|
Four Color Theorem
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions share a common boundary of non-zero length (i.e., not merely a corner where three or more regions meet). It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubts remain. The theorem is a stronger version of the five color theorem, which can be shown using a significantly simpler argument. Although the weaker five color theorem was proven already in the 1800s, the four color theorem resisted until 1976 when it was proven by Kenneth Appel and Wolfgang Haken in a computer-aided proof. This came after many false proofs and mis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four Colour Map Example
4 (four) is a number, numeral (linguistics), numeral and numerical digit, digit. It is the natural number following 3 and preceding 5. It is a square number, the smallest semiprime and composite number, and is tetraphobia, considered unlucky in many East Asian cultures. Evolution of the Hindu-Arabic digit Brahmic numerals represented 1, 2, and 3 with as many lines. 4 was simplified by joining its four lines into a cross that looks like the modern plus sign. The Shunga Empire, Shunga would add a horizontal line on top of the digit, and the Northern Satraps, Kshatrapa and Pallava dynasty, Pallava evolved the digit to a point where the speed of writing was a secondary concern. The Arabs' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the digit less cursive, endi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an '' edge coloring'' assigns a color to each edges so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face (or region) so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
United States
The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 contiguous states border Canada to the north and Mexico to the south, with the semi-exclave of Alaska in the northwest and the archipelago of Hawaii in the Pacific Ocean. The United States asserts sovereignty over five Territories of the United States, major island territories and United States Minor Outlying Islands, various uninhabited islands in Oceania and the Caribbean. It is a megadiverse country, with the world's List of countries and dependencies by area, third-largest land area and List of countries and dependencies by population, third-largest population, exceeding 340 million. Its three Metropolitan statistical areas by population, largest metropolitan areas are New York metropolitan area, New York, Greater Los Angeles, Los Angel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alaska
Alaska ( ) is a non-contiguous U.S. state on the northwest extremity of North America. Part of the Western United States region, it is one of the two non-contiguous U.S. states, alongside Hawaii. Alaska is also considered to be the northernmost, westernmost, and easternmost (the Aleutian Islands cross the 180th meridian into the eastern hemisphere) state in the United States. It borders the Canadian territory of Yukon and the province of British Columbia to the east. It shares a western maritime border, in the Bering Strait, with Russia's Chukotka Autonomous Okrug. The Chukchi and Beaufort Seas of the Arctic Ocean lie to the north, and the Pacific Ocean lies to the south. Technically, it is a semi-exclave of the U.S., and is the largest exclave in the world. Alaska is the largest U.S. state by area, comprising more total area than the following three largest states of Texas, California, and Montana combined, and is the seventh-largest subnational division i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overseas France
Overseas France (, also ) consists of 13 France, French territories outside Europe, mostly the remnants of the French colonial empire that remained a part of the French state under various statuses after decolonisation. Most are part of the European Union. "Overseas France" is a collective name; while used in everyday life in France, it is not an administrative designation in its own right. Instead, the five Overseas departments and regions of France, overseas regions have exactly the same Administrative divisions of France, administrative status as the thirteen Metropolitan France, metropolitan regions; the five Overseas collectivity, overseas collectivities are semi-autonomous; and New Caledonia is an autonomous territory. Overseas France includes island territories in the Atlantic Ocean, Atlantic, Pacific Ocean, Pacific and Indian Ocean, Indian oceans, French Guiana on the South American continent, and several list of Antarctic and Subantarctic islands, peri-Antarctic islan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
France
France, officially the French Republic, is a country located primarily in Western Europe. Overseas France, Its overseas regions and territories include French Guiana in South America, Saint Pierre and Miquelon in the Atlantic Ocean#North Atlantic, North Atlantic, the French West Indies, and List of islands of France, many islands in Oceania and the Indian Ocean, giving it Exclusive economic zone of France, one of the largest discontiguous exclusive economic zones in the world. Metropolitan France shares borders with Belgium and Luxembourg to the north; Germany to the northeast; Switzerland to the east; Italy and Monaco to the southeast; Andorra and Spain to the south; and a maritime border with the United Kingdom to the northwest. Its metropolitan area extends from the Rhine to the Atlantic Ocean and from the Mediterranean Sea to the English Channel and the North Sea. Its Regions of France, eighteen integral regions—five of which are overseas—span a combined area of and hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaliningrad Oblast
Kaliningrad Oblast () is the westernmost federal subjects of Russia, federal subject of the Russian Federation. It is a Enclave and exclave, semi-exclave on the Baltic Sea within the Baltic region of Prussia (region), Prussia, surrounded by Poland to the south and Lithuania to the north and east. The largest city and administrative centre is the city of Kaliningrad. The port city of Baltiysk is Russia's only port on the Baltic Sea that remains ice-free in winter. Kaliningrad Oblast had a population of roughly one million in the 2021 Russian census. It has an area of . Various peoples, including Lithuanians, Germans, and Polish people, Poles, lived on the land which is now Kaliningrad. The territory was formerly the northern part of East Prussia. With the defeat of Nazi Germany in World War II, the territory was annexed to the Russian Soviet Federative Socialist Republic, Russian SFSR by the Soviet Union. Following the Aftermath of World War II, post-war migration and Flight and e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Azerbaijan
Azerbaijan, officially the Republic of Azerbaijan, is a Boundaries between the continents, transcontinental and landlocked country at the boundary of West Asia and Eastern Europe. It is a part of the South Caucasus region and is bounded by the Caspian Sea to the east, Russia's republic of Dagestan to the north, Georgia (country), Georgia to the northwest, Armenia and Turkey to the west, and Iran to the south. Baku is the capital and largest city. The territory of what is now Azerbaijan was ruled first by Caucasian Albania and later by various Persian empires. Until the 19th century, it remained part of Qajar Iran, but the Russo-Persian wars of Russo-Persian War (1804–1813), 1804–1813 and Russo-Persian War (1826–1828), 1826–1828 forced the Qajar Empire to cede its Caucasian territories to the Russian Empire; the treaties of Treaty of Gulistan, Gulistan in 1813 and Treaty of Turkmenchay, Turkmenchay in 1828 defined the border between Russia and Iran. The region north o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nakhchivan Autonomous Republic
The Nakhchivan Autonomous Republic (, ) is a landlocked country, landlocked Enclave and exclave, exclave of the Azerbaijan, Republic of Azerbaijan. The region covers Official portal of Nakhchivan Autonomous RepublicNakhchivan Autonomous Republic with a population of 459,600. It is bordered by Armenia to the east and north, Iran to the southwest, and Turkey to the west. It is the sole autonomous republic of Azerbaijan, governed by Supreme Assembly (Nakhchivan), its own elected legislature. The republic, especially the capital city of Nakhchivan (city), Nakhchivan, has a long history dating back to about 1500 BC. ''Nakhijevan'' was one the Provinces of the kingdom of Armenia (antiquity), cantons of the historical Armenian province of Vaspurakan in the Kingdom of Armenia (antiquity), Kingdom of Armenia. Historically, the Persians, Armenians, Mongols, and Turkic peoples, Turks all competed for the region. The area that is now Nakhchivan became part of Safavid Iran in the 16th centur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angola
Angola, officially the Republic of Angola, is a country on the west-Central Africa, central coast of Southern Africa. It is the second-largest Portuguese-speaking world, Portuguese-speaking (Lusophone) country in both total area and List of countries and dependencies by population, population and is the List of African countries by area, seventh-largest country in Africa. It is bordered by Namibia to the south, the Democratic Republic of the Congo to the north, Zambia to the east, and the Atlantic Ocean to the west. Angola has an Enclave and exclave, exclave province, the province of Cabinda Province, Cabinda, that borders the Republic of the Congo and the Democratic Republic of the Congo. The capital and most populous city is Luanda. Angola has been inhabited since the Paleolithic, Paleolithic Age. After the Bantu expansion reached the region, states were formed by the 13th century and organised into confederations. The Kingdom of Kongo ascended to achieve hegemony among the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cabinda Province
Cabinda (formerly called Portuguese Congo, ) is an exclave and province of Angola, a status that has been disputed by several political organizations in the territory. The capital city is also called Cabinda, known locally as ''Tchiowa'', ''Tsiowa'' or ''Kiowa''. The province is divided into four municipalities—Belize, Buco-Zau, Cabinda and Cacongo. Modern Cabinda is the result of a fusion of three kingdoms: N'Goyo, Loango and Kakongo. It has an area of and a population of 716,076 at the 2014 census; the latest official estimate (as at mid-2019) is 824,143. According to 1988 United States government statistics, the total population of the province was 147,200, with a near even split between rural and urban populations. At one point an estimated one third of Cabindans were refugees living in the Democratic Republic of the Congo; however, after the 2007 peace agreement, refugees started returning to their homes. Cabinda is separated from the rest of Angola by a narrow st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, an open set is a member of a given Set (mathematics), collection of Subset, subsets of a given set, a collection that has the property of containing every union (set theory), union of its members, every finite intersection (set theory), intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology (structure), topology. These conditions are very loose, and allow enormous flexibility in the choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
