6.0 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Invariant (mathematics) | 2/3 | https://en.wikipedia.org/wiki/Invariant_(mathematics) | reference | science, encyclopedia | 2026-05-05T07:24:05.176600+00:00 | kb-cron |
The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it will not be afterwards either. Given that there is a single I in the starting string MI, and one is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three).
== Invariant set == A subset S of the domain U of a mapping T: U → U is an invariant set under the mapping when
x
∈
S
⟺
T
(
x
)
∈
S
.
{\displaystyle x\in S\iff T(x)\in S.}
The elements of S are not necessarily fixed, even though the set S is fixed in the power set of U. (Some authors use the terminology setwise invariant, vs. pointwise invariant, to distinguish between these cases.) For example, a circle is an invariant subset of the plane under a rotation about the circle's center. Further, a conical surface is invariant as a set under a homothety of space. An invariant set of an operation T is also said to be stable under T. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group. In linear algebra, if a linear transformation T has an eigenvector v, then the line through 0 and v is an invariant set under T, in which case the eigenvectors span an invariant subspace which is stable under T. When T is a screw displacement, the screw axis is an invariant line, though if the pitch is non-zero, T has no fixed points. In probability theory and ergodic theory, invariant sets are usually defined via the stronger property
x
∈
S
⇔
T
(
x
)
∈
S
.
{\displaystyle x\in S\Leftrightarrow T(x)\in S.}
When the map
T
{\displaystyle T}
is measurable, invariant sets form a sigma-algebra, the invariant sigma-algebra.
== Formal statement ==
The notion of invariance is formalized in three different ways in mathematics: via group actions, presentations, and deformation.
=== Unchanged under group action === 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 example, rotation in the plane about a point leaves the point about which it rotates invariant, while translation in the plane does not leave any points invariant, but does leave all lines parallel to the direction of translation invariant as lines. Formally, define the set of lines in the plane P as L(P); then a rigid motion of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action. More importantly, one may define a function on a set, such as "radius of a circle in the plane", and then ask if this function is invariant under a group action, such as rigid motions. Dual to the notion of invariants are coinvariants, also known as orbits, which formalizes the notion of congruence: objects that can be taken to each other by a group action. For example, under the group of rigid motions of the plane, the perimeter of a triangle is an invariant, while the set of triangles congruent to a given triangle is a coinvariant. These are connected as follows: invariants are constant on coinvariants (for example, congruent triangles have the same perimeter), while two objects that agree in the value of one invariant may or may not be congruent (for example, two triangles with the same perimeter need not be congruent). In classification problems, one might seek to find a complete set of invariants, such that if two objects have the same values for this set of invariants, then they are congruent. For example, triangles such that all three sides are equal are congruent under rigid motions, via SSS congruence, and thus the lengths of all three sides form a complete set of invariants for triangles. The three angle measures of a triangle are also invariant under rigid motions, but do not form a complete set as incongruent triangles can share the same angle measures. However, if one allows scaling in addition to rigid motions, then the AAA similarity criterion shows that this is a complete set of invariants.
=== Independent of presentation === Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying topological space (the manifold) – as different cell complexes give the same underlying manifold, one may ask if the function is independent of choice of presentation, in which case it is an intrinsically defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation, and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense. The most common examples are:
The presentation of a manifold in terms of coordinate charts – invariants must be unchanged under change of coordinates. Various manifold decompositions, as discussed for Euler characteristic. Invariants of a presentation of a group.