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The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. [3] Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important ...
Invariant theory is a branch of abstract algebra dealing with actions of groups ... "Über die vollen Invariantensysteme (On Full Invariant Systems)", Math. Annalen ...
In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor are the coefficients of the ...
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 .
Klein's j-invariant in the complex plane. In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for special linear group SL(2, Z) defined on the upper half-plane of complex numbers.
The scalar product of a vector and a covector is invariant, because one has components that vary with the base change, and the other has components that vary oppositely, and the two effects cancel out. One thus says that covectors are dual to vectors. Thus, to summarize:
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically, for functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set.
In mathematics, particularly in topology and knot theory, Arnold invariants are invariants introduced by Vladimir Arnold in 1994 [1] for studying the topology and geometry of plane curves. The three main invariants— J + {\displaystyle J^{+}} , J − {\displaystyle J^{-}} , and S t {\displaystyle St} —provide ways to classify and understand ...