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We fully generalize a previously-developed computational geometry tool [1] to perform large-scale simulations of arbitrary two-dimensional faceted surfaces $z = h(x,y)$. Our method uses a three-component facet/edge/junction storage model, which by naturally mirroring the intrinsic surface structure allows both rapid simulation and easy extraction of geometrical statistics. The bulk of this paper is a comprehensive treatment of topological events, which are detected and performed explicitly. In addition, we also give a careful analysis of the subtle pitfalls associated with time-stepping schemes for systems with topological changes. The method is demonstrated using a simple facet dynamics on surfaces with three different symmetries. Appendices detail the reconnection of holes left by facet removal and a strategy for dealing with the inherent kinematic non-uniqueness displayed by several topological events. [1] S.A. Norris and S.J. Watson, Acta Mat. 55 (2007) p. 6444
A mean-field theory is developed for the scale-invariant length distributions observed during the coarsening of one-dimensional faceted surfaces. This theory closely follows the Lifshitz-Slyozov-Wagner theory of Ostwald ripening in two-phase systems
In 2017, Lienert and Tumulka proved Borns rule on arbitrary Cauchy surfaces in Minkowski space-time assuming Borns rule and a corresponding collapse rule on horizontal surfaces relative to a fixed Lorentz frame, as well as a given unitary time evolut
This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations. 2) Space o
We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of
In recent years it has been recognized that the hyperbolic numbers (an extension of complex numbers, defined as z=x+h*y with h*h=1 and x,y real numbers) can be associated to space-time geometry as stated by the Lorentz transformations of special rela