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  2. Schur's lemma - Wikipedia

    en.wikipedia.org/wiki/Schur's_lemma

    In mathematics, Schur's lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras.In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a linear map from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0.

  3. Schur's lemma (Riemannian geometry) - Wikipedia

    en.wikipedia.org/wiki/Schur's_lemma_(Riemannian...

    In Riemannian geometry, Schur's lemma is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially a one-step calculation, which has only one input: the second Bianchi identity.

  4. Summation by parts - Wikipedia

    en.wikipedia.org/wiki/Summation_by_parts

    It is used to prove Kronecker's lemma, which in turn, is used to prove a version of the strong law of large numbers under variance constraints. It may be used to prove Nicomachus's theorem that the sum of the first n {\displaystyle n} cubes equals the square of the sum of the first n {\displaystyle n} positive integers.

  5. Lemma (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Lemma_(mathematics)

    In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem".

  6. Thom's first isotopy lemma - Wikipedia

    en.wikipedia.org/wiki/Thom's_first_isotopy_lemma

    The lemma is also valid for the stratification that satisfies Bekka's condition (C), which is weaker than Whitney's condition (B). [5] (The significance of this is that the consequences of the first isotopy lemma cannot imply Whitney’s condition (B).) Thom's second isotopy lemma is a family version of the first isotopy lemma.

  7. Jordan's lemma - Wikipedia

    en.wikipedia.org/wiki/Jordan's_lemma

    The path C is the concatenation of the paths C 1 and C 2.. Jordan's lemma yields a simple way to calculate the integral along the real axis of functions f(z) = e i a z g(z) holomorphic on the upper half-plane and continuous on the closed upper half-plane, except possibly at a finite number of non-real points z 1, z 2, …, z n.

  8. Schur–Weyl duality - Wikipedia

    en.wikipedia.org/wiki/Schur–Weyl_duality

    Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finite-dimensional representations of the general linear and symmetric groups. . Schur–Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each oth

  9. Titu's lemma - Wikipedia

    en.wikipedia.org/wiki/Titu's_Lemma

    In mathematics, the following inequality is known as Titu's lemma, Bergström's inequality, Engel's form or Sedrakyan's inequality, respectively, referring to the article About the applications of one useful inequality of Nairi Sedrakyan published in 1997, [1] to the book Problem-solving strategies of Arthur Engel published in 1998 and to the book Mathematical Olympiad Treasures of Titu ...