By Hopf H.

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**Extra info for A New Proof of the Lefschetz Formula on Invariant Points**

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5) 1,2 i=1 Λ1,3 = cos θ1 , where ri is the hemisphere radius, θ1 and φ1 are the polar and azimuthal angles respectively of the first reflection point R 1 . The subscript i of ri denotes “inner radius” and it is not a summation index. 5) are combined as 3 r0,i + ξ1 k −1 1 k1,i − ri Λ1,i eˆi = 0. 6) i=1 Because the basis vectors eˆi are independent of each other, the above relations are only satisfied when each coefficients 45 A. Reflection Points on the Surface of a Resonator of eˆi vanish independently, r0,i + ξ1 k −1 1 k1,i − ri Λ1,i = 0, i = 1, 2, 3.

2, the temperature of the two hemispheres would rise indefinitely over time. This does not happen with ordinary conductors. This suggests that Boyer’s conducting material, of which his sphere is made, is completely hypothetical. Precisely because of this material assumption, Boyer’s Casimir force is repulsive. For the moment, let us relax the stringent Boyer’s material property for the hemispheres to that of ordinary conductors. For the hemispheres made of ordinary conducting materials, there would result a series of reflections in one hemisphere cavity due to those radiations entering the cavity from nearby hemisphere.

5 are assumed to be simple cubical. Normally, the dimension of conductors considered in Casimir force experiment is in the ranges of microns. When this is compared with the size of the laboratory boundaries such as the walls, the walls of the laboratory can be treated as a set of infinite parallel plates and the vacuum-fields inside the the laboratory can be treated as simple plane waves with impunity. The presence of laboratory boundaries induce reflection of energy flow similar to that between the two parallel plate arrangement.