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11 Feb 2014: It follows that for such q, gq λI is positive definite. ... The set of all such t is closed andcontained in [a,b], so it has a minimum value t0.

4 Dec 2014: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

6 Mar 2014: Show that for any Hecke character χ, thereexist continuous χv : Kv C (v ΣK) such that. ... πv7. )= 1. Show that there is a unique Hecke character χ of K such that. •

22 Sep 2014: homotopy equivalences f : B B such that. θ f = θ. ... Homology stability with respect to higher handles. A further immediateconsequence of Theorem B is an analogous homological stability theorem for gluingon to W a θ-cobordism K : W P such that

3 Nov 2014: b) Let C1c ([0, 1]) be the set of functions f C1([0, 1]) such that f(x) = 0 for x in some neighborhood ofthe end points 0 and 1. ... If Sis a subset of C([a,b]) such that every sequence in S has a subsequence converging uniformly to a functionin S (i.e.

10 Nov 2014: linearfunctional θ V such that θ(x) 6= θ(y).(b) Suppose that V is finite dimensional. ... If no such subspaceexists, show that there exists a basis 〈v,α(v),.

7 Feb 2014: That is, for everya < b and every t there should exist x (a,b) such that f(x) = t. ... 14. Let f : R R be a function that has the intermediate value property: that is, iff(a) < c < f(b) then there exists x (a,b) such that f(x) =

20 Oct 2014: On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)?

25 Nov 2014: n=1x3n? 14. Let S PN be such that if A, B S then A B or B A. ... Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T?

10 Oct 2014: a) Let a G. Show that there is a positive integer n such that an = e, the identity element.(The least such positive n is the order of a.). ... b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the

5 Mar 2014: Deducethe more general result that if f ′(a) 6= f ′(b) and z lies between f ′(a) and f ′(b), then thereexists c (a,b) such that f ′(c) = z. ... f(a) < 0 and f(b) > 0 then there exists c (a,b)such that f(c) = 0.

17 Oct 2014: 1. Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

15 Jan 2014: In this and the followingquestions, you are not 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

11 Nov 2014: linearfunctional θ V such that θ(x) 6= θ(y).(b) Suppose that V is finite dimensional. ... If no such subspaceexists, show that there exists a basis 〈v,α(v),.

18 Feb 2014: b) The model is to be reparametrised in such a way that it can be specialised to (i)two parallel lines for the two groups, (ii) two lines with the same ... 8. (a) Let A be a pp non-singular matrix and let b Rp.

20 Oct 2014: metric relation R defined on X such that for any k elements x1,. ... Prove that there existu,v Zp such that. |A (uB v)| αβp αβp, where uB v denotes the set{ub v : b B}.

5 Mar 2014: Prove that f is. integrable. What is 10f(x)dx? 6. Let a < b and let f : [a,b] R be a Riemann integrable function such that f(x) 0for every ... 1, 1], such that f(x) = g(x) h(x) for every x.

20 Oct 2014: Prove that there are. subsets A X and B G such that |η(A,B) δαβ| θ, where θ > 0 depends on γ andδ only. ... x1,. ,xk such that no xi xj with i,j distinct is a quadratic residue mod p.

16 Jul 2014: singularities is a real manifold Bwith an open subset B0 B, with := B B0 of codimension 2, such that B0 has the structure of an integral affine manifold.We also allow ... X Spec k[[t]]. such that the special fibre satisfies. X0 =. σPmax. Pσ. and the

10 Mar 2014: n}, let the non-negative numbers A(n)k andthe ‘nodes’ x. (n)k [a,b] be given such that for each polynomial P , the error. ... such that for every f C([1, 1]), there exists a unique series.

19 Dec 2014: if B = {min(Xα) : α < κ} is stationary.(iii) If {Xα : α < κ} is pairwise disjoint, then there exists a [κ]κ such that. ... Prove that there exists a family F of2κ stationary subsets of S such that if A 6= B F , then A B and B A arestationary in

19 Dec 2014: i) Prove that 20 = 1 implies the following assertion SD :there exist sets A and B such that R2 = AB and Al and Bm are countablefor every horizontal line l and ... exist x1 and x2 such that x1 6 f(x2) and x2 6 f(x1).

18 Sep 2014: Variables, such as x, y, z, a, b, c, or finite tuples of variables: x = (x1,. , ... is true then there exists a homomorphism η : GΣ Mi such that the diagramGΣ.

18 Sep 2014: To see this, choose δ such that, for x anend-point of a base b, d(x, bo S) δ. ... Let c b bean interval with end-point x such that c0 S =. Choose δ depending on Kaccording to the previous claim, and let y, k K be such that:.

16 Jul 2014: Mark Gross Mirror symmetry and the SYZ conjecture. Definition. A tropical affine manifold with singularities is a real (C 0) manifoldB along with an open subset B0 B such that := B ... For each integral point v of a two-face, we can choose anopen

2 Dec 2014: If ϕ AutK(L) where L = K(t), show that there exist a, b, c, d K withad 6= bc such that ϕ(t) = (at b)/(ct d), and conversely that ... a) such that f splits over L = Q(. a)( 3α).

18 Sep 2014: Thence it also extends to M̃. Since M is compact there exists positive such that for all p M, B(p)is ball-like (homeomorphic to a ball in Tp(M) ... 19. Proof: By the definition of the connected sum, there exists an embedded 2-sphere F in M such that M F

14 Oct 2014: 8. Let A1, A2, A3,. be sets such that A1 A2. ... c there is an x with a < x < b such that f(x) = c.

3 Feb 2014: 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.

19 Dec 2014: Let M be a CTM such that (T is an ever-branching Suslin tree )M. ... λ < κ. Prove that if | A |= κ and x Aimplies | x |< λ, then there exists B A such that | B |= κ and (r)(x B)(x 6=y xy = r).

18 Jul 2014: such that if Φ X5(K) is non-singular then CΦ has Jacobian. ... We define an action. of GL5 GL5 on the space of such matrices via.

2 Jan 2014: Hyland. existence of such a situation Lambek and Scott exploit an elegant categorical reduc-. ... for a theory in the language of the pure λ-calculus: such a theory corresponds to a.

14 Jan 2014: Show that T can be written Tz = Az Bz̄for unique A,B C. ... Can such a map be therestriction to the strip of a Möbius map?

18 Sep 2014: Proof. Let C be such that the Γ-translates of B(x0,C) cover X. ... BecauseB is compact, there is an > 0 such that Bp() is CAT(0) for every p B.

20 Oct 2014: such as this. (1) The condition about the number of quadruples a b = c d can be interpreted as. ... Then there exist sets A X and B Y of density α and β such thatη(A,B) αβδ(1 c).

15 Apr 2014: 37 ζ. 57 ζ. 67 ) =. 7. Definition 3.4. An invariant of degree m is a homogeneous polynomial I =. I(a,b,c) of degree m such that I g = I ... variables a,b,c such that v g = gv, respectively v g = gT v, for all g G.

26 Nov 2014: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

13 Oct 2014: tion for lattices of such small determinant is typically infeasible. 8 WATKINS, DONNELLY, ELKIES, FISHER, GRANVILLE & ROGERS. ... For instance, Rubinstein [51, (1.15)] reports the biasfor various curves in terms of such an integrated quantity.

26 Nov 2014: 2) Let K be a field containing an n-th primitive root of unity for some n > 1.Let a,b K such that the polynomials f(t) = tn a and ... Rn = R such that R1 is 1-step constructible from S, andfor each 1 i n 1, Ri1 is 1-step constructible from S {R1,.

16 Apr 2014: We can find minimalprimes p′A A, p′B B such that p′ = (p′A,p′B). ... the universal lifting under ϕ, and such that the action of ρ|IFṽ on gr.