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30 Aug 2019: f(r, s) elements such thatA B X 6= for all A A and B B. ... ii) Let A and B be non-empty cross-Sperner families of subsets of [n], meaning that Aand B are such that if A A and B B then neither A

30 Aug 2019: 5. Let L1/Q,L2/Q be quadratic field extensions such that L1 = Q(a),L2 = Q(. ... Let a,b K such that f(X) = Xn a and g(X) = Xn bare irreducible.

30 Aug 2019: C B of positive measure, there is n > 1 such that µ(T nA B) 6= 0and µ(T nA C) 6= 0. ... Let (X, B, µ, T) be a measure preserving system. We write P for the collection ofsets A B such that hµ(T, ξA) = 0, where ξA = {A, XA}.

19 Jun 2019: Show that there is an increasing function f : S R with f (0) = 0 such that the pointsf (X)(X Π) form a Poisson process of unit rate on R, and ... Explain carefully what is meant by a Poisson process of such lines L.

19 Jun 2019: b) Let σ : R R>0 and b : R R be locally Lipschitz and such that there existsa positive constant K so that. ... b) Show that if M Mc,loc is such that M (0) = 0 and [M ]t is deterministic,strictly increasing and such that [M ] = 0, then M is a Gaussian

30 Aug 2019: Which antichains in P([n])have size exactly. ( nn/2. )? (i) Let A P([n]) be a family of sets such that, for each A A, there do not existsets B, ... C A with B, C 6= A such that A B and A C.

19 Jun 2019: Show that there exists a uniqueconformal isomorphism gK : H H such that gK (z) z 0 as z. ... Denote by Φ the unique conformal isomorphismU H such that Φ(z)z 0 as z.

30 Aug 2019: b) Let N be a model. Prove that for every A N there is a model M such that. ... You may assume standard properties of algebraicclosure.]. (b) Show that for every N U such that |N| > |L| ω the following are equivalent:.

30 Aug 2019: the set of all sextuples such that x1 x2 x3 = x4 x5 x6. ... Ex,a,bf(x)f(x a)f(x b)f(x a b)ω2ab| > c. Prove that there exists r ZN such that.

19 Jun 2019: b) Show that there is at most one nondecreasing adapted process A (up to indistinguishab-ility) such that M2 A is a continuous local martingale. ... b) Use (a) to show that for any a < b such that 0 < a < x < b,.

19 Jun 2019: Suppose further that the Xi are symmetric (in that Xi and Xi have the samedistribution), and that there exists c R such that P(|Xi| 6 c) = 1 for all i. ... c) Let (Xn : n > 0) be a martingale such that E(X2n) 6 M < for all n.

17 Jun 2019: Show that there exist x and y such that. f(λa (1 λ)b. )= ... Show thatf is bounded on [a, b], and that there exist c, d [a, b] such that for all x [a, b],f (c) f (x) f (d).

19 Jun 2019: b) State the strong Markov property of such a process (Xn)n>0. ... Suppose there exists a bijection T : S S such that, for allx, y S,.

30 Aug 2019: 2. 1. In this question, IC stands for the axiom “there is an inaccessible cardinal”, i.e., aregular cardinal κ such that for all λ < κ, we have |(λ)| < κ. ... b) there is a p G such that p σ τ.

30 Aug 2019: 1([a, b]), and suppose that |f| > ǫ > 0 on C[a,b].Show that there is some T (depending only on a, b, and ǫ) such that if x ... H. ′′. k,r and k 1-handles H′′. k1,1,. , H′′. k1,s such thatA′′1 intersects B. ′′.

30 Aug 2019: If thereis an open ball B such that. supB. u = sup. ... 2 for all x and ξ Rn and some constant λ (0, ). (a) Prove that for any c R, for any x0 B and R > 0 such that B2R(x0) B

19 Jun 2019: Prove that f satisfies the assumptions of Kakutani’s fixed-pointtheorem. (b) Prove that there exists a p P such that z(p) (RL+) 6= , whereRL+ = {v RL | v 6 0 ... that for every k > 0 and every ε > 0, there exists an N > 0 such that.

17 Jun 2019: By consideringpairs such as [M1, M2], or otherwise, show that for a, b R3,. ... matrices A, B such that C = [A, B].

19 Jun 2019: In 3-SAT we are given such a formula and asked to say whether thereexists an assignment of the variables (to ‘true’ or ‘false’) such that the formula is true.Show ... Let. =1. 2(n 1)[(n 1)U ](n1) , P = {x Rn : Ax > b e}.

30 Aug 2019: b) Let X be a continuous local martingale such that Xt X almost surely. ... d) Let M be a continuous local martingale such that. 〈M〉t = t.