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23 Nov 2016: Mich. 2016 NUMBERS AND SETS—EXAMPLES 4 PAR. 1. The symmetric difference A4B of two sets A and B is defined to be the set of elements x such that xis ... 11. Let S be a collection of subsets of N such that for all A, B S we have A B or B A.

13 May 2016: If φ : (Sn,) (B,B) , show that we can find a mapΦ : X E such that p Φ = φ π and Φ(Sn1 0) =. If we define φ =Φ|Sn11 : Sn1 ... Show that there are maps Ap : TpM TpM such that ifv,w TpM, ω(v,w) = 〈Apv,w〉.

15 Feb 2016: 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

17 Oct 2016: ii) Let α be such that |K(α) : K| is odd. ... and let β = α4.Find. rationals a, b, c such that β = a bα cα2.

14 Oct 2016: c) Let x be a non-zero rational number. Show that there is a rational number y such thatxy = 1. ... d) Show that if a and b are natural numbers such that a is a factor of b then a 6 b.

13 May 2016: p. k, where k is the smallest value ofn such that xn 6= yn. ... Hint:first find a subsequence (fni ) such that fni (x) converges for all x Q [0, 1].)Deduce that the closure of B in (C[0, 1], d) is compact.

27 Oct 2016: b) If (fn) does not converge uniformly, show that there is a convergent sequence (xm) x [0, 1] such that (fn(xn)) does not converge to f(x). ... a) The space Cb(R) of bounded continuous functions on R.(b) The space C0(R) of continuous functions f : R R

28 Oct 2016: 7. A map π : E B is called a covering map if there is an open cover {Uα} of B such that π1(Uα)is a disjoint union. ... H(B) H(E) such that π π! is multiplication by N.

23 Nov 2016: Mich. 2016 NUMBERS AND SETS—EXAMPLES 4 PAR. 1. The symmetric difference A4B of two sets A and B is defined to be the set of elements x such that xis ... 11. Let S be a collection of subsets of N such that for all A, B S we have A B or B A.

21 Nov 2016: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 15. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

26 Oct 2016: 4. (a) Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linearfunctional θ V such that θ(x) 6= ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.

14 Oct 2016: c) Let x be a non-zero rational number. Show that there is a rational number y such thatxy = 1. ... d) Show that if a and b are natural numbers such that a is a factor of b then a 6 b.

22 Oct 2016: 6. Let X, Y be manifolds. A submersion is defined to be a smooth map f : X Y such thatdfx is surjective for all x X. ... zero outside the ball of radius b > a, and such that f(Rn) [0, 1].These functions are very useful for gluings.

11 Nov 2016: Let x0 and 0 < ρ < ρ̄ be such that the ball B(x0, ρ̄). ... where b Rd and c R. Show that there exists a constant C such thatB(x0,ρ).

12 Dec 2016: 7. For K C nonempty and compact, find a Hilbert space H and T B(H ) such that σ(T ) = K. ... sense for any z0 U there exists an open neighbourhood N U of z0 such that f can be represented on.

14 Oct 2016: 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.

14 Oct 2016: 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 exponent of G.

15 Feb 2016: 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

29 Feb 2016: 8. By a selection function for a set a, we mean a function s: Pa Pa such that s(b) b for allb a, both inclusions being strict unless b is ... First show that there exists β < α such that |f(β) f(γ)| < 1 for allγ > β.].

28 Nov 2016: 2) Convert the following CFG to CNF:. S aSbb | T, T bTaa | S |. (3) Give a CFG for each of the following CFL’s, and then transform each such CFG intoCNF:. ... b) Show that the language L := {anbncn | n 1} is not a CFL.

27 Oct 2016: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε.

1 Feb 2016: Show that for any b C, there exists a sequence of points zn B(a; r)with zn 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.

8 Feb 2016: p], and deducethat if f,g : P P are order-preserving maps such that f(x) g(x) for all x P , then µ(f) µ(g).[Consider the restriction of ... The largest such c is usually denoted ab; this defines a binary operationonL, which is called Heyting implication

18 Jan 2016: 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?

8 Feb 2016: iv) For each of the following classes of structures, either write down a universal axiomatization ofthe class or show using (ii) that no such axiomatization exists. ... For those which are not universal, eitherwrite down an inductive axiomatization or

8 Jan 2016: to V4 V3; 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

21 Nov 2016: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 15. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

26 Oct 2016: 4. (a) Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linearfunctional θ V such that θ(x) 6= ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.

16 May 2016: p. k, where k is the smallest value ofn such that xn 6= yn. ... Hint:first find a subsequence (fni ) such that fni (x) converges for all x Q [0, 1].)Deduce that the closure of B in (C[0, 1], d) is compact.

24 Oct 2016: 1and such that |fn(x)|  g(x), x 2 X , for some g : X! ... R such that E|g(X)| < 1, where Xis any real random variable taking values in X.

23 Feb 2016: If f has poles on the boundary we can find some b C such that no poles lieon the translate b P, and integrate around this curve instead.(iii/iv) Same ... But it is easy to see that the number of such pairs(a,b) equals the right hand side of the dimension

23 Mar 2016: d e p e n d e n t of B 1 ,. , B), such t h a t. a[BI,.Bn] e l. e m = ( , ' ¢ 1. ... Indeed, suppose t h a t B = ( B 1 ,. , B,, b l ,. , b,) is an arbitrary E-algebra, then for each i such t h a t t h e orbit

23 Mar 2016: d e p e n d e n t of B 1 ,. , B), such t h a t. a[BI,.Bn] e l. e m = ( , ' ¢ 1. ... Indeed, suppose t h a t B = ( B 1 ,. , B,, b l ,. , b,) is an arbitrary E-algebra, then for each i such t h a t t h e orbit

3 Feb 2016: The two properties are closely related inthe above examples as all such finitely-living observers must necessarily enter theblack hole region B. ... Black holes in general relativity 3. hole region B of such perturbations of Kerr?

18 Aug 2016: Then thereexists λ K such that. (1)i pf(B{i}) = λaifor all 1 i n. ... Then there existsλ K such that. (1)ij pf(B{i,j}) = λ(a1ia2j a1ja2i)for all 1 i < j n.

27 Apr 2016: a) For each v Q, qv 6 1 mod λm1. There are exactly r places of Q such that qv 1 mod p.(b) For every v / S Q we have ... Consider matrices in A,B GL2(R) such that A =(α 0 β.

25 Apr 2016: Moreover, if x is a basic solution,then there is a basis B such that xN = 0 and ABxB = b, and if x is a basic feasiblesolution, there is a basis ... B such that xN = 0, ABxB = b, and xB > 0.

16 Mar 2016: Then we have ρc0 = AρA1 (ν r)|, and hence (by [CHT08, Lemma2.1.9]) we can find B GLn(S) such that ρc0 = BρB1 (ν r)|. ... The pairs (φ, (av)vT ) and(φ′, (a′v)vT ) define equivalent T-framed deformations if and only if there exists b gnr such

29 Apr 2016: Such a curve is canonicallyembedded as a quartic curve in P2k, which explains the title of this paper. ... The map δ can be written down explicitly as follows: given A J(k), choose B J(ks) such that [2](B) = A.Then the cohomology class δ(A) is

25 May 2016: condition implies that there is at least one such y for given x. ... Proposition 4.4. The topology j such that Eff j ' Sets is the double negationtopology.

25 May 2016: condition implies that there is at least one such y for given x. ... Proposition 4.4. The topology j such that Eff j ' Sets is the double negationtopology.

30 Apr 2016: Let fgs W s 2. ;/g be a family of metrics such that. ... Let U be any openneighborhood of M in N such that any g-geodesic.