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12 Jul 2023: ii) Let A,B P(n) be such that if A A and B B then A 6 B and B 6 A.Show that. ... ii) Let r > 2, and let F X(2r) be such that if A, B, C are three sets in F thenAB 6 C.

9 Nov 2023: dx = hξdξ hθdθ hφdφ. [5]. (b) Find a suitable form for g(ξ) such that g(0) = 1 and the coordinate system (ξ,θ,φ)is orthogonal. ... b) A function f(x) is constrained such that f(1) = f(1) = 0.

7 Jul 2023: Show thatfor each a R and h > 0 there exists b (a, a nh) such that. ... c) Show that if a function g : [a,b] R is integrable, then there exists a sequenceof continuous functions ϕn : [a,b] R such that.

12 Jul 2023: b) Define a sequence of theories {Ti ; i N} and a theory T such that T0 = ZFC ICand (assuming the consistency of all involved theories). ... a) Construct a transitive set M with κ M and |M| = κ such that.

9 Nov 2023: Provided Mij is not singular,the matrix inverse (M1)ij is the rank-2 tensor such that Mij(M1)jk = δik. ... The modulooperation k mod r = s means that s is the smallest non-negative integer such thatk s is divisible by r.] [Hint: Rewrite U as a rotation

7 Jul 2023: Prove that for every positive integer d there exist integers a and b such that4a2 9b2 1 is divisible by d. ... radially such that its moment of inertia is γMa2 for some constant γ.

12 Jul 2023: Give an example of a nontrivialopen immersion V Y such that Y is connected and affine and V is also affine. ... Let ϕ : X Y be a morphism ofschemes. Suppose that {Ui} is an open cover of Y such that ϕ1(Ui) Ui is properfor all i.

9 Nov 2023: g(x) = x for 1 < x 6 1. [2]. (e) Using your results from parts (c) and (d) above, deduce integers A and B such that. ... a) Find the values of the parameters α,β,γ such that. Ac = b. (†).

7 Jul 2023: Suppose now that A and B are square complex matrices such that [B,A] commuteswith A, i.e. ... iii) an annular region centred at the origin? (b) Consider 2π-periodic functions f(θ) such that.

12 Jul 2023: Show that p(x̄) is realised in some N such that M 4 N. ... b) Assume that the language of T is countable. Show that every model (that is,every M U such that |M| < |U|) is homogeneous.

9 Nov 2023: 6]. 12R. Consider the function. h(x, y) = aex bey cxdy ,. where a, b, c and d are non-zero real constants such that ac > 0 and bd > 0. ... the symmetry axis at an angular speed of ω(r) such that thelinear speed at any point is given by v(r) = rω(r).

7 Jul 2023: State a necessary condition on B for such an A0 to exist. ... x = ρ cos φ, y = ρ sin φ. Find scalars hρ,hφ and unit vectors eρ,eφ such that dx = hρeρ dρ hφeφ dφ ezdz.

12 Jul 2023: b. Let (A, m) be an artinian local ring such that |A/m| <. Prove that |A| <. c. Let f : Bm Bn be an injective B-module homomorphism, where B is a ... Write Hf for the set of ring homomorphismsg : B k such that g (a) = f (a) for all a A.

12 Jul 2023: Include inyour answer a definition of the generator A G(M,ω), and of its domain, and givethe precise characterization of the set of generators of such semigroups. ... b) Prove that the domain Dom (A) is a dense subspace of X.

22 Aug 2023: The double-angle formulae, such as tan 2x = 2 tan x/(1 tan2 x) are worth knowing, asare the basic formulae for sin(A B), cos(A B) and tan(A ... B), the Pythagoras-type identities, suchas sec2 x = 1 tan2 x, and a few special values, such as sin π/4 = 1/.

12 Jul 2023: b) Suppose {pi}Ni=1 are probability distributions on Rn such that pi satisfies a ci-Poincaré inequality. ... b) A non-negative function f : Xn R is called weakly-self-bounding if there existfunctions fi : Xn1 R such that for all x Xn,.

7 Jul 2023: vx, and state the Cauchy–Riemann equations. (b) Suppose there are real constants a, b and c such that a2 b2 6= 0 and. ... In(f) I(f), n. [Hint: Use the Weierstrass theorem: for any > 0 there exists n N and a polynomialpn Pn such that |f(x)pn(x)| < ,

12 Jul 2023: P(Y n > a. )6 1a2. (b) Let Y be a standard normal random variable. ... Mtd= |Bt|. (b) Show that almost surely there is a unique time M [0, 1] such that.

7 Jul 2023: Find an example of such a ring that does not satisfy conditions(i)-(iv). ... is a smooth surface in R3. Suppose a < b and r > 0 are such that for all a 6 a′ < b′ 6 b we have.

12 Jul 2023: Part III, Paper 124. 3. 3 (i) Let a,b,c and d be four indeterminate variables. ... three rank-1 quadratic forms), such that the three polynomials ac, bd, andad bc all belong to their linear span.

12 Jul 2023: a) Give the definition of the previsible σ-algebra P. (b) Give the definition of a simple process. ... c) Suppose that u is a bounded Borel function on R2 such that the following is true.

7 Jul 2023: Show thatfor each a R and h > 0 there exists b (a,a nh) such that. ... Prove that for every positive integer d there exist integers a and b such that4a2 9b2 1 is divisible by d.

12 Jul 2023: Now assume that there exists at least one martingale deflator. (b) Let ξ0 be a real number and ξ1 a random variable such that E(ξ1Y1) < ξ0Y0 for everymartingale deflator Y. ... Show that there exists aportfolio H Rn such that. H P0 = ξ0 and H P1 =

12 Jul 2023: b) Derive the eccentricity distribution of stars n(e), defined such that n(e) de is thefraction of stars with eccentricity in the interval (e,e de). ... By comparing thisexpression with the one from (a), demonstrate that the DF of that model can be

12 Jul 2023: You may assume that for every bounded bilinearfrom ϕ: HH R, there is a unique operator T X such that ϕ = θT and ‖T‖ = ‖ϕ‖.]Deduce that B(H) is a ... For every x A there exist unique hermitian elements h,k A such that x = hik.

7 Jul 2023: Suppose now that A and B are square complex matrices such that [B,A] commuteswith A, i.e. ... iii) an annular region centred at the origin? (b) Consider 2π-periodic functions f(θ) such that.

12 Jul 2023: 5/R5), butO(V a7/R7). (c) Cartesian coordinates are defined for the problem in part (b) such that V =(V, 0, 0), with V > 0, and that the centres of ... Using the results in part (b), and explaining any further approximations, show that.

7 Jul 2023: Show that F is increasing and concave. (b) Given a constant m and a random variable Z such that E(Z) = 0, define afunction G by. ... TµνXµXν > 0. for all timelike vectors Xµ. Find the most general constraints on A and B such that ()satisfies the

12 Jul 2023: Consider waves that are generated by a distant source with a frequency ω2 > f2 andwith a group velocity vector, cg, that points down and to the right, such that cg ẑ < ... Show that under this assumption, for given k there are constants A(k) and

22 May 2023: Let ν be a measure on M such that ν λ and λ ν. ... Q,P) such that Q1 = θ and the transformedHamiltonian does not depend on Q2.

7 Jul 2023: b) With (xn) and x as above, prove that there exists another sequence (x̃k) in Hsuch that ‖x̃k x‖H 0 and such that each x̃k is a convex combination of ... random variables such that. E|Y1| =. Show that lim supn |Y1 Yn|/n = almost surely.(b) Show

12 Jul 2023: b) State the usually prescribed conditions on vector b in (a) which ensure that theassociated normalized state |b〉 in the HHL algorithm can be efficiently prepared. ... Find An such thatVnWn = exp(iAn). Is VnWn a Clifford operation?

7 Jul 2023: b) For such a pair of integers (a,N), the modular exponential function f : Z ZN ,is defined as f : k 7 ak mod N, where ZN := {0, 1,. ... Showthat we can find a matrix M of size 2n 2n such that.

7 Jul 2023: Show that there is a greatest ordinal β such that. ωβ 6 α. ... d) One can trivially construct other such pointwise Euclidean invariants by applyingfunctions of curvature and torsion, e.g.

29 Oct 2023: 2 (a, b) such that1 theorymark for. explainingconvergence. andidentifying. k > 12. ... 2. Choose k = 32, such that f(x) =6 sin(x)3x10. 7.

7 Jul 2023: 1, and another point q P1 such that π1(q) has exactly 3 irreducible components. ... Prove that ifwA. G vB, then there is some word u W such that A G uB.