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14 May 2012: Lemma. Let S := {s A | grs T}. Then(a) S is an Ore set in A.(b) There is a natural Zariskian filtration on AS such that grAS = (grA)T. ... b) If A has a Zariskian filtration such that the associated graded ring grA isAuslander–Gorenstein, then A is

28 Nov 2012: An immersion is a smooth map f : X Y such that dfx is injective forall x X. ... A submersion is a smooth map f : X Y such that dfx is surjectivefor all x X.

28 Nov 2012: vd of V such that v1,. ,vr is a basis of W and the. ... vd of V such that v1,. ,vr is a basis of W the matrices.

19 Jul 2012: Without going into the details of such adefinition (see Section 1.3), let us state:. ... 6) Γααβ = 0. In such coordinates, the Einstein vacuum equations take the form.

27 Nov 2012: Exercise: Draw on top of this picture such a vector! 2.3 The tangent bundle. ... Moreover, ifT+ < , then given any compact subset K U, there exists a tK such thatγ(tk,T+) K =. The maxaimality statement is simply the following: If γ̃ : (a,b) is

24 Feb 2012: Given two disjoint Diophantine sets A and B, constructa Diophantine set C such that. ... Furthermore, there is a number t such thatt > yb and Lb,t(yb) = L(yb) 6= La(yb) = La,t(yb) for all b < a.

27 Nov 2012: N 1}. There are only N such rotations about c so all of them mustlie in G+. ... Otherwise, there are vectors v ΛZw1.Each such vector can be written as.