Algebra, Analysis and Differential Equations II

Transcription

1 SECOND PUBLIC EXAMINATION Honour School of Mathematics Part A: Paper AC2 Honour School of Mathematics and Statistics Part A: Paper AC2 Algebra, Analysis and Differential Equations II TRINITY TERM 2012 Tuesday 19 June, 9.30am to 12.30pm You may hand in attempts to any number of questions. Answers to four questions will count towards the total mark for the paper: the best question from each section and best remaining question. Start each question in a new booklet. For each section, attach the answers to questions from that section together to make a bundle, and indicate on the front sheet the numbers of the questions in that bundle. You should hand in three bundles. Do not turn this page until you are told that you may do so Page 1 of 6

2 Algebra 1. Let V be a real vector space of finite dimension n 1 and let T : V V be a linear transformation. (a) Define the minimal polynomial m T of T and the characteristic polynomial χ T of T, explaining why the latter is well defined. (b) An element v of V is called cyclic for T if V is spanned by its subset {v, T (v), T 2 (v),...}. Show that if v is cyclic for T then there is some k 1 such that B = {v, T (v), T 2 (v),..., T k 1 (v)} is a linearly independent subset of V and T k (v) lies in the span of B. Deduce that B is a basis for V and hence that k = n. By considering the matrix of T with respect to the basis B, prove (without using the Cayley Hamilton Theorem) that m T = χ T. (c) Now let V = R 3 and define T 1 : V V and T 2 : V V by T j (v) = A j v for j = 1, 2 where A 1 = and A 2 = Is there an element v R 3 which is cyclic for T 1? Is there an element v R 3 which is cyclic for T 2? Justify your answers briefly. 2. Let V be a finite-dimensional complex inner product space with inner product,. (a) What is an orthonormal basis for V? Describe the Gram Schmidt process for obtaining an orthonormal basis from any basis {e 1,..., e n } for V. (b) Let R : V V and S : V V be linear transformations, and let {e 1,..., e n } be an orthonormal basis for V. Show that for all u, v V if and only if Su, v = u, Rv Re j = n e j, Se i e i i=1 for j = 1,..., n, and deduce that there exists a unique linear transformation S : V V with the property that Su, v = u, S v for all u, v V. [You may assume that for v 1, v 2 V if e k, v 1 = e k, v 2 for k = 1,..., n then v 1 = v 2.] Prove that (S ) = S and that (ST ) = T S for any linear transformation T : V V. (c) Let u be an eigenvector of S S with eigenvalue λ. Prove that Su 2 = λ u 2 where u 2 = u, u, and deduce that the eigenvalues of S S are real and non-negative. Let λ and λ + be the smallest and greatest eigenvalues of S S. Prove that if v V then λ v Sv λ + v. [You may assume that if T : V V is linear and T = T then V has an orthonormal basis consisting of eigenvectors of T.] Page 2 of 6

3 3. Let R be a commutative ring with identity. (a) What is a unit in R? What is an irreducible element of R? What is a prime element of R? (b) What does it mean to say that R is a Euclidean ring? Prove that if I is an ideal in a Euclidean ring R then there is some a R such that I = ar is generated by a, and show that if a is irreducible in R then I is a maximal ideal. Explain briefly why if F is a field then the polynomial ring F [X] is Euclidean. (c) Show that if n 1 is a positive integer and I n is the ideal (X n 2)Q[X] in Q[X] then K n = Q[X]/I n is a field. What is the dimension of K n as a vector space over Q? Show also that if I = (X 2 + 1)Q[X] is the ideal in Q[X] generated by X then L = Q[X]/I is a field. Are L and K 2 isomorphic as fields? Justify your answer. [You may assume the Gauss Lemma, Eisenstein s criterion and the division algorithm for polynomials over a field, and also that the quotient of a commutative ring with identity by a maximal ideal is a field.] Page 3 of 6 Turn Over

4 Analysis 4. In the following let U C be open, simply connected and non-empty. Let B 1 (0) = {z C : z < 1}. (a) State Cauchy s Integral Formula for a holomorphic function f : U C. (b) Let f : U C be holomorphic and assume that B 1 (0) U. Prove the following statements. (i) If f attains a local maximum at a point z 0 B 1 (0), then f is constant on B 1 (0). (ii) f attains its maximum value in B 1 (0) on the boundary B 1 (0). (c) Let f : B 1 (0) C be holomorphic with f(0) = 0 and f(z) < 1 for all z B 1 (0). (i) Prove that f(z) z for all z B 1 (0) and f (0) 1. (ii) Furthermore, show that if f (0) = 1, or if there exists z 0 B 1 (0) \ {0} such that f(z 0 ) = z 0, then f(z) = wz for some w C with w = (a) State without proof Laurent s theorem about expansions of holomorphic functions in an annulus A = {z C : R < z a < S}, including an integral expression for the coefficients in the expansion. (b) Let Ω be an open subset of C, and z 0 Ω. What does it mean to say for a complexvalued function f : Ω\{z 0 } C that (i) z 0 is an isolated singularity; (ii) z 0 is a removable singularity; (iii) z 0 is a pole of order n N; (iv) z 0 is an essential isolated singularity? (c) Compute the Laurent expansion for f(z) = 1 z 2 (z 1) in A 1 = {z : 0 < z < 1} and A 2 = {z : z > 1} respectively. (d) For s (0, ) let B s (0) = {z C : z < s}. Assume that f : B s (0)\{0} C is a holomorphic function. (i) Prove that if there exists r (0, s) and m > 0 such that f(z) < m for all 0 < z < r, then 0 is a removable singularity. (ii) Prove that if 0 is an essential singularity for f, then for all r (0, s) the set f ( B r (0)\{0} ) is dense in C; in other words, for all ε > 0 and w C there exists z B r (0)\{0} such that f(z) w < ε. [Standard properties of holomorphic functions may be used without proof provided they are clearly stated.] 6. (a) Compute for a > 1 the integral π 0 1 a + cos(t) dt. (b) Using an appropriate holomorphic branch of the logarithm function and a suitable keyhole contour, or otherwise, show that 0 log x ( x ) π2 = x 2 2. Page 4 of 6

5 Differential Equations 7. (a) What is a critical point of the plane autonomous system dt = f(x, y), dy M = dt = g(x, y)? Suppose that is such a critical point, explain how the matrix M given by f f x y g g x y is used to analyse the linear stability of the system near the critical point. Assume that M has nonzero distinct eigenvalues λ 1 and λ 2. Under what conditions on λ 1 and λ 2 is the critical point (i) a stable node, (ii) an unstable node, (iii) a saddle, (iv) a centre, (v) a stable spiral, (vi) an unstable spiral? (b) Find and classify the critical points of the system dt = y(x2 1), (c) State the Bendixson Dulac Theorem. Show that the system dy dt = (x + y 3)(y 1). dt = y + x5 5 x3, dy dt = x + x2 y 3 3, has a centre at the origin but that there are no closed orbits lying inside the circle whose equation is x 2 + y 2 = (a) Consider the boundary-value problem d 2 y dy + 5y = f(x), y(0) = 0 = y(π/2), where f is continuous on [0, π/2]. Describe carefully how to use the method of Variation of Parameters to construct the Green s function for this problem, justifying the choices you make. (b) In the particular case f(x) = e 2x show that the solution of the problem in part (a) is y(x) = e 2x (1 sin x cos x). (c) Now suppose the boundary conditions of the problem in part (a) are changed to y(0) = 0 = y(π), and f(x) is a continuous function on [0, π]. By using the solution obtained from the Green s function in part (a) as a particular integral, or otherwise, show that there is no solution of this problem unless f satisfies the condition π 0 f(t)e 2t sin t dt = 0. If f does satisfy this condition, what is the general solution of this problem? Page 5 of 6 Turn Over

6 9. (a) Write down the equations determining the characteristics of the first-order, quasi-linear partial differential equation (b) P (x, y, z) z + Q(x, y, z) z x y = R(x, y, z). What is a solution surface for the equation? Explain how the characteristics may be used to produce a solution surface to the equation, given data (x(t), y(t), z(t)). (i) Find the characteristics for the particular equation z x + z z y = 2. (ii) Find the solution of this equation in parametric form, given that z = 1 on x+y = 0 for x 0. (iii) Show that the projection into the (x, y)-plane of each characteristic is a parabola tangent to the line x + y = 1. (iv) Where is the Jacobian J = (x,y) equal to zero? (s,t) (v) What is the explicit form of the solution? (vi) In what region of the plane is the solution uniquely determined by the data? Page 6 of 6 End of Last Page