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22 Nov 2005: 2 ANALYSIS II EXAMPLES 3. 7. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in a neighbourhood of(a,b), and D2f = f/y exists ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that D1f = f/x exists and is

23 Jan 2015: 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 ... Z21 Z2np,. where the Zi are independent and Zi N(λi,σ2) for some λi such that.

14 May 2003: Proof. Applying the hypothesis about b to the bilinear map g:UxV-->U@V, we obtain a linear map i:X-->U@V such that g=ib. ... Similarly, applying the universal property of g to the bilinear map b, we obtain a linear map j:U@V-->X such that b=jg.

10 Mar 2021: There is a distinguished uniformizer t B. dR. such that F kBdR = tkBdR and GQ acts on t through the -adic cyclotomic character. ... Such a functor is known to exist on the subcategory of smooth quasi-projective varieties [Fon92, 6.7].

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.

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

22 Mar 2018: 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.

8 May 2007: b) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... b) (x, y, z) (x′, y′, z′) x = x′, y = y′, z = 1 z′.

24 Apr 2017: 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.

31 Aug 2015: 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.

14 May 2007: Is there always an orientable such surface, i.e. one in which“turning clockwise” makes sense globally on the surface? ... b) What is the smallest genus (number of holes) of such a surface with boundary thetrefoil knot, which is the knot which lies on

22 Nov 2005: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 9. [Tripos IB 96401(b), modified] (i) Suppose that (X,d) is a nonempty complete metric space, andf : X X a continuous map such that, for any x,y X, the

15 Jun 2000: We must convert a into a new number a' in such a way that p divides a'b and does not divide a'. ... Then we can find H and T such that HaTb=1, and we know that we will now find that p divides b if we continue the argument.

21 May 2005: a,b] such that fn(xn) does not converge to f(x). [Hint: Bolzano–Weierstrass.]. ... every point of [0,1] (here we interpret f ′(0) as the ‘one-sided derivative’ limh0+((f(h) f(0))/h),and similarly for f′(1)), but such that f ′ is

18 Oct 2005: ii) Give an example of a sequence of uniformly continuous functions fn on R, such that fn con-. ... a,b] such that fn(xn) does not converge to f(x). 12. (i) Suppose f is defined and differentiable on a (bounded or unbounded) interval E R, andthat its

21 May 2005: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 10. [Tripos IB 96401(b), modified] (i) Suppose that (X,d) is a nonempty complete metric space, andf : X X a continuous map such that, for any x,y X, the

15 Oct 2010: 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

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.

7 Mar 2024: b) Find a condition on 𝑠 such that 𝛿𝑥 𝐻𝑠(R𝑛). c) Show that 𝐻𝑡 (R𝑛) is continuously embedded in 𝐻𝑠(R𝑛) for 𝑠 < 𝑡. ... Δ2𝑢 𝑢 = 𝑓. b) Show further that there exists 𝐶 > 0 such that ‖𝑢‖𝐻 𝑠4

18 Oct 2011: 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