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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) =

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

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.

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

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.

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).

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.

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;

21 Aug 2008: B. 85]) such that:. (i) the base category B has nite products,. ... x:t is not necessarily. dened in U. For example, in any c-pca U where there are elements a; b 2 U such that a b is.

19 Aug 2008: C(B, AX ) = C(B, A)X. V -natural in B. For instance, taking V to be P oset, this allows us not only toconsider objects such as AA in a ... structures on categories, such as finiteproduct or finite coproduct, typically with many others too.

21 Aug 2008: B. 85]) such that:. (i) the base category B has nite products,. ... x:t is not necessarily. dened in U. For example, in any c-pca U where there are elements a; b 2 U such that a b is.

1 Feb 2008: Let γ : [a,b] C be a (continuous) curve. Suppose we have a = a0 < a1 < <an1 < an = b such that for each 0 i < m the restriction γi of γ to ... U is starlike if there exists a0 U such that for every b U the line segementfrom a0 to b lies in U.

27 Aug 2008: 6. Secondary reduction rules. 5 The Categorical ModelMuch work has been done on providing such (categorical) models of IntuitionisticLinear Logic. ... coalgebracarries naturally the structure of a commutative comonoid in such a way that coal-gebra maps

27 Aug 2008: 6. Secondary reduction rules. 5 The Categorical ModelMuch work has been done on providing such (categorical) models of IntuitionisticLinear Logic. ... coalgebracarries naturally the structure of a commutative comonoid in such a way that coal-gebra maps

19 Aug 2008: The first idea is to generalisefrom the category B to a category BA, parameterised by a small categoryA, such that B 1 = B. ... Xn] :(. i[n] Ai) // Set be the unique presheaf such that for all i [n],.

8 Feb 2008: Show that such an enemy can find our common key if K = 3. ... n) is the output from a LFSR of length d. (c) The period of (xn)n>0 is the least integer r > 1 such that xnr = xn for all sufficiently largen.

20 Feb 2008: A such that M and M′′ belong to B, then M′ also belongs to B[17, 12.4.2]. ... module and as such is killed by some power of the central element z.

22 Jan 2008: If further such an action is a partial function then the automatonis deterministic. ... See [7]and [8] for ramifications of the theory of such special traced monoidal categories.) WritePar for the traced monoidal category just described.

13 Aug 2008: logical operations.Such subcollections arise naturally when the PCA (A, ) has some additionalstructure respected by application. ... An extensional assembly is an object (X, | = |) of the topos S[PER] such that.

29 Sep 2008: 2-natural in C such that the diagram. [C,D]kB- [[B,C], [B,D]]. kA- [[A, [B,C]], [A, [B,D]]]. ... Note that what wecall pseudo-maps [3] call morphisms.) We write f = (f, f̄) : A B for such apseudo-map, the 2-cell usually being understood.

20 Jan 2008: b) Let N Zn be a submodule of rank n. Show that there is a matrix A Mn(Z) such thatA(Zn) = N and the index [Zn : N] of N in ... 8) An integral domain R is called a euclidian domain if there is a map N : R{0} Z>0 such thatfor all a,b R, b 6= 0, there are

2 Dec 2008: Does there exist such a bijection f which isorder-preserving (that is, satisfies f(x) < f(y) whenever x < y)? ... Let S be a collection of subsets of N such that for every A, B S we have eitherA B or B A.

27 Aug 2008: Wecan dene a procedure called normalisation which can systematically eliminate such de-tours from a deduction. ... Much work has been done on providing such (categorical) models ofIntuitionistic Linear Logic.

27 Aug 2008: Wecan dene a procedure called normalisation which can systematically eliminate such de-tours from a deduction. ... Much work has been done on providing such (categorical) models ofIntuitionistic Linear Logic.

13 Aug 2008: Thus concretely maps A B of Dial(I) consist of maps f : I U V andF : I U Y X in T such that α(i,u,F(i,u,y)) ... or equivalently. (f,F) : (U X•) V U• Y •. such that an entailment which I write. (

20 Feb 2008: Assume there is θ OK such that 1, θ,. , θn1 is anintegral basis of OK. ... dK| (. nn. n! )2 ( π4. )n. Deduce that |dK| > 1 for every number field K 6= Q.(b) Show that there are constants A > 1, c > 1, such that for every

28 Nov 2008: Then there exists a basis B for V such that. [ψ]B =. . 0 11 0 0. 0 11 0 0. 0 0 1. ... For each α End(V ) there exists a unique α End(V ) such that.

13 May 2008: A language L over vocabulary W is a set of strings, such that L W. ... input. A string is rejected only if there is no such sequence.

5 Feb 2008: For each sentence xφ(x) true in M pick the first such x. ... 3. There is a formula σ(x) Σ such that (σ(cm)) Tm1.

16 Oct 2008: We say A, B Matm,n(F ) are equivalentif there exist invertible matrices P and Q such that B = Q1AP. ... s} such that, if we delete each yij from Y and replace it by xj , theresulting set is still linearly independent.