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27 Jun 2024: b) Using Functional Principal Component Analysis, give, for some K N, a Kdimensional test of the above hypothesis, and find its properties under the nulland alternative hypotheses. ... Let C1() = λ1⟨e1, ⟩e1, andC2() = λ2⟨e2, ⟩e2 for some λ1,

3 Jul 2024: b) Assume that B is in the future lightcone of A, that C is in the future lightconeof B and that |xC xA| < |xC xB| in some inertial frame S. ... x, y) R2 : ax by c = 0}. for some a, b, c R with (a, b) ̸= (0, 0).

27 Jun 2024: Give necessary and sufficient conditions, in terms of thesubdifferential of f, to have proxf(y) = x for some x Rn. ... for some a Rn and b R to be specified. Use parts (a) and (c) to explain how theproximal operator of h can be evaluated.

3 Jul 2024: c) Consider the linear transformation A : R3 R3 defined by A : b 7 a bfor some non-zero vector a. ... ii) Let (an) be a sequence of positive real numbers. If an1/an L for some L R,show that a.

27 Jun 2024: sin2 12x. Hence prove that 1πT |Fn(t)|dt = 1. (b) Prove that, if f Lipα for some 0 < α 1, i.e. ... b) Given f C[1,1], let p Pn be a polynomial such that, for some n 2 pointst1 < t2 < < tn2 in [1,1], we have.

3 Jul 2024: 2A Differential EquationsA real-valued function f(x) is differentiable on some interval (a,a). ... From now on, assume that p = q = 1/2. (iii) Let T be the random number of steps taken by the random walk until it first hitsa or b for some

27 Jun 2024: u|k2,α;′ C (|u|0;1 |f|k,α;1)for any ′ 1 and some constant C > 0 independent of f or u. ... Bρ(z)B1 |Dv| M for. some constant M and all balls Bρ(z), thenB1e.

27 Jun 2024: b) Suppose we are given an m-qubit unitary U that performs the map. ... 2πiϕj with ϕj = cj/2m for some. m-bit integers 0 cj < 2m, where m = O(log n).

3 Jul 2024: x,y) R2 : ax by c = 0}. for some a,b,c R with (a,b) ̸= (0,0). ... c) Consider the linear transformation A : R3 R3 defined by A : b 7 a bfor some non-zero vector a.

3 Jul 2024: f(x) = a.x b :=n. i=1. aixi b for some a {0,1}n andb {0,1}. ... b) Let M = (Σ,Q,P) be a register machine such that fM,1 = χL for some languageL W.

3 Jul 2024: Suppose. that A = I pmB for some m N and some k k matrix B with integer entries, andthat An = I for some n N. ... B with integer entries, and An = Ifor some n N, then in fact A = I.

3 Jul 2024: Is every infinite cardinal equal to some α? Justifyyour answer. Prove that α α = α for every ordinal α. ... Prove that if Q = αOL for. some α OL, then P = NL/K(α)OK.

4 Jul 2024: b) Focus now on the low-momentum behaviour, where the l = 0 mode dominates.For some choice of potential V (r),. ... b) Let M = (Σ,Q,P) be a register machine such that fM,1 = χL for some languageL W.