Math Music and Mathematics; From Pythagoras to Fractals

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References 1. K. -M. 1983a Solitary and periodic August 20, 2008 19:18 WSPC - Proceedings Trim Size: 9in x 6in FACMproc Nonlinear 3D Free Surface Flows 43 gravity-capillary waves of finite amplitude J. Fluid Mech. 134, 205-219. 2. -M. J. 2002 Do true elevation gravity-capillary solitary waves exist? A numerical investigation J. Fluid Mech. 454, 403-417. 3. Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary–gravity wav es. Ann. Rev. Fluid Mech. 31 301–346. 4. -M. & Dias, F. 1992 Gravity-capillary solitary waves in water of infinite depth and related free-surface flows.

L. WK (τj ) = (24) k=1 Also, let J = max{j : 1 ≤ j ≤ L; τj ≤ η} for some prefixed η > 0. Basically, we would like to pursue the distributional features of the partial sequence: {WK (τj ), j ≤ J}. Corresponding to the known points τ1 < · · · < τJ , let us consider the partial process WK (τj ), j = 1, . . , J as defined above. 05) but can be approximated very well through the above Poisson process result. This allows a multi-stage multiple hypothesis testing procedure, illustrated in detail in Sen (2008).

9. 261. 26 Fig. 10. 34 Values of a ˜ versus α. only half of the wave is shown. 5 40 20 y 0 Ŧ20 Ŧ40 Fig. 11. 35. 5 Ŧ30 10 Ŧ20 Ŧ10 0 10 20 30 y 0 x Fig. 12. 35. 2. 16 4. 2 and 3 was to define a modified problem involving a parameter (denoted by ). -M. 5 Ŧ30 10 Ŧ20 Ŧ10 0 10 20 30 y 0 x Fig. 13. 345. Only half of the wave is shown. were first calculated with = 0. Then the solutions of the original problem were found by taking numerically the limit → 0. This method has been used successfully by the author to investigate other free boundary problems.

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