Search

10 Jan 2017: b) Find all 1-dimensional representations of G.(c) Let ψ : Fp C be a non-trivial 1-dimensional representation of the cyclic group. ... d) Prove that the collection of representations constructed in (b) and (c) gives a com-plete list of all irreducible

27 Nov 2023: In all the questions that follow, X is an n by p design matrix with full column rank and P is theorthogonal projection onto the column space of X. ... Show that for all v Rn, v2 ΠW v2 ΠV v2.

10 Nov 2016: 3. Show that the function f : Rn R given by f(v) = ‖v‖2 is differentiable at all nonzerov Rn. ... Show that f is continuousat (0, 0) and that it has directional derivatives in all directions there.

25 Oct 2007: b) Let R([0, 1]. )denote the vector space of all integrable functions on [0, 1]. ... 10. (a) Show that the space is complete. Show also that c0 = {x : xn 0}, the vector subspace of consisting of all sequences converging to 0, is complete.

1 Nov 2021: Show that M M is trivial.(b) Show that for all n we have TSn R = Rn1 over Sn. ... and Q7. compute HndR(RPn) for all n. 9. Let Σ be a compact 2-manifold-with-boundary, let F : Σ R2 be a smooth map, and let α bethe

19 Jan 2021: tβn. are joined by an edge if, after renumbering:. (a) αi is isotopic to βi for all i > 1;. ... b) if Sα2,.,αn is a one-holed torus then i(α1,β1) = 1;.

8 Nov 2012: Show that all functions f from Rwith metric d′ to R with metric d are continuous. ... denote the vector space of all (Riemann) integrable functions on [0, 1].

27 Nov 2023: In all the questions that follow, X is an n by p design matrix with full column rank and Pis the orthogonal projection onto the column space of X. ... 2. An n-vector is called constant, if all its entries are the same.

13 Oct 2021: foreach γ C there exist elements α(γ) A and β(γ) B, such that for all γ we have:. ... multiplication. An R-linear derivation OX,p R is an R-linearmap d : OX,p R such that d(fg) = d(f)g(p) f(p)d(g) for all f

10 Oct 2016: 0, 1] |. 10 f(x) dx = 0}? 5. Let 0 be the set of real sequences (xn) such that all but finitely many xn are 0. ... a) Show that |ϕ(s) ϕ(t)| |s t| for all s,t R.(b) Define f(x) =. n=0. (34. )nφ(4nx). Prove that f is well-defined and continuous.

21 Nov 2007: 5. Let Mn denote the space of nn real matrices with the operator norm ‖‖. Show that ‖AB‖ ‖A‖‖B‖for all A, B Mn. 6. Define f : Mn Mn by f ... Deduce the result that if f : U Rm isdifferentiable on U with Df|x = 0 for all x U then f is

30 Nov 2015: for some K and all t [a,b], x,y BR(x0).We showed in lecture that for each t0 [a,b], there is a unique f C([a,b]; ... tt0F(s,f(s)) ds, t [a,b]. Assuming that F. extends to all of [a,b]Rn as a continuous function, show that this f is in fact the

13 Nov 2018: Supposethat there is a constant K such that ‖F(t,x) F(t,y)‖ K‖xy‖ for all t [0, 1] and all x,y B. ... assumethat there exist constants K and α (0, 1) such that ‖F(t,x)F(t,y)‖ K‖xy‖α for all t [0, 1] and allx,y B.

21 Nov 2013: such that f(g(x)) = g(f(x)) for all x X. Show that g has a fixedpoint. ... x, y) for all x, y X with x 6= y, but such that f has no fixed point.

10 Jan 2020: tβn. are joined by an edge if, after renumbering:. 2. (a) αi is isotopic to βi for all i > 1;. ... b) if Sα2,.,αn is a one-holed torus then i(α1,β1) = 1;.

7 Mar 2015: i.e. T(xy) = 0 for all y K implies x = 0).(b) Show that the sequence xn = T(α. ... n) is the output from a LFSR of length d. (c) The period of (xn)n>0 is the least integer r > 1 such that xnr = xn for all sufficiently largen.

27 Oct 2006: 0|f|. is a normed space. b) Is this norm equivalent to the uniform norm ‖ ‖? c) Is the space of all (Riemann) integrable functions with norm ‖f‖ = 1. ... iii) {x : there exists N such that xn = 0 for all n > N} in l,.

10 Oct 2018: a (b c) = (a b) (a c). for all a, b, c L). ... ts=1 nsas where all the as are in Ai and the ns are integers.

21 Nov 2016: 10. Let f : Rn Rn be a C1 map. Suppose that there is some constant µ < 1 such that‖Df|x I‖op µ for all x Rn. ... Showthat ‖x y‖ (1 µ)1‖f(x) f(y)‖ for all x,y Rn. Deduce that f is injective andthat f(Rn) is a closed subset of Rn.

29 Oct 2009: Is it Lipschitz equivalent. to the uniform norm? (b) Let R[0, 1] denote the vector space of all integrable functions on [0, 1]. ... Usingthe fact that all norms on a finite-dimensional space are Lipschitz equivalent, deduce that α iscontinuous.