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In this example, where n=2, the red 1-simplex has vertices which are labelled by the same number with opposite signs. Tucker's lemma states that for such a triangulation at least one such 1-simplex must exist. In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker.
Ax–Grothendieck theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem ; Büchi-Elgot-Trakhtenbrot theorem (mathematical logic) Cantor–Bernstein–Schröder theorem (set theory, cardinal numbers) Cantor's theorem (set theory, Cantor's diagonal argument) Church–Rosser theorem (lambda calculus)
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".
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.
Next, we must show that if the theorem is true for a = k, then it is also true for a = k + 1. For this inductive step, we need the following lemma. Lemma. For any integers x and y and for any prime p, (x + y) p ≡ x p + y p (mod p). The lemma is a case of the freshman's dream. Leaving the proof for later on, we proceed with the induction.
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.
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
The theorem is named after the Austrian mathematician Otto Schreier who proved it in 1928. It provides an elegant proof of the Jordan–Hölder theorem. It is often proved using the Zassenhaus lemma. Baumslag (2006) gives a short proof by intersecting the terms in one subnormal series with those in the other series.