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6 Nov 2023: aa. Explicitly write down aregister machine M0 (giving all program lines in detail) that performs this operation. ... ii) The machines M and M′ are function equivalent if for all k, they define the same partial functionson Wk, i.e., if fM,k = fM′,k.

23 Oct 2017: 11. Show that A = {(xn) 1 | |xn| 1/n2 for all n} is a sequentially compact subset of(1,‖‖1), but that B = {(xn) 1 | |xn| 1/n for all n} ... Hint: show that Φ(I) contains all points of the form (a/2n,b/2n), wherea,b Z, 0 a,b 2n.).

21 Feb 2008: Deduce that traces of all words in A and B andtheir inverses (i.e. ... of all elements of the group generated by A and B) aredetermined by the three numbers {tr(A), tr(B), tr(AB)}.

10 Nov 2006: iii) X = Q, d(x,x) = 0 for all x, otherwise d(x,y) = 3n if x y = 3na/b where a,b are prime to 3(and n may be ... 0). Show that all functions f : (R,d′) (R,d) are continuous. What are thecontinuous functions from (R,d) (R,d′).

28 Nov 2006: a) has all directional derivatives at zero but is not continuous at zero. ... b) has all directional derivatives at zero, is continuous everywhere, but is not differentiable at.

23 Nov 2018: K and all t [a,b], x,y BR(x0). We showed in lecture that for each t0 [a,b], there isa unique f C([a,b]; BR(x0)) solving the ... continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I).

20 Feb 2009: αN has area (N 2)π. αj. Showthat, for every N > 3 and every α with 0 < α < (1 2N )π, there is a regular N -gon with all anglesequal to ... 15. The quaternions Q consist of all 2 2 complex matrices.

10 Jan 2017: 7 Let ρ : G GL(V ) be a representation of G of dimension d.(a) Compute the dimension of SnV and ΛnV for all n.(b) Let g G and let ... constituent of IndGKχ, then all |G : K| conjugates of χ are constituents of ResGKφ.

29 Oct 2018: Deduce. that ‖ 10f(x) dx‖2. 10‖f(x)‖2 dx. (b) Find all continuous f : [0, 1] Rn satisfying ‖ 10f(x) dx‖ =. 10‖f(x)‖dx regardless of the norm ‖ ‖. ... denote the vector space of all bounded Riemann integrable functions on [0, 1].

26 Nov 2012: continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I). ... b) Suppose instead we assume only that for all x, y X at least one of the three.

14 Nov 2008: Show that all functions f from Rwith metric d′ to R with metric d are continuous. ... Show that f has a fixed point. (b) Suppose instead we assume only that for all x, y X we have d(f (x), f (y)) Kd(x, y), ord(f

8 Nov 2007: Show that all functions f from Rwith metric d′ to R with metric d are continuous. ... Show that f has a fixed point. (b) Suppose instead we assume only that for all x, y X we have d(f (x), f (y)) kd(x, y), ord(f

29 Nov 2013: continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I). ... Show that f has a fixed point.(b) Suppose instead we assume only that for all x, y X at least one of the threedistances d(f(x),f(y)),

4 Feb 2010: 5. Find all of the Möbius transformations that commute with Mk for a fixed k. ... Are these all inversions? 9. How many square roots of a Möbius transformation are there?

14 Oct 2021: Y := {v V : (v,u) = 0 for all u Y}.(a) Show that dim Y dim Y = dim V. ... y,λη) for all (x,ξ) M and λ [0,).

8 Nov 2006: 5. Find all of the Möbius transformations that commute with Mk for a fixed k. ... Are these all inversions? 9. How many square roots of a Möbius transformation are there?

28 Jan 2009: The spherical circlewith centre C and radius r is the set of all points on the sphere at distance r from C. ... Show that two triangles that have all three angles the same are congruent.

23 Jan 2015: Show that for all v Rn, ‖v‖2 ‖ΠWv‖2 ‖ΠV v‖2. (b) Consider the linear model (1) but where only the first p0 components of β are non-zero. ... Now prove that if A1,. ,Am and B1,. ,Bm are all independent real-valuedrandom variables and A1 B1,.

17 Jul 2023: All the other edges of the network are of the form (ai, bj) for some i, j = 1,. , ... A is connected to all ai and B is connected to all ci with capacity 1 edges.The capacities of all other edges is infinite.

3 Nov 2014: denote the vector space of all (Riemann) integrable functions on [0, 1]. ... sequences (xk) such that xk = 0 for all but a finite number of k.).