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22 Nov 2017: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 13. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

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.

11 Sep 2017: b) If ι′ : Y X is another such inclusion and ιOY = ι′OY ′ as coadmissibleÙDX-modules, then Y = Y ′. 1.3. Structure of this paper. As in our previous paper [2] ... Lemma. Suppose that f1,. ,fr B are such that L fi A for each 1 6 i 6 rand there

15 Mar 2017: Let A be an nn matrix over F. The following are equivalent(a) there is a matrix B such that BA = In;(b) there is a matrix C such that AC = ... Thus (a) holds if and only if α is anisomorphism. Similarly (b) holds if and only if there exists γ such that

25 Oct 2017: 1We can actually exhibit such a pair, and using only elementary methods, at the cost of alittle bit more work. ... We say such a formula is maximal.). That feature invitesthe simplification that we showed there.

13 Jun 2017: It turns out that we can send each element b to the set ofmaximal filters F such that b F. ... A Logic is a theory closed under substitution. A Logic that contains (to takea pertinant example) A (B A) must contain all substitution instances ofit, such as:

13 Sep 2017: X! 1. T X. , for X 2 C,. such that the following conditions hold:. • ... f,g. to be the unique 2-cell. such that. g. f. g f//.

11 Oct 2017: To divide a number a by a number b means to find, if possible, anumber x such that bx = a. ... If such a number exists it is denoted[sic] by a/b. H.