following a hearing held on 12 March 2021, Rusesabagina stated his intention to no longer engage in the court process due to his belief that justice would not be achieved, and accusing the court of not respecting his rights to a fair trial. Rusesabagina did not attend subsequent hearings, with the presiding judge Antoine Muhima ruling that the trial would continue. In July 2021, the court announced that the verdict of the trial would be promulgated on 20 August 2021.


title: Paul Rusesabagina

text: On 20 September 2021, he was found guilty of terrorism-related charges. During the court proceeding, he denounced president Paul Kagame and reported that he had been abducted from exile to stand trial in Rwanda. Following his conviction, Rusesabagina was given a 25-year prison sentence. Human rights advocates stated that they believed the charges were politically motivated due to Rusesabagina's criticism of Kagame. On 24 January 2022, prosecutors in the Kigali court sought life imprisonment for Rusesabagina in the trial.


title: Paul Rusesabagina

text: Rusesabagina's lawyers have argued that the arrest was motivated by Rusesabagina's outspoken criticism of the Rwandan government, in line with other arrests and disappearances of dissidents under the presidency of Paul Kagame. They have also argued that his flight to Rwanda was illegal under international law and constitutes extraordinary rendition.


title: Paul Rusesabagina

text: Human Rights Watch called Rusesabagina's trial "flawed" and an example of Rwanda's overreach and manipulation of the country's justice system. Rwandan authorities repeatedly violated Rusesabagina's due process rights during the trial and interfered with his right to counsel. HRW accused Rwanda of violating international criminal laws by kidnapping Rusesabagina.


title: Paul Rusesabagina

text: Rusesabagina's adopted daughter Carine Kanimba have protested against his arrest, calling it politically motivated. Kanimba claimed: "What they're accusing him of is all made up. There is no evidence to what they're claiming...We know this is a wrongful arrest".


title: Paul Rusesabagina

text: Rusesabagina' family is a part of the Bring Our Families Home campaign which advocates to bring home wrongful detainees and hostages. His image is featured in a 15-foot mural in Georgetown (Washington, D.C.) along with other Americans wrongfully detained abroad.


title: Paul Rusesabagina

text: Human rights activists and celebrities have called for the release of Rusesabagina. Don Cheadle has called for his release. Others who have supported Rusesabagina include Joaquin Phoenix, Doc Rivers, Chiwetel Ejiofor, Chris Evans, Scarlett Johansson, Mark Ruffalo, and Sophie Okonedo.


title: Paul Rusesabagina

text: Rwanda is hosting the 2022 Commonwealth Heads of Government Meeting (CHGM 2022) and there have been calls for Commonwealth governments to pressure CHGM 2022 host Rwanda to free human rights defender Paul Rusesabagina.


title: Paul Rusesabagina

text: The European Union criticised the arrest in February 2021 by adopting a resolution (2021/2543[RSP]) condemning the arrest. The Rwandan parliament criticised the EU resolution, saying that it gave an "unrealistic and baseless characterisation" of his arrest.


title: Paul Rusesabagina

text: The United States Government considers Rusesabagina to be "wrongfully detained." U.S. House of Representatives passed a resolution authored by Congressman Joaquin Castro (TX-20) and Congresswoman Young Kim (CA-39) that calls on the Rwandan government to release Paul Rusesabagina.


title: Paul Rusesabagina

text: The UN Working Group on Arbitrary Detention found on 18 March 2022 that he had been illegally kidnapped, tortured, and sentenced after an unfair trial. The Working Group stated that: "It is clear on the facts that Mr. Rusesabagina has been targeted by the Government on account of his work as a human rights defender, because of his criticism of the Government on a broad range of human rights issues, including unfair elections and a lack of democracy, freedom of speech, freedom of association and freedom of the press. He has also challenged cases of arbitrary detention, torture and extrajudicial killings.’’ The Working Group called on the Government “to release Mr. Rusesabagina immediately and accord him an enforceable right to compensation and other reparations, in accordance with international law’’ and “to ensure a full and independent investigation of the circumstances surrounding the arbitrary deprivation of liberty of Mr. Rusesabagina and to take appropriate measures against those responsible for the violation of his rights.”Rusesabagina’s family welcomed the Working Group’s finding, and were “hopeful that the world will listen to this call by the United Nations and put pressure on Rwanda to immediately free our father and husband”.


title: Paul Rusesabagina

text: Some of Rusesabagina's family members are suing charter airline GainJet for not disclosing the true destination of the flight to Rusesabagina and aiding in his kidnapping.


title: Paul Rusesabagina

text: Rusesabagina's family filed a lawsuit against the Rwandan government and high-ranking officials in Rwanda alleging that they conspired to "facilitate and execute an elaborate plot to lure" Rusesabagina from his home in San Antonio to Rwanda "where he would be tortured and illegally detained for the remainder of his life".


title: Paul Rusesabagina

text: Rusesabagina's story was first told in Philip Gourevitch's book "We Wish to Inform You That Tomorrow We Will Be Killed with Our Families", which was published in 1998.


title: Paul Rusesabagina

text: His autobiography "An Ordinary Man" (written with Tom Zoellner ) was published by Zach Bell in April 2006.


title: Paul Rusesabagina

text: In 2008, Alfred Ndahiro – a close advisor to Paul Kagame – and Rwandan scholar Privat Rutazibwa wrote "Hotel Rwanda: Or the Tutsi Genocide as Seen by Hollwood" ()"." In 2014, Edouard Kayihura, who hid in the Hotel during the 100 days of genocide, wrote "Inside the Hotel Rwanda: The Surprising True Story ... and Why It Matters Today" (written with Kerry Zukus )"." Both have been critical of Rusesabagina.


title: Paul Rusesabagina

text: Rusesabagina's work during the genocide is dramatized in the 2004 movie "Hotel Rwanda", he is portrayed by Don Cheadle. Cheadle's performance was met with critical acclaim and the actor was nominated for the Academy Award for Best Actor, Golden Globe Award for Best Actor – Motion Picture Drama and Screen Actors Guild Award for Outstanding Performance by a Male Actor in a Leading Role.


title: Hauts-de-France

text: Hauts-de-France (; ; , also "Upper France") is the northernmost region of France, created by the territorial reform of French regions in 2014, from a merger of Nord-Pas-de-Calais and Picardy. Its prefecture is Lille. The new region came into existence on 1 January 2016, after regional elections in December 2015. The Conseil d'État approved Hauts-de-France as the name of the region on 28 September 2016, effective the following 30 September.


title: Hauts-de-France

text: With 6,009,976 inhabitants (as of 1 January 2015) and a population density of 189 inhabitants/km, it is the third most populous region in France and the second most densely populated in metropolitan France after its southern neighbour Île-de-France. It is bordered by Belgium to the north.


title: Hauts-de-France

text: The region's interim name "Nord-Pas-de-Calais-Picardie" was a hyphenated placename, created by hyphenating the merged regions' names—"Nord-Pas-de-Calais" and "Picardie"—in alphabetical order.


title: Hauts-de-France

text: On 14 March 2016, well ahead of the 1 July deadline, the regional council decided on "Hauts-de-France" as the region's permanent name. The provisional name of the region was retired on 30 September 2016, when the new name of the region, Hauts-de-France, took effect.


title: Hauts-de-France

text: The region covers an area of more than . It borders Belgium (Flanders and Wallonia) to the northeast, the North Sea to the north, the English Channel to the west, as well as the French regions of Grand Est to the east-southeast, Île-de-France to the south, and Normandy to the west-southwest. It is connected to the United Kingdom (England) via the Channel Tunnel.


title: Hauts-de-France

text: The Gross domestic product (GDP) of the region was 161.7 billion euros in 2018, accounting for 6.9% of French economic output. GDP per capita adjusted for purchasing power was 24,200 euros or 80% of the EU27 average in the same year. The GDP per employee was 101% of the EU average.


title: Sogdia

text: Sogdia (Sogdian: ) or Sogdiana was an ancient Iranian civilization between the Amu Darya and the Syr Darya, and in present-day Uzbekistan, Turkmenistan, Tajikistan, Kazakhstan, and Kyrgyzstan. Sogdiana was also a province of the Achaemenid Empire, and listed on the Behistun Inscription of Darius the Great. Sogdiana was first conquered by Cyrus the Great, the founder of the Achaemenid Empire, and then was annexed by the Macedonian ruler Alexander the Great in 328 BC. It would continue to change hands under the Seleucid Empire, the Greco-Bactrian Kingdom, the Kushan Empire, the Sasanian Empire, the Hephthalite Empire, the Western Turkic Khaganate and the Muslim conquest of Transoxiana.


title: Sogdia

text: The Sogdian city-states, although never politically united, were centered on the city of Samarkand. Sogdian, an Eastern Iranian language, is no longer spoken, but a descendant of one of its dialects, Yaghnobi, is still spoken by the Yaghnobis of Tajikistan. It was widely spoken in Central Asia as a "lingua franca" and served as one of the First Turkic Khaganate's court languages for writing documents.


title: Sogdia

text: Sogdians also lived in Imperial China and rose to prominence in the military and government of the Chinese Tang dynasty (618–907 AD). Sogdian merchants and diplomats travelled as far west as the Byzantine Empire. They played an important part as middlemen in the trade route of the Silk Road. While originally following the faiths of Zoroastrianism, Manichaeism, Buddhism and, to a lesser extent, Nestorian Christianity from West Asia, the gradual conversion to Islam among the Sogdians and their descendants began with the Muslim conquest of Transoxiana in the 8th century. The Sogdian conversion to Islam was virtually complete by the end of the Samanid Empire in 999, coinciding with the decline of the Sogdian language, as it was largely supplanted by Persian.


title: Sogdia

text: Sogdiana lay north of Bactria, east of Khwarezm, and southeast of Kangju between the Oxus (Amu Darya) and the Jaxartes (Syr Darya), including the fertile valley of the Zeravshan (called the Polytimetus by the ancient Greeks). Sogdian territory corresponds to the modern regions of Samarkand and Bukhara in modern Uzbekistan, as well as the Sughd region of modern Tajikistan. In the High Middle Ages, Sogdian cities included sites stretching towards Issyk Kul, such as that at the archeological site of Suyab.


title: Sogdia

text: Oswald Szemerényi devotes a thorough discussion to the etymologies of ancient ethnic words for the Scythians in his work "Four Old Iranian Ethnic Names: Scythian – Skudra – Sogdian – Saka". In it, the names provided by the Greek historian Herodotus and the names of his title, except Saka, as well as many other words for "Scythian", such as Assyrian "Aškuz" and Greek "Skuthēs", descend from *skeud-, an ancient Indo-European root meaning "propel, shoot" (cf. English shoot). *skud- is the zero-grade; that is, a variant in which the -e- is not present. The restored Scythian name is *Skuda (archer), which among the Pontic or Royal Scythians became *Skula, in which the d has been regularly replaced by an l. According to Szemerényi, Sogdiana (; ; ; ; ; ) was named from the Skuda form. Starting from the names of the province given in Old Persian inscriptions, Sugda and Suguda, and the knowledge derived from Middle Sogdian that Old Persian -gd- applied to Sogdian was pronounced as voiced fricatives, -γδ-, Szemerényi arrives at *Suγδa as an Old Sogdian endonym. Applying sound changes apparent in other Sogdian words and inherent in Indo-European, he traces the development of *Suγδa from Skuda, "archer", as follows: Skuda > *Sukuda by anaptyxis > *Sukuδa > *Sukδa (syncope) > *Suγδa (assimilation).


title: Sogdia

text: Sogdiana possessed a Bronze Age urban culture: original Bronze Age towns appear in the archaeological record beginning with the settlement at Sarazm, Tajikistan, spanning as far back as the 4th millennium BC, and then at Kök Tepe, near modern-day Bulungur, Uzbekistan, from at least the 15th century BC.


title: Sogdia

text: This original culture was gradually displaced by the Indo-European migrations of the Iron Age, forming the Andronovo culture (c. 2000–1450 BC), which included Eastern Iranian speaking peoples such as the historical Sogdians. In the Avesta the Sogdians are mentioned as living in the lands created by Ahura Mazda for the Iranian peoples.


title: Sogdia

text: Achaemenid ruler Cyrus the Great conquered Sogdiana while campaigning in Central Asia in 546–539 BC, a fact mentioned by the ancient Greek historian Herodotus in his "Histories". Darius I introduced the Aramaic writing system and coin currency to Central Asia, in addition to incorporating Sogdians into his standing army as regular soldiers and cavalrymen. A contingent of Sogdian soldiers fought in the main army of Xerxes I during his second, ultimately-failed invasion of Greece in 480 BC. A Persian inscription from Susa claims that the palace there was adorned with lapis lazuli and carnelian originating from Sogdiana.


title: Sogdia

text: During this period of Persian rule, the western half of Asia Minor was part of the Greek civilization. As the Achaemenids conquered it, they met persistent resistance and revolt. One of their solutions was to ethnically cleanse rebelling regions, relocating those who survived to the far side of the empire. Thus Sogdiana came to have a significant Greek population.


title: Sogdia

text: Given the absence of any named satraps (i.e. Achaemenid provincial governors) for Sogdiana in historical records, modern scholarship has concluded that Sogdiana was governed from the satrapy of nearby Bactria. The satraps were often relatives of the ruling Persian kings, especially sons who were not designated as the heir apparent. Sogdiana likely remained under Persian control until roughly 400 BC, during the reign of Artaxerxes II. Rebellious states of the Persian Empire took advantage of the weak Artaxerxes II, and some, such as Egypt, were able to regain their independence. Persia's massive loss of Central Asian territory is widely attributed to the ruler's lack of control. However, unlike Egypt, which was quickly recaptured by the Persian Empire, Sogdiana remained independent until it was conquered by Alexander the Great. When the latter invaded the Persian Empire, Pharasmanes, an already independent king of Khwarezm, allied with the Macedonians and sent troops to Alexander in 329 BC for his war against the Scythians of the Black Sea region (even though this anticipated campaign never materialized).


title: Sogdia

text: During the Achaemenid period (550–330 BC), the Sogdians lived as a nomadic people much like the neighboring Yuezhi, who spoke Bactrian, an Indo-Iranian language closely related to Sogdian, and were already engaging in overland trade. Some of them had also gradually settled the land to engage in agriculture. Similar to how the Yuezhi offered tributary gifts of jade to the emperors of China, the Sogdians are recorded in Persian records as submitting precious gifts of lapis lazuli and carnelian to Darius I, the Persian king of kings. Although the Sogdians were at times independent and living outside the boundaries of large empires, they never formed a great empire of their own like the Yuezhi, who established the Kushan Empire (30–375 AD) of Central and South Asia.


title: Sogdia

text: A now-independent and warlike Sogdiana formed a border region insulating the Achaemenid Persians from the nomadic Scythians to the north and east. It was led at first by Bessus, the Achaemenid satrap of Bactria. After assassinating Darius III in his flight from the Macedonian Greek army, he became claimant to the Achaemenid throne. The Sogdian Rock or Rock of Ariamazes, a fortress in Sogdiana, was captured in 327 BC by the forces of Alexander the Great, the "basileus" of Macedonian Greece, and conqueror of the Persian Achaemenid Empire. Oxyartes, a Sogdian nobleman of Bactria, had hoped to keep his daughter Roxana safe at the fortress of the Sogdian Rock, yet after its fall Roxana was soon wed to Alexander as one of his several wives. Roxana, a Sogdian whose name "Roshanak" means "little star", was the mother of Alexander IV of Macedon, who inherited his late father's throne in 323 BC (although the empire was soon divided in the Wars of the Diadochi).


title: Sogdia

text: After an extended campaign putting down Sogdian resistance and founding military outposts manned by his Macedonian veterans, Alexander united Sogdiana with Bactria into one satrapy. The Sogdian nobleman and warlord Spitamenes (370–328 BC), allied with Scythian tribes, led an uprising against Alexander's forces. This revolt was put down by Alexander and his generals Amyntas, Craterus, and Coenus, with the aid of native Bactrian and Sogdian troops. With the Scythian and Sogdian rebels defeated, Spitamenes was allegedly betrayed by his own wife and beheaded. Pursuant with his own marriage to Roxana, Alexander encouraged his men to marry Sogdian women in order to discourage further revolt. This included Apama, daughter of the rebel Spitamenes, who wed Seleucus I Nicator and bore him a son and future heir to the Seleucid throne. According to the Roman historian Appian, Seleucus I named three new Hellenistic cities in Asia after her (see "Apamea").


title: Sogdia

text: The military power of the Sogdians never recovered. Subsequently, Sogdiana formed part of the Hellenistic Greco-Bactrian Kingdom, a breakaway state from the Seleucid Empire founded in 248 BC by Diodotus I, for roughly a century. Euthydemus I, a former satrap of Sogdiana, seems to have held the Sogdian territory as a rival claimant to the Greco-Bactrian throne; his coins were later copied locally and bore Aramaic inscriptions. The Greco-Bactrian king Eucratides I may have recovered sovereignty of Sogdia temporarily.


title: Sogdia

text: Finally Sogdia was occupied by nomads when the Sakas overran the Greco-Bactrian kingdom around 145 BC, soon followed by the Yuezhi, the nomadic predecessors of the Kushans. From then until about 40 BC the Yuezhi tepidly minted coins imitating and still bearing the images of the Greco-Bactrian kings Eucratides I and Heliocles I.


title: Sogdia

text: The Yuezhis were visited in Transoxiana by a Chinese mission, led by Zhang Qian in 126 BC, which sought an offensive alliance with the Yuezhi against the Xiongnu. Zhang Qian, who spent a year in Transoxiana and Bactria, wrote a detailed account in the "Shiji", which gives considerable insight into the situation in Central Asia at the time. The request for an alliance was denied by the son of the slain Yuezhi king, who preferred to maintain peace in Transoxiana rather than seek revenge.


title: Sogdia

text: From the 1st century AD, the Yuezhi morphed into the powerful Kushan Empire, covering an area from Sogdia to eastern India. The Kushan Empire became the center of the profitable Central Asian commerce. They began minting unique coins bearing the faces of their own rulers. They are related to have collaborated militarily with the Chinese against nomadic incursion, particularly when they allied with the Han dynasty general Ban Chao against the Sogdians in 84, when the latter were trying to support a revolt by the king of Kashgar.


title: Sogdia

text: Historical knowledge about Sogdia is somewhat hazy during the period of the Parthian Empire (247 BC – 224 AD) in Persia. The subsequent Sasanian Empire of Persia conquered and incorporated Sogdia as a satrapy in 260, an inscription dating to the reign of Shapur I claiming "Sogdia, to the mountains of Tashkent" as his territory, and noting that its limits formed the northeastern Sasanian borderlands with the Kushan Empire. However, by the 5th century the region was captured by the rival Hephthalite Empire.


title: Sogdia

text: The Hephthalites conquered the territory of Sogdiana, and incorporated it into their Empire, around 479 AD, as this is the date of the last known independent embassy of the Sogdians to China.


title: Sogdia

text: The Hephthalites may have built major fortified Hippodamian cities (rectangular walls with an orthogonal network of streets) in Sogdiana, such as Bukhara and Panjikent, as they had also in Herat, continuing the city-building efforts of the Kidarites. The Hephthalites probably ruled over a confederation of local rulers or governors, linked through alliance agreements. One of these vassals may have been Asbar, ruler of Vardanzi, who also minted his own coinage during the period.


title: Sogdia

text: The wealth of the Sasanian ransoms and tributes to the Hephthalites may have been reinvested in Sogdia, possibly explaining the prosperity of the region from that time. Sogdia, at the center of a new Silk Road between China to the Sasanian Empire and the Byzantine Empire became extremely prosperous under its nomadic elites. The Hephthalites took on the role of major intermediary on the Silk Road, after their great predecessor the Kushans, and contracted local Sogdians to carry on the trade of silk and other luxury goods between the China Empire and the Sasanian Empire.


title: Sogdia

text: Because of the Hephthalite occupation of Sogdia, the original coinage of Sogdia came to be flooded by the influx of Sasanian coins received as a tribute to the Hephthalites. This coinage then spread along the Silk Road. The symbol of the Hephthalites appears on the residual coinage of Samarkand, probably as a consequence of the Hephthalite control of Sogdia, and becomes prominent in Sogdian coinage from 500 to 700 AD, including in the coinage of their indigenous successors the Ikhshids (642-755 AD), ending with the Muslim conquest of Transoxiana.


title: Sogdia

text: The Turks of the First Turkic Khaganate and the Sasanians under Khosrow I allied against the Hephthalites and defeated them after an eight-day battle near Qarshi, the Battle of Bukhara, perhaps in 557. The Turks retained the area north of the Oxus, including all of Sogdia, while the Sasanians obtained the areas south of it. The Turks fragmented in 581, and the Western Turkic Khaganate took over in Sogdia.


title: Sogdia

text: Archaeological remains suggest that the Turks probably became the main trading partners of the Sogdians, as appears from the tomb of the Sogdian trader An Jia. The Turks also appear in great numbers in the Afrasiab murals of Samarkand, where they are probably shown attending the reception by the local Sogdian ruler Varkhuman in the 7th century AD. These paintings suggest that Sogdia was a very cosmopolitan environment at that time, as delegates of various nations, including Chinese and Korean delegates, are also shown. From around 650, China led the conquest of the Western Turks, and the Sogdian rulers such as Varkhuman as well as the Western Turks all became nominal vassals of China, as part of the Anxi Protectorate of the Tang dynasty, until the Muslim conquest of Transoxiana.


title: Sogdia

text: Qutayba ibn Muslim (669–716), Governor of Greater Khorasan under the Umayyad Caliphate (661–750), initiated the Muslim conquest of Sogdia during the early 8th century, with the local ruler of Balkh offering him aid as an Umayyad ally. However, when his successor al-Jarrah ibn Abdallah governed Khorasan (717–719), many native Sogdians, who had converted to Islam, began to revolt when they were no longer exempt from paying the tax on non-Muslims, the "jizya," because of a new law stating that proof of circumcision and literacy in the Quran was necessary for new converts. With the aid of the Turkic Turgesh, the Sogdians were able to expel the Umayyad Arab garrison from Samarkand, and Umayyad attempts to restore power there were rebuffed until the arrival of Sa'id ibn Amr al-Harashi (fl. 720–735). The Sogdian ruler (i.e. "ikhshid") of Samarkand, Gurak, who had previously overthrown the pro-Umayyad Sogdian ruler Tarkhun in 710, decided that resistance against al-Harashi's large Arab force was pointless, and thereafter persuaded his followers to declare allegiance to the Umayyad governor. Divashtich (r. 706–722), the Sogdian ruler of Panjakent, led his forces to the Zarafshan Range (near modern Zarafshan, Tajikistan), whereas the Sogdians following Karzanj, the ruler of Pai (modern Kattakurgan, Uzbekistan), fled to the Principality of Farghana, where their ruler at-Tar (or Alutar) promised them safety and refuge from the Umayyads. However, at-Tar secretly informed al-Harashi of the Sogdians hiding in Khujand, who were then slaughtered by al-Harashi's forces after their arrival.


title: Sogdia

text: The Umayyads fell in 750 to the Abbasid Caliphate, which quickly asserted itself in Central Asia after winning the Battle of Talas (along the Talas River in modern Talas Oblast, Kyrgyzstan) in 751, against the Chinese Tang Dynasty. This conflict incidentally introduced Chinese papermaking to the Islamic world. The cultural consequences and political ramifications of this battle meant the retreat of the Chinese empire from Central Asia. It also allowed for the rise of the Samanid Empire (819–999), a Persian state centered at Bukhara (in what is now modern Uzbekistan) that nominally observed the Abbasids as their overlords, yet retained a great deal of autonomy and upheld the mercantile legacy of the Sogdians. Yet the Sogdian language gradually declined in favor of the Persian language of the Samanids (the ancestor to the modern Tajik language), the spoken language of renowned poets and intellectuals of the age such as Ferdowsi (940–1020). So too did the original religions of the Sogdians decline; Zoroastrianism, Buddhism, Manichaeism, and Nestorian Christianity disappeared in the region by the end of the Samanid period. The Samanids were also responsible for converting the surrounding Turkic peoples to Islam, which presaged the conquest of their empire in 999 by an Islamic Turkic power, the Kara-Khanid Khanate (840–1212).


title: Sogdia

text: During the early 13th century, Khwarezmia was invaded by the early Mongol Empire and its ruler Genghis Khan destroyed the once vibrant cities of Bukhara and Samarkand. However, in 1370, Samarkand saw a revival as the capital of the Timurid Empire. The Turko-Mongol ruler Timur forcefully brought artisans and intellectuals from across Asia to Samarkand, transforming it not only into a trade hub but also one of the most important cities of the Islamic world.


title: Sogdia

text: Most merchants did not travel the entire Silk Road, but would trade goods through middlemen based in oasis towns, such as Khotan or Dunhuang. The Sogdians, however, established a trading network across the 1500 miles from Sogdiana to China. In fact, the Sogdians turned their energies to trade so thoroughly that the Saka of the Kingdom of Khotan called all merchants "suli", "Sogdian", whatever their culture or ethnicity. The Sogdians had learnt to become expert traders from the Kushans, together with whom they initially controlled trade in the Ferghana Valley and Kangju during the 'birth' of the Silk Road. Later, they became the primary middlemen after the demise of the Kushan Empire.


title: Sogdia

text: Unlike the empires of antiquity, the Sogdian region was not a territory confined within fixed borders, but rather a network of city-states, from one oasis to another, linking Sogdiana to Byzantium, India, Indochina and China. Sogdian contacts with China were initiated by the embassy of the Chinese explorer Zhang Qian during the reign of Emperor Wu (r. 141–87 BC) of the former Han dynasty. Zhang wrote a report of his visit to the Western Regions in Central Asia and named the area of Sogdiana as "Kangju".


title: Sogdia

text: Following Zhang Qian's embassy and report, commercial Chinese relations with Central Asia and Sogdiana flourished, as many Chinese missions were sent throughout the 1st century BC. In his "Shiji" published in 94 BC, Chinese historian Sima Qian remarked that "the largest of these embassies to foreign states numbered several hundred persons, while even the smaller parties included over 100 members ... In the course of one year anywhere from five to six to over ten parties would be sent out." In terms of the silk trade, the Sogdians also served as middlemen between the Chinese Han Empire and the Parthian Empire of the Middle East and West Asia. Sogdians played a major role in facilitating trade between China and Central Asia along the Silk Roads as late as the 10th century, their language serving as a "lingua franca" for Asian trade as far back as the 4th century.


title: Sogdia

text: Subsequent to their domination by Alexander the Great, the Sogdians from the city of Marakanda (Samarkand) became dominant as traveling merchants, occupying a key position along the ancient Silk Road. They played an active role in the spread of faiths such as Manicheism, Zoroastrianism, and Buddhism along the Silk Road. The Chinese "Sui Shu" ("Book of Sui") describes Sogdians as "skilled merchants" who attracted many foreign traders to their land to engage in commerce. They were described by the Chinese as born merchants, learning their commercial skills at an early age. It appears from sources, such as documents found by Sir Aurel Stein and others, that by the 4th century they may have monopolized trade between India and China. A letter written by Sogdian merchants dated 313 AD and found in the ruins of a watchtower in Gansu, was intended to be sent to merchants in Samarkand, warning them that after Liu Cong of Han Zhao sacked Luoyang and the Jin emperor fled the capital, there was no worthwhile business there for Indian and Sogdian merchants. Furthermore, in 568 AD, a Turko-Sogdian delegation travelled to the Roman emperor in Constantinople to obtain permission to trade and in the following years commercial activity between the states flourished. Put simply, the Sogdians dominated trade along the Silk Road from the 2nd century BC until the 10th century.


title: Sogdia

text: Suyab and Talas in modern-day Kyrgyzstan were the main Sogdian centers in the north that dominated the caravan routes of the 6th to 8th centuries. Their commercial interests were protected by the resurgent military power of the Göktürks, whose empire was built on the political power of the Ashina clan and economic clout of the Sogdians. Sogdian trade, with some interruptions, continued into the 9th century. For instance, camels, women, girls, silver, and gold were seized from Sogdia during a raid by Qapaghan Qaghan (692–716), ruler of the Second Turkic Khaganate. In the 10th century, Sogdiana was incorporated into the Uighur Empire, which until 840 encompassed northern Central Asia. This khaganate obtained enormous deliveries of silk from Tang China in exchange for horses, in turn relying on the Sogdians to sell much of this silk further west. Peter B. Golden writes that the Uyghurs not only adopted the writing system and religious faiths of the Sogdians, such as Manichaeism, Buddhism, and Christianity, but also looked to the Sogdians as "mentors", while gradually replacing them in their roles as Silk Road traders and purveyors of culture. Muslim geographers of the 10th century drew upon Sogdian records dating to 750–840. After the end of the Uyghur Empire, Sogdian trade underwent a crisis. Following the Muslim conquest of Transoxiana in the 8th century, the Samanids resumed trade on the northwestern road leading to the Khazars and the Urals and the northeastern one toward the nearby Turkic tribes.


title: Sogdia

text: During the 5th and 6th century, many Sogdians took up residence in the Hexi Corridor, where they retained autonomy in terms of governance and had a designated official administrator known as a "Sabao", which suggests their importance to the socioeconomic structure of China. The Sogdian influence on trade in China is also made apparent by a Chinese document which lists taxes paid on caravan trade in the Turpan region and shows that twenty-nine out of the thirty-five commercial transactions involved Sogdian merchants, and in thirteen of those cases both the buyer and the seller were Sogdian. Trade goods brought to China included grapes, alfalfa, and Sassanian silverware, as well as glass containers, Mediterranean coral, brass Buddhist images, Roman wool cloth, and Baltic amber. These were exchanged for Chinese paper, copper, and silk. In the 7th century, the Chinese Buddhist pilgrim Xuanzang noted with approval that Sogdian boys were taught to read and write at the age of five, though their skill was turned to trade, disappointing the scholarly Xuanzang. He also recorded the Sogdians working in other capacities such as farmers, carpetweavers, glassmakers, and woodcarvers.


title: Sogdia

text: Shortly after the smuggling of silkworm eggs into the Byzantine Empire from China by Nestorian Christian monks, the 6th-century Byzantine historian Menander Protector writes of how the Sogdians attempted to establish a direct trade of Chinese silk with the Byzantine Empire. After forming an alliance with the Sasanian ruler Khosrow I to defeat the Hephthalite Empire, Istämi, the Göktürk ruler of the First Turkic Khaganate, was approached by Sogdian merchants requesting permission to seek an audience with the Sassanid king of kings for the privilege of traveling through Persian territories in order to trade with the Byzantines. Istämi refused the first request, but when he sanctioned the second one and had the Sogdian embassy sent to the Sassanid king, the latter had the members of the embassy poisoned. Maniah, a Sogdian diplomat, convinced Istämi to send an embassy directly to Byzantium's capital Constantinople, which arrived in 568 and offered not only silk as a gift to Byzantine ruler Justin II, but also proposed an alliance against Sassanid Persia. Justin II agreed and sent an embassy to the Turkic Khaganate, ensuring the direct silk trade desired by the Sogdians.


title: Sogdia

text: It appears, however, that direct trade with the Sogdians remained limited in light of the small amount of Roman and Byzantine coins found in Central Asian and Chinese archaeological sites belonging to this era. Although Roman embassies apparently reached Han China from 166 AD onwards, and the ancient Romans imported Han Chinese silk while the Han-dynasty Chinese imported Roman glasswares as discovered in their tombs, Valerie Hansen (2012) wrote that no Roman coins from the Roman Republic (507–27 BC) or the Principate (27 BC – 330 AD) era of the Roman Empire have been found in China. However, Warwick Ball (2016) upends this notion by pointing to a hoard of sixteen Roman coins found at Xi'an, China (formerly Chang'an), dated to the reigns of various emperors from Tiberius (14–37 AD) to Aurelian (270–275 AD). The earliest gold "solidus" coins from the Eastern Roman Empire found in China date to the reign of Byzantine emperor Theodosius II (r. 408–450) and altogether only forty-eight of them have been found (compared to thirteen-hundred silver coins) in Xinjiang and the rest of China. The use of silver coins in Turfan persisted long after the Tang campaign against Karakhoja and Chinese conquest of 640, with a gradual adoption of Chinese bronze coinage over the course of the 7th century. The fact that these Eastern Roman coins were almost always found with Sasanian Persian silver coins and Eastern Roman gold coins were used more as ceremonial objects like talismans, confirms the pre-eminent importance of Greater Iran in Chinese Silk Road commerce of Central Asia compared to Eastern Rome.


title: Sogdia

text: The Kizil Caves near Kucha, mid-way in the Tarim Basin, record many scenes of traders from Central Asia in the 5-6th century: these combine influence from the Eastern Iran sphere, at that time occupied by the Sasanian Empire and the Hephthalites, with strong Sogdian cultural elements. Sogdia, at the center of a new Silk Road between China to the Sasanian Empire and the Byzantine Empire became extremely prosperous around that time.


title: Sogdia

text: The style of this period in Kizil is characterized by strong Iranian-Sogdian elements probably brought with intense Sogdian-Tocharian trade, the influence of which is especially apparent in the Central-Asian caftans with Sogdian textile designs, as well as Sogdian longswords of many of the figures. Other characteristic Sogdian designs are animals, such as ducks, within pearl medallions.


title: Sogdia

text: Aside from the Sogdians of Central Asia who acted as middlemen in the Silk Road trade, other Sogdians settled down in China for generations. Many Sogdians lived in Luoyang, capital of the Jin Dynasty (266–420), but fled following the collapse of the Jin Dynasty's control over northern China in 311 AD and the rise of northern nomadic tribes. Still, some Sogdians continued living in Gansu. A community of Sogdians remained in the Northern Liang capital of Wuwei, but when the Northern Liang were defeated by the Northern Wei in 439 AD, many Sogdians were forcibly relocated to the Northern Wei capital of Datong, thereby fostering exchanges and trade for the new dynasty. Numerous have been found in Northern Wei tombs, such as the tomb of Feng Hetu.


title: Sogdia

text: Other Sogdians came from the west and took positions in Chinese society. The "Bei shi" describes how a Sogdian came from Anxi (western Sogdiana or Parthia) to China and became a "sabao" (薩保, from Sanskrit "sarthavaha", meaning caravan leader) who lived in Jiuquan during the Northern Wei (386 – 535 AD), and was the ancestor of An Tugen, a man who rose from a common merchant to become a top ranking minister of state for the Northern Qi (550 – 577 AD). Valerie Hansen asserts that around this time and extending into the Tang Dynasty (618 – 907 AD), the Sogdians "became the most influential of the non-Chinese groups resident in China". Two different types of Sogdians came to China, envoys and merchants. Sogdian envoys settled, marrying Chinese women, purchasing land, with newcomers living there permanently instead of returning to their homelands in Sogdiana. They were concentrated in large numbers around Luoyang and Chang'an, and also Xiangyang in present-day Hubei, building Zoroastrian temples to service their communities once they reached the threshold of roughly 100 households. From the Northern Qi to Tang periods, the leaders of these communities, the "sabao", were incorporated into the official hierarchy of state officials.


title: Sogdia

text: During the 6-7th centuries AD, Sogdian families living in China created important tombs with funerary epitaphs explaining the history of their illustrious houses. Their burial practices blended both Chinese forms such as carved funerary beds with Zoroastrian sensibilities in mind, such as separating the body from both the earth and water. are among the most lavish of the period in this country, and are only inferior to Imperial tombs, suggesting that the Sogdian "Sabao" were among the wealthiest members of the population.


title: Sogdia

text: In addition to being merchants, monks, and government officials, Sogdians also served as soldiers in the Tang military. An Lushan, whose father was Sogdian and mother a Gokturk, rose to the position of a military governor ("jiedushi") in the northeast before leading the An Lushan Rebellion (755 – 763 AD), which split the loyalties of the Sogdians in China. The An Lushan rebellion was supported by many Sogdians, and in its aftermath many of them were slain or changed their names to escape their Sogdian heritage, so that little is known about the Sogdian presence in North China since that time. The former Yan rebel general Gao Juren of Goguryeo descent ordered a mass slaughter of West Asian (Central Asian) Sogdians in Fanyang, also known as Jicheng (Beijing), in Youzhou identifying them through their big noses and lances were used to impale their children when he rebelled against the rebel Yan emperor Shi Chaoyi and defeated rival Yan dynasty forces under the Turk Ashina Chengqing, High nosed Sogdians were slaughtered in Youzhou in 761. Youzhou had Linzhou, another "protected" prefecture attached to it and Sogdians lived there in great numbers. because Gao Juren, like Tian Shengong wanted to defect to the Tang dynasty and wanted them to pubicly recognize and acknowledge him as a regional warlord and offered the slaughter of the Central Asian Hu "barbarians" as a blood sacrifice for the Tang court to acknowledge his alleigance without him giving up territory. according to the book, "History of An Lushan" (安祿山史記). Another source says the slaughter of the Hu barbarians serving Ashina Chengqing was done by Gao Juren in Fanyang in order to deprive him of his support base, since the Tiele, Tongluo, Sogdians and Turks were all Hu and supported the Turk Ashina Chengqing against the Mohe, Xi, Khitan and Goguryeo origin soldiers led by Gao Juren. Gao Juren was later killed by Li Huaixian, who was loyal to Shi Chaoyi. A massacre of foreign Arab and Persian Muslim merchants by former Yan rebel general Tian Shengong happened during the An Lushan rebellion in the Yangzhou massacre (760), since Tian Shengong was defecting to the Tang dynasty and wanted them to publicly recognized and acknowledge him, and the Tang court portrayed the war as between rebel hu barbarians of the Yan against Han Chinese of the Tang dynasty, Tian Shengong slaughtered foreigners as a blood sacrifice to prove he was loyal to the Han Chinese Tang dynasty state and for them to recognize him as a regional warlord without him giving up territory, and he killed other foreign Hu barbarian ethnicities as well whose ethnic groups were not specified, not only Arabs and Persians since it was directed against all foreigners.


title: Sogdia

text: Sogdians continued as active traders in China following the defeat of the rebellion, but many of them were compelled to hide their ethnic identity. A prominent case was An Chongzhang, Minister of War, and Duke of Liang who, in 756, asked Emperor Suzong of Tang to allow him to change his name to Li Baoyu because of his shame in sharing the same surname with the rebel leader. This change of surnames was enacted retroactively for all of his family members, so that his ancestors would also be bestowed the surname Li.


title: Sogdia

text: The Nestorian Christians like the Bactrian Priest Yisi of Balkh helped the Tang dynasty general Guo Ziyi militarily crush the An Lushan rebellion, with Yisi personally acting as a military commander and Yisi and the Nestorian Church of the East were rewarded by the Tang dynasty with titles and positions as described in the Nestorian Stele.


title: Sogdia

text: Amoghavajra used his rituals against An Lushan while staying in Chang'an when it was occupied in 756 while the Tang dynasty crown prince and Xuanzong emperor had retreated to Sichuan. Amoghavajra's rituals were explicitly intended to introduced death, disaster and disease against An Lushan. As a result of Amoghavajrya's assistance in crushing An Lushan, Estoteric Buddhism became the official state Buddhist sect supported by the Tang dynasty, "Imperial Buddhism" with state funding and backing for writing scriptures, and constructing monasteries and temples. The disciples of Amoghavajra did ceremonies for the state and emperor. Tang dynasty Emperor Suzong was crowned as cakravartin by Amoghavajra after victory against An Lushan in 759 and he had invoked the Acala vidyaraja against An Lushan. The Tang dynasty crown prince Li Heng (later Suzong) also received important strategic military information from Chang'an when it was occupied by An Lushan though secret message sent by Amoghavajra.


title: Sogdia

text: Epitaphs were found dating from the Tang dynasty of a Christian couple in Luoyang of a Nestorian Christian Sogdian woman, who Lady An (安氏) who died in 821 and her Nestorian Christian Han Chinese husband, Hua Xian (花献) who died in 827. These Han Chinese Christian men may have married Sogdian Christian women because of a lack of Han Chinese women belonging to the Christian religion, limiting their choice of spouses among the same ethnicity. Another epitaph in Luoyang of a Nestorian Christian Sogdian woman also surnamed An was discovered and she was put in her tomb by her military officer son on 22 January, 815. This Sogdian woman's husband was surnamed He (和) and he was a Han Chinese man and the family was indicated to be multiethnic on the epitaph pillar. In Luoyang, the mixed raced sons of Nestorian Christian Sogdian women and Han Chinese men has many career paths available for them. Neither their mixed ethnicity nor their faith were barriers and they were able to become civil officials, a military officers and openly celebrated their Christian religion and support Christian monasteries.


title: Sogdia

text: During the Tang and subsequent Five Dynasties and Song Dynasty, a large community of Sogdians also existed in the multicultural "entrepôt" of Dunhuang, Gansu, a major center of Buddhist learning and home to the Buddhist Mogao Caves. Although Dunhuang and the Hexi Corridor were captured by the Tibetan Empire after the An Lushan Rebellion, in 848 the ethnic Han Chinese general Zhang Yichao (799–872) managed to wrestle control of the region from the Tibetans during their civil war, establishing the Guiyi Circuit under Emperor Xuānzong of Tang (r. 846–859). Although the region occasionally fell under the rule of different states, it retained its multilingual nature as evidenced by an abundance of manuscripts (religious and secular) in Chinese and Tibetan, but also Sogdian, Khotanese (another Eastern Iranian language native to the region), Uyghur, and Sanskrit.


title: Sogdia

text: There were nine prominent Sogdian clans (昭武九姓). The names of these clans have been deduced from the Chinese surnames listed in a Tang-era Dunhuang manuscript (Pelliot chinois 3319V). Each "clan" name refers to a different city-state as the Sogdian used the name of their hometown as their Chinese surname. Of these the most common Sogdian surname throughout China was Shí (石, generally given to those from Chach, modern Tashkent). The following surnames also appear frequently on Dunhuang manuscripts and registers: Shǐ (史, from Kesh, modern Shahrisabz), An (安, from Bukhara), Mi (米, from Panjakent), Kāng (康, from Samarkand), Cáo (曹, from Kabudhan, north of the Zeravshan River), and Hé (何, from Kushaniyah). Confucius is said to have expressed a desire to live among the "nine tribes" which may have been a reference to the Sogdian community.


title: Sogdia

text: The influence of Sinicized and multilingual Sogdians during this "Guiyijun" (歸義軍) period (c. 850 – c. 1000 AD) of Dunhuang is evident in a large number of manuscripts written in Chinese characters from left to right instead of vertically, mirroring the direction of how the Sogdian alphabet is read. Sogdians of Dunhuang also commonly formed and joined lay associations among their local communities, convening at Sogdian-owned taverns in scheduled meetings mentioned in their epistolary letters. Sogdians living in Turfan under the Tang dynasty and Gaochang Kingdom engaged in a variety of occupations that included: farming, military service, painting, leather crafting and selling products such as iron goods. The Sogdians had been migrating to Turfan since the 4th century, yet the pace of migration began to climb steadily with the Muslim conquest of Persia and Fall of the Sasanian Empire in 651, followed by the Islamic conquest of Samarkand in 712.


title: Sogdia

text: The 6th century is thought to be the peak of Sogdian culture, judging by its highly developed artistic tradition. By this point, the Sogdians were entrenched in their role as the central Asian traveling and trading merchants, transferring goods, culture and religion. During the Middle Ages, the valley of the Zarafshan around Samarkand retained its Sogdian name, Samarkand. According to the "Encyclopædia Britannica", medieval Arab geographers considered it one of the four fairest regions of the world. Where the Sogdians moved in considerable numbers, their language made a considerable impact. For instance, during China's Han dynasty, the native name of the Tarim Basin city-state of Loulan was "Kroraina," possibly from Greek due to nearby Hellenistic influence. However, centuries later in 664 AD, the Tang Chinese Buddhist monk Xuanzang labelled it as "Nafupo" (納縛溥), which according to Dr. Hisao Matsuda is a transliteration of the Sogdian word "Navapa" meaning "new water."


title: Sogdia

text: The Afrasiab paintings of the 6th to 7th centuries in Samarkand, Uzbekistan, offer a rare surviving example of Sogdian art. The paintings, showing scenes of daily life and events such as the arrival of foreign ambassadors, are located within the ruins of aristocratic homes. It is unclear if any of these palatial residences served as the official palace of the rulers of Samarkand. The oldest surviving Sogdian monumental wall murals date to the 5th century and are located at Panjakent, Tajikistan. In addition to revealing aspects of their social and political lives, Sogdian art has also been instrumental in aiding historians' understanding of their religious beliefs. For instance, it is clear that Buddhist Sogdians incorporated some of their own Iranian deities into their version of the Buddhist Pantheon. At Zhetysu, Sogdian gilded bronze plaques on a Buddhist temple show a pairing of a male and female deity with outstretched hands holding a miniature camel, a common non-Buddhist image similarly found in the paintings of Samarkand and Panjakent.


title: Sogdia

text: The Sogdians spoke an Eastern Iranian language called Sogdian, closely related to Bactrian, Khwarazmian, and the Khotanese Saka language, widely spoken Eastern Iranian languages of Central Asia in ancient times. Sogdian was also prominent in the oasis city-state of Turfan in the Tarim Basin region of Northwest China (in modern Xinjiang). Judging by the Sogdian Bugut inscription of Mongolia written c. 581, the Sogdian language was also an official language of the First Turkic Khaganate established by the Gokturks.


title: Sogdia

text: Sogdian was written largely in three scripts: the Sogdian alphabet, the Syriac alphabet, and the Manichaean alphabet, each derived from the Aramaic alphabet, which had been widely used in both the Achaemenid and Parthian empires of ancient Iran. The Sogdian alphabet formed the basis of the Old Uyghur alphabet of the 8th century, which in turn was used to create the Mongolian script of the early Mongol Empire during the 13th century. Later in 1599, the Jurchen leader Nurhaci decided to convert the Mongolian alphabet to make it suitable for the Manchu people.


title: Sogdia

text: The Yaghnobi people living in the Sughd province of Tajikistan still speak a descendant of the Sogdian language. Yaghnobi is largely a continuation of the medieval Sogdian dialect from the Osrushana region of the western Fergana Valley. The great majority of the Sogdian people assimilated with other local groups such as the Bactrians, Chorasmians, and in particular with Persians, and came to speak Persian. In 819, the Persians founded the Samanid Empire in the region. They are among the ancestors of the modern Tajiks. Numerous Sogdian cognates can be found in the modern Tajik language, although the latter is a Western Iranian language.


title: Sogdia

text: Early medieval Sogdian costumes can be divided in two periods: Hephtalitic (5th and 6th centuries) and Turkic (7th and early 8th centuries). The latter did not become common immediately after the political dominance of the Gökturks but only in c. 620 when, especially following Western Turkic Khagan Ton-jazbgu's reforms, Sogd was Turkized and the local nobility was officially included in the Khaganate's administration.


title: Sogdia

text: For both sexes clothes were tight-fitted, and narrow waists and wrists were appreciated. The silhouettes for grown men and young girls emphasized wide shoulders and narrowed to the waist; the silhouettes for female aristocrats were more complicated. The Sogdian clothing underwent a thorough process of Islamization in the ensuing centuries, with few of the original elements remaining. In their stead, turbans, kaftans, and sleeved coats became more common.


title: Sogdia

text: The Sogdians practiced a variety of religious faiths. However, Zoroastrianism was most likely their main religion, as demonstrated by material evidence, such as the discovery in Samarkand, Panjakent and Er-Kurgan of murals depicting votaries making offerings before fire altars and ossuaries holding the bones of the dead - in accordance with Zoroastrian ritual. At Turfan, Sogdian burials shared similar features with traditional Chinese practices, yet they still retained essential Zoroastrian rituals, such as allowing the bodies to be picked clean by scavengers before burying the bones in ossuaries. They also sacrificed animals to Zoroastrian deities, including the supreme deity Ahura Mazda. Zoroastrianism remained the dominant religion among Sogdians until after the Islamic conquest, when they gradually converted to Islam, as is shown by Richard Bulliet's "conversion curve".


title: Sogdia

text: The Sogdian religious texts found in China and dating to the Northern Dynasties, Sui, and Tang are mostly Buddhist (translated from Chinese sources), Manichaean, and Nestorian Christian, with only a small minority of Zoroastrian texts. But, tombs of Sogdian merchants in China dated to the last third of the 6th century show predominantly Zoroastrian motifs or Zoroastrian-Manichaean syncretism, while archaeological remains from Sogdiana appear fairly Iranian and conservatively Zoroastrian.


title: Sogdia

text: However, the Sogdians epitomized the religious plurality found along the trade routes. The largest body of Sogdian texts are Buddhist, and Sogdians were among the principal translators of Buddhist sutras into Chinese. However, Buddhism did not take root in Sogdiana itself. Additionally, the Bulayiq monastery to the north of Turpan contained Sogdian Christian texts, and there are numerous Manichaean texts in Sogdiana from nearby Qocho. The reconversion of Sogdians from Buddhism to Zoroastrianism coincided with the adoption of Zoroastrianism by the Sassanid Empire of Persia. From the 4th century onwards, Sogdian Buddhist pilgrims left behind evidence of their travels along the steep cliffs of the Indus River and Hunza Valley. It was here that they carved images of the Buddha and holy stupas in addition to their full names, in hopes that the Buddha would grant them his protection.


title: Sogdia

text: The Sogdians also practiced Manichaeism, the faith of Mani, which they spread among the Uyghurs. The Uyghur Khaganate (744–840 AD) developed close ties to Tang China once it had aided the Tang in suppressing the rebellion of An Lushan and his Göktürk successor Shi Siming, establishing an annual trade relationship of one million bolts of Chinese silk for one hundred thousand horses. The Uyghurs relied on Sogdian merchants to sell much of this silk further west along the Silk Road, a symbiotic relationship that led many Uyghurs to adopt Manichaeism from the Sogdians. However, evidence of Manichaean liturgical and canonical texts of Sogdian origin remains fragmentary and sparse compared to their corpus of Buddhist writings. The Uyghurs were also followers of Buddhism. For instance, they can be seen wearing silk robes in the "praṇidhi" scenes of the Uyghur Bezeklik Buddhist murals of Xinjiang, China, particularly Scene 6 from Temple 9 showing .


title: Sogdia

text: In addition to Puranic cults, there were five Hindu deities known to have been worshipped in Sogdiana. These were Brahma, Indra, Mahadeva (Shiva), Narayana, and Vaishravana; the gods Brahma, Indra, and Shiva were known by their Sogdian names Zravan, Adbad and Veshparkar, respectively. Durga, a mother goddess in Shaktism, may be represented in Sogdian art as a four-armed goddess riding atop a lion. As seen in an 8th-century mural from Panjakent, portable fire altars can be "associated" with Mahadeva-Veshparkar, Brahma-Zravan, and Indra-Abdab, according to Braja Bihārī Kumar.


title: Sogdia

text: Among the Sogdian Christians known in China from inscriptions and texts were An Yena, a Christian from An country (Bukhara). Mi Jifen a Christian from Mi country (Maymurgh), Kang Zhitong, a Sogdian Christian cleric from Kang country (Samarkand), Mi Xuanqing a Sogdian Christian cleric from Mi country (Maymurgh), Mi Xuanying, a Sogdian Christian cleric from Mi country (Maymurgh), An Qingsu, a Sogdian Christian monk from An country (Bukhara).


title: Sogdia

text: When visiting Yuan-era Zhenjiang, Jiangsu, China during the late 13th century, the Venetian explorer and merchant Marco Polo noted that a large number of Christian churches had been built there. His claim is confirmed by a Chinese text of the 14th century explaining how a Sogdian named Mar-Sargis from Samarkand founded six Nestorian Christian churches there, in addition to one in Hangzhou during the second half of the 13th century. Nestorian Christianity had existed in China earlier during the Tang Dynasty when a Persian monk named Alopen came to Chang'an in 653 to proselytize, as described in a dual Chinese and Syriac language inscription from Chang'an (modern Xi'an), dated to the year 781. Within the Syriac inscription is a list of priests and monks, one of whom is named Gabriel, the archdeacon of "Xumdan" and "Sarag", the Sogdian names for the Chinese capital cities Chang'an and Luoyang, respectively. In regards to textual material, the earliest Christian gospel texts translated into Sogdian coincide with the reign of the Sasanian Persian monarch Yazdegerd II (r. 438–457), and were translated from the "Peshitta", the standard version of the Bible in Syriac Christianity.


title: Sogdia

text: Slavery existed in China since ancient times, although during the Han dynasty the proportion of slaves to the overall population was roughly 1%, far lower than the estimate for the contemporary Greco-Roman world (estimated at about 15% of the entire population). During the Tang period, slaves were not allowed to marry a commoner's daughter, were not allowed to have sexual relations with any female member of their master's family, and although fornication with female slaves was forbidden in the Tang code of law, it was widely practiced. Manumission was also permitted when a slave woman gave birth to her master's son, which allowed for her elevation to the legal status of a commoner, yet she could only live as a concubine and not as the wife of her former master.


title: Sogdia

text: Sogdian and Chinese merchants regularly traded in slaves in and around Turpan during the Tang dynasty. Turpan under Tang dynasty rule was a center of major commercial activity between Chinese and Sogdian merchants. There were many inns in Turpan. Some provided Sogdian sex workers with an opportunity to service the Silk Road merchants, since the official histories report that there were markets in women at Kucha and Khotan. The Sogdian-language contract buried at the Astana graveyard demonstrates that at least one Chinese man bought a Sogdian girl in 639 AD. One of the archaeologists who excavated the Astana site, Wu Zhen, contends that, although many households along the Silk Road bought individual slaves, as demonstrated in the earlier documents from Niya, the Turpan documents point to a massive escalation in the volume of the slave trade. In 639 a female Sogdian slave was sold to a Chinese man, as recorded in an Astana cemetery legal document written in Sogdian. Khotan and Kucha were places where women were commonly sold, with ample evidence of the slave trade in Turfan thanks to contemporary textual sources that have survived. In Tang poetry Sogdian girls also frequently appear as serving maids in the taverns and inns of the capital Chang'an.


title: Sogdia

text: Sogdian slave girls and their Chinese male owners made up the majority of Sogdian female-Chinese male pairings, while free Sogdian women were the most common spouse of Sogdian men. A smaller number of Chinese women were paired with elite Sogdian men. Sogdian man-and-woman pairings made up eighteen out of twenty-one marriages according to existing documents.


title: Sogdia

text: A document dated 731 AD reveals that precisely forty bolts of silk were paid to a certain Mi Lushan, a slave dealing Sogdian, by a Chinese man named Tang Rong (唐榮) of Chang'an, for the purchase of an eleven-year-old girl. A person from Xizhou, a Tokharistani (i.e. Bactrian), and three Sogdians verified the sale of the girl.


title: Sogdia

text: Central Asians like Sogdians were called "Hu" (胡) by the Chinese during the Tang dynasty. Central Asian "Hu" women were stereotyped as barmaids or dancers by Han in China. Han Chinese men engaged in mostly extra-marital sexual relationships with them as the "Hu" women in China mostly occupied positions where sexual services were sold to patrons like singers, maids, slaves and prostitutes. Han men did not want to legally marry them unless they had no choice such as if they were on the frontier or in exile since the Han men would be socially disadvantaged and have to marry non-Han. The task of taking care of herd animals like sheep and cattle was given to "Hu" slaves in China.


title: Sogdia

text: In 1916, the French Sinologist and historian Paul Pelliot used Tang Chinese manuscripts excavated from Dunhuang, Gansu to identify an ancient Sogdian colony south of Lop Nur in Xinjiang (Northwest China), which he argued was the base for the spread of Buddhism and Nestorian Christianity in China. In 1926, Japanese scholar Kuwabara compiled evidence for Sogdians in Chinese historical sources, and by 1933, Chinese historian Xiang Da published his "Tang Chang'an and Central Asian Culture", detailing the Sogdian influence on Chinese social religious life in the Tang-era Chinese capital city. The Canadian Sinologist Edwin G. Pulleyblank published an article in 1952, demonstrating the presence of a Sogdian colony founded in Six Hu Prefectures of the Ordos Loop during the Chinese Tang period, composed of Sogdians and Turkic peoples who migrated from the Mongolian steppe. The Japanese historian Ikeda on wrote an article in 1965, outlining the history of the Sogdians inhabiting Dunhuang from the beginning of the 7th century, analyzing lists of their Sinicized names and the role of Zoroastrianism and Buddhism in their religious life. Yoshida Yutaka and Kageyama Etsuko, Japanese ethnographers and linguists of the Sogdian language, were able to reconstruct Sogdian names from forty-five different Chinese transliterations, noting that these were common in Turfan whereas Sogdians living closer to the center of Chinese civilization for generations adopted traditional Chinese names.


title: Series (mathematics)

text: In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.


title: Series (mathematics)

text: For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. Zeno concluded that Achilles could "never" reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.


title: Series (mathematics)

text: In modern terminology, any (ordered) infinite sequence formula_1 of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like


title: Series (mathematics)

text: The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as tends to infinity (if the limit exists) of the finite sums of the first terms of the series, which are called the th partial sums of the series. That is,


title: Series (mathematics)

text: When this limit exists, one says that the series is convergent or summable, or that the sequence formula_1 is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent.


title: Series (mathematics)

text: The notation formula_6 denotes both the series—that is the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generalization of the similar convention of denoting by formula_7 both the addition—the process of adding—and its result—the "sum" of and .


title: Series (mathematics)

text: Generally, the terms of a series come from a ring, often the field formula_8 of the real numbers or the field formula_9 of the complex numbers. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product.


title: Series (mathematics)

text: An infinite series or simply a series is an infinite sum, represented by an infinite expression of the form


title: Series (mathematics)

text: where formula_11 is any ordered sequence of terms, such as numbers, functions, or anything else that can be added (an abelian group). This is an expression that is obtained from the list of terms formula_12 by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as


title: Series (mathematics)

text: If an abelian group of terms has a concept of limit (e.g., if it is a metric space), then some series, the convergent series, can be interpreted as having a value in , called the "sum of the series". This includes the common cases from calculus, in which the group is the field of real numbers or the field of complex numbers. Given a series formula_14, its th partial sum is


title: Series (mathematics)

text: By definition, the series formula_16 "converges" to the limit (or simply "sums" to ), if the sequence of its partial sums has a limit . In this case, one usually writes


title: Series (mathematics)

text: A series is said to be "convergent" if it converges to some limit, or "divergent" when it does not. The value of this limit, if it exists, is then the value of the series.


title: Series (mathematics)

text: A series is said to converge or to "be convergent" when the sequence of partial sums has a finite limit. If the limit of is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the value (or sum) of the series


title: Series (mathematics)

text: An easy way that an infinite series can converge is if all the are zero for sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.


title: Series (mathematics)

text: Working out the properties of the series that converge, even if infinitely many terms are nonzero, is the essence of the study of series. Consider the example


title: Series (mathematics)

text: It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is "equal" to 2 (although it is), but it does prove that it is "at most" 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted , it can be seen that


title: Series (mathematics)

text: The idiom can be extended to other, equivalent notions of series. For instance, a recurring decimal, as in


title: Series (mathematics)

text: Since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, the decimal expansion 0.111... can be identified with 1/9. This leads to an argument that , which only relies on the fact that the limit laws for series preserve the arithmetic operations; for more detail on this argument, see 0.999...


title: Series (mathematics)

text: Partial summation takes as input a sequence, ("a"), and gives as output another sequence, ("S"). It is thus a unary operation on sequences. Further, this function is linear, and thus is a linear operator on the vector space of sequences, denoted Σ. The inverse operator is the finite difference operator, denoted Δ. These behave as discrete analogues of integration and differentiation, only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence (1, 1, 1, ...) has series (1, 2, 3, 4, ...) as its partial summation, which is analogous to the fact that formula_52


title: Series (mathematics)

text: Series are classified not only by whether they converge or diverge, but also by the properties of the terms a (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term a (whether it is a real number, arithmetic progression, trigonometric function); etc.


title: Series (mathematics)

text: When "a" is a non-negative real number for every "n", the sequence "S" of partial sums is non-decreasing. It follows that a series Σ"a" with non-negative terms converges if and only if the sequence "S" of partial sums is bounded.


title: Series (mathematics)

text: and a telescopic sum argument implies that the partial sums are bounded by 2. The exact value of the original series is the Basel problem.


title: Series (mathematics)

text: When you group a series reordering of the series does not happen, so Riemann series theorem does not apply. A new series will have its partial sums as subsequence of original series, which means if the original series converges, so does the new series. But for divergent series that is not true, for example 1-1+1-1+... grouped every two elements will create 0+0+0+... series, which is convergent. On the other hand, divergence of the new series means the original series can be only divergent which is sometimes useful, like in Oresme proof.


title: Series (mathematics)

text: converges. This is sufficient to guarantee not only that the original series converges to a limit, but also that any reordering of it converges to the same limit.


title: Series (mathematics)

text: A series of real or complex numbers is said to be conditionally convergent (or semi-convergent) if it is convergent but not absolutely convergent. A famous example is the alternating series


title: Series (mathematics)

text: which is convergent (and its sum is equal to formula_58), but the series formed by taking the absolute value of each term is the divergent harmonic series. The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the formula_59 are real and formula_60 is any real number, that one can find a reordering so that the reordered series converges with sum equal to formula_60.


title: Series (mathematics)

text: where the partial sums formula_63 are bounded, formula_64 has bounded variation, and formula_65 exists:


title: Series (mathematics)

text: then the series formula_67 is convergent. This applies to the point-wise convergence of many trigonometric series, as in


title: Series (mathematics)

text: with formula_69. Abel's method consists in writing formula_70, and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series formula_67 to the absolutely convergent series


title: Series (mathematics)

text: The evaluation of truncation errors is an important procedure in numerical analysis (especially validated numerics and computer-assisted proof).


title: Series (mathematics)

text: When conditions of the alternating series test are satisfied by formula_73, there is an exact error evaluation. Set formula_74 to be the partial sum formula_75 of the given alternating series formula_60. Then the next inequality holds:


title: Series (mathematics)

text: Taylor's theorem is a statement that includes the evaluation of the error term when the Taylor series is truncated.


title: Series (mathematics)

text: By using the ratio, we can obtain the evaluation of the error term when the hypergeometric series is truncated.


title: Series (mathematics)

text: converges pointwise on a set "E", if the series converges for each "x" in "E" as an ordinary series of real or complex numbers. Equivalently, the partial sums


title: Series (mathematics)

text: A stronger notion of convergence of a series of functions is the uniform convergence. A series converges uniformly if it converges pointwise to the function "ƒ"("x"), and the error in approximating the limit by the "N"th partial sum,


title: Series (mathematics)

text: Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if the "ƒ" are integrable on a closed and bounded interval "I" and converge uniformly, then the series is also integrable on "I" and can be integrated term-by-term. Tests for uniform convergence include the Weierstrass' M-test, Abel's uniform convergence test, Dini's test, and the Cauchy criterion.


title: Series (mathematics)

text: More sophisticated types of convergence of a series of functions can also be defined. In measure theory, for instance, a series of functions converges almost everywhere if it converges pointwise except on a certain set of measure zero. Other modes of convergence depend on a different metric space structure on the space of functions under consideration. For instance, a series of functions converges in mean on a set "E" to a limit function "ƒ" provided


title: Series (mathematics)

text: The Taylor series at a point "c" of a function is a power series that, in many cases, converges to the function in a neighborhood of "c". For example, the series


title: Series (mathematics)

text: Unless it converges only at "x"="c", such a series converges on a certain open disc of convergence centered at the point "c" in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this disc is known as the radius of convergence, and can in principle be determined from the asymptotics of the coefficients "a". The convergence is uniform on closed and bounded (that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets.


title: Series (mathematics)

text: Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.


title: Series (mathematics)

text: While many uses of power series refer to their sums, it is also possible to treat power series as "formal sums", meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself is of interest, rather than the convergence of the series. Formal power series are used in combinatorics to describe and study sequences that are otherwise difficult to handle, for example, using the method of generating functions. The Hilbert–Poincaré series is a formal power series used to study graded algebras.


title: Series (mathematics)

text: Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such as addition, multiplication, derivative, antiderivative for power series "formally", treating the symbol "+" as if it corresponded to addition. In the most common setting, the terms come from a commutative ring, so that the formal power series can be added term-by-term and multiplied via the Cauchy product. In this case the algebra of formal power series is the total algebra of the monoid of natural numbers over the underlying term ring. If the underlying term ring is a differential algebra, then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.


title: Series (mathematics)

text: Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents. A Laurent series is thus any series of the form


title: Series (mathematics)

text: If such a series converges, then in general it does so in an annulus rather than a disc, and possibly some boundary points. The series converges uniformly on compact subsets of the interior of the annulus of convergence.


title: Series (mathematics)

text: where "s" is a complex number. For example, if all "a" are equal to 1, then the Dirichlet series is the Riemann zeta function


title: Series (mathematics)

text: Like the zeta function, Dirichlet series in general play an important role in analytic number theory. Generally a Dirichlet series converges if the real part of "s" is greater than a number called the abscissa of convergence. In many cases, a Dirichlet series can be extended to an analytic function outside the domain of convergence by analytic continuation. For example, the Dirichlet series for the zeta function converges absolutely when Re("s") > 1, but the zeta function can be extended to a holomorphic function defined on formula_134 with a simple pole at 1.


title: Series (mathematics)

text: A series of functions in which the terms are trigonometric functions is called a trigonometric series:


title: Series (mathematics)

text: method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of π.


title: Series (mathematics)

text: In the 17th century, James Gregory worked in the new decimal system on infinite series and published several Maclaurin series. In 1715, a general method for constructing the Taylor series for all functions for which they exist was provided by Brook Taylor. Leonhard Euler in the 18th century, developed the theory of hypergeometric series and q-series.


title: Series (mathematics)

text: The investigation of the validity of infinite series is considered to begin with Gauss in the 19th century. Euler had already considered the hypergeometric series


title: Series (mathematics)

text: on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.


title: Series (mathematics)

text: Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms "convergence" and "divergence" had been introduced long before by Gregory (1668). Leonhard Euler and Gauss had given various criteria, and Colin Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form.


title: Series (mathematics)

text: corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of formula_138 and formula_139. He showed the necessity of considering the subject of continuity in questions of convergence.


title: Series (mathematics)

text: the same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose


title: Series (mathematics)

text: (1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Chebyshev (1852), and Arndt


title: Series (mathematics)

text: General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various


title: Series (mathematics)

text: DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.


title: Series (mathematics)

text: The theory of uniform convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it


title: Series (mathematics)

text: doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform


title: Series (mathematics)

text: A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not absolutely convergent.


title: Series (mathematics)

text: Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlömilch ("Zeitschrift", Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function


title: Series (mathematics)

text: Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into


title: Series (mathematics)

text: multiples of "x", a problem which he embodied in his "Théorie analytique de la chaleur" (1822). Euler had already given the formulas for determining the coefficients in the series;


title: Series (mathematics)

text: scientific manner (see convergence of Fourier series). Dirichlet's treatment ("Crelle", 1829), of trigonometric series was the subject of criticism and improvement by


title: Series (mathematics)

text: Asymptotic series, otherwise asymptotic expansions, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge, but they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers.


title: Series (mathematics)

text: Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Summability methods include Cesàro summation, ("C","k") summation, Abel summation, and Borel summation, in increasing order of generality (and hence applicable to increasingly divergent series).


title: Series (mathematics)

text: A variety of general results concerning possible summability methods are known. The Silverman–Toeplitz theorem characterizes "matrix summability methods", which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general method for summing a divergent series is non-constructive, and concerns Banach limits.


title: Series (mathematics)

text: Definitions may be given for sums over an arbitrary index set formula_141 There are two main differences with the usual notion of series: first, there is no specific order given on the set formula_142; second, this set formula_142 may be uncountable. The notion of convergence needs to be strengthened, because the concept of conditional convergence depends on the ordering of the index set.


title: Series (mathematics)

text: If formula_144 is a function from an index set formula_142 to a set formula_146 then the "series" associated to formula_147 is the formal sum of the elements formula_148 over the index elements formula_149 denoted by the


title: Series (mathematics)

text: When the index set is the natural numbers formula_151 the function formula_152 is a sequence denoted by formula_153 A series indexed on the natural numbers is an ordered formal sum and so we rewrite formula_154 as formula_155 in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers


title: Series (mathematics)

text: When the supremum is finite then the set of formula_159 such that formula_160 is countable. Indeed, for every formula_161 the cardinality formula_162 of the set formula_163 is finite because


title: Series (mathematics)

text: If formula_142 is countably infinite and enumerated as formula_166 then the above defined sum satisfies


title: Series (mathematics)

text: Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the counting measure, which accounts for the many similarities between the two constructions.


title: Series (mathematics)

text: Let formula_169 be a map, also denoted by formula_170 from some non-empty set formula_142 into a Hausdorff abelian topological group formula_172


title: Series (mathematics)

text: Let formula_173 be the collection of all finite subsets of formula_174 with formula_173 viewed as a directed set, ordered under inclusion formula_176 with union as join.


title: Series (mathematics)

text: The family formula_170 is said to be if the following limit, which is denoted by formula_178 and is called the of formula_170 exists in formula_180


title: Series (mathematics)

text: Saying that the sum formula_182 is the limit of finite partial sums means that for every neighborhood formula_183 of the origin in formula_184 there exists a finite subset formula_185 of formula_142 such that


title: Series (mathematics)

text: Because formula_173 is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net.


title: Series (mathematics)

text: For every neighborhood formula_189 of the origin in formula_184 there is a smaller neighborhood formula_183 such that formula_192 It follows that the finite partial sums of an unconditionally summable family formula_170 form a , that is, for every neighborhood formula_189 of the origin in formula_184 there exists a finite subset formula_185 of formula_142 such that


title: Series (mathematics)

text: When formula_203 is complete, a family formula_204 is unconditionally summable in formula_203 if and only if the finite sums satisfy the latter Cauchy net condition. When formula_203 is complete and formula_170 is unconditionally summable in formula_184 then for every subset formula_209 the corresponding subfamily formula_210 is also unconditionally summable in formula_172


title: Series (mathematics)

text: When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group formula_212


title: Series (mathematics)

text: If a family formula_204 in formula_203 is unconditionally summable then for every neighborhood formula_189 of the origin in formula_184 there is a finite subset formula_217 such that formula_199 for every index formula_219 not in formula_220 If formula_203 is a first-countable space then it follows that the set of formula_159 such that formula_223 is countable. This need not be true in a general abelian topological group (see examples below).


title: Series (mathematics)

text: Suppose that formula_224 If a family formula_225 is unconditionally summable in a Hausdorff abelian topological group formula_184 then the series in the usual sense converges and has the same sum,


title: Series (mathematics)

text: By nature, the definition of unconditional summability is insensitive to the order of the summation. When formula_228 is unconditionally summable, then the series remains convergent after any permutation formula_229 of the set formula_230 of indices, with the same sum,


title: Series (mathematics)

text: Conversely, if every permutation of a series formula_228 converges, then the series is unconditionally convergent. When formula_203 is complete then unconditional convergence is also equivalent to the fact that all subseries are convergent; if formula_203 is a Banach space, this is equivalent to say that for every sequence of signs formula_235, the series


title: Series (mathematics)

text: If formula_203 is a topological vector space (TVS) and formula_239 is a (possibly uncountable) family in formula_203 then this family is summable if the limit formula_241 of the net formula_242 exists in formula_184 where formula_173 is the directed set of all finite subsets of formula_142 directed by inclusion formula_176 and formula_247


title: Series (mathematics)

text: It is called absolutely summable if in addition, for every continuous seminorm formula_248 on formula_184 the family formula_250 is summable.


title: Series (mathematics)

text: If formula_203 is a normable space and if formula_239 is an absolutely summable family in formula_184 then necessarily all but a countable collection of formula_254’s are zero. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms.


title: Series (mathematics)

text: If formula_255 is a sequence of elements of a normed space formula_203 and if formula_257 then the series formula_258 converges to formula_139 in formula_203 if the sequence of partial sums of the series formula_261 converges to formula_139 in formula_203; to wit,


title: Series (mathematics)

text: Specifically, in this case, formula_258 converges to formula_139 if the sequence of partial sums converges to formula_267


title: Series (mathematics)

text: If a countable series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem of ).