|
Contingent Cone
In mathematics, the paratingent cone and contingent cone were introduced by , and are closely related to tangent cones. Definition Let S be a nonempty subset of a Real number, real normed vector space (X, \, \cdot\, ). # Let some \bar \in \operatorname(S) be a point in the Closure (topology), closure of S. An element h \in X is called a ''tangent'' (or ''tangent vector'') to S at \bar, if there is a sequence (x_n)_ of elements x_n \in S and a sequence (\lambda_n)_ of positive real numbers \lambda_n > 0 such that \bar = \lim_ x_n and h = \lim_ \lambda_n (x_n - \bar). # The set T(S,\bar) of all tangents to S at \bar is called the ''contingent cone'' (or the ''Bouligand tangent cone'') to S at \bar. An equivalent definition is given in terms of a distance function and the limit infimum. As before, let (X, \, \cdot \, ) be a normed vector space and take some nonempty set S \subset X. For each x \in X, let the ''distance function'' to S be :d_S(x) := \inf\. Then, the ''contingent con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Cone
In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities. Definitions in nonlinear analysis In nonlinear analysis, there are many definitions for a tangent cone, including the adjacent cone, Bouligand's contingent cone, and the Clarke tangent cone. These three cones coincide for a convex set, but they can differ on more general sets. Clarke tangent cone Let A be a nonempty closed subset of the Banach space X. The Clarke's tangent cone to A at x_0\in A, denoted by \widehat_A(x_0) consists of all vectors v\in X, such that for any sequence \_\subset\mathbb tending to zero, and any sequence \_\subset A tending to x_0, there exists a sequence \_\subset X tending to v, such that for all n\ge 1 holds x_n+t_nv_n\in A Clarke's tangent cone is always subset of the corresponding contingent cone (and coincides with it, when the set in question is convex). It has the important property of being a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Vector Space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real (physical) world. A norm is a real-valued function defined on the vector space that is commonly denoted x\mapsto \, x\, , and has the following properties: #It is nonnegative, meaning that \, x\, \geq 0 for every vector x. #It is positive on nonzero vectors, that is, \, x\, = 0 \text x = 0. # For every vector x, and every scalar \alpha, \, \alpha x\, = , \alpha, \, \, x\, . # The triangle inequality holds; that is, for every vectors x and y, \, x+y\, \leq \, x\, + \, y\, . A norm induces a distance, called its , by the formula d(x,y) = \, y-x\, . which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (topology)
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of all closed sets containing . Intuitively, the closure can be thought of as all the points that are either in or "near" . A point which is in the closure of is a point of closure of . The notion of closure is in many ways dual to the notion of interior. Definitions Point of closure For S as a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point can be x itself). This definition generalizes to any subset S of a metric space X. Fully expressed, for X as a metric space with metric d, x is a point of closure of S if for every r > 0 there exists some s \in S such that the distance d(x, s) < r ( is allowed). Another way to express this is to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.png "Click for more on -> Tangent Cone")
