Invariant (math)

An invariant is a property of a certain class ( set ) of mathematical objects that remains unchanged during transformations of a certain type.

Content

Definition

Let be $A$ ${\ displaystyle A}$ - many and $G$ ${\ displaystyle G}$ Is the set of mappings from A to A. A mapping f from the set A to the set B is called an invariant for G if, for any $a\in A$ ${\ displaystyle a \ in A}$ and $g\in G$ ${\ displaystyle g \ in G}$ identity holds $f(a)=f(g(a))$ ${\ displaystyle f (a) = f (g (a))}$ .

The concept of an invariant is one of the most important in mathematics, since the study of an invariant is directly related to the problems of classifying objects of one type or another. Essentially, the goal of any mathematical classification is to construct some complete system of invariants (possibly the simplest), that is, a system that separates any two nonequivalent objects from the considered set. ^[one]

Invariants are used in various fields of mathematics, such as geometry , topology, and algebra . The discovery of invariants is an important step in the process of classifying mathematical objects.

Examples

The area of the triangle is invariant with respect to the isometries of Euclidean space .
The power of a set is invariant with respect to bijections .
The determinant , trace , eigenvectors and eigenvalues of the matrix are invariant with respect to the choice of basis.
In the theory of differential equations, an invariant is a function that depends on the desired function, the value of which is constant ( first integral ).
The Lebesgue measure is invariant under shifts.
Singular numbers of the matrix are invariant under orthogonal transformations .
The theory of invariants is engaged in the search for invariant polynomials (or simply “invariants” ) and the study of the algebra formed by them for the case of linear representations of algebraic groups, as well as the actions of algebraic groups on algebraic varieties.
Topological invariant - see Glossary of General Topology Terms .
Invariant problems represent a large class of problems in olympiad mathematics .
The Hardwiger number and the chromatic number are invariants of the graph when renumbering its vertices.
Elliptic Curve Invariant - Number $J(E)=1728{\frac {4a^{3}}{4a^{3}+27b^{2}}}{\pmod {p}}$ ${\ displaystyle J (E) = 1728 {\ frac {4a ^ {3}} {4a ^ {3} + 27b ^ {2}}} {\ pmod {p}}}$ . See GOST R 34.10-2001 .

Notes

↑ V.L.Popov. Invariant // Mathematical Encyclopedia. - M .: Soviet Encyclopedia, 1979. - T. 2 . - S. 526 .

Literature

Dieudonne J. , Carroll J., Mumford D. Geometric theory of invariants. - M .: Mir, 1974.- 278 p.

[1] V.L.Popov. Invariant // Mathematical Encyclopedia. - M .: Soviet Encyclopedia, 1979. - T. 2 . - S. 526 .