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3 Dec 2014: If f : B′ B, let E′ = {(v,b) E B′ |π(v) = f(b)}, and let p : E′ B be the obvious projection. ... His known as the Hopf invariant. (b) Show that H is surjective for n = 2, 4.

10 Oct 2014: Show that f : Q Q withf ( a. b) = b.

22 Sep 2014: Secondly, let hAut(θ) denote the grouplike topological monoid of self-homotopy. equivalences of B over BO(2n), i.e. ... homotopy equivalences f : B B such that. θ f = θ.

27 Oct 2014: Concretely we have. Objects Over b B: x E(b′) with u : b′ b.Maps From y with v : c′ c to x with u : b′ b :. c′v. - c. y ... Bf. - A. '. a A, b B.IdA(f (b), a). -. B. wwwwwwwwwwwwwwwwwwwwwwwwf. -. -. A. wwwwwwwwwwwwwwwwwwwwwwww. The Sceptics

11 Feb 2014: Now let γ : [a,b] B(p) be a smooth path. If Lg(γ) and L(γ) are the lengthsof γ with respect to g and the Euclidean metric, then. ... theorem implies theremust be some point t [a,b] with d(γ(t)) = /2.

22 Nov 2014: Show that the linear map A : R2 R2peserves the equivalence relation (a,b) (a′,b′) (a a′,b b′) Z2, and soinduces a continuous map fA from the torus T to

24 Jan 2014: likelihood estimate of θ is (2a b)/(b 2c). 4. (a) Let X1,. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).

24 Nov 2014: Show that the linear map A : R2 R2peserves the equivalence relation (a,b) (a′,b′) (a a′,b b′) Z2, and soinduces a continuous map fA from the torus T to

24 Jan 2014: 5. Let f : B B be a continuous function from the open disc B = {(x,y) R2 :x2 y2 < 1} to itself. ... 11. Suppose f : R R has the intermediate value property that if a,b R andf(a) < c < f(b), then f(y) = c for some y between a and b.

4 Feb 2014: possible area is an arc of a circle passing through A and B. ... φ(s,v) = α(s) r(n(s) cos v b(s) sin v)where n = n(s) and b = b(s) are the normal and binormal vectors of α.