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Schoenflies
Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. Schoenflies was born in Landsberg an der Warthe (modern Gorzów Wielkopolski, Gorzów, Poland). Arthur Schoenflies married Emma Levin (1868–1939) in 1896. He studied under Ernst Kummer and Karl Weierstrass, and was influenced by Felix Klein. The Schoenflies problem is to prove that an (n - 1)-sphere in Euclidean ''n''-space bounds a topological ball, however embedded. This question is much more subtle than it initially appears. He studied at the University of Berlin from 1870 to 1875. He obtained a doctorate in 1877, and in 1878 he was a teacher at a school in Berlin. In 1880, he went to Colmar to teach. Schoenflies was a frequent contributor to Klein's encyclopedia: In 1898 he wrote on set theory, in 1902 on kinematics, and on projective geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan–Schoenflies Theorem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Moritz Schoenflies, Arthur Schoenflies. For Camille Jordan, Jordan curves in the Plane (geometry), plane it is often referred to as the Jordan–Schoenflies theorem. Original formulation The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane (mathematics), plane separate the plane into two regions, one (the "inside") bounded set, bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane. An alternative statement is that if C \subset \mathbb R^2 is a simple closed curve, then there is a homeomorphism f : \mathbb R^2 \to \mathbb R^2 such that f(C) is the unit circle in the plane. Elementary proofs can be found in , , and . The result can first be proved for polygons when the homeomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schoenflies Problem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem. Original formulation The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane. An alternative statement is that if C \subset \mathbb R^2 is a simple closed curve, then there is a homeomorphism f : \mathbb R^2 \to \mathbb R^2 such that f(C) is the unit circle in the plane. Elementary proofs can be found in , , and . The result can first be proved for polygons when the homeomorphism can be taken to be piecewise linear and the identity map off some compact set; the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schoenflies Displacement
Schoenflies (or Schönflies) displacement (or motion) named after Arthur Moritz Schoenflies is a rigid body motion consisting of linear motion in three dimensional space plus one orientation around an axis with fixed direction. In robotic manipulation this is a common motion as many pick and place operations require moving an object from one plane and placing it with a different orientation onto another parallel plane (''e.g.'', placement of components on a circuit board). These robots are commonly called Schoenflies-motion generators. Because the SCARA manipulator was one of the first manipulators providing similar motion, this is often referred to as SCARA-type motion. Today, many robotic manipulators, including some with parallel kinematic architecture, are used in industry for applications ranging from the manufacture of electronics to food processing and packaging industry. See also * Articulated robot * Parallel manipulator * SCARA * Delta robot A delta robot is a type ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schoenflies Notation
The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule, the notation is often sufficient and commonly used for spectroscopy. However, in crystallography, there is additional translational symmetry, and point groups are not enough to describe the full symmetry of crystals, so the full space group is usually used instead. The naming of full space groups usually follows another common convention, the Hermann–Mauguin notation, also known as the international notation. Although Schoenflies notation without superscripts is a pure point group notation, optionally, superscripts can be added to further specify individual space groups. However, for space groups, the connection to the underlying symmetry elements is much more clear in Hermann–Mauguin notation, so the latter n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frankfurt Main Cemetery
The Frankfurt Main Cemetery (German: ''Hauptfriedhof'') is the largest cemetery in Frankfurt am Main, Germany. It was opened in 1828. The cemetery is located directly adjacent to two Jewish cemeteries—the Old Jewish Cemetery, Frankfurt, Old Jewish Cemetery (opened together with the Main Cemetery in 1828) and the (opened in 1928)—and together they form one of the largest cemetery areas in Germany. The cemetery is noted for its many monumental graves, its garden architecture and as the site of the graves of many notable individuals. History The Frankfurt Main Cemetery was planned as the replacement of St. Peter's Cemetery, which had been the main cemetery of the city since the 16th century. At the ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching mathematics, physics, botany and gymnastics. He later received an honorary doctorate and became professor of mathematics in Berlin. Among many other contributions, Weierstrass formalized the definition of the continuity of a function, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals. Biography Weierstrass was born into a Roman Catholic family in Ostenfelde, a village near Ennigerloh, in the Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst both of whom were catholic Rhinelanders. His int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre (geometry), centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. spherical Earth, The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in any direction, so mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gorzów Wielkopolski
Gorzów Wielkopolski (; german: Landsberg an der Warthe) often abbreviated to Gorzów Wlkp. or simply Gorzów, is a city in western Poland, on the Warta river. It is the second largest city in the Lubusz Voivodeship with 120,087 inhabitants (December 2021) and one of its two capitals with a seat of a voivode, with the other being Zielona Góra. Around Gorzów, there are two large forest areas: Gorzów Woods to the north, where the Barlinek-Gorzów Landscape Park is situated, and Noteć Woods to the southeast. The biggest oil fields in Poland are located near Gorzów. Etymology The pre-1945 German name ''Landsberg an der Warthe'', dating back to 1257, derived from the German words ''land'' or 'state' and ''berg'' or 'mountain' combined with ''Warthe''the German name for the river Warta. The Polish name Gorzów, written as Gorzew, is known from Polish maps and historical books dating back to the 19th century or perhaps earlier.Henryk M. Wozniak, Gazeta Zachodnia "Gorzów tak - Wie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colmar
Colmar (, ; Alsatian: ' ; German during 1871–1918 and 1940–1945: ') is a city and commune in the Haut-Rhin department and Grand Est region of north-eastern France. The third-largest commune in Alsace (after Strasbourg and Mulhouse), it is the seat of the prefecture of the Haut-Rhin department and of the subprefecture of the Colmar-Ribeauvillé arrondissement. The city is renowned for its well-preserved old town, its numerous architectural landmarks, and its museums, among which is the Unterlinden Museum, which houses the ''Isenheim Altarpiece''. Colmar is situated on the Alsatian Wine Route and considers itself to be the "capital of Alsatian wine" ('). History Colmar was first mentioned by Charlemagne in his chronicle about Saxon wars. This was the location where the Carolingian Emperor Charles the Fat held a diet in 884. Colmar was granted the status of a free imperial city by Emperor Frederick II in 1226. In 1354 it joined the Décapole city league.G. Köbler, ''H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Berlin
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative of Wilhelm von Humboldt, Johann Gottlieb Fichte and Friedrich Ernst Daniel Schleiermacher as the University of Berlin () in 1809, and opened in 1810, making it the oldest of Berlin's four universities. From 1828 until its closure in 1945, it was named Friedrich Wilhelm University (german: Friedrich-Wilhelms-Universität). During the Cold War, the university found itself in East Berlin and was ''de facto'' split in two when the Free University of Berlin opened in West Berlin. The university received its current name in honour of Alexander and Wilhelm von Humboldt in 1949. The university is divided into nine faculties including its medical school shared with the Freie Universität Berlin. The university has a student enrollment of around ... [...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, structure, space, models, and 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 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 mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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