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21 Jan 2009: B[p] := {bp : b B}. in B provided that B is reduced.Let t be a positive integer. ... r. for all v V.Let B = Sym(V k) and let P be a graded prime ideal of B.

13 Aug 2009: Let us try some examples. Let L be the set of strings over the alphabet {a, b} which have the same number of as as bs. ... Let L be the set of strings over the alphabet {a, b} which have an even number of as and an even number of bs.

21 Apr 2009: b) Let π be the true probability that a one-litre water sample fromthis type of site contains pathogenic material. ... Show thatv is also an eigenvector of A1. (b) Let.  =. [2 11 2. ]. i. Find the eigenvalues of  and derive an eigenvector

24 Jun 2009: Let A K be a finite setof sources, and let B B, G F. ... Let B B, G F, let A Kbe a finite set of sources, and write ψA = ψA(B,G) for the colouringgiven in the last section.

28 Jan 2009: to. 4. Let A, B be two distinct points in the Euclidean plane. ... Let be a spherical triangle with sides of length a,b,c and angles α,β,γ.

18 Nov 2009: 4) And here a really hard one: Two logicians meet, let us call them A and B. ... B: I do not know the numbers. A: I do not know the numbers.

30 Jan 2009: to. 4. Let A, B be two distinct points in the Euclidean plane. ... Let be a spherical triangle with sides of length a,b,c and angles α,β,γ.

23 Oct 2009: What if we replace ‘injective’ by ‘surjective’ passim? 5. Let A = {1, 2, 3} and B = {1, 2, 3, 4, 5}? ... Let R be the relation on Z‘ aRb if b = a 3 ’. How many equivalence relations on Z contain R? 8. Let A be a set of n positive integers.

13 Oct 2009: a) How do we know that R(G) exists? (b) Let Ik be a set of k independent edges (so |Ik| = 2k). ... b) Let A be a collection of subsets of [s 1].

2 Dec 2009: 3) Let a, b be positive integers. Let. x =1. a 1b+ 1. ... b) Let f : [0, 1] R be a bounded Riemann integrable function.

8 Jul 2009: Denote the first digit of Wm by a {0, 1}, and let b = 1 a be the complementarydigit. ... Proof of Theorem 1(b). Let. δp,q = vpq up,q, p,q 0.

11 Nov 2009: Show that P(X b) = P(X) for every b Fp K.Now suppose that P does not have a root in K, and let L/K be a splitting field ... 5. (i) Let f K(X). Show that K(X) = K(f) if and only if f = (aX b)/(cX d)for some a, b, c, d K with ad bc

15 Oct 2009: Let x [a, b]. Applying Theorem 1 ′to u′k and f on [x0, x] (or on [x, x0] if x < x0) we obtain. ... F (x) F (x0) = x. x0. f (x)dx. Differentiation in x gives F ′(x) = f (x) for x [a, b] as f is continuous.(2) Let ε > 0.

2 Oct 2009: 2 )dx 0. 3.8 Let u and v be differentiable functions on [a, b] with continuous derivatives u′. ... b. a. u′(x)v(x)dx. 3.10 Let (E, E) and (G, G) be measurable spaces and let f : E G be a measurablefunction.

13 Aug 2009: Let L be the language of palindromes over {a, b}. It isn't regular, but there is no obvious bomb. ... n. b. n. : n }. 2.1.1.2 The language {ww : w Σ} is not regular.

13 Aug 2009: Let w. 1. and w. 2. be two words having no characters in common. ... c. Question 2. Now let L. 1. and L. 2. be regular languages.

2 Oct 2009: cn, } are. independent. 1.8 Let B be a Borel subset of the interval [0, 1]. ... 1.9 Let (E, E, µ) be a measure space. Call a subset N E null if N B for someB E with µ(B) = 0.

30 Oct 2009: 5. Let A and B be n n matrices over a field F. ... 14. Let A be an n m matrix. Prove that if B is an m n matrix then.

4 Feb 2009: Let ξ1, ξ2 , and writeAk = η(ξk ). Let B = A1 A2 = {b1, b2,. , ... Let a (respectively, c) be the number of edgesof the form 〈x , z〉 (respectively, 〈y, z〉) with z A, let b (respectively, d) be thenumber of edges of the form 〈x ,

16 Oct 2009: 2.4. Theorem. Let Q Jr1s (n 1) be open, dense and invariant. ... of the form6 Z =. [b ca b. ]. Let us consider the map given by.