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Dual Cone
Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. Dual cone In a vector space The dual cone ''C'' of a subset ''C'' in a linear space ''X'' over the real numbers, reals, e.g. Euclidean space R''n'', with dual space ''X'' is the set :C^* = \left \, where \langle y, x \rangle is the dual system, duality pairing between ''X'' and ''X'', i.e. \langle y, x\rangle = y(x). ''C'' is always a convex cone, even if ''C'' is neither convex set, convex nor a linear cone, cone. In a topological vector space If ''X'' is a topological vector space over the real or complex numbers, then the dual cone of a subset ''C'' ⊆ ''X'' is the following set of continuous linear functionals on ''X'': :C^ := \left\, which is the polar set, polar of the set -''C''. No matter what ''C'' is, C^ will be a convex cone. If ''C'' ⊆ then C^ = X^. In a Hilbert space (internal dual cone) Alternatively, many authors define the dual cone in the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Cone Illustration
Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual number, a number system used in automatic differentiation * Dual (grammatical number), a grammatical category used in some languages * Dual county, a Gaelic games county which competes in both Gaelic football and hurling * Dual diagnosis, a psychiatric diagnosis of co-occurrence of substance abuse and a mental problem * Dual fertilization, simultaneous application of a P-type and N-type fertilizer * Dual impedance, electrical circuits that are the dual of each other * Dual SIM cellphone supporting use of two SIMs * Aerochute International Dual a two-seat Australian powered parachute design Acronyms and other uses * Dual (brand), a manufacturer of Hifi equipment * DUAL (cognitive architecture), an artificial intelligence design model * DUAL algorithm, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, mathematical formulation of quantum mechanics, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, convex minimization, a subdomain of optimization (mathematics), optimization theory. Convex sets A subset C \subseteq X of some vector space X is if it satisfies any of the following equivalent conditions: #If 0 \leq r \leq 1 is real and x, y \in C then r x + (1 - r) y \in C. #If 0 < r < 1 is real and with then Throughout, will be a map valued in the Extended real number line, extended real numbers with a Domain of a function, domain that is a convex subset of some vector space. The map is a if holds for any real and any |
Polar Set
In functional and convex analysis, and related disciplines of mathematics, the polar set A^ is a special convex set associated to any subset A of a vector space X, lying in the dual space X^. The bipolar of a subset is the polar of A^\circ, but lies in X (not X^). Definitions There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis. In each case, the definition describes a duality between certain subsets of a pairing of vector spaces \langle X, Y \rangle over the real or complex numbers (X and Y are often topological vector spaces (TVSs)). If X is a vector space over the field \mathbb then unless indicated otherwise, Y will usually, but not always, be some vector space of linear functionals on X and the dual pairing \langle \cdot, \cdot \rangle : X \times Y \to \mathbb will be the bilinear () defined by \langle x, f \rangle := f(x). If X is a topological vector space then the space Y will usually, but no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipolar Theorem
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem. Preliminaries Suppose that X is a topological vector space (TVS) with a continuous dual space X^ and let \left\langle x, x^ \right\rangle := x^(x) for all x \in X and x^ \in X^. The convex hull of a set A, denoted by \operatorname A, is the smallest convex set containing A. The convex balanced hull of a set A is the smallest convex balanced set containing A. The polar of a subset A \subseteq X is defined to be: A^\circ := \left\. while the prepolar of a subset B \subseteq X^ is: ^ B := \left\. The bipolar of a subset A \subseteq X, often denoted by A^ is the set A^ := ^\left(A^\right) = \left\. Statement in functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Cone Illustration
Polar may refer to: Geography * Geographical pole, either of the two points on Earth where its axis of rotation intersects its surface ** Polar climate, the climate common in polar regions ** Polar regions of Earth, locations within the polar circles, referred to as the Arctic and Antarctic Places * Polar, Wisconsin, town in Langlade County, Wisconsin, United States ** Polar (community), Wisconsin, unincorporated community in Langlade County, Wisconsin, United States Arts, entertainment and media * ''Polar'' (webcomic), a webcomic and series of graphic novels by Víctor Santos * ''Polar'' (film), a 2019 Netflix film adaption of the above comic series * ''Polar'', a 2002 novel by T. R. Pearson Music * Polar Music, a record label * Polar Studios, music studio of ABBA in Sweden * ''Polar'' (album), second album by the High Water Marks * ''Polars'' (album), an album by the Dutch metal band, Textures Brands and enterprises * Polar Air Cargo, an American airline * Polar Ai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Semidefinite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \mathbf^* M \mathbf is positive for every nonzero complex column vector \mathbf, where \mathbf^* denotes the conjugate transpose of \mathbf. Positive semi-definite matrices are defined similarly, except that the scalars \mathbf^\mathsf M \mathbf and \mathbf^* M \mathbf are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''. Some authors use more general definitions of definiteness, permitting the matrices to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthant
In geometry, an orthant or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutually orthogonal half-spaces. By independent selections of half-space signs, there are 2''n'' orthants in ''n''-dimensional space. More specifically, a closed orthant in R''n'' is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities: :ε1''x''1 ≥ 0 ε2''x''2 ≥ 0 · · · ε''n''''x''''n'' ≥ 0, where each ε''i'' is +1 or −1. Similarly, an open orthant in R''n'' is a subset defined by a system of strict inequalities :ε1''x''1 > 0 ε2''x''2 > 0 · ·& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the pictu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane Separation Theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces. A related result is the supporting hyperplane theorem. In the context of support-vector machines, the ''optimally separating hyperplane'' or ''maximum-margin hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is Closure (mathematics), closed under the limit of a sequence, limit operation. This should not be confused with closed manifold. Sets that are both open and closed and are called clopen sets. Definition Given a topological space (X, \tau), the following statements are equivalent: # a set A \subseteq X is in X. # A^c = X \setminus A is an open subset of (X, \tau); that is, A^ \in \tau. # A is equal to its Closure (topology), closure in X. # A contains all of its limit points. # A contains all of its Boundary (topology), boundary points. An alternative characterization (mathematics), characterization of closed sets is available via sequences and Net (mathematics), net ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supporting Hyperplane
In geometry, a supporting hyperplane of a Set (mathematics), set S in Euclidean space \mathbb R^n is a hyperplane that has both of the following two properties: * S is entirely contained in one of the two closed set, closed Half-space (geometry), half-spaces bounded by the hyperplane, * S has at least one boundary-point on the hyperplane. Here, a closed half-space is the half-space that includes the points within the hyperplane. Supporting hyperplane theorem This theorem states that if S is a convex set in the topological vector space X=\mathbb^n, and x_0 is a point on the boundary (topology), boundary of S, then there exists a supporting hyperplane containing x_0. If x^* \in X^* \backslash \ (X^* is the dual space of X, x^* is a nonzero linear functional) such that x^*\left(x_0\right) \geq x^*(x) for all x \in S, then :H = \ defines a supporting hyperplane. Conversely, if S is a closed set with nonempty interior (topology), interior such that every point on the boundary has a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
