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20 Jan 2003: Homotopy 19Definition of homotopy 19Definition of simply-connected 19Definition of star domains 19Proposition 4.5 Winding number and homotopy 20. ... This clearly establishes the result. 20. 5 CAUCHY’S THEOREM. Let T be a closed triangle that lies

19 Feb 2003: 20. and Φc(x,x) = 0 for all x,y,z X. Hence. Φc(p1,pn1) Φc(p1,q1) Φc(q1,p2) Φc(pn,qn) Φc(qn,pn1).

4 Mar 2003: r 3δ2t2, r 3ωδ2t2, r 3ω2δ2t21These calculations were performed using Version 2.0.20 (beta). ... 3t2 = ((2 ω)/2)u2 (6 4ω)uv (7 3ω)v2+ (14 6ω)uw (52 72ω)vw (20 52ω)w2.

11 Mar 2003: 20 G. CONTRERAS, J.-M. GAMBAUDO, R. ITURRIAGA, AND G. P. PATERNAIN.

19 May 2003: The reader is warned that Lemma 1.20 does not generalise in the obviousway to direct sums of more than two subspaces. ... α : V V satisfies its own characteristic equation. 20. We bring together the results of the corollary as a theorem.

26 Jun 2003: Paterson’s recent paper[20] gives axioms for an ArrowLoop operation which are thesame as our definition of a premonoidal trace; our resultsthus prove an equivalence between ArrowLoop and a ... 20] R. Paterson. A new notation for arrows. In Proceed-ings

17 Jul 2003: 19). Up to this action, the irrational cusps (0 : a1 : a2 :. ) satisfy. rank(ai j )n1i, j=0 2. (20). Indeed (20) may be viewed both as the condition for (0 : ... Essentially bysolving a second order recurrence relation, we find that the solutions to (20)

25 Jul 2003: Introductory Course: Fourier Analysis and its many uses. Solutions - Exercises 16.1-16.5, 16.9, 16.12, 16.13, 16.15, 16.16, 16.17, 16.20,16.21, ... Exercise 16.20. i) This is the Cauchy Schwarz inequality for a = (aN , aN1,.

2 Aug 2003: 20. (iii) K̃n(t) 0 for all t.(iv) K̃n is a (multidimensional) trigonometric polynomial.(v) If f : Tm C is continuous then.

21 Aug 2003: The axioms of a premonoidal trace are shown in Figure 3, where we followJeffrey [20] in using a heavy line to indicate the sequencing of effects in K (and thatline runs ... This may also help establish a connection with the partial trace operationused by