autoevaluate/autoeval-staging-eval-squad_v2-squad_v2-38b250-14916076

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Provided is a universal Ku band LNB (9.75/10.600 GHz) which is fitted at the end of the dish and pointed at the correct satellite constellation; most digital receivers will receive the free to air channels. Some broadcasts are free-to-air and unencrypted, some are encrypted but do not require a monthly subscription (known as free-to-view), some are encrypted and require a monthly subscription, and some are pay-per-view services. To view the encrypted content a VideoGuard UK equipped receiver (all of which are dedicated to the Sky service, and cannot be used to decrypt other services) needs to be used. Unofficial CAMs are now available to view the service, although use of them breaks the user's contract with Sky and invalidates the user's rights to use the card.	What is the universal band that digital recievers will receive free to air channels on?	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At the beginning of the 20th century, important advancement in geological science was facilitated by the ability to obtain accurate absolute dates to geologic events using radioactive isotopes and other methods. This changed the understanding of geologic time. Previously, geologists could only use fossils and stratigraphic correlation to date sections of rock relative to one another. With isotopic dates it became possible to assign absolute ages to rock units, and these absolute dates could be applied to fossil sequences in which there was datable material, converting the old relative ages into new absolute ages.	What did using fossils help change for scientists?	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However, attempting to reconcile electromagnetic theory with two observations, the photoelectric effect, and the nonexistence of the ultraviolet catastrophe, proved troublesome. Through the work of leading theoretical physicists, a new theory of electromagnetism was developed using quantum mechanics. This final modification to electromagnetic theory ultimately led to quantum electrodynamics (or QED), which fully describes all electromagnetic phenomena as being mediated by wave–particles known as photons. In QED, photons are the fundamental exchange particle, which described all interactions relating to electromagnetism including the electromagnetic force.[Note 4]	What was used to create a new electromagnetic theory to reconcile the troubles with electromagnetic theory as it used to stand?	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Packet mode communication may be implemented with or without intermediate forwarding nodes (packet switches or routers). Packets are normally forwarded by intermediate network nodes asynchronously using first-in, first-out buffering, but may be forwarded according to some scheduling discipline for fair queuing, traffic shaping, or for differentiated or guaranteed quality of service, such as weighted fair queuing or leaky bucket. In case of a shared physical medium (such as radio or 10BASE5), the packets may be delivered according to a multiple access scheme.	If there is no radio or 10BASE5, how are the packets delivered?	[ "" ]	[ -1 ]	[ "5a66797c846392001a1e1c4b" ]	[ "Packet_switching" ]	[ 11.046875, -3.484375, -4.66015625, -5.4453125, -3.873046875, -4.01953125, -5.6328125, -4.4296875, -5.55078125, -6.2421875, -5.671875, -4.35546875, -5.90625, -5.05859375, -4.86328125, -5.93359375, -6.859375, -1.7734375, -0.88037109375, -3.033203125, -1.6328125, -2.931640625, -3.373046875, -1.962890625, -2.33984375, -3.71875, -2.1484375, -1.0341796875, -1.7109375, -3.58984375, -2.291015625, -2.86328125, -1.509765625, -2.7734375, -4.08203125, -2.337890625, -3.560546875, -3.57421875, -2.40625, -1.212890625, -3.087890625, -1.4267578125, -0.90966796875, -3.0703125, -1.4482421875, -0.484130859375, -1.8154296875, -1.8486328125, -1.3193359375, -2.98828125, -3.150390625, -2.953125, -2.69140625, -1.330078125, -0.96923828125, -3.73046875, -3.16796875, -3.3828125, -1.05859375, -3.75390625, -3.158203125, -1.7265625, -2.896484375, -3.185546875, -2.818359375, -2.001953125, -2.806640625, -1.0673828125, -3.216796875, -0.75341796875, 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In ring theory, the notion of number is generally replaced with that of ideal. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.	Which theorem can be simplified to the Lasker–Noether theorem?	[ "The fundamental theorem of arithmetic", "The fundamental theorem of arithmetic", "fundamental theorem of arithmetic", "theorem of arithmetic" ]	[ 413, 413, 417, 429 ]	[ "57299ec43f37b31900478510", "57299ec43f37b31900478510", "57299ec43f37b31900478510", "57299ec43f37b31900478510" ]	[ "Prime_number", "Prime_number", "Prime_number", "Prime_number" ]	[ 5.8671875, -3.876953125, -5.359375, -5.9140625, -6.12109375, -4.97265625, -5.1640625, -3.119140625, -2.802734375, -5.34375, -5.41796875, -4.21484375, -5.484375, -6.08984375, -5.5, -7.07421875, -2.1328125, -2.314453125, -0.51171875, -2.5546875, -3.640625, -2.19921875, -2.04296875, -3.640625, -1.6923828125, -4.0703125, -3.3203125, -3.125, -3.806640625, -3.2890625, -3.501953125, -1.5927734375, -2.91796875, -0.8291015625, -2.435546875, -3.779296875, -3.205078125, -0.96923828125, -3.44140625, -1.48828125, -3.125, -3.9921875, -2.400390625, -3.046875, -3.36328125, -1.7998046875, -1.3154296875, -2.23046875, -2.568359375, -3.412109375, 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Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.	What are two examples of different types of reduction?	[ "Karp reductions and Levin reductions", "Cook reductions, Karp reductions" ]	[ 469, 452 ]	[ "56e1c9bfe3433e1400423195", "56e1c9bfe3433e1400423195" ]	[ "Computational_complexity_theory", "Computational_complexity_theory" ]	[ 4.9140625, -4.55859375, -5.7734375, -5.4375, -5.8671875, -5.90625, -5.42578125, -5.98828125, -5.9140625, -5.8203125, -6.6484375, -1.2265625, -2.640625, -1.201171875, -3.072265625, -4.234375, -2.91796875, -3.353515625, -2.68359375, -2.63671875, -3.970703125, -2.57421875, -2.19921875, -3.7421875, -2.4296875, -2.359375, -3.578125, -2.65234375, -2.041015625, -4.0390625, -2.666015625, -2.861328125, -4.078125, -3.19140625, -2.56640625, -3.482421875, -2.810546875, -2.4296875, -2.7734375, -2.072265625, -2.802734375, -3.978515625, -3.1328125, -2.744140625, -3.478515625, -2.92578125, -3.82421875, -3.361328125, -2.5, -4.2890625, -3.4140625, -2.771484375, -3.66796875, -3.6796875, -3.791015625, -3.8359375, -1.697265625, -3.26953125, -2.400390625, -2.50390625, -3.443359375, 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In 1096, Crusaders passing by the siege of Amalfi were joined by Bohemond of Taranto and his nephew Tancred with an army of Italo-Normans. Bohemond was the de facto leader of the Crusade during its passage through Asia Minor. After the successful Siege of Antioch in 1097, Bohemond began carving out an independent principality around that city. Tancred was instrumental in the conquest of Jerusalem and he worked for the expansion of the Crusader kingdom in Transjordan and the region of Galilee.[citation needed]	When did Tancred lay siege to Antioch?	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On 1 July 1851, writs were issued for the election of the first Victorian Legislative Council, and the absolute independence of Victoria from New South Wales was established proclaiming a new Colony of Victoria. Days later, still in 1851 gold was discovered near Ballarat, and subsequently at Bendigo. Later discoveries occurred at many sites across Victoria. This triggered one of the largest gold rushes the world has ever seen. The colony grew rapidly in both population and economic power. In ten years the population of Victoria increased sevenfold from 76,000 to 540,000. All sorts of gold records were produced including the "richest shallow alluvial goldfield in the world" and the largest gold nugget. Victoria produced in the decade 1851–1860 20 million ounces of gold, one third of the world's output[citation needed].	How much gold did Victoria produce in 1860?	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In October 2010, the open-access scientific journal PLoS Pathogens published a paper by a multinational team who undertook a new investigation into the role of Yersinia pestis in the Black Death following the disputed identification by Drancourt and Raoult in 1998. They assessed the presence of DNA/RNA with Polymerase Chain Reaction (PCR) techniques for Y. pestis from the tooth sockets in human skeletons from mass graves in northern, central and southern Europe that were associated archaeologically with the Black Death and subsequent resurgences. The authors concluded that this new research, together with prior analyses from the south of France and Germany, ". . . ends the debate about the etiology of the Black Death, and unambiguously demonstrates that Y. pestis was the causative agent of the epidemic plague that devastated Europe during the Middle Ages".	What scientific journal in France published a prior analysis of the Black Death?	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The first European to travel the length of the Amazon River was Francisco de Orellana in 1542. The BBC's Unnatural Histories presents evidence that Orellana, rather than exaggerating his claims as previously thought, was correct in his observations that a complex civilization was flourishing along the Amazon in the 1540s. It is believed that the civilization was later devastated by the spread of diseases from Europe, such as smallpox. Since the 1970s, numerous geoglyphs have been discovered on deforested land dating between AD 0–1250, furthering claims about Pre-Columbian civilizations. Ondemar Dias is accredited with first discovering the geoglyphs in 1977 and Alceu Ranzi with furthering their discovery after flying over Acre. The BBC's Unnatural Histories presented evidence that the Amazon rainforest, rather than being a pristine wilderness, has been shaped by man for at least 11,000 years through practices such as forest gardening and terra preta.	The BBC's documentary presented evidence that the Amazon had been a pristine wilderness for at least how long?	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The immune system is a system of many biological structures and processes within an organism that protects against disease. To function properly, an immune system must detect a wide variety of agents, known as pathogens, from viruses to parasitic worms, and distinguish them from the organism's own healthy tissue. In many species, the immune system can be classified into subsystems, such as the innate immune system versus the adaptive immune system, or humoral immunity versus cell-mediated immunity. In humans, the blood–brain barrier, blood–cerebrospinal fluid barrier, and similar fluid–brain barriers separate the peripheral immune system from the neuroimmune system which protects the brain.	What connects the peripheral immune system to the neuroimmune system?	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While acknowledging the central role economic growth can potentially play in human development, poverty reduction and the achievement of the Millennium Development Goals, it is becoming widely understood amongst the development community that special efforts must be made to ensure poorer sections of society are able to participate in economic growth. The effect of economic growth on poverty reduction – the growth elasticity of poverty – can depend on the existing level of inequality. For instance, with low inequality a country with a growth rate of 2% per head and 40% of its population living in poverty, can halve poverty in ten years, but a country with high inequality would take nearly 60 years to achieve the same reduction. In the words of the Secretary General of the United Nations Ban Ki-Moon: "While economic growth is necessary, it is not sufficient for progress on reducing poverty."	What isn't economic growth sufficient for progress on?	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Each year, the southern California area has about 10,000 earthquakes. Nearly all of them are so small that they are not felt. Only several hundred are greater than magnitude 3.0, and only about 15–20 are greater than magnitude 4.0. The magnitude 6.7 1994 Northridge earthquake was particularly destructive, causing a substantial number of deaths, injuries, and structural collapses. It caused the most property damage of any earthquake in U.S. history, estimated at over $20 billion.	How many earthquakes does southern California experience in a year?	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The plague disease, caused by Yersinia pestis, is enzootic (commonly present) in populations of fleas carried by ground rodents, including marmots, in various areas including Central Asia, Kurdistan, Western Asia, Northern India and Uganda. Nestorian graves dating to 1338–39 near Lake Issyk Kul in Kyrgyzstan have inscriptions referring to plague and are thought by many epidemiologists to mark the outbreak of the epidemic, from which it could easily have spread to China and India. In October 2010, medical geneticists suggested that all three of the great waves of the plague originated in China. In China, the 13th century Mongol conquest caused a decline in farming and trading. However, economic recovery had been observed at the beginning of the 14th century. In the 1330s a large number of natural disasters and plagues led to widespread famine, starting in 1331, with a deadly plague arriving soon after. Epidemics that may have included plague killed an estimated 25 million Chinese and other Asians during the 15 years before it reached Constantinople in 1347.	What is the term for a disease that is not commonly present?	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In ring theory, the notion of number is generally replaced with that of ideal. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.	What type of ideals generalize Noetherian arithmetic?	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Provided is a universal Ku band LNB (9.75/10.600 GHz) which is fitted at the end of the dish and pointed at the correct satellite constellation; most digital receivers will receive the free to air channels. Some broadcasts are free-to-air and unencrypted, some are encrypted but do not require a monthly subscription (known as free-to-view), some are encrypted and require a monthly subscription, and some are pay-per-view services. To view the encrypted content a VideoGuard UK equipped receiver (all of which are dedicated to the Sky service, and cannot be used to decrypt other services) needs to be used. Unofficial CAMs are now available to view the service, although use of them breaks the user's contract with Sky and invalidates the user's rights to use the card.

What is the universal band that digital recievers will receive free to air channels on?

[ "9.75/10.600 GHz", "universal Ku band", "Ku band" ]

[ 37, 14, 24 ]

[ "57096c95200fba1400367fbe", "57096c95200fba1400367fbe", "57096c95200fba1400367fbe" ]

[ "Sky_(United_Kingdom)", "Sky_(United_Kingdom)", "Sky_(United_Kingdom)" ]

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At the beginning of the 20th century, important advancement in geological science was facilitated by the ability to obtain accurate absolute dates to geologic events using radioactive isotopes and other methods. This changed the understanding of geologic time. Previously, geologists could only use fossils and stratigraphic correlation to date sections of rock relative to one another. With isotopic dates it became possible to assign absolute ages to rock units, and these absolute dates could be applied to fossil sequences in which there was datable material, converting the old relative ages into new absolute ages.

What did using fossils help change for scientists?

[ "" ]

[ -1 ]

[ "5a57f487770dc0001aeefefd" ]

[ "Geology" ]

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However, attempting to reconcile electromagnetic theory with two observations, the photoelectric effect, and the nonexistence of the ultraviolet catastrophe, proved troublesome. Through the work of leading theoretical physicists, a new theory of electromagnetism was developed using quantum mechanics. This final modification to electromagnetic theory ultimately led to quantum electrodynamics (or QED), which fully describes all electromagnetic phenomena as being mediated by wave–particles known as photons. In QED, photons are the fundamental exchange particle, which described all interactions relating to electromagnetism including the electromagnetic force.[Note 4]

What was used to create a new electromagnetic theory to reconcile the troubles with electromagnetic theory as it used to stand?

[ "quantum mechanics", "the work of leading theoretical physicists", "quantum mechanics", "quantum mechanics" ]

[ 283, 186, 283, 283 ]

[ "573792ee1c456719005744ba", "573792ee1c456719005744ba", "573792ee1c456719005744ba", "573792ee1c456719005744ba" ]

[ "Force", "Force", "Force", "Force" ]

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Packet mode communication may be implemented with or without intermediate forwarding nodes (packet switches or routers). Packets are normally forwarded by intermediate network nodes asynchronously using first-in, first-out buffering, but may be forwarded according to some scheduling discipline for fair queuing, traffic shaping, or for differentiated or guaranteed quality of service, such as weighted fair queuing or leaky bucket. In case of a shared physical medium (such as radio or 10BASE5), the packets may be delivered according to a multiple access scheme.

If there is no radio or 10BASE5, how are the packets delivered?

[ "" ]

[ -1 ]

[ "5a66797c846392001a1e1c4b" ]

[ "Packet_switching" ]

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In ring theory, the notion of number is generally replaced with that of ideal. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.

Which theorem can be simplified to the Lasker–Noether theorem?

[ "The fundamental theorem of arithmetic", "The fundamental theorem of arithmetic", "fundamental theorem of arithmetic", "theorem of arithmetic" ]

[ 413, 413, 417, 429 ]

[ "57299ec43f37b31900478510", "57299ec43f37b31900478510", "57299ec43f37b31900478510", "57299ec43f37b31900478510" ]

[ "Prime_number", "Prime_number", "Prime_number", "Prime_number" ]

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Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.

What are two examples of different types of reduction?

[ "Karp reductions and Levin reductions", "Cook reductions, Karp reductions" ]

[ 469, 452 ]

[ "56e1c9bfe3433e1400423195", "56e1c9bfe3433e1400423195" ]

[ "Computational_complexity_theory", "Computational_complexity_theory" ]

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In 1096, Crusaders passing by the siege of Amalfi were joined by Bohemond of Taranto and his nephew Tancred with an army of Italo-Normans. Bohemond was the de facto leader of the Crusade during its passage through Asia Minor. After the successful Siege of Antioch in 1097, Bohemond began carving out an independent principality around that city. Tancred was instrumental in the conquest of Jerusalem and he worked for the expansion of the Crusader kingdom in Transjordan and the region of Galilee.[citation needed]

When did Tancred lay siege to Antioch?

[ "" ]

[ -1 ]

[ "5ad4017a604f3c001a3ffd1f" ]

[ "Normans" ]

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On 1 July 1851, writs were issued for the election of the first Victorian Legislative Council, and the absolute independence of Victoria from New South Wales was established proclaiming a new Colony of Victoria. Days later, still in 1851 gold was discovered near Ballarat, and subsequently at Bendigo. Later discoveries occurred at many sites across Victoria. This triggered one of the largest gold rushes the world has ever seen. The colony grew rapidly in both population and economic power. In ten years the population of Victoria increased sevenfold from 76,000 to 540,000. All sorts of gold records were produced including the "richest shallow alluvial goldfield in the world" and the largest gold nugget. Victoria produced in the decade 1851–1860 20 million ounces of gold, one third of the world's output[citation needed].

How much gold did Victoria produce in 1860?

[ "" ]

[ -1 ]

[ "5a63787868151a001a9222fe" ]

[ "Victoria_(Australia)" ]

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In October 2010, the open-access scientific journal PLoS Pathogens published a paper by a multinational team who undertook a new investigation into the role of Yersinia pestis in the Black Death following the disputed identification by Drancourt and Raoult in 1998. They assessed the presence of DNA/RNA with Polymerase Chain Reaction (PCR) techniques for Y. pestis from the tooth sockets in human skeletons from mass graves in northern, central and southern Europe that were associated archaeologically with the Black Death and subsequent resurgences. The authors concluded that this new research, together with prior analyses from the south of France and Germany, ". . . ends the debate about the etiology of the Black Death, and unambiguously demonstrates that Y. pestis was the causative agent of the epidemic plague that devastated Europe during the Middle Ages".

What scientific journal in France published a prior analysis of the Black Death?

[ "" ]

[ -1 ]

[ "5a2eca47a83784001a7d2484" ]

[ "Black_Death" ]

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The first European to travel the length of the Amazon River was Francisco de Orellana in 1542. The BBC's Unnatural Histories presents evidence that Orellana, rather than exaggerating his claims as previously thought, was correct in his observations that a complex civilization was flourishing along the Amazon in the 1540s. It is believed that the civilization was later devastated by the spread of diseases from Europe, such as smallpox. Since the 1970s, numerous geoglyphs have been discovered on deforested land dating between AD 0–1250, furthering claims about Pre-Columbian civilizations. Ondemar Dias is accredited with first discovering the geoglyphs in 1977 and Alceu Ranzi with furthering their discovery after flying over Acre. The BBC's Unnatural Histories presented evidence that the Amazon rainforest, rather than being a pristine wilderness, has been shaped by man for at least 11,000 years through practices such as forest gardening and terra preta.

The BBC's documentary presented evidence that the Amazon had been a pristine wilderness for at least how long?

[ "" ]

[ -1 ]

[ "5a0c7b88f5590b0018dab422" ]

[ "Amazon_rainforest" ]

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The immune system is a system of many biological structures and processes within an organism that protects against disease. To function properly, an immune system must detect a wide variety of agents, known as pathogens, from viruses to parasitic worms, and distinguish them from the organism's own healthy tissue. In many species, the immune system can be classified into subsystems, such as the innate immune system versus the adaptive immune system, or humoral immunity versus cell-mediated immunity. In humans, the blood–brain barrier, blood–cerebrospinal fluid barrier, and similar fluid–brain barriers separate the peripheral immune system from the neuroimmune system which protects the brain.

What connects the peripheral immune system to the neuroimmune system?

[ "" ]

[ -1 ]

[ "5ad49c34ba00c4001a268d9e" ]

[ "Immune_system" ]

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While acknowledging the central role economic growth can potentially play in human development, poverty reduction and the achievement of the Millennium Development Goals, it is becoming widely understood amongst the development community that special efforts must be made to ensure poorer sections of society are able to participate in economic growth. The effect of economic growth on poverty reduction – the growth elasticity of poverty – can depend on the existing level of inequality. For instance, with low inequality a country with a growth rate of 2% per head and 40% of its population living in poverty, can halve poverty in ten years, but a country with high inequality would take nearly 60 years to achieve the same reduction. In the words of the Secretary General of the United Nations Ban Ki-Moon: "While economic growth is necessary, it is not sufficient for progress on reducing poverty."

What isn't economic growth sufficient for progress on?

[ "reducing poverty", "reducing poverty", "reducing poverty" ]

[ 884, 884, 884 ]

[ "572a1a5c6aef051400155288", "572a1a5c6aef051400155288", "572a1a5c6aef051400155288" ]

[ "Economic_inequality", "Economic_inequality", "Economic_inequality" ]

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Each year, the southern California area has about 10,000 earthquakes. Nearly all of them are so small that they are not felt. Only several hundred are greater than magnitude 3.0, and only about 15–20 are greater than magnitude 4.0. The magnitude 6.7 1994 Northridge earthquake was particularly destructive, causing a substantial number of deaths, injuries, and structural collapses. It caused the most property damage of any earthquake in U.S. history, estimated at over $20 billion.

How many earthquakes does southern California experience in a year?

[ "10,000", "10,000", "10,000" ]

[ 50, 50, 50 ]

[ "5705fec152bb89140068977a", "5705fec152bb89140068977a", "5705fec152bb89140068977a" ]

[ "Southern_California", "Southern_California", "Southern_California" ]

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The plague disease, caused by Yersinia pestis, is enzootic (commonly present) in populations of fleas carried by ground rodents, including marmots, in various areas including Central Asia, Kurdistan, Western Asia, Northern India and Uganda. Nestorian graves dating to 1338–39 near Lake Issyk Kul in Kyrgyzstan have inscriptions referring to plague and are thought by many epidemiologists to mark the outbreak of the epidemic, from which it could easily have spread to China and India. In October 2010, medical geneticists suggested that all three of the great waves of the plague originated in China. In China, the 13th century Mongol conquest caused a decline in farming and trading. However, economic recovery had been observed at the beginning of the 14th century. In the 1330s a large number of natural disasters and plagues led to widespread famine, starting in 1331, with a deadly plague arriving soon after. Epidemics that may have included plague killed an estimated 25 million Chinese and other Asians during the 15 years before it reached Constantinople in 1347.

What is the term for a disease that is not commonly present?

[ "" ]

[ -1 ]

[ "5a2eb95ca83784001a7d242d" ]

[ "Black_Death" ]

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In ring theory, the notion of number is generally replaced with that of ideal. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.

What type of ideals generalize Noetherian arithmetic?

[ "" ]

[ -1 ]

[ "5a89473c3b2508001a72a599" ]

[ "Prime_number" ]

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Datasets:

autoevaluate
/

autoeval-staging-eval-squad_v2-squad_v2-38b250-14916076

Dataset Card for AutoTrain Evaluator

Contributions