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Applied Mathematics Qualifying Exam University of British Columbia September 4, 2009. Part I 1. (a) Let I= 1 0 0 1 , J= 0 −1 1 0 , and verify that for all real numbers x1 , y1 , x2 , y2 we have (x1 I + y1 J)(x2 I + y2 J) = (x1 x2 − y1 y2 )I + (x1 y2 + x2 y1 )J. (b) Find An for any integer n, if √ 3/4 1/4 − . A= √ 3/4 1/4 2. Prove that the equation XY − Y X = In has no solution (where X, Y are unknown real n × n-matrices, and In is the identity matrix). 3. A nonzero matrix A is called nilpotent if there exists a positive integer n such that An = 0. Two matrices A and B are called similar if they can be obtained from one another by a change of basis. a) Prove that if A is nilpotent and B is similar to A, then B is also nilpotent. b) Find a set of representatives of all equivalence classes of nilpotent 3 × 3-matrices with complex entries, where we declare two matrices equivalent if they are similar. (You may want to solve this question for 2 × 2-matrices first). 4. Define the Fourier transform pair to be: Z ∞ Z ∞ 1 ˆ = f(k) fˆ(k)eikx dx. f (x)e−ikx dx and f (x) = 2π −∞ −∞ ˆ (a) Use contour integration to calculate the Fourier transform f(k) for f (x) = 1 , (x2 + a2 )2 where a ∈ R is a constant. (b) Calculate the inverse Fourier transform f (x) for fˆ(k) = where a ∈ R is a constant. 1 1 , (k 2 + a2 )2 5. Use a keyhole-shaped contour to evaluate the integral Z ∞ dx √ I= . x(x2 + 1) 0 6. Let D be the circle of radius 4 centred at the point (0, 5) in the x − y plane. Find a function φ(x, y) that satisfies the following restrictions: • φ is harmonic in the upper half-plane exterior to D; • φ = 1 on D; • φ = 0 on the x-axis. Hint: Consider a conformal map of the form w = 2 z+α z+β . Part II 1. Suppose f is a continuous function on R such that |f (x) − f (y)| ≥ |x − y| for all x and y. Show that the range of f is all of R. 2. For every a ∈ R, determine whether the integral ZZ (x4 + y 2 )a dA D is finite, where D is the square {(x, y) | −1 ≤ x ≤ 1, −1 ≤ y ≤ 1}. 3. Let S be the finite solid region bounded by the plane z = 0 and the surface z = 1−x2 −y 2 . Find the flux of the vector field V = xyi+xzj+(1−z−yz)k outward through the surface of S. 4. (a) Find the general solution of the homogeneous linear system x0 = x + y, y 0 = y. (b) Solve the initial value problem √ x0 = x + y + et 1 + t, y0 = y + x(0) = 0, et , 1 + t2 t > −1, y(0) = 1. 5. Consider the system of ordinary differential equations in the (x, y) plane x0 = (a − π 2 )x − 3x3 − 6xy 2 , y 0 = (a − 4π 2 )y − 6x2 y − 3y 3 , (1) where a is a real constant. Throughout this question, assume that 0 < a < 7π 2 . (a) Find all equilibria (i.e. critical points or constant solutions or steady states), and determine the linearized stability and type of each equilibrium. You may wish to consider different cases. (b) Let V (x, y) = 12 (x2 + y 2 ). Show that for all solutions (x(t), y(t)) of (1) with sufficiently large distance from the origin, the expression V (x(t), y(t)) is a decreasing function of t. Discuss the behaviour as t → ∞, of solutions of (1). 6. (a) Solve the initial boundary value problem ut = uxx + au, u(0, t) = 0, 0 < x < 1, u(1, t) = 0, u(x, 0) = g(x), t > 0, t > 0, 0 ≤ x ≤ 1, where a is a positive constant, and g(x) is a continuous function defined for 0 ≤ x ≤ 1, with g(0) = g(1) = 0. Describe the solution’s behaviour as t → ∞. You may wish to consider different cases of a and g(x). 3 R∞ 2 ˆ (b) Let t > 0 be fixed, and let f(k) = −∞ eikx e−x /(4t) dx, −∞ < ˆ k < ∞. Show that fˆ satisfies the differential equation fˆ0 = −2tk f, then solve this differential equation to find an explicit formula for fˆ(k) in terms ofpelementary functions. You may use the fact that R ∞ −αx2 dx = π/α for any constant α > 0. −∞ e (c) Solve the initial boundary value problem ut = uxx + au, −∞ < x < ∞, lim u(x, t) = 0, t > 0, lim ux (x, t) = 0, x→±∞ x→±∞ u(x, 0) = g(x), t > 0, −∞ < x < ∞, where a is a positive constant, and g(x) is a continuous function defined for −∞ < x < ∞, with limx→±∞ g(x) = limx→±∞ g 0 (x) = 0. Express the solution as a single integral over a spatial variable. Describe the solution’s behaviour as t → ∞. You may wish to consider different cases of a and g(x). 4