The transfer matrix in four-dimensional CDT

The Causal Dynamical Triangulation model of quantum gravity (CDT) has a transfer matrix, relating spatial geometries at adjacent (discrete lattice) times. The transfer matrix uniquely determines the theory. We show that the measurements of the scale factor of the (CDT) universe are well described by an effective transfer matrix where the matrix elements are labeled only by the scale factor. Using computer simulations we determine the effective transfer matrix elements and show how they relate to an effective minisuperspace action at all scales.Comment: 32 pages, 19 figure

(2+1)-Dimensional Quantum Gravity as the Continuum Limit of Causal Dynamical Triangulations

We perform a non-perturbative sum over geometries in a (2+1)-dimensional quantum gravity model given in terms of Causal Dynamical Triangulations. Inspired by the concept of triangulations of product type introduced previously, we impose an additional notion of order on the discrete, causal geometries. This simplifies the combinatorial problem of counting geometries just enough to enable us to calculate the transfer matrix between boundary states labelled by the area of the spatial universe, as well as the corresponding quantum Hamiltonian of the continuum theory. This is the first time in dimension larger than two that a Hamiltonian has been derived from such a model by mainly analytical means, and opens the way for a better understanding of scaling and renormalization issues.Comment: 38 pages, 13 figure

Only distances are required to reconstruct submanifolds

In this paper, we give the first algorithm that outputs a faithful reconstruction of a submanifold of Euclidean space without maintaining or even constructing complicated data structures such as Voronoi diagrams or Delaunay complexes. Our algorithm uses the witness complex and relies on the stability of power protection, a notion introduced in this paper. The complexity of the algorithm depends exponentially on the intrinsic dimension of the manifold, rather than the dimension of ambient space, and linearly on the dimension of the ambient space. Another interesting feature of this work is that no explicit coordinates of the points in the point sample is needed. The algorithm only needs the distance matrix as input, i.e., only distance between points in the point sample as input.Comment: Major revision, 16 figures, 47 page

Characterization and surface reconstruction of objects in tomographic images of composite materials

Dissertação para obtenção do Grau de Mestre em Engenharia InformáticaIn the scope of the project Tomo-GPU supported by FCT / MCTES the aim is to build an interactive graphical environment that allows a Materials specialist to define their own programs for analysis of 3D tomographic images. This project aims to build a tool to characterize and investigate the identified objects, where the user can define search criteria such as size, orientation, bounding boxes, among others. All this processing will be done on a desktop computer equipped with a graphics card with some processing power. On the proposed solution the modules for characterizing objects, received from the identification phase, will be implemented using some existing software libraries, most notably the CGAL library. The characterization modules with bigger execution times will be implemented using OpenCL and GPUs. With this work the characterization and reconstruction of objects and their research can now be done on conventional machines, using GPUs to accelerate the most time-consuming computations. After the conclusion of this thesis, new tools that will help to improve the current development cycle of new materials will be available for Materials Science specialists

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

A Spectral Perspective on Neumann-Zagier

We provide a new topological interpretation of the symplectic properties of gluing equations for triangulations of hyperbolic 3-manifolds, first discovered by Neumann and Zagier. We also extend the symplectic properties to more general gluings of PGL(2,C) flat connections on the boundaries of 3-manifolds with topological ideal triangulations, proving that gluing is a K_2 symplectic reduction of PGL(2,C) moduli spaces. Recently, such symplectic properties have been central in constructing quantum PGL(2,C) invariants of 3-manifolds. Our methods adapt the spectral network construction of Gaiotto-Moore-Neitzke to relate framed flat PGL(2,C) connections on the boundary C of a 3-manifold to flat GL(1,C) connections on a double branched cover S -> C of the boundary. Then moduli spaces of both PGL(2,C) connections on C and GL(1,C) connections on S gain coordinates labelled by the first homology of S, and inherit symplectic properties from the intersection form on homology.Comment: 53 + 12 page