 By C.A. Reiter, W.R. Jones

This booklet might be of curiosity to arithmetic scientists operating within the components of linear algebra, summary algebra, quantity concept, numerical research, operations learn and mathematical modelling

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Example text

KCOt(Kt) < 33 Continuity Put d0 = min \h(s) - h(t)\ > 0. )(g;s) i < e. do2 R3 (0 < 9 < ir) be a curve segment defined by ( - 1 to E^(hg) Z J_^ r* i ( i — dsdt 2 J^yihe^-he^W (t-s) of the subarc rn = hg((- — ,—^—\ U [—^,— )) (n = 0,1,2, •••).

1 , . 14) -^—)dsdt. 15) Since the integrand is non-negative and not identically zero E(a\h) for any knot h when 1 < a < 3. >0 (2) When 3 < a < 4 F («> m _ ff ( l WW2 y . , . i (3) When 4 < a < 5 F(«)(hs = 1J ff ( i L _ "l ft// ( sa)l22 yy^xsAl^)-^)! 01 d(S)*)« 24d(s7t) ~ a(h"{s),h<-3\s)) 24d(s,i)°- 3 dsdt. E^(h) for a > 5 is given similarly. We call a the index of E^a>. The integrand of E^(h) is divided into the principal term \h(s) — h(t)\~a and the counter term to cancel the blow up of the integral.

This implies t h a t if a < 2 then E^ takes a finite value even for a "singular knot" with a self-intersection, and hence E^ is not self-repulsive. 1)). Let h b e a knot with \h"\ = 1 a n d b = E^a\h) (b > 0). Let s , i b e points in S 1 = [ 0 , 1 ] / - with 0 —5. The self-repulsiveness of E^a> 27 h(s + u)