MATHEMATICAL TRIPOS Part II PAPER 3 Before you begin read ...

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6SECTION II11I Number TheoryDefine equivalence of binary quadratic forms and show that equivalent forms havethe same discriminant.Show that an integer n is properly represented by a binary quadratic form ofdiscriminant d if and only if x 2 ≡ d mod 4n is soluble in integers. Which primes arerepresented by a form of discriminant −20?What does it mean for a positive definite form to be reduced? Find all reducedforms of discriminant −20. For each member of your list find the primes less than 100represented by the form.12F Topics in AnalysisSuppose that x 0 , x 1 , . . . , x n ∈ [−1, 1] are distinct points. Let f be an infinitelydifferentiable real-valued function on an open interval containing [−1, 1]. Let p be theunique polynomial of degree at most n such that f(x r ) = p(x r ) for r = 0, 1, . . . , n. Showthat for each x ∈ [−1, 1] there is some ξ ∈ (−1, 1) such thatNow take x r = cos 2r+12n+2π. Show thatf(x) − p(x) = f (n+1) (ξ)(n + 1)! (x − x 0) . . . (x − x n ) .|f(x) − p(x)| 12 n sup |f (n+1) (ξ)|(n + 1)! ξ∈[−1,1]for all x ∈ [−1, 1]. Deduce that there is a polynomial p of degree at most n such that∣ 1 ∣∣∣∣3 + x − p(x) 14 n+1for all x ∈ [−1, 1].Part II, Paper 3
13A Mathematical BiologyAn activator-inhibitor system is described by the equationswhere a, b, d > 0.∂u∂t∂v∂t7= ∂2 u∂x 2 + u − uv + au2 ,= d ∂2 v∂x 2 + u2 − buv ,Find and sketch the range of a, b for which the spatially homogeneous system hasa stable stationary solution with u > 0 and v > 0.Considering spatial perturbations of the form cos(kx) about the solution foundabove, find conditions for the system to be unstable. Sketch this region in the (d, b) planefor fixed a ∈ (0, 1).Find k c , the critical wavenumber at the onset of the instability, in terms of a and b.14C Dynamical SystemsLet f : I → I be a continuous map of an interval I ⊂ R. Explain what is meantby the statements (a) f has a horseshoe and (b) f is chaotic according to Glendinning’sdefinition of chaos.Assume that f has a 3-cycle {x 0 , x 1 , x 2 } with x 1 = f(x 0 ), x 2 = f(x 1 ), x 0 = f(x 2 ),x 0 < x 1 < x 2 . Prove that f 2 has a horseshoe. [You may assume the Intermediate ValueTheorem.]Represent the effect of f on the intervals I a = [x 0 , x 1 ] and I b = [x 1 , x 2 ] by means ofa directed graph. Explain how the existence of the 3-cycle corresponds to this graph.The map g : I → I has a 4-cycle {x 0 , x 1 , x 2 , x 3 } with x 1 = g(x 0 ), x 2 = g(x 1 ),x 3 = g(x 2 ) and x 0 = g(x 3 ). If x 0 < x 3 < x 2 < x 1 is g necessarily chaotic? [You may usea suitable directed graph as part of your argument.]How does your answer change if x 0 < x 2 < x 1 < x 3 ?Part II, Paper 3[TURN OVER
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