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24 Nov 2003: 17. Let S be a nested collection of subsets of N - that is, for every A,B S we have eitherA B or B A. ... 3. 2 100. 3. 2 is irrational. 19. Construct a function f : R R that takes every value on every interval - in otherwords, such that for every a < b

29 Sep 2003: a) f, D1f, D2f are continuous in R2;(b) D12f and D21f exist at every point in R2, and are continuous except at (0, 0);(c) D12f(0, 0) = 1 and ... 8. Let E R2 be an open set that contains the rectangle [a,b] [c,d].

7 Oct 2003: 4 Wellfounded Trees. Definition 8. We say that a lccc C has W-types if for every arrow f : B Ain C there is a diagram Pf (Wf ) Wf which is ... Theorem 11. If C has W-types then every polynomial endofunctor on C has apointwise free monad.

9 Feb 2003: Show that every eigenvalue ofA. must be either 1 or"1.c. Provide an example of an invertible 2 2 matrix A such that every entry ofA is a real. ... number and every eigenvalue ofA 2 is negative.3. Consider the following quadratic form forx x1, x2, x3 I:.

20 Jan 2003: Suppose that there exists a point X such. that every ball B(,r) contains infinitely many terms of the sequence. ... Every subsequence also converges to , so no subsequence can possibly converge in X.

26 Mar 2003: Let A be a set that is NOT. Then we can find a subset B of {1,2,.,N} with |B|> cN and a function g:B-->C such that ... k,g(k))> t for every k in B.

18 Sep 2003: its pose. B. Initialization Problem. The difficulty of nontrivial initialization, which exists in every algorithm for general motion. ... views) avoids the problem of local minima which exists in every general motion estimation using.

5 Nov 2003: A relation R on a set A is reflexive if xRx for every x in A. ... element of $A$ is an element of $B$ and every element of $B$ is an element of $A$.

19 Feb 2003: Mather’s Lipschitz. 3A curve γ : [a,b] X is called absolutely continuous if for every > 0 there exists δ > 0so that for each finite collection of pairwise disjoint open intervals ... 1. u+ is Lipschitz, and (x,du+(x)) UΣ for almost every x X;.

19 May 2003: Thedepth setting was 4cm, giving a B-scan resolution of 0.1mm/pixel. Every effort was made toensure that the sweeps were correctly registered at the time of acquisition: the arm ... The table indicates thatwe can expect to arrive at the same alignment,