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1 Oct 2012: Let Bun(V,θγd)denote the space of such maps. Recall from [1] that the cobordism category Cθ isa topological category roughly defined as follows. ... In dimension two, this can be improved. Theorem B. Let θ : X BO(2) be a tangential structure such

10 Dec 2012: encourage mathematical investigation without any examination emphasis. 1. Calculate d = (a, b) and find integers x and y such that d = ax by when. ... 11. Let A be the group (Z/65520Z). Determine the least positive integer n such thatgn = 1 for all g in A

8 Feb 2012: 14. Exhibit a subset X of R2 such that (a) any two points x, y X can be joined bya continuous path γ : [0, 1] X and (b) for x 6= ... A regular polyhedron is one in which all two-dimensional faces are congruent regularpolygons, and such that for each

22 Nov 2012: Show that, for such a map,. dimH f (M ) 61α. ... Prove that the limit set for such a group is totallydisconnected.

21 Nov 2012: 12. Let S be a collection of subsets of N such that for every A, B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f (x) = c.

1 Oct 2012: Suppose W is a (n1)-connected 2n-manifold such that πn(W) πn(BO) is surjective. ... Theorems A and B follow fromtwo observations about the monoid E: firstly that the submonoid E′ consisting ofthose path components represented by manifolds W such that

29 Oct 2012: On the other hand, show that if fn does not. converge uniformly to f then we can find a convergent sequence xm x in [a, b] such that fn(xn) 6 ... valued functions (fn) on R such that fn converges pointwise to a function f which is continuous but not.

12 Nov 2012: such that θ(x) 6= θ(y).(b) Suppose that V is finite dimensional. ... b) Suppose AB = BA. Show there exists an invertible matrix P such that P AP 1 and P BP 1 are both uppertriangular.

10 Oct 2012: show that for every β L we can find a, b, c, d K such that β = a bαc dα. ... 1.16. Let K, L be subfields of a field M such that M/K is finite.

31 Jan 2012: ii) Find an example of a finite lattice L which admits at least one homomorphism L 2,but does not have enough such homomorphisms to separate its elements. ... The largest such c is usually denoted a b; this defines a binaryoperation on L, which is called

30 Oct 2012: ii) Find an example of a finite lattice L which admits at least one homomorphism L 2,but does not have enough such homomorphisms to separate its elements. ... b. (The largest such c is usually denoted a b; this defines a binaryoperation on L, which is

29 Feb 2012: 8. By a selection function for a set a, we mean a function s : Pa Pa such that s(b) bfor all b a, both inclusions being strict unless b is ... First show that there exists β < α such that|f (β) f (γ)| < 1 for all γ > β.].

8 Feb 2012: 14. Exhibit a subset X of R2 such that (a) any two points x, y X can be joined bya continuous path γ : [0, 1] X and (b) for x 6= ... A regular polyhedron is one in which all two-dimensional faces are congruent regularpolygons, and such that for each

30 Oct 2012: 8. By a selection function for a set a, we mean a function s : Pa Pa such that s(b) bfor all b a, both inclusions being strict unless b is ... First show that there exists β < α such that|f (β) f (γ)| < 1 for all γ > β.].

21 Feb 2012: Show that for any b C, there exists a sequence of points zn Ba(R) withzn 6= a such that zn a and f (zn) b as n. ... Find such a sequence when f (z) = e1/z, a = 0 and b = 2.

23 Jun 2012: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y.

22 Nov 2012: K. σ. ::uuuuuuuuu. As σ(oK) Rn is a lattice, there exists R > 0 such that the image of B in Rncontains a fundamental domain for σ(oK) i.e. ... B Rn/σ(oK). For such R,. v| Kv = B oK. (b) x arbitrary.

24 Jul 2012: Theorem 137 (Chebychev) There exist constants A and B with 0 < A B such that. ... if and only if there exist constants A′ and B′ with 0 < A B such thatA′n β(n) B′n.

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.

26 Apr 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.

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

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.