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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 ...
In theoretical physics, an invariant is an observable of a physical system which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition .
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 .
A real tensor in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part.
The FPP is a topological invariant, i.e., it is preserved by any homeomorphism. The FPP is also preserved by any retraction. According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP, and convexity is not even a topological property, so it makes sense ...
Invariant (computer science), an expression whose value doesn't change during program execution Loop invariant, a property of a program loop that is true before (and after) each iteration; A data type in method overriding that is neither covariant nor contravariant; Class invariant, an invariant used to constrain objects of a class
A scalar (also called type-0 or rank-0 tensor) is an object that does not vary with the change in basis. An example of a physical observable that is a scalar is the mass of a particle. The single, scalar value of mass is independent to changes in basis vectors and consequently is called invariant.
The set of T-invariant subspaces of V is sometimes called the invariant-subspace lattice of T and written Lat(T). As the name suggests, it is a lattice, with meets and joins given by (respectively) set intersection and linear span. A minimal element in Lat(T) in said to be a minimal invariant subspace.