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17 Jun 2019: 1 (mod p), where p is prime. (b) Suppose that p is an odd prime. ... b) The order of a (mod p) is the least positive integer d such that ad 1 (mod p).Suppose now that ax 1 (mod p); what can you say about

17 Jun 2019: or otherwise, show that there is a subinterval [a′, b′] [a, b] such that. ... integrable. Let f be a function on [a, b] such that |f| is Riemann integrable.

30 Aug 2019: and Mµ. (i) Prove that if there exists (an extension) Φ HomFSn(Mλ, Mµ) such that Φ|Sλ = φ. ... Let n and k be integers such that 0 6 k 6 n/2.

30 Aug 2019: c) Let u W1,p(Rn). Defining u as in part b), show that u is classically differentiableat almost every point x Rn, and at each such point the classical derivative ... You may assume that there exists K > 0 such that for any u H1(U) the inequality:.

30 Aug 2019: You may assume Landau’slemma concerning Dirichlet series with non-negative coefficients. (b) Deduce that, for fixed a and q such that (a,q) = 1,. ... e) Suppose that there exists some absolute ǫ > 0 such that the bound.

19 Jun 2019: The Lebesgue measure on ([0, 1) , B([0, 1))) is denoted by λ. ... b) Show that there exists a constant C > 0 such that E [σa] = Ca2 for every a > 0.

17 Jun 2019: instructed to do so by the Invigilator. 2. SECTION I. 1E Numbers and Sets(a) Find all integers x and y such that. ... defined by. m n a, b N such that m divides na and n divides mb.

30 Aug 2019: d) A spacetime admits a Killing vector field V a and contains a Maxwell field F such thatLV F = 0. ... e) Explain why such solutions can be interpreted as polarized gravitational waves.[Hint: linearize.].

30 Aug 2019: One of the estimates that can be used to obtain this resultis the following: Given initial data such that. ... and locally bounded functionf : R>0 R>0 depending only on D and E such that the following bound is obeyedas long as the solution exists:.

21 Aug 2019: The die is thrown n times. By using Chebyshev’s inequality, find an n such that. ... where a and b are constants. Determine the value of α such that the change of variable u = xα transforms ()into a differential equation with constant coefficients.

17 Jun 2019: 2 6 20 1 10 6 4. . . Find a matrix P such that B = P 1AP is diagonal. ... or otherwise, show that there is a subinterval [α, β] [a, b] such that.

17 Jun 2019: Show that there exists a non-standard pair of dice A and B such thatwhen thrown. ... Paper 2 [TURN OVER. 6. 12F Probability. Let A1, A2,. , Ar be events such that Ai Aj = for i 6= j.

17 Jun 2019: n, and there exists l [0, ] such that |an1/an| l as n , thenthe radius of convergence is 1/l [0, ]. (i) Use the ratio test with an = n2. We have ... IVT: If f : [a, b] R is continuous and c R is such that f(a) 6 c 6f(b) or f(b) 6 c 6 f(a) then there

17 Jun 2019: Compute all possible limiting values of y as t. Find a non-zero value of y(0) such that y(t) = y(0) for all t. ... Find theprobability density function of X. 4F ProbabilityLet A, B be events in the sample space such that 0 < P(A) < 1 and 0 < P(B) < 1.

30 Aug 2019: Let R be a real 22 matrix. with positive determinant such that |Rx| = |x| for all x R2. ... f : [a, b] R attains a global maximum; that is, there is a real number c [a, b] such thatf(c) > f(x) for all x [a, b].

17 Jun 2019: iii) Let y = Φ(x). Find the components Aij of the matrix A such that yi = Aijxj.When Φ is invertible, find the components of the matrix B such that xi = ... b) A scalar λ and a non-zero vector v such that.

17 Jun 2019: Justify your answer. (b) Find an example of such a G and T with H 6= G. ... Explain why we can find an M such that. |f (j)(x)| 6 M.

30 Aug 2019: w 0. Prove that. there is a sequence (xn) in 3BX and a subsequence (zkn) of (zn) such that xnw 0 and. ... Let K = σ(T) be the spectrumof T. Prove that there is a unique unital -homomorphism f 7 f(T): C(K) B(H)such that u(T) = T , where

19 Jun 2019: a) A survival dataset is such that no individuals are censored in the interval a < t < b. ... b) Define and interpret the Schoenfeld residual. Show that their sum over the observedevent times is zero.

30 Aug 2019: for some constant K <. Show that there exists a bounded measurable function g on[0, 1] such that. ... 2nα supk{0,1,.,2n1}. |ξ(k1)2n ξk2n|. (b) Hence show that there is a continuous random process (Xt)t[0,1] such thatXt = ξt almost surely, for all t