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For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite group has finite composition length, and every simple abelian group is cyclic of prime order.
However, this regularity described by short-range order does not necessarily apply to a larger area. [1] Examples of materials with short range order include amorphous materials such as wax, glass and liquids [2] as well as the collagen fibrils of the stroma in the cornea. [3] Besides ordering of atoms, short-range ordering of vacancies are ...
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} :
Conversely, a strict partial order < on may be converted to a non-strict partial order by adjoining all relationships of that form; that is, := < is a non-strict partial order. Thus, if ≤ {\displaystyle \leq } is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by a < b if a ≤ b and a ...
The converse of this implication leads to functions that are order-reflecting, i.e. functions f as above for which f(a) ≤ f(b) implies a ≤ b. On the other hand, a function may also be order-reversing or antitone, if a ≤ b implies f(a) ≥ f(b). An order-embedding is a function f between orders that is both order-preserving and order ...
As with Adobe Acrobat, Nitro PDF Pro's reader is free; but unlike Adobe's free reader, Nitro's free reader allows PDF creation (via a virtual printer driver, or by specifying a filename in the reader's interface, or by drag-'n-drop of a file to Nitro PDF Reader's Windows desktop icon); Ghostscript not needed. PagePlus: Proprietary: No
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