• 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
  • Thumbnail for Invariant (mathematics)
    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
  • Thumbnail for Projection (linear algebra)
    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
  • 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
  • Thumbnail for Standard deviation
    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
  • 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
  • Thumbnail for Lie algebra
    ]\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
  • Thumbnail for Quaternion
    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
  • Thumbnail for Singular value decomposition
    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
  • Thumbnail for Lorentz group
    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
  • Thumbnail for 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
  • Thumbnail for Spinor
    {\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
  • Thumbnail for Noether's theorem
    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
  • Thumbnail for Measure (mathematics)
    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