probability theory and ergodic theory, the invariant sigma-algebra is a sigma-algebra formed by sets which are invariant under a group action or dynamical system...

the map T {\displaystyle T} is measurable, invariant sets form a sigma-algebra, the invariant sigma-algebra. The notion of invariance is formalized in...

{\displaystyle (X,\Sigma )} is said to be invariant under f {\displaystyle f} if, for every measurable set A {\displaystyle A} in Σ , {\displaystyle \Sigma ,} μ (...

In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)...

viewpoint, the universal enveloping algebra of the Lie algebra of a Lie group may be identified with the algebra of left-invariant differential operators on the...

mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure...

otherwise, a kind of invariant delta function on isomorphism classes of intervals. Its powers in the incidence algebra are the other invariant delta functions...

The scale-invariant feature transform (SIFT) is a computer vision algorithm to detect, describe, and match local features in images, invented by David...

geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is...

v_{n}\mapsto v_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (n)}} defines a linear endomorphism of Tn(V). A symmetric tensor is a tensor that is invariant under...

In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of...

represented in mathematical texts and equations by the lowercase Greek letter σ (sigma), for the population standard deviation, or the Latin letter s, for the...

σ ) μ {\displaystyle ({\mathcal {J}}^{\rho \sigma })^{\mu }} . Similarly, since the equation is invariant under translations, there is a tensor of conserved...

]\otimes \mathrm {id} )\circ (\mathrm {id} +\sigma +\sigma ^{2})=0.} Affine Lie algebra Automorphism of a Lie algebra Frobenius integrability theorem (the integrability...

In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken...

-i\sigma _{1}=-\sigma _{2}\sigma _{3},\quad \mathbf {j} \mapsto -i\sigma _{2}=-\sigma _{3}\sigma _{1},\quad \mathbf {k} \mapsto -i\sigma _{3}=-\sigma _{1}\sigma...

spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) to physics. Spacetime algebra provides...

Choquet theory Hewitt–Savage zero–one law Krein–Milman theorem Invariant sigma-algebra Spiegelhalter, D. 2024 [1] Why probability doesn't exist (but it's...

is invariant under unitary equivalence, this shows ‖ M ‖ = | ∑ i σ i 2 {\displaystyle \|\mathbf {M} \|={\sqrt {{\vphantom {\bigg |}}\sum _{i}\sigma _{i}^{2}}}}...

abelian von Neumann algebras on separable Hilbert spaces, particularly since they are completely classifiable by simple invariants. Though there is a theory...

In mathematical physics, the Dirac algebra is the Clifford algebra Cl 1 , 3 ( C ) {\displaystyle {\text{Cl}}_{1,3}(\mathbb {C} )} . This was introduced...

which provides a convenient alternative to the left-invariant vector field for calculating the Lie algebra. We can write down a set of six generators: Vector...

}}{\begin{pmatrix}I_{2}&\sigma ^{2}\\\sigma ^{2}&-I_{2}\end{pmatrix}}~.} The Dirac algebra can be regarded as a complexification of the real algebra Cl1,3( R {\displaystyle...

polynomials. Commutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory. Central to the development of these subjects...

even number of reflections, and the value of 1 otherwise. Algebraically, the Dickson invariant can be defined as D(f) = rank(I − f) modulo 2, where I is...

{\displaystyle {\mathcal {L}}={1 \over 2}g(\partial ^{\mu }\Sigma ,\partial _{\mu }\Sigma )-V(\Sigma )} where we have used a + − − − metric signature and the...

{\displaystyle (-1)\{a_{1}\sigma _{1}+a_{2}\sigma _{2}\}\,(-1)=a_{1}\sigma _{1}+a_{2}\sigma _{2}} but a spinor rotation of 180°. The Clifford algebra Cℓ3,0( R {\displaystyle...

corresponding statements with respect to columns. The determinant is invariant under matrix similarity. This implies that, given a linear endomorphism...

of Noether's theorem for action I stipulates for the invariants: If an integral I is invariant under a continuous group Gρ with ρ parameters, then ρ...

Σ {\displaystyle \Sigma } a σ-algebra over X . {\displaystyle X.} A set function μ {\displaystyle \mu } from Σ {\displaystyle \Sigma } to the extended...