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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.

13 Aug 2009: Let's have some examples. Let Σ be the alphabet {a, b}. ... a. n. b. n. : n } is the set of all strings that consist of a's followed by the same number of b's.

7 Dec 2009: Show that f is continuous at (a,b). 6. Let Mn = Mn(R) be the space of n n real matrices (it can be identified with Rn2). ... 14. Let U R2 be an open set that contains a rectangle [a,b][c,d].

14 Oct 2009: Let (fn) be a sequence of continuous real-valued functions on a closed, bounded interval [a, b],and suppose that fn converges pointwise to a continuous function f. ... 12. Let. n=1 an be an absolutely convergent series of real numbers.

24 Nov 2009: 3. Let G be a graph and v G. Must κ(G v) κ(G)? ... 9. Let G be a k-connected graph (k 2), and let x1, x2,. ,

12 Feb 2009: 9) Give a proof of the Weierstrass approximation theorem by completing the following argument.Let 0 < a < b < 1, and f : [a, b] R be the continuous function we wish to ... 12) Let be a bounded, open subset of Rn and g : (a, b) R be a function such that.

14 Oct 2009: harmonic oscillator). Angular momentum: raising and lowering operators. L̂2|a, b〉 = a|a, b〉, L̂z |a, b〉 = b|a, b〉. ... Let us then define operators L̂ = L̂x iL̂y. Since [L̂2, L̂i ] = 0, L̂2(L̂|a, b〉) = L̂L̂2|a, b〉 = a(L̂|a, b〉),i

13 Nov 2009: Give an example to show that this is not so for 4 4 complex matrices.(b) Let A be a 55 complex matrix with A4 = A2 6= A. ... Let A, B V. Prove that A B if and only if Ao Bo.Show that A = V if and only if Ao = {0}.

19 Oct 2009: 2) Let P(w0,w1,. ,ws; z) be a polynomial in the s1 complex variables w0,w1,. ... 12) Let D C be an open disc and u a harmonic function on D.

9 Mar 2009: 3) Let a, b be positive integers. Let. x =1. a 1b+ 1. ... 3) For each n = 1, 2,. , let fn : [0, 1] R be a bounded function.

26 Feb 2009: n}, let the non-negative numbers A(n)k and the“nodes” x. (n)k [a, b] be given such that for each polynomial P , the error. ... for each continuous function f on [a, b]. (2) Let Tj be the jth Chebychev polynomial.

8 Dec 2009: Show that f is continuous at (a,b). 6. Let Mn = Mn(R) be the space of n n real matrices (it can be identified with Rn2). ... 14. Let U R2 be an open set that contains a rectangle [a,b][c,d].

21 Jan 2009: Show thatf (x) = 0 for all x. 8. Let f : [a, b] R be bounded. ... 10. Let f : [a, b] R be continuous on[a, b] and differentiable on(a, b).

14 Oct 2009: Let (fn) be a sequence of continuous real-valued functions on a closed, bounded interval [a, b],and suppose that fn converges pointwise to a continuous function f. ... 12. Let. n=1 an be an absolutely convergent series of real numbers.

4 Aug 2009: Let f(a) = cand f(b) = d. Then the map f : [a,b] [c,d] is bijective and f1 is continuouson [c,d]. ... Let f(a) = c and f(b) = d. Then the mapf : [a,b] [c,d] is bijective and f1 is differentiable on [c,d] with.

23 Mar 2009: As R is a PID we have (a, b) = d for some d R. ... Proof. Let R be a PID and suppose we have an ascending chain.

9 Jan 2009: E B) = (B)E (E)B E(B) B(E). Position vector: let r or ri denote the position vector, and r = |r|. ... φ) dl = φ(b) φ(a). 1. Curl in curvilinear coordinates (ξ1, ξ2, ξ3):.

13 Aug 2009: Theorem 12.5. Let B be a set of first category in T. ... Let. Kn = En B(0, 2n1) = {e x : ‖x‖ 2n1}.

9 Apr 2009: 34. 4.2 Faraday’s law of induction. Let C be either(a) a fixed closed geometrical curve, or(b) a physical, possibly moving circuit. ... Let S be a surface bounded by C = S. Faraday’s experimental law, valid in both the contexts(a) and (b), with an

26 Oct 2009: 1. Let (x(m)) and (y(m)) be sequences in Rn converging to x and y respectively. ... Is it Lipschitz equivalent. to the uniform norm? (b) Let R[0, 1] denote the vector space of all integrable functions on [0, 1].

18 Feb 2009: per unit length. Let the unit normal n to S point from the negative () to the positive () sideof S. ... Let C be the circumference ofthe lower plate, S2 the lower plate itself, and S1 a surface bounded by C but lying entirely outsidethe region between

28 Jan 2009: 2. THE SPHERE. 2.1 The geometry of the sphere. Let S = S2 = {x R3 : ||x|| = 1} be the unit sphere. ... Let be a spherical triangle with angles α,β,γ and sidelengths a,b,c.

18 Oct 2009: 6. (i) Let L/K be a finite extension whose degree is prime. ... 14. Let L/K be an extension, and x, y L transcendental over K.

16 Jan 2009: Consider a point charge q at rest at O, and let V be the sphere of radius r centredat O. ... n E|. =1. ǫ0σ (23). nE|. = 0. (24). b). Let S carry current density s per unit length (charge crossing unit length in S in unittime).

16 Mar 2009: 9. Let σ = (σ1,σ2) : (a,b) {(x,y) R2 : y > 0} be a unit speed curve in the upper half-plane thatdoes not intersect itself and maps the open interval ... 11. Let R be a surface in R3 that is closed and bounded.

29 Jan 2009: For simplicity, let C define aplane loop of area S = Sn of unit normal n. ... Let B be the magnetic fields in V, and let current s per unit length flow inS.

15 Oct 2009: 11. For any S > R 1 > R let b(r) be a smooth function with b(r) = 0 if r R or r S 1, and b(r) = 1 ifS > r ... 0. [δe2δr|b′|v2 e2δr|b′′|v2 re2δr|bfv|. ]dr. For any > 0 let R be such that suprR |V (r)| < , and let δ2 < (M 2)(1 ), so that |M δ2.

12 Aug 2009: Let. H(F) = det. Fxx Fxy Fxz. Fyx Fyy Fyz. Fzx Fzy Fzz. ... The invariants c4 and c6 may be defined as follows. Let A and B be the ma-.

4 Mar 2009: 1. Let X Nd(µ,Σ), and let A be an arbitrary p d matrix. ... b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a real-valued sufficient statistic T, and.

26 Apr 2009: identically distributedwith mean rLet nrt be the number of calls in progress at time t on route r let r rr. ... a Dominant coalitions. Let N be the set of players and for subsets S N dene the value of thecoalition S to be.

20 Apr 2009: Example 2. Let W and B be independent Wiener processes, c a positive constant, a a real constant. ... Proposition 4. Let ξ be a scalar Lévy process with triple (a,σ2,µ).