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30 Aug 2019: b) Let X be a continuous local martingale such that Xt X almost surely. ... dΘ = cos(Θ)dZ. (b) Hence, show that. P(Xt ) =1. 2+.

30 Aug 2019: δ′b k2θb = 0 , θ′b. a′a θb c. 2sbδb = RaneσT(θb θγ)/k2 ,. where csb is the baryon sound speed and R is given in terms of the relative ... δγ (k, τ ) [A(k) cos(kτ /. 3) B(k) sin(kτ /. 3)). ]exp(k2/k2D) ,. where kD is a time-dependent

17 Jun 2019: 3 cos θ sin θ =. 2. in the range 0 6 θ < 2π. ... i). cos xdydx. (cos x sin x)y = 2 sin 2x.

30 Aug 2019: In longitudinal Newtonian gauge we take instead E = B = 0. ... δC = Aτ 2 =34δR , for τ 2π/k ,. δC B ln τ , δR C cos(kτ /. 3) D sin(kτ /. 3) , for τ 2π/k ,. where A, B, C, D are

30 Aug 2019: real k axis at k = k0 cos θ and k = k0 sec θ). ... φi = cos(nπy/h) exp(iωt ikx) ,. where k =k20 n2π2/h2 is diffracted by the two edges.

30 Aug 2019: a) Let V,W be closed subspaces of H. Show that if cos(. ... b) Let U and V be closed subspaces of H such that cos(.

30 Aug 2019: f (λ) = π |λ|,. find the autocovariance function. (c) Let Xt = A cos(ω0t) B sin(ω0t), where A and B are uncorrelated randomvariables A and B, each with mean 0 ... 1 n!nn. (b) Suppose we observe paired data, (X1, Y1),. , (X100, Y100).

30 Aug 2019: tik]( t)k1+. be the sequences of L1 and L-normalized B-splines, respectively. ... Prove that the algebraic polynomial of best approximation is unique. (b) Let Tn(x) = cos narccos x be the Chebyshev polynomial of degree n, and let.

30 Aug 2019: h = A(1 cos x) B. (3). (Here B is assumed to be a small constant and A grows slowly at a rate c(h+ h).). ... If H aX2 b as X and H+ a+(λ)X2 b+(λ) as X show, bymatching to (3), that.

30 Aug 2019: b) Consider the 2-dimensional problem of a plane wave. φi = eikx cos θ iky sin θ. ... φr = eikx cos θ iky sin θ. and φd is the field diffracted by the discontinuity.

17 Jun 2019: F = (sin x, sin y, sin z), G = (cos x, cos y, cos z). ... i) (2xy2 y)dx (2x x2y)dy = 0 , [5]. (ii) (2y sin x 3y4 sin x cos x)dx (4y3cos2x cos x)dy = 0.

17 Jun 2019: Let S denote a surface defined by rotating a curve. η(u) = (r a sin u, 0,b cos u) 0 u 2π ,. about the z-axis. Here a,b,r ... φ =Ua3 cos θ. 2r2. Calculate the kinetic energy of the fluid.

30 Aug 2019: Show that if we apply a fieldof the form f (t) = A(t) cos(ωt φ) the rotating wave approximation Hamiltonianis. ... cos(x) = 11tan2(x). , sin(x) = tan(x)1tan2(x). ]. (b) Assume the system is initially in the state |3〉.

17 Jun 2019: f(x) = x2 sign(x)cos π. x. , x 6= 0. (b) Is f(x) differentiable at x = 0? ... x(u) = aebu cos u , y(u) = aebu sin u ,. where the constants a, b > 0. Sketch the curve segment corresponding to the range0 6 u 6 3π.

30 Aug 2019: In the case f = cos ωt, deduce that the mean thrust is approximately equal to. ... f = f0 α. Bsin θ cos φ O. [(B)2. ],. where (θ, φ) are the polar coordinates of p with θ = 0 in the x3-direction and φ = 0in the (x1,

17 Jun 2019: ω2R. gsin θ cos θ ,. working to lowest non-trivial order in ω. ... b) Now consider a particle fired vertically upwards from P with speed v.

17 Jun 2019: 6]. (d) Use complex numbers to show that. cos 4θ = 8 cos4 θ 8 cos2 θ 1. ... ii)dy. dx y cos x = 1. 2sin 2x , with y(0) = 0.

30 Aug 2019: R = A (1 cos θ), t = B (θ sin θ), (). solves the equation of motion of the mass shell for suitable values of the constants A and. ... B. Sketch this solution for a suitable range of θ and express A and B in terms of the radius.

30 Aug 2019: that exp(iθÂ) = cos(θ)Î i sin(θ)Â. ... b) Show that a generic single-qubit quantum gate can be expressed as.

16 May 2019: 3 = 2.2. a. (3i. 1i. )2018= 21009e. 5iπ6 b. 1 eiθ = 2 cos. ( ... p q = 1) et on fixe (a,b) Z N avec a b = 1 t.q.2 cos(πr) = ab.

7 Jan 2019: x = η sin θ y = cos θ. Correspondingly, the kinetic energy is given by [3]. ... 1. 2m2. [η̇2 θ̇2 2θ̇η̇ cos θ. ]and the potential energy by [3].

30 Aug 2019: Does the model conserve energy? (b) Show that the governing equations admit the following steady solution. ... u′x = u cos(kx t), u′. y =12u sin(kx t), (†). where u is an arbitrary real constant. (b) Suppose dust particles are distributed uniformly

30 Aug 2019: whereφi = exp(ik0x cos θ0 ik0y sin θ0 iωt). is the incident potential, φ(x,y) exp(iωt) is the scattered potential and k0 = ω/c0. ... Γf(k) exp(ikr cos θ γr|sin θ|)dk. (2k0πr. )1/2f(k0 cos θ)|sin θ|exp(ik0r iπ/4).

30 Aug 2019: X(θ) = b(cos θ, sin θ, θ tan φ), 0 6 θ 6 N π, (). where N is a large integer and φ is the pitch of the helix ... h = (1 α cos θ),. where θ is the angle to the downward vertical.

30 Aug 2019: The centre of the sphere is at theorigin. By considering the special form of the solution u(x) = v(r) cos θ in the sphericalcoordinates (r, θ, φ) (x3 = r cos ... u(x) = x3 r31(a1 a0)r3(a1 2a0). x3, x / B,. =3a0.

17 Jun 2019: 9. Find the Fourier sine series for f(x) = sin x(1 4 cos x) defined on π < x < π. ... b) Show that the substitutions. xi = ai cos(ωt) , i = 1, 2, 3,.

30 Aug 2019: The corresponding equations for the coupled baryons are. δb k2θb = 0 , θ′b a′. ... δγ (k, τ ) [A(k) cos(kτ /. 3) B(k) sin(kτ /. 3)). ]exp(k2/k2D) ,. where kD is a time-dependent damping scale. [Hint: Consider the substitution δγ = δ̃γ exp

17 Jun 2019: Which of these sequences converge uniformly on (0, 1)? (i) fn(x) = log (x 1n ),. (ii) fn(x) = cos ( xn ). Justify your answers. 4H Complex Analysis. ... A spatially uniform magnetic field B = B0(cos ωt, sin ωt, 0)is applied, with B0 and ω both constant

17 Jun 2019: T(x,y,t) =. n,m. Anm sin(nπx. a. ). cos(mπy. b. ). exp. [. (. ... u = ex[(. x2 y2). cos y 2xy siny]. ,. writing your final answer in terms of z.

17 Jun 2019: Let S denote a surface defined by rotating a curve. η(u) = (r a sin u, 0,b cos u) 0 u 2π ,. about the z-axis. Here a,b,r ... A spatially uniform magnetic field B = B0(cos ωt, sin ωt, 0)is applied, with B0 and ω both constant.

17 Jun 2019: r = (b cos θ, b sin θ, cθ),. where b and c are positive constants and the parameter θ takes values 0 6 θ 6 5π/2.Consider the two vector ... x = W tan θ cos φ, y = W tan θ sin φ, z = W,.

17 Jun 2019: Show that. E′ =E(1 β cos θ). 1 β2, cos θ′ =. cos θ β1 β cos θ. ,. where β = v/c. ... x1/n. 1 x2dx =. π. 2 cos(π/2n),. where n > 1,.

30 Aug 2019: b) The implicit solution to. 12. (dYdξ. )2 23 Y. 32 kY = 13 k. ... sin τ|cos τ =. n=0. an cos(nτ ). for some constants an.].

17 Jun 2019: 9E. (a) Show thatsinh(ix) = i sin x. and thatcosh(ix) = cos x. ... 4]. (c) Evaluate π/20. cosh x cos x dx. [10]. Paper 1 [TURN OVER.

17 Jun 2019: Φ = A0 B0µ. n=1. {[An cosh(nµ) Bn sinh(nµ). ]cos(nν). [Cn cosh(nµ) Dn sinh(nµ). ... Φ. µ. µ=a. = cos(2ν). The outer ellipse, defined by µ = b, is held at Φ(b,ν) = cos(ν).

7 Jan 2019: b) The canonical momentum conjugate to x is. px =L. ẋ= (m1 m2)ẋ m2lφ̇ cos φ. ... circular motion around the z axis with constantangular velocity. In terms of the z coordinate, the particle undergoes simpleharmonic motion, z(t) = a sin κt b cos κt

30 Oct 2019: k(x, xÕ) = –Ep(w). [cos(wx wxÕ)] –Ep(b). [cos(wx wxÕ 2b)] (3.17)= –E. ... p(w,b). [cos(wx b wxÕ b) cos(wx b wxÕ b)] (3.18)= 2–E.

30 Aug 2019: ψ1〉AB = | z〉A | z〉B|ψ2〉AB = | z〉A | z〉B|ψ3〉AB = | z〉A (cos θ| z〉B sinθ| z〉B)|ψ4〉AB = | z〉A (sinθ| z〉B cos ... and b, b′ respectively. Each measurement yields an outcome 1. The experimentis characterised by the

17 Jun 2019: i) exp(x y) dx exp(x y) dy ,. (ii) sin x sin y dx cos x cos y dy ,. (iii) 2xy3z4 dx 3x2y2z4 dy 4x2y3z3 dz. [9]. (b) Find and ... 7]. (b) Use de Moivre’s theorem to show that. cos θ cos(θ α) cos(θ nα) = sin[.

17 Jun 2019: a20. sin axx. dx. [8]. (b) Evaluate the integral 0. cos αx eβxdx (β > 0). ... 10]. (b) Using the substitution x = cos θ, find the general solution of.

17 Jun 2019: 4]. Paper 1 [TURN OVER. 6. 10E (a) By considering (cos θ i sin θ)5, express cos(5θ) in terms of cos θ. ... i) (x, y) = (a cos θ, b sin θ);. (ii) (x, y) = (a cosh θ, b sinh θ);.

17 Jun 2019: cos(A B) cos(A B)]. Paper 1 [TURN OVER. 6. 9C The functions p(x), q(x) and w(x) are defined on the interval 0 x 1. ... I = b. a. f(x,y,y′)dx ,. stationary, subject to appropriate boundary conditions.

17 Jun 2019: The inner cylinder is held at φ = 0 and the outer at φ = V cos 2θ. ... 12]. Find φ if the inner cylinder is held at φ = V and the outer at φ = V cos 2θ.

7 Sep 2019: Résoudre dans C l’équation :(z 1z 1. )n+(z 1z 1. )n= 2 cos α. ... 2 cos(. 3π5. ) 1 = 0. 2. En déduire la valeur de cos(π5).

17 Jun 2019: 4. 8E Classical DynamicsConsider the Lagrangian. L = A( θ̇2 φ̇2 sin2 θ ) B(ψ̇ φ̇ cos θ )2 C(cos θ)k ,. where A, B, C are positive constants and k ... g(t) N. j=0. aj cos jt 6 ǫ. for all t [0,π].

17 Jun 2019: Show that Amaps Π to itself and induces a rotation of Π by angle θ, where cos θ = 12 (trace (A) 1).Show that A is a reflection in Π if ... Prove that det(A I) = 4(cos θ 1). Paper 3. 5.

17 Jun 2019: a) (cosh θ sinh θ cos φ)dθ (cosh θ sin φ cos φ)dφ,. ... 6. 11A Vector Calculus. Prove that. (a b) = a b b a (b )a (a )b.

7 Sep 2019: Exercice 1.10 (?). Déterminer l’image de f : x R 7 x cos x. ... Exercice 1.17 (?). Soit f :]a, b[ R une fonction continue t.q.

30 Aug 2019: Show that these can be written as. W (n)(x) = A cos(k(n)x) B sin(k(n)x) D cosh(k(n)x) E sinh(k(n)x) ,. and ... ut = Duuxx a (b 1)u u2v , (1). vt = Dvvxx bu u2v , (2).

30 Aug 2019: Wn(x) = B sin knx D cos knx E sinh knx F cosh knx ,. where kn is the nth root of the transcendental equation cos kL cosh kL = 1. ... Wn(x) = N[(sin qn sinh qn) (cos knx cosh knx). (cos qn cosh qn) (sin knx sinh knx)] , (1).