Constructing Intrinsic Delaunay Triangulations of Submanifolds

We describe an algorithm to construct an intrinsic Delaunay triangulation of a smooth closed submanifold of Euclidean space. Using results established in a companion paper on the stability of Delaunay triangulations on $\delta$-generic point sets, we establish sampling criteria which ensure that the intrinsic Delaunay complex coincides with the restricted Delaunay complex and also with the recently introduced tangential Delaunay complex. The algorithm generates a point set that meets the required criteria while the tangential complex is being constructed. In this way the computation of geodesic distances is avoided, the runtime is only linearly dependent on the ambient dimension, and the Delaunay complexes are guaranteed to be triangulations of the manifold

A probabilistic approach to reducing the algebraic complexity of computing Delaunay triangulations

Computing Delaunay triangulations in $\mathbb{R}^d$ involves evaluating the so-called in\_sphere predicate that determines if a point $x$ lies inside, on or outside the sphere circumscribing $d+1$ points $p_0,\ldots ,p_d$. This predicate reduces to evaluating the sign of a multivariate polynomial of degree $d+2$ in the coordinates of the points $x, \, p_0,\, \ldots,\, p_d$. Despite much progress on exact geometric computing, the fact that the degree of the polynomial increases with $d$ makes the evaluation of the sign of such a polynomial problematic except in very low dimensions. In this paper, we propose a new approach that is based on the witness complex, a weak form of the Delaunay complex introduced by Carlsson and de Silva. The witness complex $\mathrm{Wit} (L,W)$ is defined from two sets $L$ and $W$ in some metric space $X$: a finite set of points $L$ on which the complex is built, and a set $W$ of witnesses that serves as an approximation of $X$. A fundamental result of de Silva states that $\mathrm{Wit}(L,W)=\mathrm{Del} (L)$ if $W=X=\mathbb{R}^d$. In this paper, we give conditions on $L$ that ensure that the witness complex and the Delaunay triangulation coincide when $W$ is a finite set, and we introduce a new perturbation scheme to compute a perturbed set $L'$ close to $L$ such that $\mathrm{Del} (L')= \mathrm{wit} (L', W)$. Our perturbation algorithm is a geometric application of the Moser-Tardos constructive proof of the Lov\'asz local lemma. The only numerical operations we use are (squared) distance comparisons (i.e., predicates of degree 2). The time-complexity of the algorithm is sublinear in $|W|$. Interestingly, although the algorithm does not compute any measure of simplex quality, a lower bound on the thickness of the output simplices can be guaranteed.Comment: 24 page

Spinning Geodesic Witten Diagrams

We present an expression for the four-point conformal blocks of symmetric traceless operators of arbitrary spin as an integral over a pair of geodesics in Anti-de Sitter space, generalizing the geodesic Witten diagram formalism of Hijano et al [arXiv:1508.00501] to arbitrary spin. As an intermediate step in the derivation, we identify a convenient basis of bulk three-point interaction vertices which give rise to all possible boundary three point structures. We highlight a direct connection between the representation of the conformal block as a geodesic Witten diagram and the shadow operator formalism.Comment: 28+6 pages, 8 figure

Sacred trauma: Language, recovery, and the face of God in Ingmar Bergman's trilogy of faith

Thesis (M.A.) University of Alaska Fairbanks, 2014This thesis examines the three films that constitute director Ingmar Bergman's first trilogy, Through a Glass Darkly, Winter Light, and The Silence. In the thesis I take a multidisciplinary approach to analyzing the films' treatments of language, trauma, and God. Drawing on the Old Testament and work of psychoanalysts dealing with trauma, I argue for the similarities and reciprocity between trauma and communion with God and the ways in which the three films illustrate these relationships. Each film functions on a reflexive level to criticize the tools of filmmaking--images, dialog, and narrative--and points to discordance between symbols and reality. Bringing in Jacques Lacan's model of the imaginary and symbolic orders, I analyze the treatment of language and trauma in the trilogy and the potential for recovery suggested by the end of each film. The thesis culminates by tracing the trilogy toward a new vision of God and his role in the human psyche.Preface. Beyond Bergman: A New Consideration of the Trilogy of Faith -- Chapter 1.The Terrible Face of God: Correlations between Trauma and Faith -- Chapter 2. Symbolic Discordance in the Trilogy -- Chapter 3. Silencing the Spoken Word -- Chapter 4. A Trilogy of Trauma -- Chapter 5. Post-Trauma: A New Vision of God -- Works Cited

Control System Design and Simulation of Spacecraft Formations

The objective of this thesis is to analyze an effective control scheme for a formation of spacecraft, implement that control, and provide a 3-D simulation. The thesis first summarizes the progress made in formation flight control schemes and then provides a theoretical framework for the control system design. Using established control design techniques, feedback laws are constructed to control both rotational and translational motion of a group of spacecraft. Computer simulations are carried out using Matlab to generate the ephemeris and attitude data, which are exported to Satellite Tool Kit (STK) to create animations. The process by which Matlab is used to produce the data and how the data are used to produce a 3-D animation in STK is explained. Finally, various formation scenarios are modeled using this same process