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My attitude is that the mathematics of the quantum theory deals primarily with the structure of the implicate pre-space and with how an explicate order of space and time emerges from it, rather than with movements of physical entities, such as particles and fields. (This is a kind of extension of what is done in general relativity, which deals ...
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.
Topological order is the notion that describes the long range entangled states: topological order = pattern of long range entanglements. Short range entangled states are trivial in the sense that they all belong to one phase. However, in the presence of symmetry, even short range entangled states are nontrivial and can belong to different phases.
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 ...
Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when sufficient energy is available, not needing control by any external agent.
Although a normal Voronoi cell is defined as the set of points closest to a single point in S, an nth-order Voronoi cell is defined as the set of points having a particular set of n points in S as its n nearest neighbors. Higher-order Voronoi diagrams also subdivide space. Higher-order Voronoi diagrams can be generated recursively.
8 is a cyclic group of order 7. The symmetries of P 1 F 7 are Möbius transformations, and the basic transformations are reflections (order 2, k ↦ −1/k), translations (order 7, k ↦ k + 1), and doubling (order 3 since 2 3 = 1, k ↦ 2k). The corresponding symmetries on the Fano plane are respectively swapping vertices, rotating the graph ...