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23 Mar 2009: d : R{0} N. such that for all a R and b 6= 0, b R we can write. ... e f2 Q. [2. ]. Pick g, h Z such that |e g|, |f h| 12 and set.

18 Feb 2009: Showthat a diffeomorphism f : X X extends to X if and only if f([b]) = [b]. ... Show that if f ′ is anyother such f , the resulting manifold M ′ is diffeomorphic to M.

15 Oct 2009: Then there is N1 such that whenever n m N1 we have n. ... k=1 uk(x0). Also there is N2 such thatwhenever n m N2 we have n.

23 Feb 2009: A graph is bipartite or 2-colourable if we can partition the set of vertices into sets B andR such that the only edges lie between vertices in different sets.(i) ... Suppose that any such formula is equivalent modulo T to aquantifier-free formula.

20 Feb 2009: Show that every such transformationis an isometry for the hyperbolic metric. ... Every such quaternion q can be written asq0 q1i q2j q3k where.

22 Jan 2009: Let P be the set of all p Q such that 2 < p2 < 3. ... c) If A and B are closed subsets of R such that 1 A, 1 B and [1, 1] A B, thenA B 6=.

7 Dec 2009: 5. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in some open ball around(a,b), and D2f = f/y exists at (a,b). ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that D1f = f/x exists and is bounded near (a,b),

23 Feb 2009: Show that for any b C, there exists a sequence of pointszn D(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.

14 Oct 2009: On the other hand, show that if fn does not converge uniformly to f , then we can find aconvergent sequence xm x in [a, b] such that fn(xn) does not ... Give an example of a sequence of uniformly continuousfunctions fn on R, such that fn converges

26 Feb 2009: n}, let the non-negative numbers A(n)k and the“nodes” x. (n)k [a, b] be given such that for each polynomial P , the error. ... b) If f : (0, 1) R is continuous and bounded, prove that there exists a sequence of polynomialsQn such that Qn are

21 Oct 2009: Let P be the set of all p Q such that 2 < p2 < 3. ... c) If A and B are closed subsets of R such that 1 A, 1 B and [1, 1] A B, thenA B 6=.

20 Feb 2009: Show that every such transformationis an isometry for the hyperbolic metric. ... Every such quaternion q can be written asq0 q1i q2j q3k where.

19 Feb 2009: 6. Let R be an integral domain. The highest common factor of non-zero elements a and b in R is an elementd in R such that d divides both a ... identified with the set Fq {} in such a way that GL2(Fq) acts on asthe group of Möbius transformations z 7 azb

12 Feb 2009: b) Choose numbers a1, b1 such that 0 < a1 < a < b < b1 < 1, and set, for x R and n = 1, 2, 3,. ,. ... 12) Let be a bounded, open subset of Rn and g : (a, b) R be a function such that.

8 Dec 2009: 5. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in some open ball around(a,b), and D2f = f/y exists at (a,b). ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that D1f = f/x exists and is bounded near (a,b),

30 Oct 2009: n. LetβJn (x) = (xx0)(xx1)(xx2). (xxn). Prove that for each x [1, 1], there exists ζ (1, 1)such that. ... Thus, for any fixed δ (0, 1/2), In,δ/In 0 asn. (b) Choose numbers a1, b1 such that 0 < a1 < a < b < b1 < 1, and set, for x R

20 Nov 2009: 13. Is there an uncountable family S PN such that AB is finite for all distinct A,B S?

21 Jan 2009: a)1z. (0 < |z| < ); (b) z1 z2. (1 < |z| < ). 18. For each C {0} and w C, such that exp(w) = , find a continuously differentiablecurve γ : [0, 1] C {0} joining ... 1 and , such that.

8 Oct 2009: 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

14 Oct 2009: On the other hand, show that if fn does not converge uniformly to f , then we can find aconvergent sequence xm x in [a, b] such that fn(xn) does not ... Give an example of a sequence of uniformly continuousfunctions fn on R, such that fn converges

12 Feb 2009: 2. According to the next problem, such bundles are classified by π1(SO(3)) = Z/2, sothere is only one nontrivial bundle. ... Show that if f ′ is anyother such f, the resulting manifold M ′ is diffeomorphic to M.

13 Nov 2009: a linear func-tional 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),.

21 Jan 2009: p] B1, and( iv ) there exist homogeneous elements y1,. , yt B such that. ... Lemma. Let (A, A1) be a Frobenius pair such that B and B1 are Gorenstein, andlet M be a finitely generated A-module.

8 Oct 2009: Theorem B. Let F be a Finsler metric on S2 such that K δ > 0. ... Since v nevervanishes we may choose a smooth angle ϕ(t) such that.

16 Oct 2009: g0,W) such that the (r 1)-jet ofPg(Σ,γ) belongs to Q. ... Then there exists aresidual subset O Gr such that for all g O:. •

9 Oct 2009: One way to think of such a relation is. that if xrb then the tokens in the multiset x trigger the token b. ... maximal but many minimal relations r : M A B such that r = f.