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7 Feb 2012: Show that f (x) = 0 for allx. 8. Let f : [a, b] R be bounded. ... 13. Let f : R R be a function which has the intermediate value property: If f (a) < c < f (b),then f (x) = c for some x between a and b.

26 Nov 2012: To complete the investigation, some biological specimens are also examined with the phase plate in position, and a comparison is. ... a). (b). 1 1. FIGURE 9. Densitometer traces of some features shown in figures 8(a) and (b) illustrating improvements in

13 Sep 2012: Thenlog f = h̄y for some y. 3. Rdf(x)dx = 1. ... with fnd f for some density f. Then:. (a) f is log-concave.

29 Oct 2012: a sequence of points in [a, b] with xm x then fn(xn) f (x). ... f. Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f is bounded.

27 Jan 2012: Xn be independent Poisson random variables, with Xi having meaniθ, for some θ > 0. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).

15 Aug 2012: It may be possible to derive some, at least, of. Random-cluster representation of the Blume–Capel model 3. ... Graham and G. R. Grimmett. as required. We make some observations based on Theorem 3.7 and the method ofproof.

6 Sep 2012: We end this section with such an example Consider the network illustrated in Figure where TCP connections send packets from node a tonode d with some connections routed via node b ... links Next suppose a short highcapacity link at which no congestion

23 Feb 2012: domain. Could there be some other Euclidean function φ making Z[3] into a Euclidean. ... Does it follows that the ring R is either a field or a polynomial ringF [X] for some field F?

8 Feb 2012: 3. An element r of a ring R is nilpotent if rn = 0 for some n. ... 6. Let A and B be elements of the power-set ring P(S), for some set S.

29 Feb 2012: α for some α.(iv) Any limit ordinal can be written in the form α. ... Given a subset b Pα withmore than one element, consider the least β belonging to some but not all members of b.].

30 Oct 2012: α for some α.(iv) Any limit ordinal can be written in the form α. ... Given a subset b Pα withmore than one element, consider the least β belonging to some but not all members of b.].

14 Mar 2012: t1. B(0,r). 1. (2πs)d/2dy ds c1Ψd(t), (10). for some positive constant c1. ... Lemma 4.1. Let (xi)ik be i.i.d. uniformly in the ball B(0,R) for some R > 0 and let (ξi(s))ik beindependent standard Brownian motions.

1 Nov 2012: 1 B. The shoulder associated with some of the (1,0) peaks is due to contamination from the meridian. ... may exert constraints on changes, possibly leading to some distortion o f the structure.

10 Oct 2012: Show that L = K(α) = {a bα |a, b K} for some α F with α2 K. ... ii) Show that if L/K is a finite extension with Q K, for which there exist only finitelymany intermediate subfields K K′ L, then L = K(α) for some α

14 May 2012: 2.4. Microlocalisation. We recall some basic results in the theory of algebraicmicrolocalisation. ... the form Ân,K for some almost commutative R-algebra A, provided that |q1| < 1.

15 Aug 2012: For subsets A and B of the vertexset of G, we write A B if there exists an open path in G joining somevertex in A to some vertex in B. ... some Akoccurs, so that (a, i) (b, j) a.s. by the remarks above.

22 Oct 2012: Show that they have the samesplitting field if and only if b = cnar for some c K and r N with gcd(r, n) = 1. ... for some divisor d > 1 of n.

3 Dec 2012: first part. 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... for some λ,µ,ν R (which should turn out in this example to be integers).

9 Jan 2012: Therefore the only othercase to consider is that x and y lie, respectively, above and below two differentactive blocks, say B(u,F (u)) and B(v,F (v)), for some ... b) Let d D, and suppose that with positive probability there exists a quasi-isometry from

19 Jan 2012: considered. Some astrophysical discs are non-self-gravitating, meaning that the internal. contribution can be neglected relative to that of external masses, most importantly the. ... mass does not vanish in general. However, some new topological