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According to the International Organization for Migration, Nigeria witnessed a dramatic increase in remittances sent home from overseas Nigerians, going from USD 2.3 billion in 2004 to 17.9 billion in 2007. The United States accounts for the largest portion of official remittances, followed by the United Kingdom, Italy, Canada, Spain and France. On the African continent, Egypt, Equatorial Guinea, Chad, Libya and South Africa are important source countries of remittance flows to Nigeria, while China is the biggest remittance-sending country in Asia.
Nigeria in recent years has been embracing industrialisation. It currently has an indigenous vehicle manufacturing company, Innoson Motors, which manufactures Rapid Transit Buses, Trucks and SUVs with an upcoming introduction of Cars. Nigeria also has few Electronic manufacturers like Zinox, the first Branded Nigerian Computer and Electronic gadgets (like tablet PCs) manufacturers. In 2013, Nigeria introduced a policy regarding import duty on vehicles to encourage local manufacturing companies in the country. In this regard, some foreign vehicle manufacturing companies like Nissan have made known their plans to have manufacturing plants in Nigeria. Ogun is considered to be the current Nigeria's industrial hub, as most factories are located in Ogun and more companies are moving there, followed by Lagos.
The Nigerian government has commissioned the overseas production and launch of four satellites. The Nigeriasat-1 was the first satellite to be built under the Nigerian government sponsorship. The satellite was launched from Russia on 27 September 2003. Nigeriasat-1 was part of the world-wide Disaster Monitoring Constellation System. The primary objectives of the Nigeriasat-1 were: to give early warning signals of environmental disaster; to help detect and control desertification in the northern part of Nigeria; to assist in demographic planning; to establish the relationship between malaria vectors and the environment that breeds malaria and to give early warning signals on future outbreaks of meningitis using remote sensing technology; to provide the technology needed to bring education to all parts of the country through distant learning; and to aid in conflict resolution and border disputes by mapping out state and International borders.
NigeriaSat-2, Nigeria's second satellite, was built as a high-resolution earth satellite by Surrey Space Technology Limited, a United Kingdom-based satellite technology company. It has 2.5-metre resolution panchromatic (very high resolution), 5-metre multispectral (high resolution, NIR red, green and red bands), and 32-metre multispectral (medium resolution, NIR red, green and red bands) antennas, with a ground receiving station in Abuja. The NigeriaSat-2 spacecraft alone was built at a cost of over £35 million. This satellite was launched into orbit from a military base in China.
NigComSat-1, a Nigerian satellite built in 2004, was Nigeria's third satellite and Africa's first communication satellite. It was launched on 13 May 2007, aboard a Chinese Long March 3B carrier rocket, from the Xichang Satellite Launch Centre in China. The spacecraft was operated by NigComSat and the Nigerian Space Agency, NASRDA. On 11 November 2008, NigComSat-1 failed in orbit after running out of power because of an anomaly in its solar array. It was based on the Chinese DFH-4 satellite bus, and carries a variety of transponders: 4 C-band; 14 Ku-band; 8 Ka-band; and 2 L-band. It was designed to provide coverage to many parts of Africa, and the Ka-band transponders would also cover Italy.
On 24 March 2009, the Nigerian Federal Ministry of Science and Technology, NigComSat Ltd. and CGWIC signed another contract for the in-orbit delivery of the NigComSat-1R satellite. NigComSat-1R was also a DFH-4 satellite, and the replacement for the failed NigComSat-1 was successfully launched into orbit by China in Xichang on December 19, 2011. The satellite according to then-Nigerian President Goodluck Jonathan which was paid for by the insurance policy on NigComSat-1 which de-orbited in 2009, would have a positive impact on national development in various sectors such as communications, internet services, health, agriculture, environmental protection and national security.
The United Nations estimates that the population in 2009 was at 154,729,000, distributed as 51.7% rural and 48.3% urban, and with a population density of 167.5 people per square kilometre. National census results in the past few decades have been disputed. The results of the most recent census were released in December 2006 and gave a population of 140,003,542. The only breakdown available was by gender: males numbered 71,709,859, females numbered 68,293,08. On June 2012, President Goodluck Jonathan said that Nigerians should limit their number of children.
Even though most ethnic groups prefer to communicate in their own languages, English as the official language is widely used for education, business transactions and for official purposes. English as a first language is used only by a small minority of the country's urban elite, and it is not spoken at all in some rural areas. Hausa is the most widely spoken of the 3 main languages spoken in Nigeria itself (Igbo, Hausa and Yoruba) but unlike the Yorubas and Igbos, the Hausas tend not to travel far outside Nigeria itself.[citation needed]
With the majority of Nigeria's populace in the rural areas, the major languages of communication in the country remain indigenous languages. Some of the largest of these, notably Yoruba and Igbo, have derived standardised languages from a number of different dialects and are widely spoken by those ethnic groups. Nigerian Pidgin English, often known simply as 'Pidgin' or 'Broken' (Broken English), is also a popular lingua franca, though with varying regional influences on dialect and slang. The pidgin English or Nigerian English is widely spoken within the Niger Delta Regions, predominately in Warri, Sapele, Port Harcourt, Agenebode, Ewu, and Benin City.
Nigeria is a religiously diverse society, with Islam and Christianity being the most widely professed religions. Nigerians are nearly equally divided into Christians and Muslims, with a tiny minority of adherents of Animism and other religions. According to one recent estimate, over 40% of Nigeria's population adheres to Islam (mainly Sunni, other branches are also present). Christianity is practised by 58% of the population (among them 74% are Protestant, 25% Roman Catholic, 1% other Christian). Adherents of Animism and other religions collectively represent 1.4% of the population.
The vast majority of Muslims in Nigeria are Sunni belonging to Maliki school of jurisprudence; however, a sizeable minority also belongs to Shafi madhhab. A large number of Sunni Muslims are members of Sufi brotherhoods. Most Sufis follow the Qadiriyya, Tijaniyyah and/or the Mouride movements. A significant Shia minority exists (see Shia in Nigeria). Some northern states have incorporated Sharia law into their previously secular legal systems, which has brought about some controversy. Kano State has sought to incorporate Sharia law into its constitution. The majority of Quranists follow the Kalo Kato or Quraniyyun movement. There are also Ahmadiyya and Mahdiyya minorities.
According to a 2001 report from The World Factbook by CIA, about 50% of Nigeria's population is Muslim, 40% are Christians and 10% adhere to local religions. But in some recent report, the Christian population is now sightly larger than the Muslim population. An 18 December 2012 report on religion and public life by the Pew Research Center stated that in 2010, 49.3 percent of Nigeria's population was Christian, 48.8 percent was Muslim, and 1.9 percent were followers of indigenous and other religions, or unaffiliated. Additionally, the 2010s census of Association of Religion Data Archives has reported that 46.5 percent of the total population is Christian, slightly bigger than the Muslim population of 45.5 percent, and that 7.7 percent are members of other religious groups.
Among Christians, the Pew Research survey found that 74% were Protestant, 25% were Catholic, and 1% belonged to other Christian denominations, including a small Orthodox Christian community. In terms of Nigeria's major ethnic groups, the Hausa ethnic group (predominant in the north) was found to be 95% Muslim and 5% Christian, the Yoruba tribe (predominant in the west) was 55% Muslim, 35% Christian and 10% adherents of other religions, while the Igbos (predominant in the east) and the Ijaw (south) were 98% Christian, with 2% practising traditional religions. The middle belt of Nigeria contains the largest number of minority ethnic groups in Nigeria, who were found to be mostly Christians and members of traditional religions, with a small proportion of Muslims.
Leading Protestant churches in the country include the Church of Nigeria of the Anglican Communion, the Assemblies of God Church, the Nigerian Baptist Convention and The Synagogue, Church Of All Nations Since the 1990s, there has been significant growth in many other churches, particularly the evangelical Protestant ones. These include the Redeemed Christian Church of God, Winners' Chapel, Christ Apostolic Church (the first Aladura Movement in Nigeria), Deeper Christian Life Ministry, Evangelical Church of West Africa, Mountain of Fire and Miracles, Christ Embassy and The Synagogue Church Of All Nations. In addition, The Church of Jesus Christ of Latter-day Saints, the Aladura Church, the Seventh-day Adventist and various indigenous churches have also experienced growth.
Health care delivery in Nigeria is a concurrent responsibility of the three tiers of government in the country, and the private sector. Nigeria has been reorganising its health system since the Bamako Initiative of 1987, which formally promoted community-based methods of increasing accessibility of drugs and health care services to the population, in part by implementing user fees. The new strategy dramatically increased accessibility through community-based healthcare reform, resulting in more efficient and equitable provision of services. A comprehensive approach strategy was extended to all areas of health care, with subsequent improvement in the health care indicators and improvement in health care efficiency and cost.
HIV/AIDS rate in Nigeria is much lower compared to the other African nations such as Kenya or South Africa whose prevalence (percentage) rates are in the double digits. As of 2012[update], the HIV prevalence rate among adults ages 15–49 was just 3.1 percent. As of 2014[update], Life expectancy in Nigeria is 52.62 years on average according to CIA, and just over half the population have access to potable water and appropriate sanitation; As of 2010[update], the Infant mortality is 8.4 deaths per 1000 live births.
Nigeria was the only country in Africa to have never eradicated polio, which it periodically exported to other African countries; Polio was cut 98% between 2009 and 2010. However, a major breakthrough came in December 2014, when it was reported that Nigeria hadn't recorded a polio case in 6 months, and on its way to be declared Polio free. In 2012, a new bone marrow donor program was launched by the University of Nigeria to help people with leukaemia, lymphoma, or sickle cell disease to find a compatible donor for a life-saving bone marrow transplant, which cures them of their conditions. Nigeria became the second African country to have successfully carried out this surgery. In the 2014 ebola outbreak, Nigeria was the first country to effectively contain and eliminate the Ebola threat that was ravaging three other countries in the West African region, the Nigerian unique method of contact tracing employed by Nigeria became an effective method later used by countries, such as the united States, when ebola threats were discovered.
Education in Nigeria is overseen by the Ministry of Education. Local authorities take responsibility for implementing policy for state-controlled public education and state schools at a regional level. The education system is divided into Kindergarten, primary education, secondary education and tertiary education. After the 1970s oil boom, tertiary education was improved so that it would reach every subregion of Nigeria. 68% of the Nigerian population is literate, and the rate for men (75.7%) is higher than that for women (60.6%).
Nigeria is home to a substantial network of organised crime, active especially in drug trafficking. Nigerian criminal groups are heavily involved in drug trafficking, shipping heroin from Asian countries to Europe and America; and cocaine from South America to Europe and South Africa. . The various Nigerian Confraternities or "campus cults" are active in both organised crime and in political violence as well as providing a network of corruption within Nigeria. As confraternities have extensive connections with political and military figures, they offer excellent alumni networking opportunities. The Supreme Vikings Confraternity, for example, boasts that twelve members of the Rivers State House of Assembly are cult members. On lower levels of society, there are the "area boys", organised gangs mostly active in Lagos who specialise in mugging and small-scale drug dealing. According to official statistics, gang violence in Lagos resulted in 273 civilians and 84 policemen killed in the period of August 2000 to May 2001.
Internationally, Nigeria is infamous for a form of bank fraud dubbed 419, a type of advance fee fraud (named after Section 419 of the Nigerian Penal Code) along with the "Nigerian scam", a form of confidence trick practised by individuals and criminal syndicates. These scams involve a complicit Nigerian bank (the laws being set up loosely to allow it) and a scammer who claims to have money he needs to obtain from that bank. The victim is talked into exchanging bank account information on the premise that the money will be transferred to him, and then he'll get to keep a cut. In reality, money is taken out instead, and/or large fees (which seem small in comparison with the imaginary wealth he awaits) are deducted. In 2003, the Nigerian Economic and Financial Crimes Commission (or EFCC) was created, ostensibly to combat this and other forms of organised financial crime.
Nigeria has also been pervaded by political corruption. It was ranked 143 out of 182 countries in Transparency International's 2011 Corruption Perceptions Index; however, it improved to 136th position in 2014. More than $400 billion were stolen from the treasury by Nigeria's leaders between 1960 and 1999. In late 2013, Nigeria's then central bank governor Lamido Sanusi informed President Goodluck Jonathan that the state oil company, NNPC had failed to remit US$20 billion of oil revenues, which it owed the state. Jonathan however dismissed the claim and replaced Sanusi for his mismanagement of the central bank's budget. A Senate committee also found Sanusi’s account to be lacking substance. After the conclusion of the NNPC's account Audit, it was announced in January 2015 that NNPC's non-remitted revenue is actually US$1.48billion, which it needs to refund back to the Government.
The Nigerian film industry is known as Nollywood (a portmanteau of Nigeria and Hollywood) and is now the 2nd-largest producer of movies in the world. Nigerian film studios are based in Lagos, Kano and Enugu, forming a major portion of the local economy of these cities. Nigerian cinema is Africa's largest movie industry in terms of both value and the number of movies produced per year. Although Nigerian films have been produced since the 1960s, the country's film industry has been aided by the rise of affordable digital filming and editing technologies.
Football is largely considered the Nigeria's national sport and the country has its own Premier League of football. Nigeria's national football team, known as the "Super Eagles", has made the World Cup on five occasions 1994, 1998, 2002, 2010, and most recently in 2014. In April 1994, the Super Eagles ranked 5th in the FIFA World Rankings, the highest ranking achieved by an African football team. They won the African Cup of Nations in 1980, 1994, and 2013, and have also hosted the U-17 & U-20 World Cup. They won the gold medal for football in the 1996 Summer Olympics (in which they beat Argentina) becoming the first African football team to win gold in Olympic Football.
The nation's cadet team from Japan '93 produced some international players notably Nwankwo Kanu, a two-time African Footballer of the year who won the European Champions League with Ajax Amsterdam and later played with Inter Milan, Arsenal, West Bromwich Albion and Portsmouth. Other players that graduated from the junior teams are Nduka Ugbade, Jonathan Akpoborie, Victor Ikpeba, Celestine Babayaro, Wilson Oruma and Taye Taiwo. Some other famous Nigerian footballers include John Obi Mikel, Obafemi Martins, Vincent Enyeama, Yakubu Aiyegbeni, Rashidi Yekini, Peter Odemwingie and Jay-Jay Okocha.
Nigeria's human rights record remains poor; According to the US Department of State, the most significant human rights problems are: use of excessive force by security forces; impunity for abuses by security forces; arbitrary arrests; prolonged pretrial detention; judicial corruption and executive influence on the judiciary; rape, torture and other cruel, inhuman or degrading treatment of prisoners, detainees and suspects; harsh and life‑threatening prison and detention centre conditions; human trafficking for the purpose of prostitution and forced labour; societal violence and vigilante killings; child labour, child abuse and child sexual exploitation; female genital mutilation (FGM); domestic violence; discrimination based on sex, ethnicity, region and religion.
Although there is some evidence of earlier inhabitation in the region of Utrecht, dating back to the Stone Age (app. 2200 BCE) and settling in the Bronze Age (app. 1800–800 BCE), the founding date of the city is usually related to the construction of a Roman fortification (castellum), probably built in around 50 CE. A series of such fortresses was built after the Roman emperor Claudius decided the empire should not expand north. To consolidate the border the limes Germanicus defense line was constructed along the main branch of the river Rhine, which at that time flowed through a more northern bed compared to today (what is now the Kromme Rijn). These fortresses were designed to house a cohort of about 500 Roman soldiers. Near the fort settlements would grow housing artisans, traders and soldiers' wives and children.
From the middle of the 3rd century Germanic tribes regularly invaded the Roman territories. Around 275 the Romans could no longer maintain the northern border and Utrecht was abandoned. Little is known about the next period 270–650. Utrecht is first spoken of again several centuries after the Romans left. Under the influence of the growing realms of the Franks, during Dagobert I's reign in the 7th century, a church was built within the walls of the Roman fortress. In ongoing border conflicts with the Frisians this first church was destroyed.
By the mid-7th century, English and Irish missionaries set out to convert the Frisians. The pope appointed their leader, Willibrordus, bishop of the Frisians. The tenure of Willibrordus is generally considered to be the beginning of the Bishopric of Utrecht. In 723, the Frankish leader Charles Martel bestowed the fortress in Utrecht and the surrounding lands as the base of the bishops. From then on Utrecht became one of the most influential seats of power for the Roman Catholic Church in the Netherlands. The archbishops of Utrecht were based at the uneasy northern border of the Carolingian Empire. In addition, the city of Utrecht had competition from the nearby trading centre Dorestad. After the fall of Dorestad around 850, Utrecht became one of the most important cities in the Netherlands. The importance of Utrecht as a centre of Christianity is illustrated by the election of the Utrecht-born Adriaan Florenszoon Boeyens as pope in 1522 (the last non-Italian pope before John Paul II).
When the Frankish rulers established the system of feudalism, the Bishops of Utrecht came to exercise worldly power as prince-bishops. The territory of the bishopric not only included the modern province of Utrecht (Nedersticht, 'lower Sticht'), but also extended to the northeast. The feudal conflict of the Middle Ages heavily affected Utrecht. The prince-bishopric was involved in almost continuous conflicts with the Counts of Holland and the Dukes of Guelders. The Veluwe region was seized by Guelders, but large areas in the modern province of Overijssel remained as the Oversticht.
Several churches and monasteries were built inside, or close to, the city of Utrecht. The most dominant of these was the Cathedral of Saint Martin, inside the old Roman fortress. The construction of the present Gothic building was begun in 1254 after an earlier romanesque construction had been badly damaged by fire. The choir and transept were finished from 1320 and were followed then by the ambitious Dom tower. The last part to be constructed was the central nave, from 1420. By that time, however, the age of the great cathedrals had come to an end and declining finances prevented the ambitious project from being finished, the construction of the central nave being suspended before the planned flying buttresses could be finished. Besides the cathedral there were four collegiate churches in Utrecht: St. Salvator's Church (demolished in the 16th century), on the Dom square, dating back to the early 8th century. Saint John (Janskerk), originating in 1040; Saint Peter, building started in 1039 and Saint Mary's church building started around 1090 (demolished in the early 19th century, cloister survives). Besides these churches the city housed St. Paul's Abbey, the 15th-century beguinage of St. Nicholas, and a 14th-century chapter house of the Teutonic Knights.
The location on the banks of the river Rhine allowed Utrecht to become an important trade centre in the Northern Netherlands. The growing town Utrecht was granted city rights by Henry V in 1122. When the main flow of the Rhine moved south, the old bed, which still flowed through the heart of the town became evermore canalized; and the wharf system was built as an inner city harbour system. On the wharfs storage facilities (werfkelders) were built, on top of which the main street, including houses was constructed. The wharfs and the cellars are accessible from a platform at water level with stairs descending from the street level to form a unique structure.[nb 2] The relations between the bishop, who controlled many lands outside of the city, and the citizens of Utrecht was not always easy. The bishop, for example dammed the Kromme Rijn at Wijk bij Duurstede to protect his estates from flooding. This threatened shipping for the city and led the city of Utrecht to commission a canal to ensure access to the town for shipping trade: the Vaartse Rijn, connecting Utrecht to the Hollandse IJssel at IJsselstein.
In 1528 the bishop lost secular power over both Neder- and Oversticht – which included the city of Utrecht – to Charles V, Holy Roman Emperor. Charles V combined the Seventeen Provinces (the current Benelux and the northern parts of France) as a personal union. This ended the prince-bishopric Utrecht, as the secular rule was now the lordship of Utrecht, with the religious power remaining with the bishop, although Charles V had gained the right to appoint new bishops. In 1559 the bishopric of Utrecht was raised to archbishopric to make it the religious center of the Northern ecclesiastical province in the Seventeen provinces.
The transition from independence to a relatively minor part of a larger union was not easily accepted. To quell uprisings Charles V was struggling to exert his power over the citizens of the city, who had struggled to gain a certain level of independence from the bishops and were not willing to cede this to their new lord. The heavily fortified castle Vredenburg was built to house a large garrison whose main task was to maintain control over the city. The castle would last less than 50 years before it was demolished in an uprising in the early stages of the Dutch Revolt.
In 1579 the northern seven provinces signed the Union of Utrecht, in which they decided to join forces against Spanish rule. The Union of Utrecht is seen as the beginning of the Dutch Republic. In 1580 the new and predominantly Protestant state abolished the bishoprics, including the archbishopric of Utrecht. The stadtholders disapproved of the independent course of the Utrecht bourgeoisie and brought the city under much more direct control of the republic; which shifted the power towards its dominant province Holland. This was the start of a long period of stagnation of trade and development in Utrecht. Utrecht remained an atypical city in the new republic with about 40% Catholic in the mid-17th century, and even more among the elite groups, who included many rural nobility and gentry with town houses there.
The fortified city temporarily fell to the French invasion in 1672 (the Disaster Year); where the French invasion was only stopped west of Utrecht at the Old Hollandic Waterline. In 1674, only two years after the French left, the centre of Utrecht was struck by a tornado. The halt to building before construction of flying buttresses in the 15th century now proved to be the undoing of the central section of the cathedral of St Martin church which collapsed; creating the current Dom square between the tower and choir. In 1713, Utrecht hosted one of the first international peace negotiations when the Treaty of Utrecht settled the War of the Spanish Succession. Since 1723 Utrecht became the centre of the non-Roman Old Catholic Churches in the world.
In the early 19th century, the role of Utrecht as a fortified town had become obsolete. The fortifications of the Nieuwe Hollandse Waterlinie were moved east of Utrecht. The town walls could now be demolished to allow for expansion. The moats remained intact and formed an important feature of the Zocher plantsoen, an English style landscape park that remains largely intact today. Growth of the city increased when, in 1843, a railway connecting Utrecht to Amsterdam was opened. After that, Utrecht gradually became the main hub of the Dutch railway network. With the industrial revolution finally gathering speed in the Netherlands and the ramparts taken down, Utrecht began to grow far beyond the medieval centre. In 1853, the Dutch government allowed the bishopric of Utrecht to be reinstated by Rome, and Utrecht became the centre of Dutch Catholicism once more. From the 1880s onward neighbourhoods such as Oudwijk, Wittevrouwen, Vogelenbuurt to the East, and Lombok to the West were developed. New middle class residential areas, such as Tuindorp and Oog in Al, were built in the 1920s and 1930s. During this period, several Jugendstil houses and office buildings were built, followed by Rietveld who built the Rietveld Schröder House (1924), and Dudok's construction of the city theater (1941).
The area surrounding Utrecht Centraal railway station and the station itself were developed following modernist ideas of the 1960s, in a brutalist style. This led to the construction of the shopping mall Hoog Catharijne (nl), music centre Vredenburg (Hertzberger, 1979), and conversion of part of the ancient canal structure into a highway (Catherijnebaan). Protest against further modernisation of the city centre followed even before the last buildings were finalised. In the early 21st century the whole area is being redeveloped. The music redeveloped music centre opened in 2014 where the original Vredenburg concert and rock and jazz halls are brought together in a single building.
About 69% of the population is of Dutch ancestry. Approximately 10% of the population consists of immigrants from Western countries, while 21% of the population is of non-Western origin (9% Moroccan, 5% Turkish, 3% Surinamese and Dutch Caribbean and 5% of other countries). Some of the city's boroughs have a relatively high percentage of originally non-Dutch inhabitants – i.e. Kanaleneiland 83% and Overvecht 57%. Like Rotterdam, Amsterdam, The Hague and other large Dutch cities, Utrecht faces some socio-economic problems. About 38% percent of its population either earns a minimum income or is dependent on social welfare (17% of all households). Boroughs such as Kanaleneiland, Overvecht and Hoograven consist primarily of high-rise housing developments, and are known for relatively high poverty and crime rate.
Utrecht is the centre of a densely populated area, which makes concise definitions of its agglomeration difficult, and somewhat arbitrary. The smaller Utrecht agglomeration of continuously built up areas counts some 420,000 inhabitants and includes Nieuwegein, IJsselstein and Maarssen. It is sometimes argued that the close by municipalities De Bilt, Zeist, Houten, Vianen, Driebergen-Rijsenburg (Utrechtse Heuvelrug), and Bunnik should also be counted towards the Utrecht agglomeration, bringing the total to 640,000 inhabitants. The larger region, including slightly more remote towns such as Woerden and Amersfoort counts up to 820,000 inhabitants.
Utrecht's cityscape is dominated by the Dom Tower, the tallest belfry in the Netherlands and originally part of the Cathedral of Saint Martin. An ongoing debate is over whether any building in or near the centre of town should surpass the Dom Tower in height (112 m). Nevertheless, some tall buildings are now being constructed that will become part of the skyline of Utrecht. The second tallest building of the city, the Rabobank-tower, was completed in 2010 and stands 105 m (344.49 ft) tall. Two antennas will increase that height to 120 m (393.70 ft). Two other buildings were constructed around the Nieuw Galgenwaard stadium (2007). These buildings, the 'Kantoortoren Galghenwert' and 'Apollo Residence', stand 85.5 and 64.5 metres high respectively.
Another landmark is the old centre and the canal structure in the inner city. The Oudegracht is a curved canal, partly following the ancient main branch of the Rhine. It is lined with the unique wharf-basement structures that create a two-level street along the canals. The inner city has largely retained its Medieval structure, and the moat ringing the old town is largely intact. Because of the role of Utrecht as a fortified city, construction outside the medieval centre and its city walls was restricted until the 19th century. Surrounding the medieval core there is a ring of late 19th- and early 20th-century neighbourhoods, with newer neighbourhoods positioned farther out. The eastern part of Utrecht remains fairly open. The Dutch Water Line, moved east of the city in the early 19th century required open lines of fire, thus prohibiting all permanent constructions until the middle of the 20th century on the east side of the city.
Utrecht Centraal is the main railway station of Utrecht. There are regular intercity services to all major Dutch cities; direct services to Schiphol Airport. Utrecht Centraal is a station on the night service, providing 7 days a week an all night service to (among others) Schiphol Airport, Amsterdam and Rotterdam. International InterCityExpress (ICE) services to Germany (and further) through Arnhem call at Utrecht Centraal. Regular local trains to all areas surrounding Utrecht also depart from Utrecht Centraal; and service several smaller stations: Utrecht Lunetten, Utrecht Vaartsche Rijn, Utrecht Overvecht, Utrecht Leidsche Rijn, Utrecht Terwijde, Utrecht Zuilen and Vleuten. A former station Utrecht Maliebaan closed in 1939 and has since been converted into the Dutch Railway Museum.
The main local and regional bus station of Utrecht is located adjacent to Utrecht Centraal railway station, at the East and West entrances. Due to large scale renovation and construction works at the railway station, the station's bus stops are changing frequently. As a general rule, westbound buses depart from the bus station on the west entrance, other buses from the east side station. Local buses in Utrecht are operated by Qbuzz – its services include a high-frequency service to the Uithof university district. The local bus fleet is one of Europe's cleanest, using only buses compliant with the Euro-VI standard as well as electric buses for inner city transport. Regional buses from the city are operated by Arriva and Connexxion.
Like most Dutch cities, Utrecht has an extensive network of cycle paths, making cycling safe and popular. 33% of journeys within the city are by bicycle, more than any other mode of transport. (Cars, for example, account for 30% of trips). Bicycles are used by young and old people, and by individuals and families. They are mostly traditional, upright, steel-framed bicycles, with few or no gears. There are also barrow bikes, for carrying shopping or small children. As thousands of bicycles are parked haphazardly in town, creating an eyesore but also impeding pedestrians, the City Council decided in 2014 to build the world's largest bicycle parking station, near the Central Railway Station. This 3-floor construction will cost an estimated 48 million Euro and will hold 12,500 bicycles. Completion is foreseen in 2018.
Utrecht is well-connected to the Dutch road network. Two of the most important major roads serve the city of Utrecht: the A12 and A2 motorways connect Amsterdam, Arnhem, The Hague and Maastricht, as well as Belgium and Germany. Other major motorways in the area are the Almere–Breda A27 and the Utrecht–Groningen A28. Due to the increasing traffic and the ancient city plan, traffic congestion is a common phenomenon in and around Utrecht, causing elevated levels of air pollutants. This has led to a passionate debate in the city about the best way to improve the city's air quality.
Production industry constitutes a small part of the economy of Utrecht. The economy of Utrecht depends for a large part on the several large institutions located in the city. It is the centre of the Dutch railroad network and the location of the head office of Nederlandse Spoorwegen. ProRail is headquartered in The De Inktpot (nl) (The Inkpot) – the largest brick building in the Netherlands (the "UFO" featured on its façade stems from an art program in 2000). Rabobank, a large bank, has its headquarters in Utrecht.
A large indoor shopping centre Hoog Catharijne (nl) is located between Utrecht Centraal railway station and the city centre. The corridors are treated as public places like streets, and the route between the station and the city centre is open all night. In 20 years from 2004, parts of Hoog Catharijne will be redeveloped as part of the renovation of the larger station area. Parts of the city's network of canals, which were filled to create the shopping center and central station area, will be recreated. The Jaarbeurs, one of the largest convention centres in the Netherlands, is located at the west side of the central railway station.
Utrecht hosts several large institutions of higher education. The most prominent of these is Utrecht University (est. 1636), the largest university of the Netherlands with 30,449 students (as of 2012). The university is partially based in the inner city as well as in the Uithof campus area, to the east of the city. According to Shanghai Jiaotong University's university ranking in 2014 it is the 57th best university in the world. Utrecht also houses the much smaller University of Humanistic Studies, which houses about 400 students.
Utrecht city has an active cultural life, and in the Netherlands is second only to Amsterdam. There are several theatres and theatre companies. The 1941 main city theatre was built by Dudok. Besides theatres there is a large number of cinemas including three arthouse cinemas. Utrecht is host to the international Early Music Festival (Festival Oude Muziek, for music before 1800) and the Netherlands Film Festival. The city has an important classical music hall Vredenburg (1979 by Herman Hertzberger). Its acoustics are considered among the best of the 20th-century original music halls.[citation needed] The original Vredenburg music hall has been redeveloped as part of the larger station area redevelopment plan and in 2014 has gained additional halls that allowed its merger with the rock club Tivoli and the SJU jazzpodium. There are several other venues for music throughout the city. Young musicians are educated in the conservatory, a department of the Utrecht School of the Arts. There is a specialised museum of automatically playing musical instruments.
There are many art galleries in Utrecht. There are also several foundations to support art and artists. Training of artists is done at the Utrecht School of the Arts. The Centraal Museum has many exhibitions on the arts, including a permanent exhibition on the works of Utrecht resident illustrator Dick Bruna, who is best known for creating Miffy ("Nijntje", in Dutch). Although street art is illegal in Utrecht, the Utrechtse Kabouter, a picture of a gnome with a red hat, became a common sight in 2004. Utrecht also houses one of the landmarks of modern architecture, the 1924 Rietveld Schröder House, which is listed on UNESCO's world heritage sites.
To promote culture Utrecht city organizes cultural Sundays. During a thematic Sunday several organisations create a program, which is open to everyone without, or with a very much reduced, admission fee. There are also initiatives for amateur artists. The city subsidises an organisation for amateur education in arts aimed at all inhabitants (Utrechts Centrum voor de Kunsten), as does the university for its staff and students. Additionally there are also several private initiatives. The city council provides coupons for discounts to inhabitants who receive welfare to be used with many of the initiatives.
Utrecht is home to the premier league (professional) football club FC Utrecht, which plays in Stadium Nieuw Galgenwaard. It is also the home of Kampong, the largest (amateur) sportsclub in the Netherlands (4,500 members), SV Kampong. Kampong features fieldhockey, soccer, cricket, tennis, squash and jeu de boules. Kampong's men and women top hockey squads play in the highest Dutch hockey league, the Rabohoofdklasse.Utrecht is also home to the baseball and Sofball club: UVV which plays in the highest Dutch baseball league: de Hoofdklasse. Utrecht's waterways are used by several rowing clubs. Viking is a large club open to the general public, and the student clubs Orca and Triton compete in the Varsity each year.
John von Neumann (/vɒn ˈnɔɪmən/; Hungarian: Neumann János Lajos, pronounced [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian-American pure and applied mathematician, physicist, inventor, computer scientist, and polymath. He made major contributions to a number of fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics, fluid dynamics and quantum statistical mechanics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics.
He was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor and the digital computer. He published 150 papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. His last work, an unfinished manuscript written while in the hospital, was later published in book form as The Computer and the Brain.
Von Neumann's mathematical analysis of the structure of self-replication preceded the discovery of the structure of DNA. In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."
During World War II he worked on the Manhattan Project with J. Robert Oppenheimer and Edward Teller, developing the mathematical models behind the explosive lenses used in the implosion-type nuclear weapon. After the war, he served on the General Advisory Committee of the United States Atomic Energy Commission, and later as one of its commissioners. He was a consultant to a number of organizations, including the United States Air Force, the Armed Forces Special Weapons Project, and the Lawrence Livermore National Laboratory. Along with theoretical physicist Edward Teller, mathematician Stanislaw Ulam, and others, he worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.
Von Neumann was born Neumann János Lajos (in Hungarian the family name comes first), Hebrew name Yonah, in Budapest, Kingdom of Hungary, which was then part of the Austro-Hungarian Empire, to wealthy Jewish parents of the Haskalah. He was the eldest of three children. He had two younger brothers: Michael, born in 1907, and Nicholas, who was born in 1911. His father, Neumann Miksa (Max Neumann) was a banker, who held a doctorate in law. He had moved to Budapest from Pécs at the end of the 1880s. Miksa's father and grandfather were both born in Ond (now part of the town of Szerencs), Zemplén County, northern Hungary. John's mother was Kann Margit (Margaret Kann); her parents were Jakab Kann and Katalin Meisels. Three generations of the Kann family lived in spacious apartments above the Kann-Heller offices in Budapest; von Neumann's family occupied an 18-room apartment on the top floor.
In 1913, his father was elevated to the nobility for his service to the Austro-Hungarian Empire by Emperor Franz Joseph. The Neumann family thus acquired the hereditary appellation Margittai, meaning of Marghita. The family had no connection with the town; the appellation was chosen in reference to Margaret, as was those chosen coat of arms depicting three marguerites. Neumann János became Margittai Neumann János (John Neumann of Marghita), which he later changed to the German Johann von Neumann.
Formal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins. Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. By the age of 8, von Neumann was familiar with differential and integral calculus, but he was particularly interested in history, reading his way through Wilhelm Oncken's Allgemeine Geschichte in Einzeldarstellungen. A copy was contained in a private library Max purchased. One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor.
Von Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911. This was one of the best schools in Budapest, part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. Despite being run by the Lutheran Church, the majority of its pupils were Jewish. The school system produced a generation noted for intellectual achievement, that included Theodore von Kármán (b. 1881), George de Hevesy (b. 1885), Leó Szilárd (b. 1898), Eugene Wigner (b. 1902), Edward Teller (b. 1908), and Paul Erdős (b. 1913). Collectively, they were sometimes known as Martians. Wigner was a year ahead of von Neumann at the Lutheran School. When asked why the Hungary of his generation had produced so many geniuses, Wigner, who won the Nobel Prize in Physics in 1963, replied that von Neumann was the only genius.
Although Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are still on display at the von Neumann archive in Budapest. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition. At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics.
Since there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a chemical engineer. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the University of Berlin, after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor's set theory. He passed his final examinations for his Ph.D. soon after graduating from ETH Zurich in 1926. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.
Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month. His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University in Princeton, New Jersey.
On New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, Marina, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.
In 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resign and become a professor at large at the University of California. His mother, brothers and in-laws followed John to the United States in 1939. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen of the United States in 1937, and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. He passed the exams easily, but was ultimately rejected because of his age. His prewar analysis is often quoted. Asked about how France would stand up to Germany he said "Oh, France won't matter."
Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especially limericks). He was a non-smoker. At Princeton he received complaints for regularly playing extremely loud German march music on his gramophone, which distracted those in neighbouring offices, including Albert Einstein, from their work. Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.
Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend. He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.
The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class.
The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the succession, hence excluding the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the method of inner models, which later became an essential instrument in set theory.
The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.
With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language. This result was sufficiently innovative as to confound the majority of mathematicians of the time.
But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consistency. However, Gödel had already discovered this consequence, now known as his second incompleteness theorem and sent von Neumann a preprint of his article containing both incompleteness theorems. Von Neumann acknowledged Gödel's priority in his next letter. He never thought much of "the American system of claiming personal priority for everything."
Von Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.
In a series of famous papers, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."
In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.
Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. The direct integral was introduced in 1949 by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.
Von Neumann worked on lattice theory between 1937 and 1939. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: "Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity." Garrett Birkhoff writes: "John von Neumann's brilliant mind blazed over lattice theory like a meteor".
Additionally, "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe."
Von Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, with his 1932 work Mathematical Foundations of Quantum Mechanics. After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics. He realized, in 1926, that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system.
The physics of quantum mechanics was thereby reduced to the mathematics of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the non-commutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger. When Heisenberg was informed von Neumann had clarified the difference between an unbounded operator that was a Self-adjoint operator and one that was merely symmetric, Heisenberg replied "Eh? What is the difference?"
Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables," as in classical statistical mechanics. In 1966, John S. Bell published a paper arguing that the proof contained a conceptual error and was therefore invalid. However, in 2010, Jeffrey Bub argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for all hidden variable theories, does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation, and that von Neumann did not claim that his proof completely ruled out hidden variable theories.
In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter (although this view was accepted by Eugene Wigner, the Von Neumann–Wigner interpretation never gained acceptance amongst the majority of physicists).
In a famous paper of 1936 with Garrett Birkhoff, the first work ever to introduce quantum logics, von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work, but in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters that are polarized perpendicularly (e.g., one horizontally and the other vertically), and therefore, a fortiori, it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession, but if the third filter is added in between the other two, the photons will, indeed, pass through. This experimental fact is translatable into logic as the non-commutativity of conjunction . It was also demonstrated that the laws of distribution of classical logic, and , are not valid for quantum theory.
Von Neumann founded the field of game theory as a mathematical discipline. Von Neumann proved his minimax theorem in 1928. This theorem establishes that in zero-sum games with perfect information (i.e. in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses, hence the name minimax. When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy that will result in the minimization of his maximum loss.
The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is, in turn, attributable to the fact that it is frequently the case, in quantum mechanics, that a pair of alternatives are semantically determinate, while each of its members are necessarily indeterminate. This latter property can be illustrated by a simple example. Suppose we are dealing with particles (such as electrons) of semi-integral spin (angular momentum) for which there are only two possible values: positive or negative. Then, a principle of indetermination establishes that the spin, relative to two different directions (e.g., x and y) results in a pair of incompatible quantities. Suppose that the state ɸ of a certain electron verifies the proposition "the spin of the electron in the x direction is positive." By the principle of indeterminacy, the value of the spin in the direction y will be completely indeterminate for ɸ. Hence, ɸ can verify neither the proposition "the spin in the direction of y is positive" nor the proposition "the spin in the direction of y is negative." Nevertheless, the disjunction of the propositions "the spin in the direction of y is positive or the spin in the direction of y is negative" must be true for ɸ. In the case of distribution, it is therefore possible to have a situation in which , while .
Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). Von Neumann improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944 Theory of Games and Economic Behavior (written with Oskar Morgenstern). Morgenstern wrote a paper on game theory and thought he would show it to von Neumann because of his interest in the subject. He read it and said to Morgenstern that he should put more in it. This was repeated a couple of times, and then von Neumann became a coauthor and the paper became 100 pages long. Then it became a book. The public interest in this work was such that The New York Times ran a front-page story. In this book, von Neumann declared that economic theory needed to use functional analytic methods, especially convex sets and topological fixed-point theorem, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions.
Von Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. Von Neumann's model of an expanding economy considered the matrix pencil A − λB with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation
along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.
Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices. The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics. This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities, complementary slackness, and saddlepoint duality. In the proceedings of a conference on von Neumann's growth model, Paul Samuelson said that many mathematicians had developed methods useful to economists, but that von Neumann was unique in having made significant contributions to economic theory itself.
Von Neumann's famous 9-page paper started life as a talk at Princeton and then became a paper in Germany, which was eventually translated into English. His interest in economics that led to that paper began as follows: When lecturing at Berlin in 1928 and 1929 he spent his summers back home in Budapest, and so did the economist Nicholas Kaldor, and they hit it off. Kaldor recommended that von Neumann read a book by the mathematical economist Léon Walras. Von Neumann found some faults in that book and corrected them, for example, replacing equations by inequalities. He noticed that Walras's General Equilibrium Theory and Walras' Law, which led to systems of simultaneous linear equations, could produce the absurd result that the profit could be maximized by producing and selling a negative quantity of a product. He replaced the equations by inequalities, introduced dynamic equilibria, among other things, and eventually produced the paper.
Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Gordan (1873), which was later popularized by Karmarkar's algorithm. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex). Von Neumann's algorithm was the first interior point method of linear programming.
Von Neumann made fundamental contributions to mathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression.
Von Neumann made fundamental contributions in exploration of problems in numerical hydrodynamics. For example, with Robert D. Richtmyer he developed an algorithm defining artificial viscosity that improved the understanding of shock waves. A problem was that when computers solved hydrodynamic or aerodynamic problems, they tried to put too many computational grid points at regions of sharp discontinuity (shock waves). The mathematics of artificial viscosity smoothed the shock transition without sacrificing basic physics. Other well known contributions to fluid dynamics included the classic flow solution to blast waves, and the co-discovery of the ZND detonation model of explosives.
Von Neumann's principal contribution to the atomic bomb was in the concept and design of the explosive lenses needed to compress the plutonium core of the Fat Man weapon that was later dropped on Nagasaki. While von Neumann did not originate the "implosion" concept, he was one of its most persistent proponents, encouraging its continued development against the instincts of many of his colleagues, who felt such a design to be unworkable. He also eventually came up with the idea of using more powerful shaped charges and less fissionable material to greatly increase the speed of "assembly".
When it turned out that there would not be enough uranium-235 to make more than one bomb, the implosive lens project was greatly expanded and von Neumann's idea was implemented. Implosion was the only method that could be used with the plutonium-239 that was available from the Hanford Site. He established the design of the explosive lenses required, but there remained concerns about "edge effects" and imperfections in the explosives. His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry. After a series of failed attempts with models, this was achieved by George Kistiakowsky, and the construction of the Trinity bomb was completed in July 1945.
Along with four other scientists and various military personnel, von Neumann was included in the target selection committee responsible for choosing the Japanese cities of Hiroshima and Nagasaki as the first targets of the atomic bomb. Von Neumann oversaw computations related to the expected size of the bomb blasts, estimated death tolls, and the distance above the ground at which the bombs should be detonated for optimum shock wave propagation and thus maximum effect. The cultural capital Kyoto, which had been spared the bombing inflicted upon militarily significant cities, was von Neumann's first choice, a selection seconded by Manhattan Project leader General Leslie Groves. However, this target was dismissed by Secretary of War Henry L. Stimson.
On July 16, 1945, with numerous other Manhattan Project personnel, von Neumann was an eyewitness to the first atomic bomb blast, code named Trinity, conducted as a test of the implosion method device, at the bombing range near Alamogordo Army Airfield, 35 miles (56 km) southeast of Socorro, New Mexico. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to 5 kilotons of TNT (21 TJ) but Enrico Fermi produced a more accurate estimate of 10 kilotons by dropping scraps of torn-up paper as the shock wave passed his location and watching how far they scattered. The actual power of the explosion had been between 20 and 22 kilotons. It was in von Neumann's 1944 papers that the expression "kilotons" appeared for the first time. After the war, Robert Oppenheimer remarked that the physicists involved in the Manhattan project had "known sin". Von Neumann's response was that "sometimes someone confesses a sin in order to take credit for it."
Von Neumann continued unperturbed in his work and became, along with Edward Teller, one of those who sustained the hydrogen bomb project. He then collaborated with Klaus Fuchs on further development of the bomb, and in 1946 the two filed a secret patent on "Improvement in Methods and Means for Utilizing Nuclear Energy", which outlined a scheme for using a fission bomb to compress fusion fuel to initiate nuclear fusion. The Fuchs–von Neumann patent used radiation implosion, but not in the same way as is used in what became the final hydrogen bomb design, the Teller–Ulam design. Their work was, however, incorporated into the "George" shot of Operation Greenhouse, which was instructive in testing out concepts that went into the final design. The Fuchs–von Neumann work was passed on, by Fuchs, to the Soviet Union as part of his nuclear espionage, but it was not used in the Soviets' own, independent development of the Teller–Ulam design. The historian Jeremy Bernstein has pointed out that ironically, "John von Neumann and Klaus Fuchs, produced a brilliant invention in 1946 that could have changed the whole course of the development of the hydrogen bomb, but was not fully understood until after the bomb had been successfully made."
In 1950, von Neumann became a consultant to the Weapons Systems Evaluation Group (WSEG), whose function was to advise the Joint Chiefs of Staff and the United States Secretary of Defense on the development and use of new technologies. He also became an adviser to the Armed Forces Special Weapons Project (AFSWP), which was responsible for the military aspects on nuclear weapons.Over the following two years, he also became a consultant to the Central Intelligence Agency (CIA), a member of the influential General Advisory Committee of the Atomic Energy Commission, a consultant to the newly established Lawrence Livermore National Laboratory, and a member of the Scientific Advisory Group of the United States Air Force.
In 1955, von Neumann became a commissioner of the AEC. He accepted this position and used it to further the production of compact hydrogen bombs suitable for Intercontinental ballistic missile delivery. He involved himself in correcting the severe shortage of tritium and lithium 6 needed for these compact weapons, and he argued against settling for the intermediate range missiles that the Army wanted. He was adamant that H-bombs delivered into the heart of enemy territory by an ICBM would be the most effective weapon possible, and that the relative inaccuracy of the missile wouldn't be a problem with an H-bomb. He said the Russians would probably be building a similar weapon system, which turned out to be the case. Despite his disagreement with Oppenheimer over the need for a crash program to develop the hydrogen bomb, he testified on the latter's behalf at the 1954 Oppenheimer security hearing, at which he asserted that Oppenheimer was loyal, and praised him for his helpfulness once the program went ahead.

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