82 research outputs found
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
Bioeconomic Model of Rainbow Trout (\u3cem\u3eOncorhynchus mykiss\u3c/em\u3e) and Humpback Chub (\u3cem\u3eGila cypha\u3c/em\u3e) Management in the Grand Canyon
The Colorado River, from Glen Canyon Dam (GCD) to the Little Colorado River (LCR) confluence, includes both non-native Rainbow Trout (Oncorhynchus mykiss) and endangered native Humpback Chub (Gila cypha). While both Rainbow Trout and Humpback Chub are valued fish species in this system, Rainbow Trout can have a negative effect on Humpback Chub survival. We developed a bioeconomic model to determine management actions that minimize the costs of controlling Rainbow Trout abundance subject to achieving Humpback Chub population goals. The model is compartmentalized into population and management components. The population component characterizes the stylized dynamics of Rainbow Trout and Humpback Chub from GCD to the LCR confluence within the Colorado River. The management component of the model identifies Rainbow Trout mechanical removal strategies that achieve average annual juvenile Humpback Chub survival targets while minimizing management costs. This research is an interdisciplinary effort combining biological models and economic methods to address federal, state and tribal stakeholder resource goals related to Rainbow Trout and Humpback Chub management in this complex social-ecological system
- …