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...
10 KB (1,461 words) - 11:53, 27 July 2024
the map T {\displaystyle T} is measurable, invariant sets form a sigma-algebra, the invariant sigma-algebra. The notion of invariance is formalized in...
24 KB (2,787 words) - 13:39, 3 April 2025
{\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 (...
5 KB (852 words) - 19:30, 14 March 2025
In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)...
34 KB (5,806 words) - 14:46, 17 February 2025
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...
51 KB (8,954 words) - 11:11, 9 February 2025
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...
65 KB (9,287 words) - 07:33, 12 May 2025
otherwise, a kind of invariant delta function on isomorphism classes of intervals. Its powers in the incidence algebra are the other invariant delta functions...
18 KB (3,019 words) - 16:28, 26 May 2025
The scale-invariant feature transform (SIFT) is a computer vision algorithm to detect, describe, and match local features in images, invented by David...
69 KB (9,232 words) - 19:22, 19 April 2025
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is...
93 KB (13,801 words) - 22:00, 13 April 2025
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...
13 KB (2,050 words) - 23:04, 2 March 2025
Donaldson–Thomas theory (redirect from Donaldson–Thomas invariant)
In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of...
10 KB (1,324 words) - 19:46, 17 December 2024
Standard deviation (redirect from Sigma interval)
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...
59 KB (8,233 words) - 19:16, 23 April 2025
Dirac equation (section The algebra of physical space)
σ ) μ {\displaystyle ({\mathcal {J}}^{\rho \sigma })^{\mu }} . Similarly, since the equation is invariant under translations, there is a tensor of conserved...
79 KB (13,120 words) - 13:01, 16 May 2025
]\otimes \mathrm {id} )\circ (\mathrm {id} +\sigma +\sigma ^{2})=0.} Affine Lie algebra Automorphism of a Lie algebra Frobenius integrability theorem (the integrability...
61 KB (10,477 words) - 22:23, 2 April 2025
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...
22 KB (4,143 words) - 19:08, 31 January 2025
Quaternion (category Composition algebras)
-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...
96 KB (12,665 words) - 22:22, 26 May 2025
spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) to physics. Spacetime algebra provides...
48 KB (7,332 words) - 19:32, 1 May 2025
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...
18 KB (2,571 words) - 11:09, 17 April 2025
Singular value decomposition (category Linear algebra)
is invariant under unitary equivalence, this shows ‖ M ‖ = | ∑ i σ i 2 {\displaystyle \|\mathbf {M} \|={\sqrt {{\vphantom {\bigg |}}\sum _{i}\sigma _{i}^{2}}}}...
91 KB (14,584 words) - 06:04, 19 May 2025
abelian von Neumann algebras on separable Hilbert spaces, particularly since they are completely classifiable by simple invariants. Though there is a theory...
10 KB (1,551 words) - 19:33, 9 February 2025
In mathematical physics, the Dirac algebra is the Clifford algebra Cl 1 , 3 ( C ) {\displaystyle {\text{Cl}}_{1,3}(\mathbb {C} )} . This was introduced...
15 KB (2,342 words) - 23:31, 7 April 2025
Lorentz group (redirect from Lorentzian algebra)
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...
66 KB (9,875 words) - 01:47, 3 May 2025
Gamma matrices (category Clifford algebras)
}}{\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...
61 KB (7,254 words) - 20:44, 10 May 2025
Ring theory (category Algebraic structures)
polynomials. Commutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory. Central to the development of these subjects...
24 KB (3,093 words) - 14:02, 18 May 2025
Orthogonal group (redirect from Invariant theory of the orthogonal group)
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...
56 KB (7,881 words) - 20:44, 2 May 2025
{\displaystyle {\mathcal {L}}={1 \over 2}g(\partial ^{\mu }\Sigma ,\partial _{\mu }\Sigma )-V(\Sigma )} where we have used a + − − − metric signature and the...
10 KB (1,287 words) - 19:13, 31 January 2025
Spinor (section Exterior algebra construction)
{\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...
72 KB (9,924 words) - 15:56, 26 May 2025
Determinant (category Linear algebra)
corresponding statements with respect to columns. The determinant is invariant under matrix similarity. This implies that, given a linear endomorphism...
91 KB (14,375 words) - 14:49, 9 May 2025
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...
71 KB (11,790 words) - 13:59, 23 May 2025
Measure (mathematics) (section Sigma-finite measures)
Σ {\displaystyle \Sigma } a σ-algebra over X . {\displaystyle X.} A set function μ {\displaystyle \mu } from Σ {\displaystyle \Sigma } to the extended...
35 KB (5,559 words) - 12:21, 2 May 2025