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Almgren Regularity Theorem
In geometric measure theory, a field of mathematics, the Almgren regularity theorem, proved by , states that the singular set of a mass-minimizing surface has codimension at least 2. Almgren's proof of this was 955 pages long. Within the proof many new ideas are introduced, such as monotonicity of a ''frequency function'' and the use of a ''center manifold'' to perform a more intricate blow-up procedure. A streamlined and more accessible proof of Almgren's regularity theorem, following the same ideas as Almgren, was given by Camillo De Lellis and Emanuele Spadaro Emanuele is the Italian form of Manuel. People with the name include: * Carlo Emanuele Buscaglia (1915–1944), Italian aviator * Emanuele Basile (1949–1980), captain of Carabinieri * Emanuele Belardi (born 1977), Italian football player * E ... in a series of three papers.De Lellis, Camillo; Spadaro, Emanuele Regularity of area minimizing currents III: blow-up. Ann. of Math. (2) 183 (2016), no. 2, 577–617. Refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Measure Theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth. History Geometric measure theory was born out of the desire to solve Plateau's problem (named after Joseph Plateau) which asks if for every smooth closed curve in \mathbb^3 there exists a surface of least area among all surfaces whose boundary equals the given curve. Such surfaces mimic soap films. The problem had remained open since it was posed in 1760 by Lagrange. It was solved independently in the 1930s by Jesse Douglas and Tibor Radó under certain topological restrictions. In 1960 Herbert Federer and Wendell Fleming used the theory of currents with which they were able to solve the orientable Plateau's problem analytically without topological restrictions, thus sparking geometric measu ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Set
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth. Definition A plane curve defined by an implicit equation :F(x,y)=0, where is a smooth function is said to be ''singular'' at a point if the Taylor series of has order at least at this point. The reason for this is that, in differential calculus, the tangent at the point of such a curve is defined by the equation :(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0)=0, whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension. Definition Codimension is a ''relative'' concept: it is only defined for one object ''inside'' another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector ''sub''space. If ''W'' is a linear subspace of a finite-dimensional vector space ''V'', then the codimension of ''W'' in ''V'' is the difference between the dimensions: :\operatorname(W) = \dim(V) - \dim(W). It is the complement of the dimension of ''W,'' in that, with the dimension of ''W,'' it adds up to the dimension of the ambient space ''V:'' :\dim(W) + \operatorname(W) = \dim(V). Simila ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Camillo De Lellis
Camillo De Lellis (born 11 June 1976) is an Italian mathematician who is active in the fields of calculus of variations, hyperbolic systems of conservation laws, geometric measure theory and fluid dynamics. He is a permanent faculty member in the School of Mathematics at the Institute for Advanced Study. He is also one of the two managing editors of Inventiones Mathematicae. Biography Prior joining the faculty of the Institute for Advanced Study, De Lellis was a professor of mathematics at the University of Zurich from 2004 to 2018. Before this, he was a postdoctoral researcher at ETH Zurich and at the Max Planck Institute for Mathematics in the Sciences. He received his PhD in mathematics from the Scuola Normale Superiore at Pisa, under the guidance of Luigi Ambrosio in 2002. Scientific activity De Lellis has given a number of remarkable contributions in different fields related to partial differential equations. In geometric measure theory he has been interested in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emanuele Spadaro
Emanuele is the Italian form of Manuel. People with the name include: * Carlo Emanuele Buscaglia (1915–1944), Italian aviator * Emanuele Basile (1949–1980), captain of Carabinieri * Emanuele Belardi (born 1977), Italian football player * Emanuele Calaiò (born 1982), Italian football player * Emanuele Canonica (born 1971), Italian professional golfer * Emanuele Chiapasco (1930–2021), Italian baseball player and entrepreneur * Emanuele Crialese (born 1965), Italian film screenwriter and director * Emanuele d'Astorga (1681–1736), Italian composer * Emanuele Filiberto, 2nd Duke of Aosta (1869–1931), eldest son of Amadeo I of Spain * Emanuele Filiberto, Prince of Venice and Piedmont (born 1972), member of the House of Savoy * Emanuele Filippini (born 1973), Italian football player * Emanuele Gianturco (1857–1907), Italian legal scholar and politician * Emanuele Idini (born 1970), retired freestyle swimmer * Emanuele Luzzati (1921–2007), Italian painter, produc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
World Scientific
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, along with 135 journals in various fields. In 1995, World Scientific co-founded the London-based Imperial College Press together with the Imperial College of Science, Technology and Medicine. Company structure The company head office is in Singapore. The Chairman and Editor-in-Chief is Dr Phua Kok Khoo, while the Managing Director is Doreen Liu. The company was co-founded by them in 1981. Imperial College Press In 1995 the company co-founded Imperial College Press, specializing in engineering, medicine and information technology, with Imperial College London. In 2006, World Scientific assumed full ownership of Imperial College Press, under a license granted by the university. Finally, in August 2016, ICP was fully incorporated into World Scientific under the new impr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Measure Theory
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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