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30 Aug 2019: u = R (u1 cos t u2 sin t). where ui = ûieikix for distinct vectors k1, k2 with |k1| = |k2| = k. ... Show that b takes the form. b = R. j=1,2. eikjx(pj cos t qj sin t).

30 Aug 2019: Find the order of ω(g, δ) for the Weierstrass function. g(x) :=. k=012k. cos 3kx. using the fact that, for 3m n < 3m1, the polynomial of best approximation of ... degreen to g is the partial sum tn(x) =. mk=0. 12k. cos 3kx. 4 Let (Ni,k)ni=0 be the

17 Jun 2019: n! nn. ;. (b). n>1. 1. n (log n)2;. (c). n>1. ... f (x) = x2 sign(x). cos. π. x. , x 6= 0.

17 Jun 2019: 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. ... Show that if g G stabilizes the point y X , then there exists an h G withhgh1 B.

30 Aug 2019: b) Let. f(x) =. k=0. ak cos 5kx, ak > 0,. ... tn(x) =m. k=0. ak cos 5kx. is the best approximant to f from Tn and find the value of En(f) in terms of ak.

17 Jun 2019: Show that the following are. equivalent. (i) a1b1ab H for all a,b G. ... y. 0. Q(x,y′)dy′. or otherwise, show that V is conservative. (b) Now let V =(. 1 cos(2πx 2πy),2 cos(2πx 2πy)). Show that there existsa smooth function

30 Aug 2019: a) q0 > 1/2H0 t = q0(2q0 1)3/2 [θ sin θ] ,. where the development angle θ is defined by. 1 cos θ =(. 2q0 1q0. )a. a0. (b) q0 = ... a) m,0 = 1, Λ,0 = 0;. (b) m,0 = 0, Λ,0 = 1;.

17 Jun 2019: 4]. (b) Find the eigenvalues of the matrixcos 2θ sin 2θ 0sin 2θ cos 2θ 00 0 2. ... 11 cos 2θ. dθ. [6]. (c) Using your results from (b), or otherwise, evaluate the definite integral π/20.

17 Jun 2019: Give a reason for your answer. [8]. (b) For what real values of λ does the equation cos θ sin θ 0sin θ cos θ 00 0 5. ... a)d2ydx2. 4y = cos x ,[6]. (b)d2ydx2. 4y = cos2 x ,[6].

17 Jun 2019: Ψ(r,θ) =. =0. (ar. br1. )P(cos θ). Here, P(cos θ) is the th Legendre polynomial, i.e., the solution of the differentialequation. ... d. dx. ((1 x2)dP. dx. ) ( 1)P = 0 ,. with x = cos θ, which is regular at x = 1.

17 Jun 2019: 3 5i. 5 4i. [2]. 6 Give the first three terms of the Taylor expansion of the function f (x) = cos xabout the point x = π/3. ... A B cos θ. where A > B > 0. By rewriting the equation of this curve in terms of cartesiancoordinates (x, y), or otherwise,

30 Aug 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,. ,

17 Jun 2019: 2 cos θ| > 1. [2]. 10 Show that the probability distribution. ... g(x) =1. 3+. 4. π2. n=1. (1)nn2. cos(πnx). [6]. (b) By considering.

7 Jan 2019: L =1. 2m[ȧ2x ȧ. 2y l. 2θ̇2 2lθ̇(ȧx cos θ ȧy sin θ)] mg(ay l cos θ). ... lcos(θ1) cos(ωt),. A. (TURN OVER for continuation of question 1. 2.

17 Jun 2019: d. dξ. (. (1 ξ2)dP. dξ. ). λP = 0,. where ξ = cos θ and λ = n(n 1), which is non-singular at ξ = 1.[8]. ... You may use the formulae P0 (cos(θ)) = 1, P2 (cos(θ)) = (3 cos2(θ) 1)/2.

17 Jun 2019: Hence express y2(x) in terms of y1(x) and W(x). (b) Verify that y1(x) = cos(xγ) is a solution of. ... xy. ). = α. (. 5 cos t2 cos t sin t. ).

30 Aug 2019: α =|k3|. (k21 k22)1/2. |σ13 cos φ σ23 sin φ|a2 b2. ... as t. What is the physical interpretation of α? Now assume that a and b are independent Gaussian random variables with zeromean and variance Γ2 and that σ13 cos φ σ23

17 Jun 2019: n=1. (. an rn. bnrn. )(. An cos(nθ) Bn sin(nθ). ). [4]. Find a solution Φ(r, θ) which is bounded inside the disc r 6 R and such that. ... I =. π. π. d θ. a b cos θ c sin θ.[5].

17 Jun 2019: Prove the following inequalities:. | cos(ex) cos(ey)| 6 |x y| for x, y > 0. ... g(x) = cos(|x|), h(x) = x|x|. (d) Determine the points at which the following function from R to R is continuous:.

30 Aug 2019: z = b(x,y) = cos (y) exp (ikx) with the real part understood,. ... Deduce that. ψ′. t=. ikN2U0Kf. cos (y) exp(Kz) exp (ikx). with the real part understood.

17 Jun 2019: has polar equationAr = 1 ε cos θ with A > 0 and 0 < ε < 1.]. ... S half of the light is concentrated into a narrow conewhose semi-angle α is given by cos α = v/c.

30 Aug 2019: 1/2. (b) Assuming a functional form φ(r) = A cos(q0z), where z is an arbitrary direction,find for nonzero B the resulting mean-field free energy per unit volume ... H =. q. G(q)φqφq B. V. q1,q2,q3. (q1 q2) φq1φq2φq3φq1q2q3 (2).

17 Jun 2019: Ψ(r, φ) = A0. n=1. rn[An cos(nφ) Bn sin(nφ). ]. (b) Assume Ψ satisfies the boundary condition Ψ(R, φ) = f(φ) for 0 6 φ < 2π. ... cos x. x4 x2 1dx. [10]. Natural Sciences IB, Paper 2 [TURN OVER.

30 Aug 2019: 0. )and pressure-anomaly field. p f(x) exp(imz) b(z) (†)where b(z) is an arbitrary function. ... ψ′ = Re[eikx sin(y). {B t cos(mz) C (z H) sin(mz).

30 Aug 2019: cos(. πyL. ). , h < z < 0. 0, H < z < h.(7). ... A B) = A( B) B( A) (B )A (A )B. ]

17 Jun 2019: n=1. fn(A) = and f(B) = B. [Here fn denotes the n-fold composite of f with itself.]. ... c) Finally, show that. v2 = v1. (. 1 (A B cos θ0)l.

17 Jun 2019: Then sin x sin x1 cos x. cos x q(x) sin x = 0. ... so q(x) = 1 cos x/(1 cos x) = 1/(1 cos x)[3].

17 Jun 2019: ψ(r,θ) =. n=0. (Anrn Bnr. (n1))Pn(cos θ). Hence, find the solution of Laplace’s equation in the region a 6 r 6 b satisfying theboundary conditions {. ... ρdU. dt. a(b a)b a. cos θ,. where θ is the azimuthal angle measured from the direction of U.

30 Aug 2019: R =mfcollλ. xd cos β. (b) A cross-dispersed echelle spectrograph has a flat focal plane. ... Zmn (ρ, φ) = Rmn (ρ) cos mφ. where n is the radial degree and m is the azimuthal degree and Rmn (ρ) is a radial polynomialgiven by.

30 Aug 2019: δγ(k, τ) [. A(k) cos(kτ/3) B(k) sin(kτ/. 3)]. exp(k2/k2D) ,. where kD is a time-dependent damping scale which you should specify. ... Identify the interaction Hamiltonian density Hint. (b) A rescaling is defined to be x λx where λ > 0 is a real

17 Jun 2019: ω2R. gsin θ cos θ ,. working to lowest non-trivial order in ω. ... ω2R sin θ cos θ. gand tan φ φ[8]. (b) Now consider a particle fired vertically upwards from P with speed v.

16 Oct 2019: p2ψ > 4MglI1 cos θ. (b) A rigid body of mass M is of uniform density and its surface is defined by. ... φ : (0,2π) (a,b) R3,. where φ(u,v) = (f(v) cos u,f(v) sin u,g(v)), i.e.

17 Jun 2019: SL(2,R),. the corresponding Möbius transformation is. z 7 Az =az b. ... form. f(x) = 12a0. n=1. an cos(nx). n=1. bn sin(nx). (b) Find the general solution to.

17 Jun 2019: b) Let a > 0 and. f(x) =. {cos x for x [a, a]0 elsewhere. ... Derive Clairault’s relationthat. f(u(t). )cos. (θ(t). ). is independent of t.

30 Aug 2019: sin δ = sin γ sin θ cos γ cos θ cos η. ... cos δ sin h = sin η cos γ. cos δ cos h = sin γ cos θ cos γ sin θ cos η.

7 Jan 2019: 2. (2P1 cos(ωt ϕ) Q2. )py =. α. 2. (2P1 sin(ωt ϕ) P2. ... µ and WµWµ. The coefficients are. m2. λ[e cos α e′ sin α].

16 Oct 2019: φ : (0,2π) (a,b) R3,. where φ(u,v) = (f(v) cos u,f(v) sin u,g(v)). ... If θ(s) denotes the angle that the geodesicmakes with the parallel S {z = g(v(s))}, then show that f(v(s)) cos θ(s) isconstant in s.

17 Jun 2019: σ(u, v) =(Q(u, v) sin u, Q(u, v) cos u, v cos(u/2). ). ... J = ẋy ẏxis conserved. Write down the general solution of the equations of motion in terms ofsin ωt and cos ωt, and evaluate J in terms of the coefficients that arise

17 Jun 2019: E′0 = λ (sin θ, cos θ, 0). Show further that the electric charge density on the surface x = 0 takes the form. ... u(x) = λ sin x A π. 0. (cos x cos t cos 2x cos 2t) u(t)dt.

17 Jun 2019: E = (E0 cos(kz ωt), 0, 0). You should give the form of the magnetic field B in this case. ... Give a physical interpre-tation of this equation. Evaluate the time-averaged energy 〈E〉 and energy flux 〈I〉 of a monochromaticwave φ = cos(kx wt), and

17 Jun 2019: B = B0 cos(. ωxc cos θ. )exp. [i(. ωzc sin θ ωt. ... An electron is at rest in the presence of a magnetic field B = (B, 0, 0), andexperiences an interaction potential µσ B.

17 Jun 2019: Tn(cos t) = cos nt. for all real t. (ii) There exists a real polynomial Rn of degree n such that. ... L =m12r21 φ̇. 21. m22r22 φ̇. 22 ω. 2r1r2 cos(φ2 φ1) ,. using the particles’ cylindrical polar angles φ1 and φ2 as generalized coordinates.

17 Jun 2019: Hence obtain the identity. n=0. ett3n. (3n)!=. 13. 23. e32 t cos. ... Φ(x, y) =(. B0k. )cos kx eky,. referred to cartesian coordinates (x, y, z).

17 Jun 2019: E = (E0 cos(kz ωt), 0, 0). You should give the form of the magnetic field B in this case. ... Consider the surface of revolution:. (uv. )7 f(u,v) =.  ϕ(u) cos(v)ϕ(u) sin(v)u.

11 Nov 2019: p2ψ > 4MglI1 cos θ. (b) A rigid body of mass M is of uniform density and its surface is defined by. ... φ : (0,2π) (a,b) R3,. where φ(u,v) = (f(v) cos u,f(v) sin u,g(v)).

17 Jun 2019: x2 =a0. 2+. n=1. an cos nx ,. where an, n = 1, 2,. , ... Let the energy eigenstates of even parity be given by. ψeven(x) =. . . . A cos λx B sin λx a < x < 0A cos λx B sin λx 0 < x <

6 May 2019: h = h0[1 a cos cosh b sin sinh ], (4.1). ... where =. p. 2/P(4.2). and. a = cosh sin cos sinh cos sin cosh sinh. ,

19 Jul 2019: along the a = (sin, 0, cos) axis or along the b = (sin, 0, cos) axis. ... hEPR|Az B |EPRi = sin2. 2 cos2. 2= cos. This means that if we take the four operators A2, B1, A1 and B2 to be spin operators,.

17 Jun 2019: α. sin2 θ+ mgl cos θ. is a constant of the motion, where α is a non-negative constant. ... Suppose that E = E0(0 , 1 , 0) and B = E0(0 , cos θ , sin θ), where 0 θ < π/2.

16 Jul 2019: Compute the Moore-Penrose inverse of the operator K : L2([a,b]) R with. ... b) Show that the functions ϕpq C() with. ϕpq(x1,x2) := vpwq cos (πqx1) cos (πpx2).