By Kupiainen M., Sjogreen B.

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**Additional info for A Cartesian Embedded Boundary Method for the Compressible Navier-Stokes Equations**

**Sample text**

When this expression is summed from r = 0 t o r = n — 1 the first term gives lOn and each of the two sums involving co is zero, since 1 + co + co2 + • • • + (on~l = 0. This proves the required result. If iv = s(cos (j) + j sin (j)} then wm = sm(cos mcj) + j sin mcj)} for all integers m by de Moivre's theorem. The complex number wm has the n nth roots One of these is sm/n , and the notation wm'n is used to mean this nth root of wm. This definition ensures that de Moivre's theorem is also true for rational powers, since ?

The part between the outer bull and outer double circles is divided into twenty equal 'sectors', numbered as shown below, and the board is hung with the 20 sector vertically above the centre so that the initial line bisects the 6 sector. A dart scores 50 in the inner bull and 25 in the outer bull, where 6 < r < 16. A dart in a sector scores the sector number, except that within the doubles ring (162 < r < 170) or trebles ring (99 < r < 107) it scores double or treble the sector number respectively.

Iii) Hence find the cartesian co-ordinates of B, C, D. ] 4 In this question r is in millimetres and 9 is in degrees. The scoring region of a dartboard is marked by six concentric circles, called inner bull, outer bull, inner treble, outer treble, inner double, outer double, with radii 6, 16, 99, 107, 162, 170 mm respectively (to the nearest mm, ignoring the thickness of the dividing wire). The part between the outer bull and outer double circles is divided into twenty equal 'sectors', numbered as shown below, and the board is hung with the 20 sector vertically above the centre so that the initial line bisects the 6 sector.

### A Cartesian Embedded Boundary Method for the Compressible Navier-Stokes Equations by Kupiainen M., Sjogreen B.

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