Search

7 Dec 2022: Use (ii) to show that there exists b K such that det(σiσjg1(b)) 6=0. ... Deduce that if y = g1(b) then {σ(y) | σ G} is a K-basis for L.Such a basis {σ(y)} is said to be a normal basis for

19 Nov 2022: 13. Let S be a collection of subsets of N such that for every A,B S we have A B or B A.Can S be uncountable? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

11 Oct 2022: The least such n iscalled the order of g.). (b) Show that there exists a positive integer n such that gn = e for all g G. ... 10. Let H and G be groups and let X G such that 〈X〉 = G.

28 Oct 2022: 2. Given (fi)iI an arbitrary collection of continuous functions [0, 1] R such thatsupiI |fi(x)| < at each x [0, 1], show that there is an interval [a,b] [0, ... 1] witha < b such that supx[a,b] supiI |fi(x)| <. 3. Let X be a closed subspace of 1. Assume

11 Oct 2022: Find a, b, c Q such thaty = a bx cx2. ... 15. Let K and L be subfields of a field M such that M/K is finite.

13 Oct 2022: 1. Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

22 Nov 2022: Let a, b K such thatthe polynomials f = Xm a, g = Xm b are irreducible. ... Use a suitable Lagrange resolvent to finda Q(ζ3) such that K = Q(ζ3, 3.

28 Oct 2022: 1. Let T B(H) with |||T ||| < 1. Show that there is R B(H) such that R2 = 1 T. ... 11. Let T B(B) compact and λ 6 (σp(T) {0}). Show that T λ is bounded below.

24 Oct 2022: Ln := {w ; there are x, y W such that |y| = n 1 and w = x1y)},. ... Show that:. (a) There is a nondeterministic automaton N with n 1 states such that L(N) = Ln and(b) if D is a deterministic automaton with fewer than 2n states,

24 Oct 2022: 5. (a) Suppose that f Matn,n(F) is such that f(AB) = f(BA) for all A,B Matn,n(F) and f(I) = n.Show that f is the ... 0. Is detA > 0? 14. Let A,B in Mn(R) such that X KerA{0} with BX ImA.

1 Feb 2022: b) We say that a measurable f : Rn! C is in weak-Lp(Rn), written f 2 Lp,w(Rn) ifthere exists a constant C such that. ... Deduce. N[. i=1. Bi. 6 3n. kX. j=1. Bij. 3. b) () Suppose {Bj : j 2 J} is an arbitrary collection of balls in Rn such that each

3 Feb 2022: 1. Let A,B F be two events such that P(B) = 0 or 1. ... Ie, at least one day d such that at least k people were born on day d.).

6 Feb 2022: Regarding T as a map from C into C by identifying R2with C in the usual way, show that there exist unique complex numbers A, B such that for everyz C, ... that there need notexist holomorphic f : U C such that u = Re (f) on all of U.

1 Feb 2022: there exists 2 C such that:. u[] =. Z. R(x)dx, for all 2 D(R). () ... Rm(x, y)dy. b) Show that there exists a sequence of smooth functions fn 2 C1c (Rn) such that Tfn!

14 Nov 2022: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

24 Oct 2022: Suppose thatα: V W is a linear map such that α(Y ) Z. ... Let f be an endomorphism of E. Show theexistence and uniqueness of (F,G) vector subspaces of E such that:(a) F G = E;(b) F,G are stable by

17 Feb 2022: 6) Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements aand b in R is an element d in R such that d divides both ... a and b, and if c divides both aand b then c divides d.

13 Jan 2022: Prove directly from the definition thatAX has an m-variate normal distribution AX Nm(Aµ,AΣAT ). 2. Let X and Y be N(0, 1) random variables such that ... a random set C(Y ) such that Pβ,σ2 (b C(Y )) = 1 α for all β,σ2.

25 Feb 2022: Can you justify that f is unique? [Note: such a function f is called the (discrete) harmonic extension of (av). ]. ... Explain why E[|T |] = ,and find constants C,α such that P(|T | = m) Cmα.

20 Jan 2022: a) Construct a continuous surjection p : T2 T2 from the torus to itself such that for every x T2, p1(x)consists of exactly two points, and such that p is ... Such a map is called acovering map.]. (b) Show that T2 also admits a covering map p̂ : T2 K to

26 Apr 2022: Show that ϕ is convex. 5. Suppose f : X R is twice-differentiable and such that 2f(x) is positive definite for allx X. ... a) Show that the unique minimizer is x = 0. (b) Find constants 0 < m < M and L > 0 such that m f ′′(x) M and |f ′′′(x)|

4 Oct 2022: b) Now let X and Y be jointly Gaussian random vectors such that Cov(X) is positivedefinite. ... Let Qbe the equivalent probability measure such that dQ. dP eX.

4 Oct 2022: n such that Ax = 0.• There exists a y Rm with (AT y)i 0 for all i = 1,. , ... Make this assumption for the rest of the problem.(b) Why must there exist an equivalent measure Q such that 1.

1 Feb 2022: b) Find a condition on s such that x 2 Hs(Rn). ... 2u u = f. b) Show further that there exists C > 0 such that kukHs4.

21 Nov 2022: Show that the following sets are computablyenumerable:. (a) {w ; there are three distinct words v such that f(w, v)};(b) {w ; there is a word v of even length such ... g(u, v) :=. {ffu,1(u)(v) if u K and. otherwiseis computable. (b) By the s-m-n theorem,

24 Oct 2022: similar to any of them? If so, which? Find a basis such that it is in Jordan Normal Form. ... Withoutappealing to Jordan Normal Form, show that there is a vector y such that (y,α(y),α2(y),. ,

19 Dec 2022: Of course, Corollary 2 is only useful if we can find such an algebra homomorphism. ... Since our proof does not rely on the details of the construction of ks, it can readily accommodatemodifications such as bulk deformations and B-fields, which we

12 Oct 2022: Mich. 2022 ANALYSIS AND TOPOLOGY—EXAMPLES 1 PAR. 1. Let (zn) be a sequence in R2 such that zn a and zn b. ... On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a,b] such that fn(xn) 6

6 Jul 2022: vd of V such that v1,. ,vr is a basis of W and the. ... vd of V such that v1,. ,vr is a basis of W the matrices.

29 Nov 2022: vd of V such that v1,. ,vr is a basis of W and the. ... vd of V such that v1,. ,vr is a basis of W the matrices.

12 Dec 2022: Then there is a point y / E such. that B(y,r) E 6= whenever r > 0. ... Since Uj1 isopen we can find a δj1 > 0 with δj1 δj/4 such that B(xj1,δj1) Uj1.

12 Dec 2022: Theorem 4.2. If f : [a,b] [a,b] is a continuous function, then we canfind a w [a,b] such that f(w) = w. ... Supposethat we can find a a0 < a1 < < an b such that, writing σ = ‖fP‖we have either.

5 Oct 2022: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε.

30 Mar 2022: In 7 we prove that such orbits admit integral representatives away from smallprimes. ... We have a Gm-action on Vgiven by λ v = λv and there is a unique Gm-action on B such that π is Gm-equivariant.

22 Feb 2022: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

7 Oct 2022: Choose then a and b such that. 1. a. def=. 1. ... 29. Proof of Lemma 3.1.1. x K , rx > 0 such that B(x,rx).