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Paul Monsky
Paul Monsky (born June 17, 1936) is an American mathematician and professor at Brandeis University. After earning a bachelor's degree from Swarthmore College, he received his Ph.D. in 1962 from the University of Chicago under the supervision of Walter Lewis Baily, Jr. He has introduced the Monsky–Washnitzer cohomology and he has worked intensively on Hilbert–Kunz functions and Hilbert–Kunz multiplicity. In 2007, Monsky and Holger Brenner gave an example showing that tight closure does not commute with localization. Monsky's theorem In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square does not have an odd equidissection. The problem was posed by Fred Richman in the '' American ..., the statement that a square cannot be divided into an odd number of equal-area triangles, is named after Monsky, who published the first proof of it in 1970. In the mid-1970s, Monsky stopped pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Research Institute Of Oberwolfach
The Oberwolfach Research Institute for Mathematics (german: Mathematisches Forschungsinstitut Oberwolfach) is a center for mathematical research in Oberwolfach, Germany. It was founded by mathematician Wilhelm Süss in 1944. It organizes weekly workshops on diverse topics where mathematicians and scientists from all over the world come to do collaborative research. The Institute is a member of the Leibniz Association, funded mainly by the German Federal Ministry of Education and Research and by the state of Baden-Württemberg. It also receives substantial funding from the ''Friends of Oberwolfach'' foundation, from the ''Oberwolfach Foundation'' and from numerous donors. History The Oberwolfach Research Institute for Mathematics (MFO) was founded as the ''Reich Institute of Mathematics'' (German: ''Reichsinstitut für Mathematik'') on 1 September 1944. It was one of several research institutes founded by the Nazis in order to further the German war effort, which at that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tight Closure
In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by . Let R be a commutative noetherian ring containing a field of characteristic p > 0. Hence p is a prime number. Let I be an ideal of R. The tight closure of I, denoted by I^*, is another ideal of R containing I. The ideal I^* is defined as follows. :z \in I^* if and only if there exists a c \in R, where c is not contained in any minimal prime ideal of R, such that c z^ \in I^ for all e \gg 0. If R is reduced, then one can instead consider all e > 0. Here I^ is used to denote the ideal of R generated by the p^e'th powers of elements of I, called the eth Frobenius power of I. An ideal is called tightly closed if I = I^*. A ring in which all ideals are tightly closed is called weakly F-regular (for Frobenius regular). A previous major open question in tight closure is whether the operation of tight closure commutes with localizati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21st-century American Mathematicians
The 1st century was the century spanning AD 1 ( I) through AD 100 ( C) according to the Julian calendar. It is often written as the or to distinguish it from the 1st century BC (or BCE) which preceded it. The 1st century is considered part of the Classical era, epoch, or historical period. The 1st century also saw the appearance of Christianity. During this period, Europe, North Africa and the Near East fell under increasing domination by the Roman Empire, which continued expanding, most notably conquering Britain under the emperor Claudius ( AD 43). The reforms introduced by Augustus during his long reign stabilized the empire after the turmoil of the previous century's civil wars. Later in the century the Julio-Claudian dynasty, which had been founded by Augustus, came to an end with the suicide of Nero in AD 68. There followed the famous Year of Four Emperors, a brief period of civil war and instability, which was finally brought to an end by Vespasian, ninth Roman empero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Living People
Related categories * :Year of birth missing (living people) / :Year of birth unknown * :Date of birth missing (living people) / :Date of birth unknown * :Place of birth missing (living people) / :Place of birth unknown * :Year of death missing / :Year of death unknown * :Date of death missing / :Date of death unknown * :Place of death missing / :Place of death unknown * :Missing middle or first names See also * :Dead people * :Template:L, which generates this category or death years, and birth year and sort keys. : {{DEFAULTSORT:Living people 21st-century people People by status ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1936 Births
Events January–February * January 20 – George V of the United Kingdom and the British Dominions and Emperor of India, dies at his Sandringham Estate. The Prince of Wales succeeds to the throne of the United Kingdom as King Edward VIII. * January 28 – Britain's King George V state funeral takes place in London and Windsor. He is buried at St George's Chapel, Windsor Castle * February 4 – Radium E (bismuth-210) becomes the first radioactive element to be made synthetically. * February 6 – The 1936 Winter Olympics, IV Olympic Winter Games open in Garmisch-Partenkirchen, Germany. * February 10–February 19, 19 – Second Italo-Ethiopian War: Battle of Amba Aradam – Italian forces gain a decisive tactical victory, effectively neutralizing the army of the Ethiopian Empire. * February 16 – 1936 Spanish general election: The left-wing Popular Front (Spain), Popular Front coalition takes a majority. * February 26 – February 26 Inci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equidissection
In geometry, an equidissection is a partition of a polygon into triangles of equal area. The study of equidissections began in the late 1960s with Monsky's theorem, which states that a square cannot be equidissected into an odd number of triangles. In fact, most polygons cannot be equidissected at all. Much of the literature is aimed at generalizing Monsky's theorem to broader classes of polygons. The general question is: Which polygons can be equidissected into how many pieces? Particular attention has been given to trapezoids, kites, regular polygons, centrally symmetric polygons, polyominos, and hypercubes. Equidissections do not have many direct applications. They are considered interesting because the results are counterintuitive at first, and for a geometry problem with such a simple definition, the theory requires some surprisingly sophisticated algebraic tools. Many of the results rely upon extending ''p''-adic valuations to the real numbers and extending Sperner's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Kunz Function
In algebra, the Hilbert–Kunz function of a local ring (''R'', ''m'') of prime characteristic ''p'' is the function :f(q) = \operatorname_R(R/m^) where ''q'' is a power of ''p'' and ''m'' 'q''/sup> is the ideal generated by the ''q''-th powers of elements of the maximal ideal ''m''. The notion was introduced by Ernst Kunz, who used it to characterize a regular ring as a Noetherian ring in which the Frobenius morphism is flat Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), .... If d is the dimension of the local ring, Monsky showed that f(q)/(q^d) is c+O(1/q) for some real constant c. This constant, the "Hilbert-Kunz" multiplicity", is greater than or equal to 1. Watanabe and Yoshida strengthened some of Kunz's results, showing that in the unmixed case, the ring is regular precisely ... [...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] |
Monsky's Theorem
In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square does not have an odd equidissection. The problem was posed by Fred Richman in the '' American Mathematical Monthly'' in 1965, and was proved by Paul Monsky in 1970. Proof Monsky's proof combines combinatorial and algebraic techniques, and in outline is as follows: #Take the square to be the unit square with vertices at (0,0), (0,1), (1,0) and (1,1). If there is a dissection into ''n'' triangles of equal area then the area of each triangle is 1/''n''. #Colour each point in the square with one of three colours, depending on the 2-adic valuation of its coordinates. #Show that a straight line can contain points of only two colours. #Use Sperner's lemma to show that every triangulation of the square into triangles meeting edge-to-edge must contain at least one triangle whose vertices have three different colours. #Conclude from t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monsky–Washnitzer Cohomology
In algebraic geometry, Monsky–Washnitzer cohomology is a ''p''-adic cohomology theory defined for non-singular affine varieties over fields of positive characteristic ''p'' introduced by , who were motivated by the work of . The idea is to lift the variety to characteristic 0, and then take a suitable subalgebra of the algebraic de Rham cohomology Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ... of . The construction was simplified by . Its extension to more general varieties is called rigid cohomology. References * * (letter to Atiyah, Oct. 14 1963) * * * {{DEFAULTSORT:Monsky-Washnitzer cohomology Algebraic geometry Cohomology theories Homological algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
