By Michio Kaku
A shiny portrait of the idea of hyperspace through a professor of theoretical physics on the urban college of recent York discusses the superstring thought and the concept that of a jigsaw-puzzle universe. Reprint. journey. NYT.
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Additional resources for Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimens ion
Example text
124) Here V(z) and R(z) are slowly varying but otherwise arbitrary functions of the variable z. By specifying these functions we find various configurations of the surfaces, ~ (r, z) = const. 124) that in this approximation p = p(z). , we assume that among the streamlines ~ (r, z) = const there is a straight line r = R. The correspond- MOROZOV AND SOWV'EV \ "" " o' \ '-... ........... -/ / / / / I r=R Fig. 11 ing pattern of streamlines is shown in Fig. 11, where the velocity V(z) is plotted along the z axis.
32 MOROZOV AND SOWV'EV Accordingly, in this case (r == const) we find a property of an ordinary gasdynamic nozzle: The sound velocity and the flow velocity are equal at the critical cross section. 58) that the signal velocity cannot be crossed at the minimum of f. For high-velocity flows we are primarily interested in the case c). » c~. 63a), we find f = const;p V l-pjp 0 • (2. 70) These equations show that if the velocity increases monotonically in the channel, then p and H decrease monotonically.
78) 4np 0 Thus, with p ,... __ 4;-~p ~ cfo_ (. _ll__ \ v- 1 y-1 Po ) (2. DV'EV we have Hmax ~ Ho l/ Pmax. 80) An exact expression for the maximum magnetic field is given in [22]. § 4. , ~ = ~ (r, ez), p = p (r, ez), where e is a small parameter. Neglecting terms in Eqs. _ 2 (-I-. ~)2 +pr2B2=U. 82) As noted above, B and U are arbitrary functions of ~ alone. 82) only contain partial derivatives with respect to r. In general, solution of this system of equations depends on the two arbitrary functions c 1 (z) and c 2 (z), so that ~ = ~ (r, c 1, c 2), p = p(r, c 1, c 2).