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A subset of a vector space is called a cone if for all real >,.A cone is called pointed if it contains the origin. A cone is convex if and only if +. The intersection of any non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone); the same is true of the union of an increasing (under set inclusion) family of cones (resp. convex cones).
Suppose (,) is an ordered vector space over the reals with an order unit whose order is Archimedean and let = [,]. Then the Minkowski functional p U {\displaystyle p_{U}} of U {\displaystyle U} (defined by p U ( x ) := inf { r > 0 : x ∈ r [ − u , u ] } {\displaystyle p_{U}(x):=\inf \left\{r>0:x\in r[-u,u]\right\}} ) is a norm called the ...
In mathematics, specifically in order theory and functional analysis, a subset of an ordered vector space is said to be order complete in if for every non-empty subset of that is order bounded in (meaning contained in an interval, which is a set of the form [,]:= {:}, for some ,), the supremum ' and the infimum both exist and are elements of .
The left and right order topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded set provides an example of a compact space that is not Hausdorff. The left order topology is the standard topology used for many set-theoretic purposes on a Boolean algebra. [clarification needed]
The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector , the state vector . If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.
The space of continuous real valued functions with compact support on a topological space with the pointwise partial order defined by when () for all , is a Riesz space. It is Archimedean, but usually does not have the principal projection property unless X {\displaystyle X} satisfies further conditions (for example, being extremally ...
The strictest form of order in a solid is lattice periodicity: a certain pattern (the arrangement of atoms in a unit cell) is repeated again and again to form a translationally invariant tiling of space. This is the defining property of a crystal. Possible symmetries have been classified in 14 Bravais lattices and 230 space groups.
Bohm, his co-worker Basil Hiley, and other physicists of Birkbeck College worked toward a model of quantum physics in which the implicate order is represented in the form of an appropriate algebra or other pregeometry. They considered spacetime itself as part of an explicate order that is connected to an implicate order that they called pre-space.