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Let ≤ be a partial order over a set X and let f: X → X be a function over X. Then a prefixed point (also spelled pre-fixed point, sometimes shortened to prefixpoint or pre-fixpoint) [citation needed] of f is any p such that f(p) ≤ p. Analogously, a postfixed point of f is any p such that p ≤ f(p). [3] The opposite usage occasionally ...
The + and invariants keep track of how curves change under these transformations and deformations. The + invariant increases by 2 when a direct self-tangency move creates new self-intersection points (and decreases by 2 when such points are eliminated), while decreases by 2 when an inverse self-tangency move creates new intersections (and increases by 2 when they are eliminated).
Firstly, if one has a group G acting on a mathematical object (or set of objects) X, then one may ask which points x are unchanged, "invariant" under the group action, or under an element g of the group. Frequently one will have a group acting on a set X, which leaves one to determine which objects in an associated set F(X) are invariant. For ...
The point z cannot lie in D, because, by passing to the limit, z would have to be a fixed point. The result for the case of fixed points implies that the maps f k {\displaystyle f_{k}} leave invariant all Euclidean disks whose hyperbolic center is located at z k {\displaystyle z_{k}} .
In the sense of universal algebra, a pointed set is a set together with a single nullary operation:, [a] which picks out the basepoint. [7] Pointed maps are the homomorphisms of these algebraic structures. The class of all pointed sets together with the class of all based maps forms a category.
In the context of a group action, this may be stated as: invariants are functions of coinvariants (equivalence classes, orbits), and a complete set of invariants characterizes the coinvariants (is a set of defining equations for the coinvariants).
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