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Original Articles

Tracking Critical Points on Evolving Curves and Surfaces

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References

  • [Heath 97] T. I. Heath (ed.). The Works of Archimedes. Cambridge: Cambridge University Press, 1897.
  • [Nowacki 02] H. Nowacki, Archimedes and ship stability, In: Passenger ship design, construction, operation and safety: Euroconference; Knossos Royal Village, Anissaras, Crete, Greece, October 15-17, 2001, 335-360 (2002). Ed: Kaklis, P.D. National Technical Univ. of Athens, Department of Naval Architecture and Marine Engineering, Athens.
  • [Arnold 98] Arnold, V.I., Ordinary differential equations, 10th printing, MIT Press, Cambridge, 1998.
  • [Blaschke 56] W. Blaschke. Kreis and Kugel, 2nd edn. Berlin: W. de Gruyter, 1956.
  • [Cayley 59] A. Cayley. “On Contour and Slope Lines.” Phi. Mag. 18 (1859), 264–268.
  • [Chen and Schmitt 92] X. Chen and F. Schmitt. “Intrinsic Surface Properties from Surface Triangulation.” LNCS 588 (1992), 739–743.
  • [Conway and Guy 69] J. H. Conway and R. Guy. “Stability of Polyhedra.” SIAM Rev. 11 (1969), 78–82.
  • [Dawson 85] R. Dawson. “Monostatic Simplexes.” Amer. Math. Monthly 92 (1985), 541–546.
  • [Dawson and Finbow 99] R. Dawson and W. Finbow. “What Shape is a Loaded Die?” Math. Intell. 22 (1999), 32–37.
  • [Damon 95] J. Damon. “Morse Theory for Solutions to the Heat Equation and Gaussian Blurring.” J. Differ. Equ. 115 (1995), 368–401.
  • [Doedel and Beyn 81] E. J. Doedel and W. J. Beyn. “Stability and Multiplicity of Solutions to Discretizations of Nonlinear Ordinary Differential Equations.” SIAM J. Sci. Comput. 2 (1981), 107–120.
  • [Domokos 14] G. Domokos. “Monotonicity of Spatial Critical Points Evolving Under Curvature-Driven Flows.” J. Nonlinear Sci. 25 (2014), 247–275.
  • [Domokos and Holmes 03] G. Domokos and P. J. Holmes. “On Nonlinear Boundary-Value Problems: Ghosts, Parasites and Discretizations.” Proc, Roy. Soc. London A 459:2034, (2003) 1535–1561, DOI: https://doi.org/10.1098/rspa.2002.1091
  • [Domokos and Lángi 18] G. Domokos and Z. Lángi. “The Evolution of Geological Shape Descriptors Under Distance-Driven Flows.” Math. Geosci. 50 (2018), 337–363, DOI: https://doi.org/10.1007/s11004-017-9723-9.
  • [Domokos and Lángi 18] G. Domokos and Z. Lángi. “The Isoperimetric Quotient of a Convex Body Decreases Monotonically under the Eikonal Abrasion Model.” Arxiv preprint https://arxiv.org/abs/1801.06796, 2018.
  • [Domokos et al. 12] G. Domokos, Z. Lángi, and T. Szabó. “On the Equilibria of Finely Discretized Curves and Surfaces.” Monatsheft für Math. 168:3–4 (2012), 321–345
  • [Domokos and Lángi 14] G. Domokos and Z. Lángi. “The Robustness of Equilibria on Convex Solids.” Mathematika 60 (2014), 337–256.
  • [Domokos et al. 94] G. Domokos, J. Papadopoulos, and A. Ruina. “Static Equilibria of Rigid Bodies: Is There Anything New?” J. Elast. 36 (1994), 59–66.
  • [Domokos 10] G. Domokos, A. A. Sipos, T. Szabo, P. L. Varkonyi. “Pebbles, Shapes and Equilibria.” Math. Geosci. 42 (2010), 29–47.
  • [Domokos et al. 14] G. Domokos, D. J. Jerolmack, A. Á. Sipos, and Á. Török. “How River Rocks Round: Resolving the Shape-Size Paradox.” PLoS ONE 9:2 (2014), e88657. DOI:https://doi.org/10.1371/journal.pone.0088657
  • [Euler 52] L. Euler. Elementa doctrinae solidorum. Demonstratio nonnullarum insignium proprietatum, quibus solida hedris planis inclusa sunt praedita, Novi comment acad. sc. imp. Petropol 4 (1752), 109–160.
  • [Grayson 87] M. Grayson. “The Heat Equation Shrinks Embedded Plane Curves to Round Points.” J. Diff. Geom. 26 (1987), 285–314.
  • [Heppes 67] A. Heppes. “A Double-Tipping Tetrahedron.” SIAM Rev. 9 (1967), 599–600.
  • [Krapivsky and Redner 07] P. L. Krapivsky and S. Redner. “Smoothing a Rock by Chipping.” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75:3 Pt 1 (2007), 031119. DOI:https://doi.org/10.1103/PhysRevE.75.031119
  • [Kuijper and Florack 04] A. Kuijper and L. Florack. “The Relevance of Non-Generic Events in Scale Space Models.” Int. J. Comp. Vis. 57 (2004), 67–84.
  • [Miller et al. 14] K. L. Miller, T. Szabó, D. J. Jerolmack, G. Domokos. “Quantifying the Significance of Abrasion and Selective Transport for Downstream Fluvial Grain Size Evolution.” J. Geophys. Res-Earth 119 (2014), 2412–2429 DOI: https://doi.org/10.1002/2014JF003156
  • [Maxwell 70] J. C. Maxwell. “On Hills and Dales.” Phil. Mag. 40 (1870), 421–427.
  • [Morse 42] M. Morse. “What is Analysis in the Large?.” Amer. Math. Monthly 49 (1942), 358–364.
  • [Niven 56] I. M. Niven. Irrational Numbers, The Carus Mathematical Monographs 11, The Mathematical Association of America. New York: John Wiley and Sons, Inc., 1956.
  • [Persson and Strang 04] P. O. Persson and G. Strang. “A Simple Mesh Generator in MATLAB.” SIAM Rev. 46:2 (2004), 329–345.
  • [Poston and Stewart 78] T. Poston and I. Stewart. Catastrophe Theory and its Applications. London: Pitman, 1978.
  • [Ray et al. 11] S. S. Ray, U. Frisch, S. Nazarenko and T. Matsumoto. “Resonance Phenomenon for the Galerkin-Truncated Burgers and Euler Equations.” Phys. Rev. E Stat. Nonlin. Soft Matter. Phys. 84 (2011), 016301.
  • [Varkonyi and Domokos 06] P. L. Varkonyi and G. Domokos. “Static Equilibria of Rigid Bodies: Dice, Pebbles and the Poincaré-Hopf Theorem.” J. Nonlin. Sci. 16 (2006), 255–281.
  • [Villani 15] C. Villani. “Birth of a Theorem.” The Bodley Head, (2015).
  • [Szabó et al. 13] T. Szabó, S. Fityus, and G. Domokos. Abrasion model of downstream changes in grain shape and size along the Williams river, Australia. J. Geophys. Res-Earth 118 (2013), 1–13. doi:https://doi.org/10.1002/jgrf.20142.
  • [Zamfirescu 95] T. Zamfirescu. “How do Convex Bodies Sit?” Mathematica 42 (1995), 179–181.

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