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By Wilhelm Blaschke, Gerrit Bol

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Extra resources for Geometrie der Gewebe;: Topologische Fragen der Differentialgeometrie, (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Band XLIX)

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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. 7) The three terms Λ1,i=1 , Λ1,i=2 and Λ1,i=3 satisfy an identity 3 Λ21,i = 1. 7), Λ21,i is computed for each i : −2 Λ21,i = [ri ] r0,i 2 + ξ12 k −2 1 k1,i 2 + 2r0,i ξ1 k −1 1 k1,i , i = 1, 2, 3. 8) and after rearrangement, one obtains 3 ξ12 k −2 1 k1,i 2 + 2ξ1 k 3 −1 3 r0,i k1,i + 1 i=1 i=1 r0,i 2 2 − [ri ] = 0.

Reflection Dynamics ✄✂✄✂✄✂ ☞☛☞☛☞☛ Lx,1 ☎✆✝ ☎✆✝ ☎✆✝ ☎✝ ✄✂✄✂ ✄✂ Rrp,1 ✁ ✁ ✁ Lx,2 ☞☛☞☛☞☛ 1 R dpl Right Plate Left Plate Left Driven Plate ✁ ✁ ✁ 2 R rp,2 3 Rlp,3 Rlp,2 ^y ^x Right Driven Plate ✟✞✟✞✟✞ ✟✞✟✞✟✞ ✆✠✡ ✠✆✡ ✠✆✡ ✠✡ R dpr ^Z axis is out of the page! : A one dimensional driven parallel plates configuration. 16) of Appendix D1. The force shown in the above expression vanishes for the one dimensional case. This is an expected result. 4) in the Appendix D1. The summation there obviously runs only once to arrive at the expression, ∂Hns /∂ki = ns + 12 c.

For that system   R˙ rp,cm,α (t0 ) = R˙ lp,3 (t0 ) + R˙ rp,2 (t0 ) , 1 pvirtual−photon = Hns , (t0 ) ,  R˙ lp,cm,α (t0 ) = R˙ rp,1 (t0 ) + R˙ lp,2 (t0 ) . c For simplicity, assuming that the impact is always only in the normal direction, R˙ rp,cm,α (t0 ) = 2 Hns ,3 (t0 ) − Hns ,2 (t0 ) , mrp c R˙ lp,cm,α (t0 ) = 2 Hns ,1 (t0 ) − Hns ,2 (t0 ) , mlp c where the differences under the magnitude symbol imply field energies from different regions counteract the other. The details of this section can be found in Appendix D2.

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