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23 Nov 2020: 3. The symmetric difference of sets A and B is A B = (A B) (B A). ... 8. Define a relation R on N by setting aRb if a divides b or b divides a.

5 Jun 2020: 2 f-Regularity. Following [7], we denote for each r 0 and B B,. ... x X and every B B satisfying ψ(B) > 0, there is a constantcB < such that,.

9 Nov 2020: lemma. Since the slope filtration descend to G u(a,b), for a a′ b′ b we have a subquotient over. ... First form the pullbackof. 0 AA B B′ C C 0by the diagonal morphism : C C C′ to get obtain an extension.

4 Mar 2020: µ(Xn,αn)w. nµ{0,1}N. Proof. Fix r0 > 0 and (B,b,β) X{0,1}r0. •,6r0. ... Define. pn = PxXn ((Xn,x) 'r (B,b)) and p = P(Y,y)µ ((Y,y) 'r (B,b)).

10 Oct 2020: 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 ... 1.8. Let B be a Borel subset of the interval [0, 1].

5 Jun 2020: ixs xs. Here B B denotes the algebra generated by B and B and xs xs. ... suces to check that. limN. ess sup supA BN. PA j B PA.

23 Nov 2020: b) Use the Mayer–Vietoris sequence to compute the homology of Σg#Sn, andhence deduce that it is homeomorphic to Sn2g. ... Show that the linear map A : R2 R2 preserves the equivalence relation (a,b) (a′,b′) (aa′,b b′) Z2,and so induces a

11 Nov 2020: We now let µ X(T) and let b G(L) such that [b] B(G,µ). ... Thenumber N (µ,b) depends on b only via its σ-conjugacy class [b] B(G).

5 Jun 2020: d sup. jP B A P BP AjP BP A. A B B Bd. ... d supjP B A P BP Aj A B B Bd.

6 May 2020: d. (p, Im c|[b′,b]. ), there exists p [x,y]. such thatd(p, Im c|[a,a′]. )= ... ii) For all (a,b) , (a′,b′) R, we have |dA (a,a′) dB (b,b′)| ε.

14 May 2020: For b R let. ϕ(b) = inf{f(x) : g(x) b, x X}. ... Assuming that ϕ(b) > for all b R, show that ϕ is convex.

4 Feb 2020: a) If f is increasing, then f ′(x) 0 for every x (a,b).(b) If f ′(x) 0 for every x (a,b), then f is increasing.(c) If f ... 12. Suppose f : [a,b] R is differentiable, and that f ′(a) < k < f ′(b).

22 Oct 2020: and for any [b] B(G,{µ}),there exists a straight representative in the admissible set of µ (see [16]). ... Fornonbasic [b] B(G,{µ}), there is another obstruction, coming from the non-uniqueness of thestraight representatives of [b].

6 Jun 2020: ω A,x X,B B(X) limnPθn(ω)(x,B) = Pθ?(x,B). (iii) the state space X is Polish. ... PθV λV b1C λ (0, 1),b < ,. and level sets of V are 1-small.(ii) (strenghtened D.A.).

8 Jun 2020: SR = {(a,c) AC, b B, (a,b) R and (b,c) S}. ... A′)r = {b B, a A′, (a,b) R} ,(B′) = {a A, b B′, (a,b) R}.

23 Jan 2020: 5. Let f : B B be a continuous function from the open disc B = {(x,y) R2 :x2 y2 < 1} to itself. ... 11. Suppose f : R R has the intermediate value property that if a,b R andf(a) < c < f(b), then f(y) = c for some y between a and b.

5 Jun 2020: Theorem 2.2:Let Φ = {Φi : i 0} be anarbitraryMarkov chain with values onX, let B B be fixedmeasurable set, and letF1,F2,. ... n 1)H(γ‖γ1 P), (32). where γ is a bivariate probability measure on(X X,B B), whose marginalsγ1 andγ2 satisfy,.

22 Oct 2020: BIs> B. B B. i0i1 >. where B B is the product of two copies of B, the diagonal, (i0, i1) a fibrationand s is a weak equivalence. ... B̃ <s′. B. B B. (i0i1)i >. which tuns B̃ into a path object for B.

9 Nov 2020: closure of A in L. Show that σ(B) = B for all σ G and that A = BG. ... Q of B which contract to P is finite. [In topological language this says the induced map.

5 Jun 2020: φ(k) = sup{|P(B|A) P(B)| : B σ(XU ), A σ(XV ), P(A) > 0|U| , |V | < , d(U,V ) k}. ... IEEE Trans. Inform. Theory, 39(3):877–886, 1993. [25] T. Wiegand and B.