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12 Feb 2008: Prove that for each x [1, 1], there existsζ (1, 1) such that. ... 14) Let be a bounded, open subset of Rn and g : (a, b) R be a function such that(i) g(, t) is bounded and continuous in for each t (a,

8 Mar 2008: Example 2.14. Given any ǫ > 0 and any K > 0 we can find A, B GL2(R)such that ‖I B‖ < ǫ but ‖I ABA1‖ > K. ... 20. 9 Characters. Recall that a character θ on a commutative Banach algebra B is non-zerolinear functional θ : B C such that θ(ab) =

19 Aug 2008: Definition 4.2 For any pseudo-monad T on a 2-category C and for any bicat-egory B, define the bicategory Cocone((C, T ), B) such that an object consistsof. • ... ᾱ. D(HA, HB). H? D(HA, αB)- D(HA, HB). D(αA, KB)? such that for every pair of

13 Aug 2008: Furthermore. (u|A(B)|B = (u|B(C)|B. is a play in B. Conversely given a play r in A ( B anda play s in B ( C such that ... B. These morphisms are uniquelyrepresented by relations F : A - B such that.

17 Nov 2008: In this and the following questions, you arenot being asked to find such an isomorphism explicitly.). ... Here O(n) denotesthe subgroup of GLn(R) preserving the standard (positive definite) symmetric bilinear form.(b) Explicitly construct such a

8 Jan 2008: v) If ‖a b‖1 = ‖a‖1 ‖b‖1, does it follow that we can find α, β F,not both zero, such that βaj = αbj for all 1 j n? ... Give reasons. (vi) If ‖ab‖ = ‖a‖ ‖b‖, does it follow that we can find α, β F,not both zero, such that

16 Jul 2008: So there exists a basis v1,. , vd for L(D1) such that L(D1Pi) has basis v1,. , ... One such involution is given in the statement of thelemma. Remark 3.6.

7 May 2008: terms each ofwhich of course must themselves evaluate to something low and each such termis of the form B(w1) B(w2). ... Here“B(x)” means the set (in the sense of 〈V,〉) of those y such that x co y).

21 Aug 2008: are those of B and to the treebelow" each such opening move, we graft onto it a copy ofthe forest of questions of A.This is a good place to consider ... We presuppose an innite collection of names whichare ranged over by symbols such as x;x0;xi;f, a;b;

3 Mar 2008: eachx Sn has a neighborhood U such that U g(U ) = for any1 6= g G. ... σ = σ0, σ1,. , σr = τ such that each σi and σi1 intersectalong an (n 1)-simplex.