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31 Jul 2023: y1(t) = A cos t , y0(t) = B sin t C sin 3t (18). ... for suitable values of the constants A, B and C [recall that cos3 θ = 34 cos θ 14 cos 3θ, and.

9 Nov 2023:  , G =. y cos(xy) by (b c)z. x cos(xy) bx bz(c b)x by. ... a) Find the general solution of the differential equation. d2y. dx2+ y = cos(kx) ,. where k is a positive integer. [10]. (b) (i) Rewrite the set of simultaneous differential equations

12 Jul 2023: i) Λ = R(φ) :=. . 1 0 0 00 cos φ sin φ 00 sin φ cos φ 00 0 0 1. ...  , ii) Λ = B(β) :=. . cosh β sinh β 0 0sinh β cosh β 0 0.

22 Aug 2023: 3, 2). 10. Basic identities. Starting from the identity sin(A B) = sin A cos B cos A sin B, use the basic properties of the trigono-metric functions (such as ... i) sin(A B) = sin A cos B cos A sin B ;. (ii) cos(A B) = cos A cos B sin A sin B ;. (iii) tan

12 Jul 2023: g(θ) =. . cos θ sin θ 0sin θ cos θ 0. ... Aµ 7 A′µ = gAµg1 (µg)g1. (b) Given that Aµ(x) g, show that A′µ(x) g.

3 Jul 2024: Show that. ψ(x) =. {Aexp(η|x|) for |x| > a,B cos(kx) for |x| < a,. ... iii) Find A(t) in terms of cos(ϕαt). You need not calculate its normalisation explicitly.

12 Jul 2023: p(m)i |i〉|θi〉(cos(θi)|0〉 sin(θi)|1〉) as an O(poly log(N)) sized circuit. of 1- and 2-qubit gates. ... Suppose wehave the n-qubit starting state |ψst〉 = sin(θ)|g〉 cos(θ)|b〉, where |g〉 G, |b〉 G andthe value of θ (0,π/2) is known.

3 Jul 2024: Given boundary conditionsu(a,θ) = 0 and u(b,θ) = cos 2θ, find u explicitly. ... b) If ⟨α(x),x⟩ = 0 for all x V , then α = 0.

7 Jul 2023: a) Prove that Tn(cos(y). )= cos(ny) for all integers n > 0. ... b) Prove that cos(π/n) is algebraic for all integers n > 1.(c) For each integer n > 1, determine whether cos(π/n) is rational or not.

7 Jul 2023: φ(r,θ) = a0 c0lnr. n=1. (anr. n cnrn) cos nθ. n=1. ... f(θ) =. n=1. An cos nθ,. for some coefficients An. (i) Solve Laplace’s equation 2φ = 0 in the annulus 1 < r < e2 with boundaryconditions:.

3 Jul 2024: T̂ =. (a b. b a. ),. where a(t) and b(t) are functions of time and a, b denote the complexconjugates, such that. ( ... Assume that w(x) = cos(πx). Show that in this case, the eigenvalues of B havenon-positive real part.

3 Jul 2024: b) Consider spatial perturbations of the form cos(kx) about the solution found inpart (a), and set λ = D1/D2. ... Ai(t) π1/2 (t)1/4 cos[23(t)3/2 14π. ]as t. ]. (c) We seek approximations to the smallest eigenvalues λ, which are close to 1.Using the

3 Jul 2024: J = J0. . sin(λz)cos(λz). 0. . with λ and J0 constant. ... Given boundary conditionsu(a,θ) = 0 and u(b,θ) = cos 2θ, find u explicitly.

7 Jul 2023: a) Prove that Tn(cos(y). )= cos(ny) for all integers n > 0. ... b) Prove that cos(π/n) is algebraic for all integers n > 1.(c) For each integer n > 1, determine whether cos(π/n) is rational or not.

4 Jul 2024: Ai(t) π1/2 (t)1/4 cos[23(t)3/2 14π. ]as t. ]. (c) We seek approximations to the smallest eigenvalues λ, which are close to 1.Using the ... S. A S. a b A A. a b b b a b.

28 Nov 2023: This book can be downloaded for free from. http://uk.mathworks.com/moler/chapters.html. (b) Matlab Guide by D.J.

7 Jul 2023: Consider the curve C in R2 defined by theequation. x2 y cos(xy) = 1. ... φ(r,θ) = a0 c0lnr. n=1. (anr. n cnrn) cos nθ. n=1.