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Additional info for Encyclopedia Dictionary of Mathematics (Section C)

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The system is not complete, but it is interpreted as a tsample space of coin tossing. Rademacher’s system is useful for constructing various counter-examples. (5) Let the binary expansion of n be n = 2’1+ 2’1+ . . + 2”~ (vl < v2 < .

Such a solution is called a general solution of the partial differential equation (8) ‘of the first order. By specializing this function cp, we obtain a particular solution of (8). Thus (i)-(iii) exhaust all the cases, and by obtaining a complete solution of (8) we can get all the solutions of (8). The number of compleie solutions may be more than 1, or it may be infinite. These complete solutions can be tr,znsformed into each other by tcontact transformations. 1191 320 D Partial Differential Moreover, they are contained in the general solutions.

D. , r. ) -=o, aan-r+l we get a solution that does not contain an arbitrary constant. Such a solution is called a singular solution of (15). (iii) If not all of the i3V/&zi vanish, there exists at least one functional relation among a,, a,, . . , u”-~+~. We assume that there exist exactly k ( < n-r) relations among a,, a,, . . , k. (16) 1,) I,, . . ,n-r+l. Method (15) We call (13) a complete solution of (15). In this case, as in the case when II = 2, we get all the solutions of (15) from a complete solution (13) of (15).

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### Encyclopedia Dictionary of Mathematics (Section C)

by George
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