AOL Instant Messenger was a service which allowed users to communicate with any user currently online. AIM had millions of users worldwide. There were also other computer programs that could be used to talk with your AIM account, for example Trillian, Pidgin, Digsby, and AIM+. AIM was shut down on December 15, 2017.
Other AOL products are AOL Spyware Protection, AOL Safety and Security Center, ICQ, AOl Computer Check-Up, AOL Explorer, AOL TopSpeed, AOL Dialer, AOL One-Click Fixes and AOL Openride.
Youssef Rzouga
Youssef Rzouga (born March 21, 1957) is a Tunisian poet. He was born in Mahdia, a Tunisian coastal city.
His first writing to be printed was a short story in the magazine "Radio et Télévision", in 1973.
Mailman
A mailman or mail carrier or postman is a person who brings people their postal mail. Normally he works for the post office of the respective country. Of course, there are also women doing this job. They are called mailwomen. For a layman, this occupation is called "letter carriers".
Fictional mailmen are traditionally shown as acting afraid of dogs, even to the point of shock.
Chinese postman problem
The Chinese postman problem is a mathematical problem of graph theory. It is also known as . Suppose there is a mailman who needs to deliver mail to a certain neighbourhood. The mailman is unwilling to walk far, so he wants to find the shortest route through the neighbourhood, that meets the following criteria:
If the graph travelled has an Eulerian path, this circuit is the ideal solution.
Alan Goldman of the U.S. National Bureau of Standards first coined the name 'Chinese Postman Problem' for this problem, as it was originally studied by the Chinese mathematician Mei-Ku Kuan in 1962.
Four color theorem
The four color theorem is a theorem of mathematics. It says that in any plane surface with regions in it (people think of them as maps), the regions can be colored with no more than four colors. Two regions that have a common border must not get the same color. They are called "adjacent" (next to each other) if they share a segment of the border, not just a point.
This was the first theorem to be proved by a computer, in a "proof by exhaustion". In proof by exhaustion, the conclusion is established by dividing it into cases, and proving each one separately. There may be a lot of cases. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because most of the cases were checked by a computer program, not by hand. The shortest known proof of the four color theorem today still has over 600 cases.
Even though the problem was first presented as a problem to color political maps of countries, mapmakers are not very interested in it. According to an article by the math historian Kenneth May , “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of map making do not mention the four-color property.”
Many simpler maps can be colored using three colors. The fourth color is needed for some maps, such as one in which one region is surrounded by an odd number of others, which touch each other in a cycle. One such example is given in the image. The five color theorem states that five colors are enough to color a map. It has a short, elementary proof and was proven in the late 19th century. Proving that four colors are all that is needed turned out to be much more difficult. Many false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852.
Intuitively, the four color theorem can be stated as 'given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two regions which are adjacent have the same color'. To be able to correctly solve the problem, it is necessary to clarify some aspects: First, all points that belong to three or more countries must be ignored. Secondly, "bizarre" maps with regions of finite area and infinite perimeter can require more than four colors.
For the purpose of the theorem every "country" has to be a simply connected region, or contiguous.
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. Because the territory of a particular country must be the same color, four colors may not be enough. 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. 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. If "A" had only three regions, six or more colors might be needed. In this way, it is possible to make maps that need an arbitrarily high number of colors. 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.