the-french-artist/wikipedia_20220301.simple_sentence_split huggingface dataset:
A similar construction also applies if a single color is used for all bodies of water, as is usual on real maps.
An easier to state version of the theorem uses graph theory.
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.
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.
Conversely any planar graph can be formed from a map in this way.
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" (; ).
The first person to name the problem was Francis Guthrie, in 1852.
He was a law student in England, at the time.
He found that he needed at least four colors to color a map of the counties of England.
Augustus de Morgan first discussed the problem, in a letter he wrote to Rowan Hamliton in August 1852.
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.
English mathematician Arthur Cayley presented the problem to the mathematical society in London, in 1878.
Within a year, Alfred Kempe found what looked like a proof  of the problem.
Eleven years later, in 1890, Percy Heawood showed that Alfred's proof was wrong.
Peter Guthrie Tait presented another attempt at a proof in 1880.
It took eleven years to show that Tait's proof did not work either.
In 1891, Julius Petersen could show this.
When he falsified Cayley's proof, Kempe also showed a proof for a problem he called Five color theorem.
The theorem says that any such map can be colored with no more than five colors.
There are two restrictions: First, any country is contiguous, there are no exclaves.
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.
Even though Kempe's proof was wrong, he used some of the ideas which would later permit a correct proof.
In the 1960s and 1970s, Heinrich Heesch developed a first sketch of a proof by computer.
Kenneth Appel and Wolfgang Haken improved this sketch in 1976 .
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.
Each case needed to be tested by a computer .
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.
Again, each case needed to be tested by computer.
In 2005, Georges Gonthier and Benjamin Werner developed a formal proof.
This was an improvement, because it allowed to use theorem-proving software, for the first time.
The software used is called Coq.
The four color theorem is the first big mathematical problem that was proved with the help of a computer.
Because the proof cannot be done by a human, some mathematicians did not recognize it as correct.
To verify the proof, it is necessary to rely on a correctly working software and hardware to validate the proof.
Because the proof was done by computer, it is also not very elegant.
Attempts that turned out to be wrong
The four color theorem has been notorious for attracting a large number of false proofs and disproofs in its long history.
At first, The New York Times refused to report on the Appel–Haken proof.
The newspaper did this as a matter of policy; it feared that the proof would be shown false like the ones before it .
The simplest counterexamples generally try to create one region which touches all the others.
This forces the remaining regions to be colored with only three colors.
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.
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.
Someone verifying the counterexample may not think that it may be needed to change the color of these regions.
This will make the counterexample look valid, even though it is not.
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.
If this were the restriction, planar graphs would require arbitrarily large numbers of colors.
Coloring political maps
In real life, many countries have exclaves or colonies.
Since they belong to the country, they need to be colored with the same color as the parent country.
This means that usually, more than four colors are needed to color such a map.
When mathematicians talk about the graph associated with the problem, they say that is not planar.
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.
It is NP-complete, which is one of the most difficult problems that exist.
The fewest number of colors needed to color a graph is known as its chromatic number.
Many of the problems that happen when trying to solve the four color theorem are related to discrete mathematics.
For this reason, methods from algebraic topology are often used.
Extension to "non-flat" maps
The four color theorem requires the "map" to be on a flat surface, what mathematicians call a plane.
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.
As an example, a torus can be colored with at most seven colors.
The Heawood conjecture gives a formula that works for all such objects, except the Klein bottle.
Articles and Books
Chechulin V. L. About а one proof of a planar's graphs 4-chromatically http://www.uresearch.psu.ru/files/articles/17_89592.doc
Mentioned in this article
:WikiBooks:Famous Theorems of Mathematics/Four color theorem
Mathematical theorems
A theorem is a proven idea in mathematics.
Theorems are proved using logic and other theorems that have already been proved.
A minor theorem that one must prove to prove a major theorem is called a lemma.
Theorems are made of two parts: hypotheses and conclusions.
Some theorems are trivial, since they directly follow from the propositions.
Other theorems are called "deep", because their proof is long and difficult.
Sometimes, such proofs involve other areas of mathematics or show connections between different areas.
A theorem might be simple to state and yet be deep.
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).
There are other theorems for which a proof is known, but cannot be easily written down.
Among the best examples are the four color theorem and the Kepler conjecture.
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.
At first, many mathematicians did not accept this form of proof, but it has become more widely accepted over the years.
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.
Many mathematical theorems can be reduced to simpler computations, including polynomial identities, trigonometric identities and hypergeometric identities.
Triceratops was a huge herbivorous ceratopsid dinosaur from the late Cretaceous.
Its name came from having three horns on its head.
They were mainly found in North America.
As adults, they grew up to 30 feet long by 9 feet tall (), and probably weighed around 5,400 kg (12,000 lb).
Fully-grown, their skull was enormous (two meters long and nearly a meter wide).
Triceratops was a low browser with a bony beak in front of its jaws.
The jaws had close-set grinding teeth.
Its defence had to stand up to attacks from taller theropods, hence the bony shield which covered its neck.
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).
Many fossils of Triceratops have been collected since the genus was first described in 1889.
There is at least one complete individual skeleton.
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".
Forty-seven complete or partial skulls were discovered in just that area during the decade 2000–2010.
Specimens showing life stages from hatchling to adult have been found.
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.
The most distinctive feature is their large skull, among the largest of all land animals.
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.
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.
Most other ceratopsids had large holes (fenestrae) in their frills, while those of Triceratops were noticeably solid.
Limbs
Triceratops species were sturdy, with strong limbs and short three-hoofed hands and four-hoofed feet.
The posture of these dinosaurs has long been the subject of some debate.
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.
This stance can be seen in paintings by Charles Knight and Rudolph Zallinger.
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.
The stance was intermediate between fully upright and fully sprawling, similar the modern rhinoceros.
Although Triceratops are commonly portrayed as herd animals, there is little evidence that they lived in herds.
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.
The remains are currently under excavation by paleontologist Peter Larson and a team from the Black Hills Institute.
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.
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.
For many years, Triceratops finds were known only from solitary individuals.
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 teeth, horn fragments, frill fragments, and other skull fragments are abundant fossils in the latest Upper Cretaceous of western North America.
It was the most dominant herbivore of the time.
In 1986, Robert Bakker estimated it as making up 5/6ths of the large dinosaur fauna at the end of the Cretaceous.
Triceratops was one of the last ceratopsian genera to appear before the Cretaceous–Paleogene extinction event.
The related Torosaurus, and the more distantly related diminutive Leptoceratops, were also present, though their remains are rarely found.
Dentition and diet
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.
The jaws were tipped with a deep, narrow beak, good for grasping and plucking.
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.
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).
The great size and numerous teeth of Triceratops suggests that they ate large volumes of fibrous plant material, such as palms and cycads.
Functions of the horns and frill
There has been much speculation over the functions of Triceratops''' head adornments.
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 were long thought to have used their horns and frills in combat with predators such as Tyrannosaurus.
The idea was discussed first in 1917 and again 70 years later by Robert Bakker.
Sternberg C.H. 1917.
Hunting dinosaurs in the badlands of the Red Deer River, Alberta, Canada.
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.
Since the Triceratops wounds healed, the Triceratops survived the encounter.
Tyrannosaurus is also known to have fed on Triceratops.
Evidence for this includes a heavily tooth-scored Triceratops ilium  and sacrum.
In addition to combat with predators using horns, Triceratops are classically shown engaging each other in combat with horns locked.
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.
The large frill also may have helped to increase body area to regulate body temperature.
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.
This observation is highly suggestive of what is now believed to be the primary function, display.
The theory of their use in sexual display was first proposed by Davitashvili in 1961 and has gained increasing acceptance since.
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.
Also, modern living creatures with such displays of horns and adornments use them in similar behavior.
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.
Paleopathology
One skull, assigned to Triceratops, has a hole in the jugal bone.
It looks like a puncture wound sustained while the animal was still alive.
This is supported by signs of healing that are present in the bone around the supposed wound.
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.
This is evidence of competition between individual dinosaurs.
Martin A.J. 2006.
Introduction to the study of dinosaurs.
2nd ed, Oxford, Blackwell.
299-300. .
Triceratops  (short summary and good color illustration)
Triceratops For Kids (a fact sheet about the Triceratops'' with activities for kids)
Smithsonian Exhibit
Triceratops Skull Picture
Ceratopsids
Dinosaurs of North America
The border between two regions is a line, where one region is next to the other.
If the regions are countries, the border can only be crossed at certain points.
To cross the border, a passport or a visa might be necessary; such as the Wagah border between Pakistan and the Republic of India.
A genus is a rank in the biological classification (or taxonomy).
It is above species, and below families.
A genus can include more than one species.
When biologists talk about a genus, they mean one or more species of animals or plants that are closely related to each other.
As with other taxa, the plural is different from other English words because it is a Latin word. '
Genus' is the singular, and 'genera' is the plural form of the word.
When printing the scientific name of an organism, the name is always in italic.
A name of species has two parts, with the genus first.
For example, in "Felis silvestris", Felis is the genus.
The genus name always begins with a capital letter.
In "Felis silvestris catus", the third word is the subspecies, which is not often used.
As a common word
In writing, genus names in Latin may be 'anglicised' to form a common name.
For example, the genus Pseudomonas is "pseudomonad" (plural: "pseudomonads").
In practice, most really common animals and plants already have a common name.
So instead of saying 'felids', or 'felines', one says 'cats' both for the family pet, and for all the cat family (Felidae).
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).
It followed the Jurassic period, and is the third and last period of the Mesozoic era.
After it came the Cainozoic era.
Though famous for its land fauna, the Cretaceous sea level rose much higher than today.
North America was partly covered by an epicontinental inland sea, and half of Britain was under water.
In the Upper Cretaceous, chalk, a type of limestone, was laid down in warm shallow seas.
Chalk is 95% to 99% made up of coccoliths, the calcium carbonate (CaCO3) plates which tiny single-celled algae (coccolithophorids) produce.
Upper Cretaceous 100.5 million years ago (mya) to 65 mya
Lower Cretaceous 145.5 mya to 100.5 mya.
A textbook is a book used for the study of a subject.
People use a textbook to learn facts and methods about a certain subject.
Textbooks sometimes have questions to test the knowledge and understanding of the learner.
A workbook is a type of textbook that has only practice questions and exercises.
Workbooks are designed not to teach but to provide practice and to highlight areas which need more learning.
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.
A textbook is usually lent to students by a school to accompany a course the school is teaching.
Sometimes, especially at university, students have to buy the textbooks they need themselves or borrow them from a library.
Most textbooks are only published in printed format.
However, some are now available online as electronic books.
An axiom is a concept in logic.
It is a statement which is assumed to be true without question, and which does not require proof.
It is also known as a postulate (as in the parallel postulate).
The axiom is to be used as the premise or starting point for further reasoning or arguments, usually in logic or in mathematics.
This means it cannot be proved within the discussion of a problem.
So inside some discussion, it is thought to be true.
There are many reasons why it has no proof.
The statement might be obvious.
This means most people think it is clearly true.
An example of an obvious axiom is the principle of contradiction.
It says that a statement and its opposite cannot both be true at the same time and place.
The statement is based on physical laws and can easily be observed.
An example is Newton's laws of motion.
They are easily observed in the physical world.
The statement is a proposition.
Here, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived.
This means the emphasis is on what happens if the axiom is true.
Whether the proposition is empirically true is not the goal of logic.
This is a more modern definition of an axiom.
Logic can be used to find theorems from the axioms.
Then those theorems can be used to make more theorems.
This is often how math works.
Axioms are important because logical arguments start with them.
Euclid's axioms
Euclid of Alexandria was a Greek mathematician.
Around the year 300 BC, he made the earliest list of axioms which we know of.
The following are just a few of them:
Two numbers that are both the same as a third number are the same number.
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.
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.
Two shapes that fill exactly the same space are the same shape.
If you divide a number by anything more than 1, the quotient (result) will be less than the original number.
Axiom of choice, an axiom in set theory
A set is an idea from mathematics.
A set has members (also called elements).
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 ).
A set cannot have the same member more than once.
Membership is the only thing that matters.
For example, there is no order or other difference among the members.
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).
What to do with sets
Element of
Various things can be put into a bag.
Later on, a good question would be if a certain thing is in the bag.
Mathematicians call this element of.
Something is an element of a set, if that thing can be found in the respective bag.
The symbol used for this is :
which means that  is in the bag  or  is an element of .
Unlike a bag, a set can contain at most one item of a given type.
So for a set of fruits, it would make no difference if there is one orange, or if there are 10 oranges.
Like a bag, a set can also be empty.
The empty set is like an empty bag: it has nothing in it.
The "empty set" is also called the null set and is represented by the symbol .
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.
In this case, the set of all American cars would be called a universe.
In other words, a universe is a collection of all the elements one wishes to consider in a given problem.
The universe is usually named .
Comparing sets
Two sets can be compared.
This is like looking into two different bags.
If they contain the same things, they are equal.
No matter, in which order these things are.
For example, if  and , the sets are the same.
Cardinality of a 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).
They do this by counting how many elements are in the set (how many items are in the bag).
For finite sets the cardinality is a simple number.
The empty set has a cardinality of 0.
The set  has a cardinality of 2.
Two sets have the same cardinality if we can pair up their elements—if we can join two elements, one from each set.
The set  and the set  have the same cardinality.
E.g, we could pair apple with sun, and orange with moon.
The order does not matter.
It is possible to pair all the elements, and none is left out.
But the set  and the set  have different cardinality.
Infinite cardinality
At times cardinality is not a number.
Sometimes a set has infinite cardinality.
The set of all integers is a set with infinite cardinality.
Some sets with infinite cardinality are bigger (have a bigger cardinality) than others.
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.
If you can count the elements of a set, it is called a countable set.
Countable sets include all sets with a finite number of members.
Countable sets also include some infinite sets, such as the natural numbers.
You can count the natural numbers with .
The natural numbers are nicknamed "the counting numbers", since they are what we usually use to count things with.
An uncountable set is an infinite set that is impossible to count.
If we try to count the elements, we will always skip some.
It does not matter what step we take.
The set of real numbers is an uncountable set.
There are many other uncountable sets, even such a small interval like .[3]
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.
We say:  is a subset of
As a formula it looks like this:
In general, when all elements of set  are also elements of set , we call   a subset of :
It is usually read " is contained in ."
Example: Every Chevrolet is an American car.
So the set of all Chevrolets is contained in the set of all American cars.
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.
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.
The union  of two sets  and  is a set that contains all the elements that are in set  or in set .
This "or" is the inclusive disjunction, so the union also contains the elements, that are in set  and in 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.
The complement of  is the universe  without all the elements of :
The universe  is the set of all things you speak about.
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.
The set difference of  and  is the set  without all the elements of :
It is also called the relative complement of  in  .
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.
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.
Most mathematicians use uppercase ITALIC (usually Roman) letters to write about sets (such as , , ).
The things that are seen as elements of sets are usually written with lowercase Roman letters.
is set which has members 1, 2, and 3.
{x | x is a natural number & x < 4}.
In spoken English, this reads: "the set of all x such that x is a natural number and x is less than four".
The symbol [ipe "|" means "such that" or "so that".
The empty set is written in a special way: ,  or .
When object a is the member of set  it is written as:
In spoken English, this reads: "a is a member of ".
Venn diagrams
To illustrate operations on sets mathematicians use Venn diagrams.
Venn diagrams use circles to show individual sets.
The universe is depicted with a rectangle.
Results of operations are shown as colored areas.
In the illustration of the intersection operation the left circle shows set  and the right circle shows set .
Special sets
Some sets are very important to mathematics.
They are used very often.
One of these is the empty set.
, denoting the set of all primes.
, denoting the set of all natural numbers.
That is to say,  = {1, 2, 3, ...}, or sometimes  = {0, 1, 2, 3, ...}.
, denoting the set of all integers (whether positive, negative or zero).
So  = {..., -2, -1, 0, 1, 2, ...}.
, denoting the set of all rational numbers (that is, the set of all proper and improper fractions).
So, , meaning all fractions  where a and b are in the set of all integers and b is not equal to 0.
For example,  and .
All integers are in this set since every integer a can be expressed as the fraction .
, denoting the set of all real numbers.
This set includes all rational numbers, together with all irrational numbers (that is, numbers which cannot be rewritten as fractions, such as   and √2).
, denoting the set of all complex numbers.
Each of these sets of numbers has an infinite number of elements, and .
Paradoxes about sets
The mathematician Bertrand Russell found that there are problems with the informal definition of sets.
He stated this in a paradox called Russell's paradox.
An easier to understand version, closer to real life, is called the Barber paradox.
The barber paradox
There is a small town somewhere.
In that town, there is a barber.
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.
We can therefore make a statement about the barber himself: The barber shaves all men that do not shave themselves.
He only shaves those men (since the others shave themselves and do not need a barber to give them a shave).
This of course raises the question: What does the barber do each morning to look clean-shaven?
Halmos, Paul R., Naive Set Theory, Princeton, N.J.: Van Nostrand (1960)
Stoll, Robert R., Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979)
Allenby, R.B.J.T, Rings, Fields and Groups, Leeds, England: Butterworth Heinemann (1991)
The Maldives are a group of islands in the Indian Ocean.
They are southwest of India.
They are made up of the Maldive and Suadive archipelagos, with over 1,200 islands.
These islands are a part of 26 atolls.
And they are divided to 20 administrative atolls.
People live on about 200 of the islands.
It has more than 1126 coral reefs.
It is the flattest country in the world.
The highest point in the Maldives is only  high above the level of the sea.
Malé is the capital, with a population of 150,000 people.
Its official religion is Islam.
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.
Because the Maldives has the lowest elevation of any country in the world, the temperature is constantly hot and often humid.
These factors set off a rush of moisture-rich air from the Indian Ocean over South Asia, resulting in the southwest monsoon.
Two seasons dominate Maldives' weather: the dry season associated with the winter northeastern monsoon and the rainy season which brings strong winds and storms.
The shift from the dry northeast monsoon to the moist southwest monsoon occurs during April and May.
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.
However, the weather patterns of Maldives do not always conform to the monsoon patterns of South Asia.
The annual rainfall averages  in the north and  in the south.
The prime minister of Australia is the head of the executive branch of the government of Australia.
The prime minister leads the cabinet of Australia and is the highest ranking minister of the Crown in the federal parliament.
The prime minister is appointed by the monarch, who is advised by the governor-general.
The current prime minister of Australia, since August 2018, is Scott Morrison.
Prime ministerial power and roles
The prime minister can choose his or her own ministers;
Can choose election dates;
Represents the country;
Spokesperson for the government, both domestically and internationally;
Advisor to the Governor-General;
Create bills in parliament
Former prime ministers
As of , there are six living former Australian prime ministers.
Australia's Prime Ministers / National Archive of Australia
Biographies of Australia's Prime Ministers  / National Museum of Australia
Official website of the Prime Minister of Australia
Australia-related lists
1192 – Minamoto no Yoritomo granted title of shogun, thereby officially establishing Kamakura shogunate, the first shogunate in the history of Japan.
May 27, 1199 Friday – John Lackland (1167–1216) became King of England upon the death of his older brother Richard the Lion Hearted.
His coronation took place in Westminster Abbey.
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..
In science, structure is a fundamental concept.
It means the arrangement of parts.
Examples are the organs in a body, or  crystals in a substance like granite.
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.
It always reflects the hierarchy of its parts.
For structuralism, see Claude Lévi-Strauss
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 began in the late 1970s.
Some of the important bands of the genre are Bad Brains, Black Flag, D.O.A, Minor Threat and Cro-Mags.
Hardcore songs often have more political lyrics (words) than songs from 1970s punk.
No one is quite sure where the term "hardcore punk" came from.
The band D.O.A. may have helped to make the term more popular with the title of their album Hardcore '81 in 1981.
The first hardcore punk song was "Out of Vogue" by the Santa Ana band The Middle Class.
This band started to play a loud type of punk with shouted words.
People in hardcore punk bands and people who listened to hardcore punk wore clothes that were of two types.
Some people wore clothes and hairstyles that were strange and a bit scary, like Mohawk hairstyles and leather jackets with metal studs.
Other people wore regular clothes like black pants and T-shirts.
At hardcore punk concerts, people did a type of dancing called moshing.
This type of dance involves people pushing each other and jumping off the stage.
In the early 1980s, hardcore bands from England began to get noticed, such as Discharge, The Exploited, and The Varukers.
Discharge played with a sound that included heavy metal guitar styles.
Hardcore led to the creation of the straight edge movement.
People who are straight edge do not use alcohol, tobacco, and other drugs.
Straight edge led to two new groups, called hardline and youth crew.
Hardcore musicians also created independent record labels in the 1980s.
Many bands had a "Do it Yourself" (DIY) attitude.
Bands would record their own albums using simple four-track recorders, and the sell the records themselves at shows or through the mail.
Hardcore influenced the creation of new styles of music, including alternative rock, grunge, alternative metal, metalcore, thrash metal and post-hardcore.
Many early hardcore punk bands had songs about liberal politics such as anarchism and socialism.
They spoke against political leaders, such as American president Ronald Reagan and British prime minister Margaret Thatcher.
While George W. Bush was President of the United States, a number of hardcore bands criticized him.
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.
Former Misfits singer Michale Graves supported George W. Bush.
How it sounds
Hardcore band singers often sing by shouting the words of the song.
Hardcore songs often sound like the singer is angry.
Hardcore bands use electric guitars, electric bass guitar, drums, and a singer.
The electric guitars in hardcore bands are distorted, noisy, and fuzzy because the guitarists are using electronic boxes called distortion pedals.
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.
Limestone is a sedimentary rock, made up mostly of the mineral calcite, a form of calcium carbonate, CaCO3.
The calcium carbonate is originally produced by living organisms.
Later, some of it goes into solution in sea water.
Limestone rocks include mainly organic remains and precipitated carbonate.
Limestones are mostly made of calcium carbonate.
It is originally made by small photosynthetic protists like certain bacteria, coccolithophores, and algae.
Then the carbonate gets used by various animals, like bivalves and the polyps which make coral reefs.
They build their shells or framework out of CaCO3 crystals.
"[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
Carbonates are rocks formed from these sources, plus any CaCO3 which is precipitated out of solution.
Some carbonate dissolves in the water.
It gets  precipitated when the physical condition of the water changes.
For example, if the concentration of CaCO3 is high, then a temperature drop, or a pH (acidity) change, can cause precipitation.
Florida Bay, the shallow lagoon behind the Florida Keys, is a model for the conditions which produced limestones on the continental shelf.
Of course, all the species were somewhat different then but, as an ecosystem, it is similar.
In Florida Bay there is a lot of a type of flowering plant called sea grass.
There is also a lot of white, milky water full of calcium carbonate.
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.
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
"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
Fossiliferous limestones
Many limestones contain fossils, and some are made up almost entirely of skeletal material.
Chalk is ~99% coccoliths; Solnhofen limestone is famous for its animal fossils; and there are many shelly limestones from different periods.
Limestone is not found much before the Cambrian period, because shelled animals had not yet evolved.
When limestone is heated to a high temperature, and squeezed at the base of a mountain, it recrystallizes into the metamorphic rock marble.
Marbles are fairly dense, relatively easy to cut, or be shaped into blocks, and some forms are white in colour.
Limestone is hard enough to resist weathering but usually soft enough to be worked (cut and carved) by stonemasons.
It is used for building material.
Limestone, particularly Portland limestone, became popular in the 19th and 20th centuries for building things such as churches, banks and houses.
However, acid rain does slowly erode limestone.
Limestone is used to make glass and cement.
It is also used to make iron in a blast furnace, to remove impurities from the iron ore.
The roots of a plant is the part that is usually buried in the soil.
Roots are usually always under the soil, though–sometimes roots can be above the ground.
This is called an aerial root.
Also, stems can sometimes be under the soil (potatoes, for example).
Roots do not have leaves, and are different from stems inside.
Plants need their roots especially for three reasons: The roots take water and nutrients, a sort of food, from the soil.
They also often store this food.
And they fix the plants to the ground.
However, most plant roots do not work at their best unless they have help from fungi.
The symbiosis between roots and fungus is known as mycorrhiza.
the taproot system: there is one very big root that goes down into the ground, and many smaller roots that come out of it
the diffuse root system: there are many roots that go in all directions
Some roots go very deep into the ground.
One root that was found in Arizona, USA, was 60 m below the surface.
Roots are also very strong.
Some tree roots can destroy stones.
Roots are not green because the cells have no chlorophyll.
Root growth
Roots grow through the whole life of the plant.
They grow longer from the tip, adding cells to the end of each root.
The root adds cells to their tips, and they grow fatter as they add cells around their tube-like bodies.
At the tip of each root, there is a small group of tough, dead, hard cells called the root cap.
The root cap is the strongest part of the root tip, and its job is to push its way through the dirt to look for moisture and nutrients and protect the plant.
Aerial roots
Roots are usually found underground, but in some cases this is not so.
In the rainforest, the air is warm and humid (it has a lot of water).
Some rainforest plants, known as epiphytes, grow right on trees.
Their roots hang down in the air or running into the moss growing on the trees.
Many aerial roots, are used to receive water and nutrient intake directly from the air, from fog, dew or humid air.
Some trees have roots that are above the ground and underground.
Mangrove trees have aerial roots (roots which come up into the air).
They exchange gases with the atmosphere, just as leaves do.
They are an adaptation to the poor level of oxygen in the waterlogged soil of the mangrove swamp.
The banyan tree has a root system that is underground, but it also has roots that start in its branches and grow down towards the ground.
These roots not only take in water and nutrients from the soil, but they also help to support the long branches of the banyan tree.
Because of this extra support, banyan tree branches can be really long.
A banyantree in Lahaina, Maui, which was planted in 1873 by a man named William Owen Smith, has branches that are so long that this single tree covers a full square block in the city.
Root systems
There are two main kinds of roots systems: taproot systems and fibrous root systems.
A taproot system has one thick main root growing down from the plant's stem, and lots of smaller secondary roots branching off from this.
A taproot system is usually deeper than it is wide.
Often, we eat taproots, like carrots and turnips.
A fibrous root system has lots of roots growing in many directions.
There is not one main root.
A fibrous root system is usually wider than deep.
Root pumping
It has been usually assumed that in regular (non-aerial) terrestrial roots, the nutrients are pumped in together with the water, and that water is pumped into the roots by osmosis.
Recent studies in large trees such as the Giant Sequoia, showed, that much of the pumping power, after initial growth, is received by hydraulic suction, due to transpiration - water evaporation from the leaves, in addition to the strong capillary action in the plant's branching xylem in the stem and leaves.
Demo (software), a demonstration version of a software program
Demo (music), the first version of a song or album in music
Piero Gobetti (; June 19, 1901, Turin – February 15, 1926, Neuilly-sur-Seine) was a young journalist, intellectual, and radical liberal person.
He was a very  active critic during the years of crisis in Italy after the First World War and during the early years of Fascist rule.
He had to leave Italy and settled down in Paris, where he soon died.
He died because his health was bad, after the Fascists had terrorized him.
Italian journalists
Manama is the capital city of Bahrain.
It is also the largest city in Bahrain, having almost 150.000 residents.
Manama has been a city since at least the 1300s, when it was mentioned in Islamic books.
It was owned by Portugal from 1521 to 1602, by Persia from 1602 to 1783, and by the Al-Khalifa family for a long time after that.
In 1971, Manama became the capital city of Bahrain.
Areas of Manama
The Diplomatic Area has most of the banks and government offices.
This area is very important, because most of Bahrain's economy is connected to banks.
Gudabiya is the oldest area of the city, and it has the building where the parliament meets.
Hoora is where most visitors go.
There are many parties in the clubs here at night.
Settlements in Bahrain
Protists are single-celled eukaryotes (which are organisms with a nucleus).
The term Protista was first used by Ernst Haeckel in 1866.
The protists are not a natural group, or clade, since they have no common origin.
Like algae or invertebrates, they are often grouped together for convenience.
The term "protist" includes microorganisms from several distantly related phyla.
Some are autotrophic (which means they make their own food by photosynthesis), and others are heterotrophic (which means they eat organic material).
Most protists are very small.
They are made up of one or a few cells at most – they are microscopic and usually invisible to the naked eye.
Some algae are protists, if they are single-celled.
Many protists are part of the plankton and are very important for the ecosystem.
The cells found in protists may be extremely complex, and are often little understood.
It is now possible to do DNA sequencing, and a number of protists have been analysed.
The results show that the Protista is not a monophyletic group.
It is paraphyletic, and not a single clade.
The taxonomy of the Protista is therefore rather confused.
Some protists cause diseases.
Plasmodium falciparum causes malaria; sleeping sickness is also caused by a protist.
An example of a single celled organism in the protist kingdom is the Paramecium or "slipper animalcule."
The Paramecium moves using its small, hair-like fibers called cilia, and eats using the cilia to sweep the food into its food vacuole.
Other protists can be amoebas, which move by extending pseudopods and flowing into them, or flowing around food particles and engulfing them.
Origin and taxonomy
Protists are not a monophyletic clade.
The term is a convenient holdall for about 20 different kinds of single-celled eukaryotes.
They differ in their cell organelles, specialised units which carry out well-defined functions, like mitochondria and plastids.
This proves they have made the transition from prokaryotes in different ways.
It is fairly clear now that all or most of these organelles have their origin in once-independent prokaryotes (bacteria or archaea), and that the eukaryote cell is a 'community of micro-organisms' working together in 'a marriage of convenience'.
Admittedly, the Protista is a collection of disparate single-celled forms, but while a more sophisticated taxonomy is in flux (changing), Protista is still a useful term.
The Apple Dumpling Gang is a Disney movie from 1975, in which three orphan children find gold during California's Gold Rush.
Tim Conway and Don Knotts also star in it as two gangsters.
1975 movies
Erato, Artaxiad Dynasty Queen of Armenia, r. 8–5 BC, 2 BC–2 AD, 6–11
Ariobarzan of Atropatene, Client King of Armenia, r. 1 BC–2 AD
Artavazd V, Client King of Armenia, r. 2–11
Tigranes V, Artaxiad Dynasty King of Armenia, r. 2–6
Ping Di, Emperor of Han Dynasty China, r. 1 BC–5 AD
Ruzi Ying, Emperor of Han Dynasty China, r. 6–9
Wang Mang, Usurper Emperor of the short-lived Xin Dynasty in China, r. 9–23
Antiochus III, King of Commagene, r. 12 BC–17 AD
Arminius, German war chief
Crimthann Nia Náir, Legendary High King of Ireland, r. 8 BC–9 AD
Cairbre Cinnchait, Legendary High King of Ireland, r. 9–14
Strato II and Strato III, co-kings of the Indo-Greek Kingdom, r. 25 BC–10 AD
Natakamani, King of Kush, r. 1 BC–20 AD
Abgar V of Edessa, King of Osroene, r. 4 BC–7 AD, 13–50
Ma'nu IV, King of Osroene, r. 7–13
Phraates V, King of the Parthian Empire, r. 2 BC–4 AD
Musa of Parthia, mother and co-ruler with Phraates V, r. 2 BC–4 AD
Orodes III, King of the Parthian Empire, r. 4–6
Vonones I, King of the Parthian Empire, r. 8–12
Artabanus of Parthia, pretender to the Parthian throne and future King of Parthia
Caesar Augustus, Roman Emperor, r. 27 BC–14 AD
Gaius Caesar, Roman general
Livy, Roman historian
Ovid, Roman poet
Quirinius, Roman nobleman and politician
Hillel the Elder, Jewish scholar and Nasi of the Sanhedrin, in office c. 31 BC–9 AD
Shammai, Jewish scholar and Av Beit Din of the Sanhedrin, in office 20 BC–20 AD
Tiberius, Roman general, statesman, and future emperor
Hyeokgeose, King of Silla, r. 57 BC–4 AD
Namhae, King of Silla, r. 4–24
1001 (MI) was .
Grand Prince Stephen I of Hungary is named the first King of Hungary by Pope Silvester II.
Canonisation of Edward the Martyr, King of England.
Robert II, King of France marries for the third time to Constance Taillefer d'Arles.
Khmer king Jayavarman V is succeeded by Udayadityavarman I and/or Suryavarman I.
Otto III, Holy Roman Emperor has Charlemagne's vault opened (see Aachen Cathedral).