Variational principles for circle patterns

A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the circles, whether it is unique and how it may be constructed. Somewhat more generally, we allow cone-like singularities in the centers and intersection points of the circles. We prove existence and uniqueness theorems for the solution of the circle pattern problem using a variational principle. The functionals (one for the euclidean, one for the hyperbolic case) are convex functions of the radii of the circles. The analogous functional for the spherical case is not convex, hence this case is treated by stereographic projection to the plane. From the existence and uniqueness of circle patterns in the sphere, we derive a strengthened version of Steinitz' theorem on the geometric realizability of abstract polyhedra. We derive the variational principles of Colin de Verdi\`ere, Br\"agger, and Rivin for circle packings and circle patterns from our variational principles. In the case of Br\"agger's and Rivin's functionals. Leibon's functional for hyperbolic circle patterns cannot be derived directly from our functionals. But we construct yet another functional from which both Leibon's and our functionals can be derived. We present Java software to compute and visualize circle patterns.Comment: PhD thesis, iv+94 pages, many figures (mostly vector graphics

Beyond Biomass: Valuing Genetic Diversity in Natural Resource Management

Strategies for increasing production of goods from working and natural systems have raised concerns that the diversity of species on which these services depend may be eroding. This loss of natural capital threatens to homogenize global food supplies and compromise the stability of human welfare. We assess the trade off between artificial augmentation of biomass and degradation of biodiversity underlying a populations' ability to adapt to shocks. Our application involves the augmentation of wild stocks of salmon. Practices in this system have generated warnings that genetic erosion may lead to a loss of the “portfolio effect” and the value of this loss is not accounted for in decision making. We construct an integrated bioeconomic model of salmon biomass and genetic diversity. Our results show how practices that homogenize natural systems can still generate positive returns. However, the substitution of more physical capital and labor for natural capital must be maintained for gains to persist, weakens the capacity for adaptation should this investment cease, and can cause substantial loss of population wildness. We apply an emerging optimization method—approximate dynamic programming—to solve the model without simplifying restrictions imposed previously

Minimal surfaces from circle patterns: Geometry from combinatorics

We suggest a new definition for discrete minimal surfaces in terms of sphere packings with orthogonally intersecting circles. These discrete minimal surfaces can be constructed from Schramm's circle patterns. We present a variational principle which allows us to construct discrete analogues of some classical minimal surfaces. The data used for the construction are purely combinatorial--the combinatorics of the curvature line pattern. A Weierstrass-type representation and an associated family are derived. We show the convergence to continuous minimal surfaces.Comment: 30 pages, many figures, some in reduced resolution. v2: Extended introduction. Minor changes in presentation. v3: revision according to the referee's suggestions, improved & expanded exposition, references added, minor mistakes correcte

Hyperbolic constant mean curvature one surfaces: Spinor representation and trinoids in hypergeometric functions

We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with three regular ends which are asymptotic to catenoid cousins (CMC-1 trinoids).Comment: 29 pages, 9 figures. v2: figures of cmc1-surfaces correcte

A unique representation of polyhedral types

It is known that for each combinatorial type of convex 3-dimensional polyhedra, there is a representative with edges tangent to the unit sphere. This representative is unique up to projective transformations that fix the unit sphere. We show that there is a unique representative (up to congruence) with edges tangent to the unit sphere such that the origin is the barycenter of the points where the edges touch the sphere.Comment: 4 pages, 2 figures. v2: belated upload of final version (of March 2004

Discrete conformal maps and ideal hyperbolic polyhedra

We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring M\"obius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to addresses the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and updated, minor changes in exposition. v3, final version: typos corrected, improved exposition, some material moved to appendice

Advanced-power-reactor design concepts and performance characteristics

Five reactor cooling concepts which allow continued reactor operation following a single rupture of the coolant system are presented for application with the APR. These concepts incorporate convective cooling, double containment, or heat pipes to ensure operation after a coolant line rupture. Based on an evaluation of several control system concepts, a molybdenum clad, beryllium oxide sliding reflector located outside the pressure vessel is recommended

A new doubly discrete analogue of smoke ring flow and the real time simulation of fluid flow

Modelling incompressible ideal fluids as a finite collection of vortex filaments is important in physics (super-fluidity, models for the onset of turbulence) as well as for numerical algorithms used in computer graphics for the real time simulation of smoke. Here we introduce a time-discrete evolution equation for arbitrary closed polygons in 3-space that is a discretisation of the localised induction approximation of filament motion. This discretisation shares with its continuum limit the property that it is a completely integrable system. We apply this polygon evolution to a significant improvement of the numerical algorithms used in Computer Graphics.Comment: 15 pages, 3 figure