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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 + 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).
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 ...
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 ...
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 determinant is therefore also called a similarity invariant. The determinant of a linear transformation: for some finite-dimensional vector space V is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of basis in V.
As it is a second order tensor, the stress deviator tensor also has a set of invariants, which can be obtained using the same procedure used to calculate the invariants of the stress tensor. It can be shown that the principal directions of the stress deviator tensor s i j {\displaystyle s_{ij}} are the same as the principal directions of the ...
Calculating the Minkowski norm squared of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the speed of light c) to the square of the particle's proper mass: = = = + | | = where = is the metric tensor of special relativity with metric signature for definiteness chosen to be (–1, 1, 1, 1).