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In linear algebra, a cone—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a real vector space that is closed under positive scalar multiplication; that is, is a cone if implies for every positive scalar .
More generally, given a vector bundle (finite-rank locally free sheaf) E on X, if R=Sym(E *) is the symmetric algebra generated by the dual of E, then the cone is the total space of E, often written just as E, and the projective cone is the projective bundle of E, which is written as ().
A right circular cone and an oblique circular cone A double cone (not shown infinitely extended) 3D model of a cone. A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex that is not contained in the base.
A cone C in a vector space X is said to be self-dual if X can be equipped with an inner product ⋅,⋅ such that the internal dual cone relative to this inner product is equal to C. [3] Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.
A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone (a cone with two nappes).It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
In operator algebra, the Koecher–Vinberg theorem is a reconstruction theorem for real Jordan algebras. It was proved independently by Max Koecher in 1957 [ 1 ] and Ernest Vinberg in 1961. [ 2 ] It provides a one-to-one correspondence between formally real Jordan algebras and so-called domains of positivity.
Then H n (R) is a simple Euclidean Jordan algebra of rank n for n ≥ 3. Let H n (C) be the space of complex self-adjoint n by n matrices with inner product (a,b) = Re Tr ab* and Jordan product a ∘ b = 1 / 2 (ab + ba). Then H n (C) is a simple Euclidean Jordan algebra of rank n for n ≥ 3.
In linear algebra, a power cone is a kind of a convex cone that is particularly important in modeling convex optimization problems. [1] [2] It is a generalization of the quadratic cone: the quadratic cone is defined using a quadratic equation (with the power 2), whereas a power cone can be defined using any power, not necessarily 2.
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