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This is the Past Exam Paper of Mathematical Tripos which includes Computational Neuroscience, Rate Coding, Temporal Coding, Action Potential, Poisson Process, Cumulative Distribution Function, Technique for Computing, Model Neuron, Definition of Noise Entropy etc. Key important points are: Analysis of Operators, Complex Inner Product, Exterior Multiplication Operator, Strong Operator Continuous Homomorphism, Positive Energy, Banach Algebra, Fourier Coefficients, Differentiable Map, Hilbert Space
Typology: Exams
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Thursday 31 May 2007 1.30 to 4.
Attempt THREE questions.
There are EIGHT questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 (a) Let V be a finite–dimensional complex inner product space and let W = Λ(V ). For v ∈ V define the exterior multiplication operator e(v) on W. Prove that e(a)e(b)∗^ + e(b)∗e(a) = (a, b)1 for a, b ∈ V. Show that the operators e(a), e(b)∗^ (a, b ∈ V ) act irreducibly on W.
(b) Let V be a finite–dimensional complex inner product space and A ⊆ End(V ) a unital *–algebra. Define the commutant A′^ of A and prove that A′′^ = A.
(c) Explain what it means for a strong operator continuous homomorphism z 7 → Uz , T → U (H) to have positive energy. If in addition S ⊂ B(H) satisfies S∗^ = S, S = Uz SU (^) z∗ for all z ∈ T, and H is irreducible for S ∪ {Uz }, prove that any operator that commutes with S necessarily commutes with any operator Uz.
2 (a) Let A be the Banach algebra Hs(S^1 ) for s > 1 /2 and let A+ and A− be the closed subalgebras given by the vanishing of negative and positive Fourier coefficients. Prove that if X ∈ GLn(A) and ‖X − I‖ < 1, then there are unique X± ∈ GLn(A±) such that X = X−X+ and X+(0) = I.
(b) Let G(H) be the group of invertible operators on the Hilbert space H of the form I + T with T trace–class. Given a differentiable map F : (a, b) → G(H), prove that f (t) = det F (t) is differentiable with
f −^1 f˙ = Tr(F −^1 F˙ ).
(c) If A, B ∈ B(H) with [A, B] trace–class, prove that eAeB^ e−Ae−B^ lies in G(H) with det(eAeB^ e−Ae−B^ ) = exp Tr (AB − BA).
(d) Prove that if f ∈ C∞(S^1 ) with f (z) =
anzn, then
det T (ef^ )T (e−f^ ) = exp
n> 0
nana−n,
where T (g) is the Toeplitz operator with symbol g.
Paper 10
