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 (...

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

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...

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...

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

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

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

-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...

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

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

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...

}}{\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...

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...

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...

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

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...

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

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

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

K_{j}\right]&=-i\hbar \epsilon _{ijk}L_{k}\end{aligned}}} The Casimir invariants of this algebra are P μ P μ {\textstyle P_{\mu }P^{\mu }} and W μ W μ {\textstyle...

{\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...

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

Arf invariant of a certain quadratic form on H 1 ( Σ , Z / 2 Z ) {\displaystyle H_{1}(\Sigma ,\mathbb {Z} /2\mathbb {Z} )} . This Arf invariant is obviously...

called the Chern–Weil homomorphism, from the algebra of G-adjoint invariant polynomials on g (Lie algebra of G) to the cohomology H ∗ ( M , R ) {\displaystyle...