By Daniel J. Velleman
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Additional info for American Mathematical Monthly, volume 117, August September 2010
Giuli, A primer on Stern’s diatomic sequence I, Fibonacci Quart. 17 (1979) 103–108. , A primer on Stern’s diatomic sequence II, Fibonacci Quart. 17 (1979) 246–248. 13. , A primer on Stern’s diatomic sequence III, Fibonacci Quart. 17 (1979) 318–320. 14. 15. R. L. Graham, D. E. Knuth, and O. , Addison-Wesley, Reading, MA, 1994. 16. R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks, and C. H. Yan, Apollonian circle packings: Number theory, J. Number Theory 100 (2003) 1–45. 1016/S0022-314X(03)00015-5 17.
Taylor, The Schr¨odinger equation on spheres, Pacific J. Math. 209 (2003) 145–155. 145 17. I. M. Vinogradov, The Method of Trigonometrical Sums in the Theory of Numbers, Dover, Mineola, NY, 2004. 18. G. B. Whitham, Linear and Nonlinear Waves, John Wiley, New York, 1974. 19. N. J. Zabusky and M. D. Kruskal, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965) 240–243. 240 20. E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev.
Of course, one can mollify the solution by convolution with a smooth, concentrated kernel, which would effectively smooth out the coarser quantized times. Since the underlying partial differential equation is linear, with constant coefficients, derivatives and integrals of solutions are also solutions. 7) with respect to x a couple of times (omitting the constant terms in order to preserve periodicity) produces a bona fide classical solution whose derivatives exhibit such quantization/fractalization behavior.
American Mathematical Monthly, volume 117, August September 2010 by Daniel J. Velleman
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