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6 Nov 2019: Define a new kernel on the interval [a,b] by,. k̃(M)(x,x0) = (b a)M. ... In particular they defined inducingvariables such that. cov(um, f (x)) = cos(wmx), (2.27).

19 Jul 2019: 161 –. by a sign, since this can be compensated by the cos function. ... Insisting that the two. cos functions agree, up to sign, gives us the condition.

17 Jun 2019: S3 =. 2. (1 00 1. )and. B(t) = B(sin α cos ωt, sin α sin ωt, cos α). ... eiωt/2 χ+ i. (ωλ. sin α sin(. 12λt. ))eiωt/2 χ. where λ (ω2 2ωB cos α B2)1/2 and.

17 Jun 2019: has polar equationAr = 1 ε cos θ with A > 0 and 0 < ε < 1.]. ... i) For B = R1A with. R1 =. 1 0 00 cos θ1 sin θ10 sin θ1 cos θ1.

19 Jul 2019: x(t) = X R sin(!B(t t0)) and y(t) = Y R cos(!B(t t0)). ... 4r. 1 cos. sin(1.27). This again gives rise to the magnetic field (1.26).

6 Feb 2019: This implies a = b,so using the formula cos θ 1 = 2 cos2(θ/2), we obtain,. ... 4.11). The antiderivative of cos(2 arctan(2x)) is arctan(2x) x, so there exist (new) realconstants a, b and c such that.

7 Jan 2019: 2θ̇2 2lθ̇(ȧx cos(θ) ȧy sin(θ)) mg(ay l cos(θ)). (b) We have. ... Veff(θ) = ml. (. g sin(θ) 12. a2. lω2 sin(θ) cos(θ). ).

7 Jan 2019: L =1. 2. [η̇2 2η̇θ̇ cos θ θ̇2. ]g. cos θ 1. ... 2. k. mη2 ,. up to a multiplicative constant. [10]. (b) Find the equations of motion for this system.

19 Jun 2019: i) Xt = t0. sgn(Bs)dBs,. (ii) Yt = eWt cos Bt,. ... t0. φ(Bs)ds}). where B is a Brownian motion starting from x.

17 Jun 2019: Thecoordinates are related by the transformation,. x = r cos 2θ, y = r sin 2θ,. ... 1]. (b) B =. (. cos θ sin θsin θ cos θ. ).

19 Jun 2019: Tt=1. xt cos(ωt) and B =1πT. Tt=1. xt sin(ωt). Show that. ... t=k1. cos(ωt) sin(ω(t k)). )]. Deduce that. |EAB| 6 1πT. T1k=1.

22 Jul 2019: u =e cos(2py). I0() 1, v = 0. (3.2a,b). Given this fixed, known velocity the concentration simply evolves via the advection–diffusion equation (2.7). ... u(x, y, t) =e cos(2py)Rc0(x,y,t). Z 1. 0e cos(2ps)Rc. 0(x,s,t) ds 1.

17 Jun 2019: 2π. 0. sin2 θdθ. a b cos θ=. 2π. b2. [a. ... sationσ(u, v) =. ((a b cos u) cos v, (a b cos u) sin v, b sin u. ),.

22 Jan 2019: ζ,fibLfib (sin2 α ρfib cos2 α) 6π μah, (26)B = [ζ,t Lt (1 ρt )ty tz hζ,fibLfib (1 ρfib ) sin α cos α]Rf l, (27). ... where. A = {ζ,t Lt (cos α[1 (1 ρt )t 2z ] h sin α(1 ρt )ty tz ) μ̃Lfib cos α 6π μah cos α}, (53)B = Lfib sin α,

7 Jan 2019: 6]An electromagnetic wave is represented by the 4-vector potential. Aµ = (0, A cos(kz ωt), A sin(kz ωt), 0). ... a) Evaluate the electric and magnetic fields. [6]. (b) Evaluate the Lagrangian density.

7 Jan 2019: x/a = θ sin φ (θ φ) cos φ ,y/a = 1 cos φ (θ φ) sin φ. ... A. (TURN OVER for continuation of question 4. 4. (d) By considering a wave solution of the form ϕ = C cos(kxωt) for auniform rod, find the dispersion relation and the

7 Jan 2019: m. (a, ,b ). z. M. (r, ,z). B B. (a) Show that, up to irrelevant constants and ignoring gravity, the Lagrangian forthe system is: [6]. ... 2k(r2 2ar cos(θ θB) (z bθB)2. ). A. (TURN OVER for continuation of question 1.

7 Jan 2019: E m2gl cos φ. [9]. 2 A harmonic oscillator is weakly perturbed by a cubic potential λx3 so thatits Hamiltonian has the form. ... 19]. (b) Find the temperature T (x, t) of the system if s(x, t) = cos(px)δ(t t0)and T (x, t < t0) = 0, for the two cases

19 Jun 2019: Xt = cos Bt and Yt = sin Bt. (c) Find the stochastic differential equation for the process (X, Y ) and write it inits standard form, i.e. ... Here σ1,. , σm, b are boundedmeasurable vector fields on Rd and B = (B1,. ,

30 Aug 2019: ln λ)2. ). (b) The integral I(σ) is defined by. I(σ) = 2π. ... 0. exp (σx(1 cos x)) dx. Find the the first two terms of the asymptotic expansion for I(σ) as σ.