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Cahit Arf
Cahit Arf (; 24 October 1910 – 26 December 1997) was a Turkish mathematician. He is known for the Arf invariant of a quadratic form in characteristic 2 (applied in knot theory and surgery theory) in topology, the Hasse–Arf theorem in ramification theory, Arf semigroups and Arf rings. Biography Cahit Arf was born on 11 October 1910 in Selanik (Thessaloniki), which was then a part of the Ottoman Empire. His family migrated to Istanbul with the outbreak of the Balkan War in 1912. The family finally settled in Ä°zmir where Cahit Arf received his primary education. Upon receiving a scholarship from the Turkish Ministry of Education he continued his education in Paris and graduated from École Normale Supérieure. Returning to Turkey, he taught mathematics at Galatasaray High School. In 1933 he joined the Mathematics Department of Istanbul University. In 1937 he went to Göttingen, where he received his PhD from the University of Göttingen and he worked with Helmut Has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Middle East Technical University
Middle East Technical University (commonly referred to as METU; in Turkish language, Turkish, ''Orta Doğu Teknik Üniversitesi'', ODTÜ) is a public university, public Institute of technology, technical university located in Ankara, Turkey. The university emphasizes research and education in engineering and natural sciences, offering about 41 undergraduate programs within 5 faculties, 105 masters and 70 doctorate programs within 5 graduate schools. The main campus of METU spans an area of , comprising, in addition to academic and auxiliary facilities, a forest area of , and the natural Lake Eymir. METU has more than 120,000 alumni worldwide. The official language of instruction at METU is English language, English. Over one third of the 1,000 highest scoring students in the Education in Turkey, national university entrance examination choose to enroll in METU; and most of its departments accept the top 0.1% of the nearly 3 million applicants. METU had the greatest share in nation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arf Ring
In mathematics, an Arf ring was defined by to be a 1-Krull dimension, dimensional commutative ring, commutative semi-local ring, semi-local Cohen–Macaulay ring, Macaulay ring (mathematics), ring satisfying some extra conditions studied by . References * * Commutative algebra {{commutative-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galatasaray High School
Galatasaray High School ( tr, Galatasaray Lisesi, french: Lycée de Galatasaray), established in what was then Constantinople and is now Istanbul, in 1481, is the oldest high school in Turkey. It is also the second-oldest Turkish educational institution after Istanbul University, which was established in 1453. The name ''Galatasaray'' means ''Galata Palace'', as the school is located at the far end of Galata, the medieval Genoese enclave above the Golden Horn in what is now the district of Beyoğlu. A highly selective school, Galatasaray High School is often compared to the likes of Eton College in England and Lycée Louis-le-Grand in France. Since it is now an Anatolian High School, access to the school is open to any student who achieves a high enough score in nationwide entrance exams; the intake therefore consists of the top-scoring 0.03% of students from across the country. Drawing on a blend of the Turkish and French school curricula, Galatasaray High School provides educ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ä°zmir
Ä°zmir ( , ; ), also spelled Izmir, is a metropolitan city in the western extremity of Anatolia, capital of the province of the same name. It is the third most populous city in Turkey, after Istanbul and Ankara and the second largest urban agglomeration on the Aegean Sea after Athens. As of the last estimation, on 31 December 2019, the city of Ä°zmir had a population of 2,965,900, while Ä°zmir Province had a total population of 4,367,251. Its built-up (or metro) area was home to 3,209,179 inhabitants extending on 9 out of 11 urban districts (all but Urla and Guzelbahce not yet agglomerated) plus Menemen and Menderes largely conurbated. It extends along the outlying waters of the Gulf of Ä°zmir and inland to the north across the Gediz River Delta; to the east along an alluvial plain created by several small streams; and to slightly more rugged terrain in the south. Ä°zmir has more than 3,000 years of recorded urban history, and up to 8,500 years of history as a human settlemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balkan War
The Balkan Wars refers to a series of two conflicts that took place in the Balkan States in 1912 and 1913. In the First Balkan War, the four Balkan States of Greece, Serbia, Montenegro and Bulgaria declared war upon the Ottoman Empire and defeated it, in the process stripping the Ottomans of its European provinces, leaving only Eastern Thrace under the Ottoman Empire's control. In the Second Balkan War, Bulgaria fought against the other four original combatants of the first war. It also faced an attack from Romania from the north. The Ottoman Empire lost the bulk of its territory in Europe. Although not involved as a combatant, Austria-Hungary became relatively weaker as a much enlarged Serbia pushed for union of the South Slavic peoples. The war set the stage for the Balkan crisis of 1914 and thus served as a "prelude to the First World War". By the early 20th century, Bulgaria, Greece, Montenegro and Serbia had achieved independence from the Ottoman Empire, but large elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramification Theory
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. In complex analysis In complex analysis, the basic model can be taken as the ''z'' → ''z''''n'' mapping in the complex plane, near ''z'' = 0. This is the standard local picture in Riemann surface theory, of ramification of order ''n''. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus. See also branch point. In algebraic topology In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The ''z'' → ''z''''n'' mapping shows this as a local p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse–Arf Theorem
In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of the upper numbering filtration of the Galois group of a finite Galois extension. A special case of it when the residue fields are finite was originally proved by Helmut Hasse, and the general result was proved by Cahit Arf. Statement Higher ramification groups The theorem deals with the upper numbered higher ramification groups of a finite abelian extension ''L''/''K''. So assume ''L''/''K'' is a finite Galois extension, and that ''v''''K'' is a discrete normalised valuation of ''K'', whose residue field has characteristic ''p'' > 0, and which admits a unique extension to ''L'', say ''w''. Denote by ''v''''L'' the associated normalised valuation ''ew'' of ''L'' and let \scriptstyle be the valuation ring of ''L'' under ''v''''L''. Let ''L''/''K'' have Galois group ''G'' and define the ''s''-th ramification group of ''L''/''K'' for any real ''s'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surgery Theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold ''M'' of dimension n=p+q+1, could be described as removing an imbedded sphere of dimension ''p'' from ''M''. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. More technically, the idea is to start with a well-understood manifold ''M'' and perform surgery on it to produce a manifold ''M''′ having some desired property, in such a way th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, Unknot, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring (mathematics), ring , often denoted , is defined to be the smallest number of times one must use the ring's identity element, multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent (group theory), exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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