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Extra resources for A New Equation of State for Fluids IV. An Equation Expressing the volume as an Explicit Function of the Pressure and Temperature

Example text

Thus, if QR meets the curve again at, say T, then the line S T is the tangent at S; see Fig. 3. This remains true even if the point T happens to be the point at infinity. In this case, the tangent at S is "parallel" to S for the denizens of the affine plane but we are still using only a ruler construction because in the projective plane the point at infinity is just like any other point in the plane; see Fig. 4. T h e o r e m GEO-1. Tangent construction. A2 = Q B1 9B2 = R {x . (y . x) = y} =:(gL)=~ A1.

Most of our experimentation involved fiddling with various combinations of just a few parameters and strategies until a proof was found--and, if the proof was very long, until a shorter proof was found. In most cases, we adjusted the parameter maxweight, the limit on the length of retained equations, and in some cases we assigned higher priority to clauses containing Skolem constants from the denial; for the (gL) problems, we usually had to adjust the demodulation strategy. The easy theorems fell in one or two searches, and the difficult ones required ten, twenty, or more attempts, usually of one to five minutes; if a well-behaved search was achieved, we would let Otter run for a day or more.

07 seconds). 2 re(x, e) = 3 4 5 6 re(e, x) = x x . (y. x m ( A , B ) r e. ( A . B ) 11 ,~(e, x)- (y. ~) = y [3 -~ 4] 17 28,27 x . m(e, y . x) = y m(e, x . y) = y . x [3 --~ 4] [3 -+ 5] 29 ~. (x. y) = y 35 re(x, y ) . ( z . y) = re(x, u ) . ( z . u) [17:28] 53 106 x. (y- c) = m(~, z). (y. z) ,~(x, y)- (re(x, z ) . y) = z [2 -~ 35] [35 ~ 29] 180,179 194 225 231 233 re(x, y ) . ( z . y) = x . (e. z) x . (e. re(x, y)) = y e. m ( x , y) = x . y re(x, y) = e . (x . y) [] [5 ~ 53, flip] [106:180] [194 -~ 29, flip] [225 -+ 29, flip] [231,6] [11 -+ 11 :(gL)] In Thm.