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In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) ... If the invariant held, it still does. 2: ×2: ×2:
Springer, T. A. (1977), Invariant Theory, New York: Springer, ISBN 0-387-08242-5 An older but still useful survey. Sturmfels, Bernd (1993), Algorithms in Invariant Theory, New York: Springer, ISBN 0-387-82445-6 A beautiful introduction to the theory of invariants of finite groups and techniques for computing them using Gröbner bases.
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).
In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor ... [2] The correspondence ...
In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation . For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping , and a difference of slopes is invariant under shear mapping .
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
In the classification of two-dimensional closed manifolds, Euler characteristic (or genus) and orientability are a complete set of invariants. Jordan normal form of a matrix is a complete invariant for matrices up to conjugation, but eigenvalues (with multiplicities) are not.
In algebra, the first and second fundamental theorems of invariant theory concern the generators and relations of the ring of invariants in the ring of polynomial functions for classical groups (roughly, the first concerns the generators and the second the relations). [1] The theorems are among the most important results of invariant theory.