Search

16 Aug 2011: Compressions and Probably Intersecting Families. Paul A. Russell. A family of sets is said to be intersecting if AB f or all A, B.

3 Feb 2011: Suppose that f ((x y)/2) (f (x) f (y))/2 for all x, y [a, b].Prove that f is continuous on (a, b). ... Which of (1)–(4) must betrue? (1) If f is increasing then f ′(x) 0 for all x (a, b).(2) If f ′(x) 0 for all x (a, b) then

15 Feb 2011: Suppose that f ((x y)/2) (f (x) f (y))/2 for all x, y [a, b].Prove that f is continuous on (a, b). ... Which of (1)–(4) must betrue? (1) If f is increasing then f ′(x) 0 for all x (a, b).(2) If f ′(x) 0 for all x (a, b) then

13 Oct 2011: Which of the following sets of functions form a vector subspace of RR?(a) The set C of continuous functions.(b) The set {f C : |f (t)| 1 for all t ... i6=j Ui = {0}. (iii) The Bi are pairwise disjoint and their union B is a basis for U =. i Ui. Give an

11 Oct 2011: 6. Show that R(s, t) 6(st2s1. )for all s, t > 2. ... Suppose that (i) |A| = 3 for each A A; and (ii) |A B| = 1for all distinct A, B A.

3 Feb 2011: 4. Find all analytic functions of the form f (z) = f (x iy) = u(x) iv(y). ... Show that there is an analyticfunction H : C C such that h(z) = exp H(z) for all z.

22 Jun 2011: The first four are maybe tricky, so makesure that you do all the integrals. ... Show that all roots lie inside the circle|z| = 3/2. (b) How many zeros does z4 12z 1 have in the annulus 2 < |z| < 3?

31 Oct 2011: 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.

5 Nov 2011: c) Set pn = (2In)1p̃n. Check that pn is a polynomial of degree 2n, and that supx[a,b] |pn(x) f (x)| < 2 for all sufficiently large n. ... supx[a,b]. |xn p(x)| supx[a,b]. |xn q(x)|. for all polynomials q of degree n 1.

13 Nov 2011: Show that f is continuous at(0, 0) and that it has directional derivatives in all directions there. ... some r > 0 and a continuous functiong : B(I; r) Mn such that g(X)2 = X for all X B(I; r).Is it possible to define a square-root