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30 Jan 2005: i. ) and (y. i. ) with all sums of the form a. ... x. r. m. (r. 1. <r. 2. <…<r. m. ) and all sums of the form b.

21 May 2005: for all x, otherwise d(x,y) = 3n if xy = 3na/b where a,b are prime. ... iii) Let φ be a continuous real-valued function on R[a,b] which satisfies the Lipschitz condition|φ(x,t) φ(y,t)| M |xy| , for all x,y R

21 May 2005: 0 for all x, otherwise d(x, y) = 3n if x y = 3na/b where a, b are prime. ... φ(x, t) φ(y, t)| M |x y| , for all x, y R and t [a, b],.

26 Oct 2005: 4. Is it true that for all positive integers a, b, c, d we have (a, b)(c, d) = (ac, bd)? ... 6. The Fibonacci numbers F1, F2, F3,. are defined by: F1 = F2 = 1, and Fn = Fn1 Fn2 for all n > 2 (so eg.

21 May 2005: Show, directly from the definition of holomorphic maps, that if f is holomorphic onR {p} then f is in fact holomorphic on all of R.(ii) Suppose that each of A = { ... alge-braically related’: there is a polynomial Q in two variables, so that Q(f(z),g(z)

21 May 2005: extends f (i.e.g(t) = f(t) for all t (1, 1)), but we cannot take D to be the disc B(0, 1).(ii) Give an example of an ... then the maximum modulus principle to show that|f(z)| |z| for all z B(0, 1).

21 May 2005: f(z)| =. Show that it cannot have poles at all integer points z = n on the real axis. ... of. 5. Prove that all entire (i.e. analytic on the whole of C) functions that are also injective.

21 May 2005: BA. B. B. B. A A D. A. DCC. AA A A. ... A) H(K′, A′) induced by f are isomorphisms (for all ), then sois the third.

21 May 2005: Note that the sum of all columns of A has all entries equal.]. ... Deduce that there is abasis for V relative to which α is represented by a matrix A with all of its diagonal entries equal to 0.(ii) Show that there are

21 May 2005: 3. Show that the function ψ(A,B) = tr(ABt) is a symmetric positive definite bilinear form on the spaceMn(R) of all n n real matrices. ... Show that there is an invertible matrix P such that P 1JP =. (. 0 InIn 0. ). Find the dimension of the space of all

21 May 2005: You should attempt these questions, but may not want to write out all parts in much detail.](a) The set C of continuous functions.(b) The set {f C : |f(t)| ... ne1}. 5. Let P denote the space of all polynomial functions R R.

21 May 2005: Suppose that for. x [a,b], |N. 0fn(x)| K, where K is constant, for all N; and suppose that gn(x) is monotonic. ... Suppose that. 0fn(x) is. uniformly convergent on [a,b], that each gn(x) is bounded on [a,b], and that gn(x) gn1(x) 0for all x [a,b].

10 Nov 2005: αβ. γ. α. β γ. Exercises: you needn’t do all the basic examples before attempting these. ... Can the 4-cube?[ The n-cube’s vertices are all 2n subsets of [n]; join a to b if |a4b| = 1. ].

21 May 2005: 10. (i) Show that the set of all subsets of a given set S is a ring with respect to the operations of symmetricdifference and intersection. ... Describe the principal idealsin this ring.(ii) Let R be any ring satisfying a2 = a for all elements a in R.

21 May 2005: xk} V (G) then there is acycle in G of length at least k 1 that contains all xi, 1 i k. ... F5) Show that if G is a regular (all degrees equal) bipartite graph then κ(G) 6= 1.

22 Nov 2005: 0 for all x, otherwise d(x,y) = 3n if x y = 3na/b where a,b are prime. ... φ(x,t) φ(y,t)| M |x y| , for all x,y R and t [a,b],.

21 May 2005: 0 for all x, otherwise d(x,y) = 3n if x y = 3na/b where a,b are prime. ... φ(x,t) φ(y,t)| M |x y| , for all x,y R and t [a,b],.

21 May 2005: Then by theanalysis above, in dx the coefficients of the “inner edges” (the 1-simplices notmaking up the edges A, B, C, D) are all zero. ... c24,25 are all zero. Supposez has coefficients a9,25,. , a24,25 for these, and let.

21 May 2005: Show(a) f′(x) exists for all real x;(b) f′(x) is NOT a continuous function of x. ... Can you find a‘simpler’ combination of derivatives that annhilates all Möbius maps?

8 Dec 2005: there exists a colour c1 such that c(F {a1}) = c1for all F B(r1)1. ... say with active coordinate set I, such. that for all a [m]n and all b, b′ L{L+}, we have c(a, b) = c(a, b′) = c′(a),say.