Bloomberg Businessweek - 10 October - Steve Jobs by Bloomberg LP

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24 CHAPTER 1 FIG. 6. 29). 3) was studied many years ago in connection with the theory of pseudospherical surfaces (Darboux [1, p. 432] Bianchi [2]). In Bianchi ([1], [2]) it is shown that a theorem of permutability (today called a nonlinear superposition— see Chaper 4) exists for this equation. 28), respectively; then there is a solution z4 which is generated from z2 through a3 and also from z3 through a2. This permutability was expressed by Bianchi ([2], p. 743) in the diagram in Fig. 6. 28) the analytic expressions for these solutions can be written, and then it can be shown that they are related by the expression which is the classical theorem of permutability .

36 CHAPTER 1 Burgers Equation. The Burgers equation arises in turbulence and wave propagation. In 190612 it was already known to possess a relationship with the linear equation q + r = 0. (This was rediscovered by Hopf and Cole some fifty years later. For a discussion of those results and a simple explanation of them see Ames [1], pp. 40b) completes the Backlundtransformation pair. Example of Clairin. The equations 5j=0 and s + qe~z = Q are transformed into one another by the Backlund transformation I2 See Forsyth, op.

31). 28) results in 4 H. T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, 1962). CLASSICAL FOUNDATIONS 25 a Riccati equation. We shall see the same phenomenon occurring in the next section and will elaborate on the idea in Chapter 4. The Korteweg-de Vries (KdV) Equation. 32) becomes after integration and the discard of an arbitrary function of integration. Here a = uxxx, while p and q are as before. 33) by using the Lie-Backlund transformation zx = u. 33). 33). 34b').

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