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The definition of a cone may be extended to higher dimensions; see convex cone. In this case, one says that a convex set C in the real vector space R n {\displaystyle \mathbb {R} ^{n}} is a cone (with apex at the origin) if for every vector x in C and every nonnegative real number a , the vector ax is in C . [ 2 ]
Cone of a circle. The original space X is in blue, and the collapsed end point v is in green.. In topology, especially algebraic topology, the cone of a topological space is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.
According to the above definition, if C is a convex cone, then C ∪ {0} is a convex cone, too. A convex cone is said to be pointed if 0 is in C, and blunt if 0 is not in C. [2] [21] Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β.
In flat spacetime, the future light cone of an event is the boundary of its causal future and its past light cone is the boundary of its causal past. In a curved spacetime, assuming spacetime is globally hyperbolic , it is still true that the future light cone of an event includes the boundary of its causal future (and similarly for the past).
Given a space X and a loop : representing an element of the fundamental group of X, we can form the mapping cone . The effect of this is to make the loop α {\displaystyle \alpha } contractible in C α {\displaystyle C_{\alpha }} , and therefore the equivalence class of α {\displaystyle \alpha } in the fundamental group of C α {\displaystyle ...
In physics, there are equations in every field to relate physical quantities to each other and perform calculations. Entire handbooks of equations can only summarize most of the full subject, else are highly specialized within a certain field. Physics is derived of formulae only.
Light cone coordinates can also be generalized to curved spacetime in general relativity. Sometimes calculations simplify using light cone coordinates. See Newman–Penrose formalism. Light cone coordinates are sometimes used to describe relativistic collisions, especially if the relative velocity is very close to the speed of light.
Let X be a projective space over some field K, and R, S be disjoint subspaces of X. Let A be an arbitrary subset of S. Then we define RA, the cone with top R and basis A, as follows : When A is empty, RA = A. When A is not empty, RA consists of all those points on a line connecting a point on R and a point on A.