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Eugen Netto
Eugen Otto Erwin Netto (30 June 1848 – 13 May 1919) was a German mathematician. He was born in Halle and died in Giessen. Netto's theorem, on the dimension-preserving properties of continuous bijections, is named for Netto. Netto published this theorem in 1878, in response to Georg Cantor's proof of the existence of discontinuous bijections between the unit interval and unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordina .... His proof was not fully rigorous, but its errors were later repaired. Works''Substitutionentheorie und ihre Anwendung auf die Algebra.''Teubner 1882.''Theory of Substitutions and Its Applications to Algebra.''Ann Arbor, Mich. 1892.''Die Determinanten.''Teubner, 1910. *''Die Determinanten.'' Teubner, 2nd edition 1925.''Lehrbuch der Combinatorik.' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
German People
, native_name_lang = de , region1 = , pop1 = 72,650,269 , region2 = , pop2 = 534,000 , region3 = , pop3 = 157,000 3,322,405 , region4 = , pop4 = 21,000 3,000,000 , region5 = , pop5 = 125,000 982,226 , region6 = , pop6 = 900,000 , region7 = , pop7 = 142,000 840,000 , region8 = , pop8 = 9,000 500,000 , region9 = , pop9 = 357,000 , region10 = , pop10 = 310,000 , region11 = , pop11 = 36,000 250,000 , region12 = , pop12 = 25,000 200,000 , region13 = , pop13 = 233,000 , region14 = , pop14 = 211,000 , region15 = , pop15 = 203,000 , region16 = , pop16 = 201,000 , region17 = , pop17 = 101,000 148,00 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagoreans, Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halle, Saxony-Anhalt
Halle (Saale), or simply Halle (; from the 15th to the 17th century: ''Hall in Sachsen''; until the beginning of the 20th century: ''Halle an der Saale'' ; from 1965 to 1995: ''Halle/Saale'') is the largest city of the German state of Saxony-Anhalt, the fifth most populous city in the area of former East Germany after ( East) Berlin, Leipzig, Dresden and Chemnitz, as well as the 31st largest city of Germany, and with around 239,000 inhabitants, it is slightly more populous than the state capital of Magdeburg. Together with Leipzig, the largest city of Saxony, Halle forms the polycentric Leipzig-Halle conurbation. Between the two cities, in Schkeuditz, lies Leipzig/Halle International Airport. The Leipzig-Halle conurbation is at the heart of the larger Central German Metropolitan Region. Halle lies in the south of Saxony-Anhalt, in the Leipzig Bay, the southernmost part of the North German Plain, on the River Saale (a tributary of the Elbe), which is the third longest r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giessen
Giessen, spelled Gießen in German (), is a town in the German state (''Bundesland'') of Hesse, capital of both the district of Giessen and the administrative region of Giessen. The population is approximately 90,000, with roughly 37,000 university students. The name comes from ''Giezzen'', as it was first referred to in 1197, which refers to the position of the town between several rivers, lakes and streams. The largest river in Giessen is the Lahn, which divides the town in two parts (west and east), roughly north of Frankfurt am Main. Giessen is also home to the University of Giessen. In 1969, the town hosted the ninth '' Hessentag'' state festival. History Giessen came into being as a moated castle in 1152 built by Count Wilhelm von Gleiberg, although the history of the community in the northeast and in today's suburb called "Wieseck" dates back to 775. The town became part of Hesse-Marburg in 1567, passing to Hesse-Darmstadt in 1604. The University of Giessen was foun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Netto's Theorem
In mathematical analysis, Netto's theorem states that continuous bijections of smooth manifolds preserve dimension. That is, there does not exist a continuous bijection between two smooth manifolds of different dimension. It is named after Eugen Netto. The case for maps from a higher-dimensional manifold to a one-dimensional manifold was proven by Jacob Lüroth in 1878, using the intermediate value theorem to show that no manifold containing a topological circle can be mapped continuously and bijectively to the real line. Both Netto in 1878, and Georg Cantor in 1879, gave faulty proofs of the general theorem. The faults were later recognized and corrected. An important special case of this theorem concerns the non-existence of continuous bijections from one-dimensional spaces, such as the real line or unit interval, to two-dimensional spaces, such as the Euclidean plane or unit square. The conditions of the theorem can be relaxed in different ways to obtain interesting classes o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein rai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation ' is most commonly reserved for the closed interval . Properties The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinate system with coordinates , a unit square is defined as a square consisting of the points where both and lie in a closed unit interval from to . That is, a unit square is the Cartesian product , where denotes the closed unit interval. Complex coordinates The unit square can also be thought of as a subset of the complex plane, the topological space formed by the complex numbers. In this view, the four corners of the unit square are at the four complex numbers , , , and . Rational distance problem It is not known whether any point in the plane is a rational distance from all four vertices of the unit square.. See also * Unit circle * Unit cube * Unit sphere In mathematics, a unit sphere is simply a sphere of radius one ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1848 Births
1848 is historically famous for the Revolutions of 1848, wave of revolutions, a series of widespread struggles for more classical liberalism, liberal governments, which broke out from Brazil to Hungary; although most failed in their immediate aims, they significantly altered the political and philosophical landscape and had major ramifications throughout the rest of the century. Ereignisblatt aus den revolutionären Märztagen 18.-19. März 1848 mit einer Barrikadenszene aus der Breiten Strasse, Berlin 01.jpg, Cheering German revolutions of 1848–49, revolutionaries in Berlin, on March 19, 1848, with the new flag of Germany Lar9 philippo 001z.jpg, French Revolution of 1848: Republican riots forced King Louis-Philippe to abdicate Zeitgenössige Lithografie der Nationalversammlung in der Paulskirche.jpg, Frankfurt Parliament, German National Assembly's meeting in St. Paul's Church Pákozdi csata.jpg, Battle of Pákozd in the Hungarian Revolution of 1848 Events January ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1919 Deaths
Events January * January 1 ** The Czechoslovak Legions occupy much of the self-proclaimed "free city" of Pressburg (now Bratislava), enforcing its incorporation into the new republic of Czechoslovakia. ** HMY ''Iolaire'' sinks off the coast of the Hebrides; 201 people, mostly servicemen returning home to Lewis and Harris, are killed. * January 2– 22 – Russian Civil War: The Red Army's Caspian-Caucasian Front begins the Northern Caucasus Operation against the White Army, but fails to make progress. * January 3 – The Faisal–Weizmann Agreement is signed by Emir Faisal (representing the Arab Kingdom of Hejaz) and Zionist leader Chaim Weizmann, for Arab–Jewish cooperation in the development of a Jewish homeland in Palestine, and an Arab nation in a large part of the Middle East. * January 5 – In Germany: ** Spartacist uprising in Berlin: The Marxist Spartacus League, with the newly formed Communist Party of Germany and the Independent Social ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
