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Polar 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 reals, e.g. Euclidean space R''n'', with dual space ''X'' is the set :C^* = \left \, where \langle y, x \rangle is the 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 nor a 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 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 context of a real Hilbert space (such as R''n'' equipped with t ...
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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 (grammatical number), a grammatical category used in some languages * Dual county, a Gaelic games county which 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, or diffusing update algorithm, used to update Internet protocol routing ta ...
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Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view 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, 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 term ''Hilbert space'' for the abstract concept that under ...
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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 minimization, a subdomain of 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 is a if holds for any real 0 is called if \operatorname f \neq \varnothing and f(x) > -\infty for x \in \operatorname f. Alternatively, this means that there exists some x in the domain of f at which f(x) \in \mathbb and f is also equal to -\infty. In words, a function is if its domain is not empty, it never takes on the value -\infty, and it also is not identically equal to +\infty. If f : \mathbb^n \to \infty, \infty/math> is a proper convex function then there exist some vector b \in \mathbb^n and some r \in \mathbb such that :f(x) \geq x \cdot b - r for every x where ...
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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^, 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 \left\langle \cdot, \cdot \right\rangle : X \times Y \to \mathbb will be the bilinear () defined by :\left\langle x, f \right\rangle := f(x). If X is a topological vector space then the space Y w ...
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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 functional a ...
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Polar Cone Illustration
Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates *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 People * Polar (musician), Norwegian electronic music producer Arts, entertainment and media Music Labels and studios * Polar Music, a record label * Polar Studios, music studio of ABBA in Sweden Albums * ''Polar'' (album), second album by the High Water Marks * ''Polars'' (album), an album by the Dutch metal band, Textures Other uses in arts, entertainment and media * ''Polar'' (webcomic), a webcomic and series of graphic novel ...
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Positive Semidefinite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz 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. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ...
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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     · ·&nb ...
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Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two vectors in the space is a Scalar (mathematics), 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 (vector space), dimension are widely used in functional analysis. Inner product spaces over the Field (mathematics), 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 ...
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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 hyp ...
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Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose 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 closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \oper ...
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Supporting Hyperplane
In geometry, a supporting hyperplane of a 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 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 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 such that every point on the boundary has a supporting hyperplane, then S is a convex set. The hyperplane in the theorem may not be uniqu ...
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