the-french-artist/wikipedia_20220301.simple_sentence_embeddings huggingface dataset:
Four color theorem 
 In the real world, this is not true: Alaska as part of the United States, Nakhchivan as part of Azerbaijan, and Kaliningrad as part of Russia are not contiguous.
Four color theorem 
 Because the territory of a particular country must be the same color, four colors may not be enough.
Four color theorem 
 For instance, consider a simplified map, such as the one shown on the left: In this map, the two regions labeled A belong to the same country, and must be the same color.
Four color theorem 
 This map then requires five colors, since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others.
Four color theorem 
 A similar construction also applies if a single color is used for all bodies of water, as is usual on real maps.
Four color theorem 
 The set of regions of a map can be represented more abstractly as an undirected graph that has a vertex for each region and an edge for every pair of regions that share a boundary segment.
Four color theorem 
 This graph is planar: it can be drawn in the plane without crossings by placing each vertex at an arbitrarily chosen location within the region to which it corresponds, and by drawing the edges as curves that lead without crossing within each region from the vertex location to each shared boundary point of the region.
Four color theorem 
 In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices have the same color, or for short, "every planar graph is four-colorable" (; ).
Four color theorem 
 Augustus de Morgan first discussed the problem, in a letter he wrote to Rowan Hamliton in August 1852.
Four color theorem 
 In the letter, de Morgan asks whether four colors are really enough to color a map, such that countries that are next to each other get different colors.
Four color theorem 
 English mathematician Arthur Cayley presented the problem to the mathematical society in London, in 1878.
Four color theorem 
 When he falsified Cayley's proof, Kempe also showed a proof for a problem he called Five color theorem.
Four color theorem 
 The second restriction is that countries need to have a common border; if they only touch in a point, they can be colored with the same color.
Four color theorem 
 Even though Kempe's proof was wrong, he used some of the ideas which would later permit a correct proof.
Four color theorem 
 They were able to reduce the number of cases that would need to be tested to 1936; a later version was made that relied on testing only 1476 cases.
Four color theorem 
 In 1996, Neil Robertson, Daniel Sanders, Paul Seymour and Robin Thomas improved the method, and reduced the number of cases to be tested to 663.
Four color theorem 
 The four color theorem is the first big mathematical problem that was proved with the help of a computer.
Four color theorem 
 To verify the proof, it is necessary to rely on a correctly working software and hardware to validate the proof.
Four color theorem 
 The four color theorem has been notorious for attracting a large number of false proofs and disproofs in its long history.
Four color theorem 
 The newspaper did this as a matter of policy; it feared that the proof would be shown false like the ones before it .
Four color theorem 
 Some proofs took a long time, till they could be falsified: For Kempe's and Tait's proofs falsifying them took over a decade.
Four color theorem 
 Because the four color theorem is true, this is always possible; however, because the person drawing the map is focused on the one large region, they do not notice that the remaining regions can in fact be colored with three colors.
Four color theorem 
 This trick can be generalized: If the colors of some regions in a map are chosen beforehand, it becomes impossible to color the remaining regions in such a way that in total, only four colors are used.
Four color theorem 
 Someone verifying the counterexample may not think that it may be needed to change the color of these regions.
Four color theorem 
 Perhaps one effect underlying this common misconception is the fact that the color restriction is not transitive: a region only has to be colored differently from regions it touches directly, not regions touching regions that it touches.
Four color theorem 
 Other false disproofs violate the assumptions of the theorem in unexpected ways, such as using a region that has multiple disconnected parts, or not allowing regions of the same color from touching at a point.
Four color theorem 
 Since they belong to the country, they need to be colored with the same color as the parent country.
Four color theorem 
 Even though it is easy to check if a graph is planar, finding the fewest number of colors needed to color it is very difficult.
Four color theorem 
 Many of the problems that happen when trying to solve the four color theorem are related to discrete mathematics.
Four color theorem 
 The four color theorem requires the "map" to be on a flat surface, what mathematicians call a plane.
Four color theorem 
 In 1890, Percy John Heawood created what is called Heawood conjecture today: It asks the same question as the four color theorem, but for any topological object.
Four color theorem 
 Chechulin V. L. About а one proof of a planar's graphs 4-chromatically http://www.uresearch.psu.ru/files/articles/17_89592.doc
Theorem 
 Sometimes, such proofs involve other areas of mathematics or show connections between different areas.
Theorem 
 An excellent example is Fermat's Last Theorem, and there are many other examples of simple yet deep theorems in number theory and combinatorics (among other areas).
Theorem 
 Both of these theorems are only known to be true by reducing them to a computational search which is then verified by a computer program.
Theorem 
 At first, many mathematicians did not accept this form of proof, but it has become more widely accepted over the years.
Theorem 
 The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved.
Theorem 
 Many mathematical theorems can be reduced to simpler computations, including polynomial identities, trigonometric identities and hypergeometric identities.
Triceratops 
 As adults, they grew up to 30 feet long by 9 feet tall (), and probably weighed around 5,400 kg (12,000 lb).
Triceratops 
 Its defence had to stand up to attacks from taller theropods, hence the bony shield which covered its neck.
Triceratops 
 Holes made by teeth have been found on the bony frill behind the horns, and on the sacrum (the part of the spine above the pelvis).
Triceratops 
 Paleontologist John Scannella observed: "It is hard to walk out into the Hell Creek Formation and not stumble upon a Triceratops weathering out of a hillside".
Triceratops 
 Forty-seven complete or partial skulls were discovered in just that area during the decade 2000–2010.
Triceratops 
 Individual Triceratops were about 7.9 to 9.0 m (26.0–29.5 ft) in length, 2.9 to 3.0 m (9.5–9.8 ft) in height, and 6.1–12.0 tonnes (13,000–26,000 lb) in weight.
Triceratops 
 The largest known skull (specimen BYU 12183) is estimated to have been  in length when complete, and could reach almost a third of the length of the entire animal.
Triceratops 
 It bore a single horn on the snout, above the nostrils, and a pair of horns approximately 1 m (3 ft) long, with one above each eye.
Triceratops 
 Most other ceratopsids had large holes (fenestrae) in their frills, while those of Triceratops were noticeably solid.
Triceratops 
 Triceratops species were sturdy, with strong limbs and short three-hoofed hands and four-hoofed feet.
Triceratops 
 Originally, it was believed that the front legs of the animal had to be sprawling at angles from the thorax, in order to better bear the weight of the head.
Triceratops 
 However, evidence of trackways, and reconstructions of skeletons show that Triceratops and other ceratopsids had an upright stance during normal locomotion, with the elbows flexed and slightly bowed out.
Triceratops 
 The stance was intermediate between fully upright and fully sprawling, similar the modern rhinoceros.
Triceratops 
 Although Triceratops are commonly portrayed as herd animals, there is little evidence that they lived in herds.
Triceratops 
 In 2012, a group of three Triceratops in relatively complete condition, each of varying sizes from a full-grown adult to a small juvenile, were found in Wyoming, near Newcastle.
Triceratops 
 The remains are currently under excavation by paleontologist Peter Larson and a team from the Black Hills Institute.
Triceratops 
 It is believed that the animals were traveling as a family unit, but it remains unknown if the group consists of a mated pair and their offspring, or two females and a juvenile they were caring for.
Triceratops 
 The remains also show signs of predation or scavenging from Tyrannosaurus, particularly on the largest specimen, with the bones of the front limbs showing breakage and puncture wounds from Tyrannosaurus teeth.
Triceratops 
 These remains are very common: one palaeontologist reported having seen 200 specimens of T. prorsus in the Hell Creek Formation of Montana, U.S. Similarly, Barnum Brown claimed to have seen over 500 skulls in the field.
Triceratops 
 Triceratops teeth, horn fragments, frill fragments, and other skull fragments are abundant fossils in the latest Upper Cretaceous of western North America.
Triceratops 
 In 1986, Robert Bakker estimated it as making up 5/6ths of the large dinosaur fauna at the end of the Cretaceous.
Triceratops 
 Triceratops was one of the last ceratopsian genera to appear before the Cretaceous–Paleogene extinction event.
Triceratops 
 The related Torosaurus, and the more distantly related diminutive Leptoceratops, were also present, though their remains are rarely found.
Triceratops 
 Triceratops were herbivorous, and because of their low head, their primary food was probably low growth, although they may have been able to knock down taller plants with their horns, beak, and bulk.
Triceratops 
 Triceratops teeth were arranged in groups called batteries, of 36 to 40 tooth columns, in each side of each jaw with 3 to 5 stacked teeth per column, depending on the size of the animal.
Triceratops 
 This gives a range of 432 to 800 teeth, of which only a fraction were in use at any given time (tooth replacement was continuous and occurred throughout the life of the animal).
Triceratops 
 The great size and numerous teeth of Triceratops suggests that they ate large volumes of fibrous plant material, such as palms and cycads.
Triceratops 
 The two main theories have revolved around use in combat, or display in courtship, with the latter thought now to be the most likely primary function.
Triceratops 
 Triceratops were long thought to have used their horns and frills in combat with predators such as Tyrannosaurus.
Triceratops 
 There is evidence that Tyrannosaurus did have aggressive head-on encounters with Triceratops, based on partially healed tyrannosaur tooth marks on a Triceratops brow horn and squamosal; the bitten horn is also broken, with new bone growth after the break.
Triceratops 
 In addition to combat with predators using horns, Triceratops are classically shown engaging each other in combat with horns locked.
Triceratops 
 While studies show that such activity would be feasible, if unlike that of present-day horned animals, there is disagreement about whether they did so.
Triceratops 
 A similar theory has been proposed regarding the plates of Stegosaurus, although this use alone would not account for the bizarre and extravagant variation seen in different members of the Ceratopsidae.
Triceratops 
 The theory of their use in sexual display was first proposed by Davitashvili in 1961 and has gained increasing acceptance since.
Triceratops 
 Evidence that visual display was important, either in courtship or in other social behavior, can be seen in the fact that horned dinosaurs differ markedly in their adornments, making each species highly distinctive.
Triceratops 
 Also, modern living creatures with such displays of horns and adornments use them in similar behavior.
Triceratops 
 A 2006 study of the smallest Triceratops skull, ascertained to be a juvenile, shows the frill and horns developed at a very early age, predating sexual development and thus probably important for visual communication and species recognition in general.
Triceratops 
 When examined closely, the hole in the bone has a diameter that is very similar to diameter of the distal end of a Triceratops horn.
Border 
 To cross the border, a passport or a visa might be necessary; such as the Wagah border between Pakistan and the Republic of India.
Genus 
 When biologists talk about a genus, they mean one or more species of animals or plants that are closely related to each other.
Genus 
 So instead of saying 'felids', or 'felines', one says 'cats' both for the family pet, and for all the cat family (Felidae).
Cretaceous 
 The Cretaceous period is the ninth and longest period in the Phanerozoic eon, lasting 80 million years from 145 to 65 million years ago (mya).
Cretaceous 
 North America was partly covered by an epicontinental inland sea, and half of Britain was under water.
Cretaceous 
 Chalk is 95% to 99% made up of coccoliths, the calcium carbonate (CaCO3) plates which tiny single-celled algae (coccolithophorids) produce.
Cretaceous 
 At its end came the famous meteorite strike which, with the volcanic flood basalts spewed out in the Deccan Plateau (India), ended so many of the dominant life-forms: see K/T extinction event.
Textbook 
 Workbooks are designed not to teach but to provide practice and to highlight areas which need more learning.
Textbook 
 A revision guide is a type of textbook that is used to remind the learner about the subject and give him/her extra practice, especially before an examination.
Textbook 
 Sometimes, especially at university, students have to buy the textbooks they need themselves or borrow them from a library.
Axiom 
 The axiom is to be used as the premise or starting point for further reasoning or arguments, usually in logic or in mathematics.
Axiom 
 Here, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived.
Axiom 
 If A and B are two numbers that are the same, and C and D are also the same, A+C is the same as B+D.
Axiom 
 If A and B are two numbers that are the same, and C and D are also the same, A-C is the same as B-D.
Axiom 
 If you divide a number by anything more than 1, the quotient (result) will be less than the original number.
Set 
 A set is defined by its members, so any two sets with the same members are the same (e.g., if set  and set  have the same members, then ).
Set 
 Anything can be a member of a set, including sets themselves (though if a set is a member of itself, paradoxes such as Russell's paradox can happen).
Set 
 So for a set of fruits, it would make no difference if there is one orange, or if there are 10 oranges.
Set 
 If we consider, say, some sets of American cars, e.g. a set of all Fords and a set of all Dodges, we may also wish to consider the whole set of American cars.
Set 
 In other words, a universe is a collection of all the elements one wishes to consider in a given problem.
Set 
 When mathematicians talk about a set, they sometimes want to know how big a set is (or what is the cardinality of the set).
Set 
 Two sets have the same cardinality if we can pair up their elements—if we can join two elements, one from each set.
Set 
 There are more real numbers than there are natural numbers, for example, which means we cannot pair up the set of integers and the set of real numbers, even if we worked forever.
Set 
 The natural numbers are nicknamed "the counting numbers", since they are what we usually use to count things with.
Set 
 If you look at the set  and the set , you can see that all elements in the first set are also in the second set.
Set 
 The intersection  of two sets  and  is a set that contains all the elements that are both in set  and in set  at the same time.
Set 
 Example: When  is the set of all cheap cars, and  is the set of all American cars, then  is the set of all cheap American cars.
Set 
 This "or" is the inclusive disjunction, so the union also contains the elements, that are in set  and in set .
Set 
 Example: When  is the set of all cheap cars, and  is the set of all American cars, then  is the set of all cars, without all expensive cars that are not from America.
Set 
 Example: When  is the set of all cars, and  is the set of all cheap cars, then C is the set of all expensive cars.
Set 
 Example: When  is the set of all cheap cars, and  is the set of all American cars, then  is the set of all expensive American cars.
Set 
 If you exchange the sets in the set difference, the result is different:In the example with the cars, the difference  is the set of all cheap cars, that are not made in America.
Set 
 Most mathematicians use uppercase ITALIC (usually Roman) letters to write about sets (such as , , ).
Set 
 Another way, called the set-builder notation, is by a statement of what is true of the members of the set, like this:
Set 
 In spoken English, this reads: "the set of all x such that x is a natural number and x is less than four".
Set 
 In the illustration of the intersection operation the left circle shows set  and the right circle shows set .
Set 
 This set includes all rational numbers, together with all irrational numbers (that is, numbers which cannot be rewritten as fractions, such as   and √2).
Set 
 The mathematician Bertrand Russell found that there are problems with the informal definition of sets.
Set 
 All the men in the town do not like beards, so they either shave themselves, or they go to the barber shop to be shaved by the barber.
Set 
 We can therefore make a statement about the barber himself: The barber shaves all men that do not shave themselves.
Set 
 He only shaves those men (since the others shave themselves and do not need a barber to give them a shave).
Set 
 If he does not shave himself, he falls in the category of those who do not shave themselves, and so, cannot be a barber.
Maldives 
 The Maldives has a tropical monsoon climate (Am) under the Köppen climate classification, which is affected by the large landmass of South Asia to the north.
Maldives 
 Because the Maldives has the lowest elevation of any country in the world, the temperature is constantly hot and often humid.
Maldives 
 These factors set off a rush of moisture-rich air from the Indian Ocean over South Asia, resulting in the southwest monsoon.
Maldives 
 Two seasons dominate Maldives' weather: the dry season associated with the winter northeastern monsoon and the rainy season which brings strong winds and storms.
Maldives 
 The shift from the dry northeast monsoon to the moist southwest monsoon occurs during April and May.
Maldives 
 During this period, the southwest winds contribute to the formation of the southwest monsoon, which reaches Maldives in the beginning of June and lasts until the end of August.
Maldives 
 However, the weather patterns of Maldives do not always conform to the monsoon patterns of South Asia.
Maldives 
 The monsoonal influence is greater in the north of the Maldives than in the south, more influenced by the equatorial currents.
Maldives 
 The highest temperature ever recorded in Malé was 39.6 °C (103.3 °F) on 1 January 2000 and the lowest was 20.4 °C (68.7 °F) on 20 March 1989 and 21 January 2017.
Prime Minister of Australia 
 The prime minister leads the cabinet of Australia and is the highest ranking minister of the Crown in the federal parliament.
1190s 
 1192 – Minamoto no Yoritomo granted title of shogun, thereby officially establishing Kamakura shogunate, the first shogunate in the history of Japan.
1190s 
 May 27, 1199 Friday – John Lackland (1167–1216) became King of England upon the death of his older brother Richard the Lion Hearted.
1190s 
 Lackland was so called, because his inheritance of the Angevin Empire did not have land for him; he was the youngest son of Henry II, so this event could not happen..
Structure 
 A structure may refer to buildings, machines and similar things made by people as well as to a rock or a mineral, atoms or sentences.
Hardcore punk 
 Hardcore punk (or hardcore) is a style of punk music that usually sounds louder, faster and more angry than earlier punk rock music from the 1970s.
Hardcore punk 
 Some of the important bands of the genre are Bad Brains, Black Flag, D.O.A, Minor Threat and Cro-Mags.
Hardcore punk 
 The band D.O.A. may have helped to make the term more popular with the title of their album Hardcore '81 in 1981.
Hardcore punk 
 People in hardcore punk bands and people who listened to hardcore punk wore clothes that were of two types.
Hardcore punk 
 Some people wore clothes and hairstyles that were strange and a bit scary, like Mohawk hairstyles and leather jackets with metal studs.
Hardcore punk 
 In the early 1980s, hardcore bands from England began to get noticed, such as Discharge, The Exploited, and The Varukers.
Hardcore punk 
 Bands would record their own albums using simple four-track recorders, and the sell the records themselves at shows or through the mail.
Hardcore punk 
 Hardcore influenced the creation of new styles of music, including alternative rock, grunge, alternative metal, metalcore, thrash metal and post-hardcore.
Hardcore punk 
 They spoke against political leaders, such as American president Ronald Reagan and British prime minister Margaret Thatcher.
Hardcore punk 
 A small number of hardcore musicians have expressed right wing views, including the band Antiseen, whose guitarist Joe Young ran for office as a Libertarian in North Carolina.
Hardcore punk 
 The electric guitars in hardcore bands are distorted, noisy, and fuzzy because the guitarists are using electronic boxes called distortion pedals.
Hardcore punk 
 Some hardcore guitarists have changed the tuning of their guitar strings so they play notes that are lower than those a normal guitar can play.
Hardcore punk 
 The drums in hardcore music play loud and fast, often characterized by use of the double-bass pedal, particularly notable in the breakdowns.
Limestone 
 Limestone is a sedimentary rock, made up mostly of the mineral calcite, a form of calcium carbonate, CaCO3.
Limestone 
 It is originally made by small photosynthetic protists like certain bacteria, coccolithophores, and algae.
Limestone 
 Then the carbonate gets used by various animals, like bivalves and the polyps which make coral reefs.
Limestone 
 "[Coccolithophores] were the first organisms to bring about a steady rain of calcium carbonate from the upper, illuminated zone of the ocean towards the deep sea".p144
Limestone 
 Carbonates are rocks formed from these sources, plus any CaCO3 which is precipitated out of solution.
Limestone 
 For example, if the concentration of CaCO3 is high, then a temperature drop, or a pH (acidity) change, can cause precipitation.
Limestone 
 Florida Bay, the shallow lagoon behind the Florida Keys, is a model for the conditions which produced limestones on the continental shelf.
Limestone 
 This is mainly formed by Melobesia, a small red alga which lives on the sea grass.p151 Finally, the organisms produce huge amounts of slime, which is a polysaccharide.
Limestone 
 There is good reason to think the slime is produced by microorganisms to control the way their carbonate skeletons are formed.p139 As a side-effect the slime limits the amount of carbonate which goes into solution.p155
Limestone 
 "You have to drill for more than 3000 feet before you reach a different rock type, and that is of Cretaceous age—some 70 million years ago".p158
Limestone 
 Chalk is ~99% coccoliths; Solnhofen limestone is famous for its animal fossils; and there are many shelly limestones from different periods.
Limestone 
 Limestone is not found much before the Cambrian period, because shelled animals had not yet evolved.
Limestone 
 When limestone is heated to a high temperature, and squeezed at the base of a mountain, it recrystallizes into the metamorphic rock marble.