Computer-Aided Analysis of Active Circuits, Vol. 67 by Ioinovici A.

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Uravneniyam, VGU, Voronezh, 197~ (in Russian). 9. , A theorem on periodic transformation of homology spheres, Ann. , 56 (1952), 68-83. 10. , To the theory of periodic mappings of spheres. The 7th summer mathematical school, IM AN USSR, Kiev, 1970 (in Russian). ll. 3 (1955), 401-40#. (in Russian). 12. L. Homotopic topology. Izd-vo MGU, Moscow, 1971 (in Russian). 13. , Calculation of the degree of equivariant mapping by spectral sequences method. 10 (1973), VGU, Voronezh, 1-12. (in Russian). lA. A .

Let f : X l ~ X 2 be an equivariant mapping of the Zk-COhOmological spheres of the dimensions m and n respectively. Let the actions T l, T2 of the group Zk be semi-free on X1, X 2. The dimensions of the sets F1, F 2 of~ the fixed points are the same and equal to r and the degree deg(flF l) @ 0. Then it is necessary that m~ n . Then we have the following diagram: m-n+2 Hr(FI)__~ where the homomorphisms ~m-r' I Hn( ) Hr(F2 ) ~n-r define the equivariance indi- ces I(Jl), I(J2) of the inclusions Jl: Fl~--*-Xl' J2 : F2----~-X2 respectively.

We shall orient ~ R e as a boundary of the domain ~ + . e. [~= O, I~I = 1 moves along 54 ~ R e in the direction of a positive orientation, then the multiplier ~ rotates counter-clockwise. Calculate the degree of the mapping ~ : Re RP in the neighbourhood of the i~finite point = c ~ o ~ R P 1. From the as~ptotics ~k ~ exp (i ~akT) at P - ~ P k it follows that the degree of the mapping ~: P R e - - * RP1 in this point P is equal to the sign of the number ak, since with the increase of ~ the multiplier ~ k ( ~ ) will rotate co~ter-clockwise at ak ~ 0 and clocMvise - at ak ~ O.

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