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Van Der Waerden
Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics. Biography Education and early career Van der Waerden learned advanced mathematics at the University of Amsterdam and the University of Göttingen, from 1919 until 1926. He was much influenced by Emmy Noether at Göttingen, Germany. Amsterdam awarded him a Ph.D. for a thesis on algebraic geometry, supervised by Hendrick de Vries. Göttingen awarded him the habilitation in 1928. In that year, at the age of 25, he accepted a professorship at the University of Groningen. At 27, Van der Waerden published his ''Moderne Algebra'', an influential two-volume treatise on abstract algebra, still cited, and perhaps the first treatise to treat the subject as a comprehensive whole. This work systematized an ample body of research by Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amsterdam
Amsterdam ( , ; ; ) is the capital of the Netherlands, capital and Municipalities of the Netherlands, largest city of the Kingdom of the Netherlands. It has a population of 933,680 in June 2024 within the city proper, 1,457,018 in the City Region of Amsterdam, urban area and 2,480,394 in the Amsterdam metropolitan area, metropolitan area. Located in the Provinces of the Netherlands, Dutch province of North Holland, Amsterdam is colloquially referred to as the "Venice of the North", for its canals of Amsterdam, large number of canals, now a World Heritage Site, UNESCO World Heritage Site. Amsterdam was founded at the mouth of the Amstel River, which was dammed to control flooding. Originally a small fishing village in the 12th century, Amsterdam became a major world port during the Dutch Golden Age of the 17th century, when the Netherlands was an economic powerhouse. Amsterdam was the leading centre for finance and trade, as well as a hub of secular art production. In the 19th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Der Waerden's Theorem
Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory. Van der Waerden's theorem states that for any given positive integers ''r'' and ''k'', there is some number ''N'' such that if the integers are colored, each with one of ''r'' different colors, then there are at least ''k'' integers in arithmetic progression whose elements are of the same color. The least such ''N'' is the Van der Waerden number ''W''(''r'', ''k''), named after the Dutch mathematician B. L. van der Waerden. This was conjectured by Pierre Joseph Henry Baudet in 1921. Waerden heard of it in 1926 and published his proof in 1927, titled ''Beweis einer Baudetschen Vermutung roof of Baudet's conjecture'. Example For example, when ''r'' = 2, you have two colors, say and . ''W''(2, 3) is bigger than 8, because you can color the integers from like this: and no three integers of the same color form an arithmetic progression. But you can't add a ninth integer to the end w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moderne Algebra
''Moderne Algebra'' is a two-volume German textbook on graduate abstract algebra by , originally based on lectures given by Emil Artin in 1926 and by from 1924 to 1928. The English translation of 1949–1950 had the title ''Modern algebra'', though a later, extensively revised edition in 1970 had the title ''Algebra''. The book was one of the first textbooks to use an abstract axiomatic approach to groups, rings, and fields, and was by far the most successful, becoming the standard reference for graduate algebra for several decades. It "had a tremendous impact, and is widely considered to be the major text on algebra in the twentieth century." In 1975 van der Waerden described the sources he drew upon to write the book. In 1997 Saunders Mac Lane recollected the book's influence: * Upon its publication it was soon clear that this was the way that algebra should be presented. * Its simple but austere style set the pattern for mathematical texts in other subjects, from Bana ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in Germany, France, Italy, Poland and some other European and non-English-speaking countries. The candidate fulfills a university's set criteria of excellence in research, teaching, and further education, which usually includes a dissertation. The degree, sometimes abbreviated ''Dr. habil''. (), ''dr hab.'' (), or ''D.Sc.'' ('' Doctor of Sciences'' in Russia and some CIS countries), is often a qualification for full professorship in those countries. In German-speaking countries it allows the degree holder to bear the title ''PD'' (for ). In a number of countries there exists an academic post of docent, appointment to which often requires such a qualification. The degree conferral is usually accompanied by a public oral defence event (a lecture or a colloquium) with one or more opponents. Habilitation is usually awarded 5–15 years after a PhD degree or its equivalent. Achieving this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germany
Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total population of over 84 million in an area of , making it the most populous member state of the European Union. It borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, and France, Luxembourg, Belgium, and the Netherlands to the west. The Capital of Germany, nation's capital and List of cities in Germany by population, most populous city is Berlin and its main financial centre is Frankfurt; the largest urban area is the Ruhr. Settlement in the territory of modern Germany began in the Lower Paleolithic, with various tribes inhabiting it from the Neolithic onward, chiefly the Celts. Various Germanic peoples, Germanic tribes have inhabited the northern parts of modern Germany since classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Göttingen
Göttingen (, ; ; ) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. According to the 2022 German census, the population of Göttingen was 124,548. Overview The origins of Göttingen lay in a village called ''Gutingi, ''first mentioned in a document in 953 AD. The city was founded northwest of this village, between 1150 and 1200 AD, and adopted its name. In Middle Ages, medieval times the city was a member of the Hanseatic League and hence a wealthy town. Today, Göttingen is famous for its old university (''Georgia Augusta'', or University of Göttingen, "Georg-August-Universität"), which was founded in 1734 (first classes in 1737) and became the most visited university of Europe. In 1837, seven professors protested against the absolute sovereignty of the House of Hanover, kings of Kingdom of Hanover, Hanover; they lost their positions, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emmy Noether
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which are fundamental in mathematical physics. Noether was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important List of women in mathematics, woman in the history of mathematics. Transcribeonlineat the MacTutor History of Mathematics Archive. As one of the leading mathematicians of her time, she developed theories of ring (mathematics), rings, field (mathematics), fields, and algebras. In physics, Noether's theorem explains the connection between Symmetry (physics), symmetry and conservation laws. in . Noether was born to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad (region), Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy to record time and formulate calendars. The earliest mathematical texts available are from Mesopotamia and Ancient Egypt, Egypt – ''Plimpton 322'' (Babylonian mathematics, Babylonian – 1900 BC),Friberg, J. (1981). "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", ''Historia Mathematica'', 8, pp. 277–318. the ' ... [...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 was Thales of Miletus (); 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's theorem. The number of known mathematicians grew when Pythagoras of Samos () 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 math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Progression Game
The arithmetic progression game is a positional game where two players alternately pick numbers, trying to occupy a complete arithmetic progression of a given size. The game is parameterized by two integers ''n'' > ''k''. The game-board is the set . The winning-sets are all the arithmetic progressions of length ''k''. In a Maker-Breaker game variant, the first player (Maker) wins by occupying a ''k''-length arithmetic progression, otherwise the second player (Breaker) wins. The game is also called the van der Waerden game, named after Van der Waerden's theorem. It says that, for any ''k'', there exists some integer ''W''(2,''k'') such that, if the integers are partitioned arbitrarily into two sets, then at least one set contains an arithmetic progression of length ''k''. This means that, if n \geq W(2,k), then Maker has a winning strategy. Unfortunately, this claim is not constructive - it does not show a specific strategy for Maker. Moreover, the current upper bound for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
