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In geometry, a centre (British English) or center (American English) (from Ancient Greek κέντρον (kéntron) 'pointy object') of an object is a point in some sense in the middle of the object. According to the specific definition of centre taken into consideration, an object might have no centre.
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. [further explanation needed] The same definition extends to any object in -dimensional Euclidean space. [1]
The kernel of the map G → G i is the i th center [1] of G (second center, third center, etc.), denoted Z i (G). [2] Concretely, the (i+1)-st center comprises the elements that commute with all elements up to an element of the i th center. Following this definition, one can define the 0th center of a group to be the identity subgroup.
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid , circumcenter , incenter and orthocenter were familiar to the ancient Greeks , and can be obtained by simple constructions .
The center of a group G consists of all those elements x in G such that xg = gx for all g in G. This is a normal subgroup of G. The similarly named notion for a semigroup is defined likewise and it is a subsemigroup. [1] [2] The center of a ring (or an associative algebra) R is the subset of R consisting of all those elements x of R such that ...
Another less common notation for the centralizer is Z(a), which parallels the notation for the center. With this latter notation, one must be careful to avoid confusion between the center of a group G, Z(G), and the centralizer of an element g in G, Z(g). The normalizer of S in the group (or semigroup) G is defined as
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The finite difference of higher orders can be defined in recursive manner as Δ n h ≡ Δ h (Δ n − 1 h) . Another equivalent definition is Δ n h ≡ [T h − I ] n . The difference operator Δ h is a linear operator, as such it satisfies Δ h [ α f + β g ](x) = α Δ h [ f ](x) + β Δ h [g](x) . It also satisfies a special Leibniz rule: