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12 Oct 2017: x E such that x Am for all large enough m N, and lim sup Anis the subset of those x E such that x belongs to Am for infinitely many ... 1. 2. 1.7. A subset E R is called Jordan measurable if for every > 0 there are two finite unions ofintervals A =. n1

13 Mar 2017: 2. for a > 0and b R, prove that such inequality cannot hold when p (2, ]. Exercise 4. Consider f L1(R) that is differentiable almost everywhere with f′ L1(R), does ... and a limit function f L2(U) such that ‖fϕ(n) f‖L2() 0 as n.

24 Apr 2017: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y.

5 Dec 2017: 2) Convert the following CFG to CNF:. S aSbb | T, T bTaa | S |. (3) Give a CFG for each of the following CFL’s, and then transform each such CFG intoCNF:. ... b) Show that the language L := {anbncn | n 1} is not a CFL.

13 Nov 2017: b) By replacing A in the construction of B with a suitably-chosen set, construct a setC N such that neither C nor NC are r.e. ... Use this to show there exists some m N such that Wm has exactly m elements.

8 Nov 2017: tiplicativity of the norm, we can find ξ Q( 3a) such that NQ( 3a)/Q(ξ) = b. ... Algorithm 3.6. (Legendre-type algorithm for solving cubic norm equations). Input: A pair of positive integers (a,b) such that b is a norm for Q( 3a)/Q.

17 Apr 2017: If A, B, U are subspaces of a vector space V such that U = (A U) (B U)then (A U) (B U) = (AB) U. ... 19. (2.4.11) The main inductive step is the following: let n = a b, and V = WnV be anobject of Crk such that gr.

27 Nov 2017: Show also that AI is minimal in the sense that if B is any set such that foreach i I there is an injection τi : Ai B and the τi commute ... A is sandwiched by B if there are elementary extensions A′ of A and B′ ofB such that B A′ B′.

26 Jul 2017: We choose φ1 such that. φ1(e1) = (0, 0), φ1(eσi ) = (1 σ. ... y4 such that. 4i=1. xibij(y1,. ,y4) V for all j = 1,. ,

7 Mar 2017: Lemma 2.1. There exist ξ,η,a1,a2,a3,a4,a6 Z[aij] such that(i) We have c4 = b. ... Lemma 3.1. There exist ξ,η,a1,a2,a3,a4,a6 Z[sijk] such that(i) We have c4 = b.