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H-principle
In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as the immersion problem, isometric immersion problem, fluid dynamics, and other areas. The theory was started by Yakov Eliashberg, Mikhail Gromov and Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. The first evidence of h-principle appeared in the Whitney–Graustein theorem. This was followed by the Nash–Kuiper isometric ''C''1 embedding theorem and the Smale–Hirsch immersion theorem. Rough idea Assume we want to find a function ''ƒ'' on R''m'' which satisfies a partial differential equation of degree ''k'', in co-ordinates (u_1,u_2,\dots,u_m). One can rewrite it as :\Psi(u_1,u_2,\dots,u_m, J^k_f)=0 where J^k_f stands for all parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Gromov (mathematician)
Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; russian: link=no, Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a professor of mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry". Biography Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His Russian father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. His mother was the cousin of World Chess Champion Mikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich. Gromov was born during World War II, and his mother, who worked as a medical doctor in the Soviet Army, had to leave the front line in order to give birth to him. When Gromov was nine years old, his mother ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yakov Eliashberg
Yakov Matveevich Eliashberg (also Yasha Eliashberg; russian: link=no, Яков Матвеевич Элиашберг; born 11 December 1946) is an American mathematician who was born in Leningrad, USSR. Education and career Eliashberg received his PhD, entitled ''Surgery of Singularities of Smooth Mappings'', from Leningrad University in 1972, under the direction of Vladimir Rokhlin. Due to the growing anti-Semitism in the Soviet Union, from 1972 to 1979 he had to work at the Syktyvkar State University in the isolated Komi Republic. In 1980 Eliashberg returned to Leningrad and applied for a visa, but his request was denied and he became a refusenik until 1987. He was cut off from mathematical life and was prevented to work in academia, but due to a friend's intercession, he managed to secure a job in industry as the head of a computer software group. In 1988 Eliashberg managed to move to the United States, and since 1989 he has been Herald L. and Caroline L. Ritch profe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash-Kuiper Theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (''C''1) embeddings and the second for embeddings that are analytic or smooth of class ''Ck'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result. The ''C''1 theorem was published in 1954, the ''Ck''-theorem in 1956. The real analytic theorem was first treat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash–Kuiper Theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (''C''1) embeddings and the second for embeddings that are analytic or smooth of class ''Ck'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result. The ''C''1 theorem was published in 1954, the ''Ck''-theorem in 1956. The real analytic theorem was first treated b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Embedding Theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (''C''1) embeddings and the second for embeddings that are analytic or smooth of class ''Ck'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result. The ''C''1 theorem was published in 1954, the ''Ck''-theorem in 1956. The real analytic theorem was first treat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Eversion
In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word ''eversion'' means "turning inside out"). Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (with possible self-intersections) without cutting or tearing it or creating any crease. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false. More precisely, let :f\colon S^2\to \R^3 be the standard embedding; then there is a regular homotopy of immersions :f_t\colon S^2\to \R^3 such that ''ƒ''0 = ''ƒ'' and ''ƒ''1 = −''ƒ''. History An existence proof for crease-free sphere eversion was first created by . It is difficult to visualize a particular example of such a turning, although some digital animations have been produced tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frame Bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E''''x''. The general linear group acts naturally on F(''E'') via a change of basis, giving the frame bundle the structure of a principal GL(''k'', R)-bundle (where ''k'' is the rank of ''E''). The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle. Definition and construction Let ''E'' → ''X'' be a real vector bundle of rank ''k'' over a topological space ''X''. A frame at a point ''x'' ∈ ''X'' is an ordered basis for the vector space ''E''''x''. Equivalently, a frame can be viewed as a linear isomorphism :p : \mathbf^k \to E_x. The set of all frames at ''x'', denoted ''F''''x'', has a natural right action by the general linear group GL(''k'', R) of invertible ''k'' × ''k'' matri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorema Egregium
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this manner (translated from Latin): :Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged. The theorem is "remarkable" because the starting ''definition'' of Gaussian curvatu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . For example, a sphere of radius has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is an ''intrinsic'' measure of curvature, depending only on distances that are measured “within” or along the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the ''Theorema egregium''. Gaussian curvature is named after Carl Friedrich Gauss, who published the ''Theorema egregium'' in 1827. Informal definition At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Map
In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous). These maps are the morphisms in the category of metric spaces, Met (Isbell 1964). They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps. Specifically, suppose that ''X'' and ''Y'' are metric spaces and ƒ is a function from ''X'' to ''Y''. Thus we have a metric map when, for any points ''x'' and ''y'' in ''X'', : d_(f(x),f(y)) \leq d_(x,y) . \! Here ''d''''X'' and ''d''''Y'' denote the metrics on ''X'' and ''Y'' respectively. Examples Let us consider the metric space ,1/2/math> with the Euclidean metric. Then the function f(x)=x^2 is a metric map, since for x\ne y, , f(x)-f(y), =, x+y, , x-y, <, x-y, . Category of metric maps The |
