168,518 research outputs found
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Radon Transform in Finite Dimensional Hilbert Space
Novel analysis of finite dimensional Hilbert space is outlined. The approach
bypasses general, inherent, difficulties present in handling angular variables
in finite dimensional problems: The finite dimensional, d, Hilbert space
operators are underpinned with finite geometry which provide intuitive
perspective to the physical operators. The analysis emphasizes a central role
for projectors of mutual unbiased bases (MUB) states, extending thereby their
use in finite dimensional quantum mechanics studies. Interrelation among the
Hilbert space operators revealed via their (finite) dual affine plane geometry
(DAPG) underpinning are displayed and utilized in formulating the finite
dimensional ubiquitous Radon transformation and its inverse illustrating phase
space-like physics encoded in lines and points of the geometry. The finite
geometry required for our study is outlined.Comment: 8page
On the Geometric Interpretation of the Nonnegative Rank
The nonnegative rank of a nonnegative matrix is the minimum number of
nonnegative rank-one factors needed to reconstruct it exactly. The problem of
determining this rank and computing the corresponding nonnegative factors is
difficult; however it has many potential applications, e.g., in data mining,
graph theory and computational geometry. In particular, it can be used to
characterize the minimal size of any extended reformulation of a given
combinatorial optimization program. In this paper, we introduce and study a
related quantity, called the restricted nonnegative rank. We show that
computing this quantity is equivalent to a problem in polyhedral combinatorics,
and fully characterize its computational complexity. This in turn sheds new
light on the nonnegative rank problem, and in particular allows us to provide
new improved lower bounds based on its geometric interpretation. We apply these
results to slack matrices and linear Euclidean distance matrices and obtain
counter-examples to two conjectures of Beasly and Laffey, namely we show that
the nonnegative rank of linear Euclidean distance matrices is not necessarily
equal to their dimension, and that the rank of a matrix is not always greater
than the nonnegative rank of its square
Fast Isogeometric Boundary Element Method based on Independent Field Approximation
An isogeometric boundary element method for problems in elasticity is
presented, which is based on an independent approximation for the geometry,
traction and displacement field. This enables a flexible choice of refinement
strategies, permits an efficient evaluation of geometry related information, a
mixed collocation scheme which deals with discontinuous tractions along
non-smooth boundaries and a significant reduction of the right hand side of the
system of equations for common boundary conditions. All these benefits are
achieved without any loss of accuracy compared to conventional isogeometric
formulations. The system matrices are approximated by means of hierarchical
matrices to reduce the computational complexity for large scale analysis. For
the required geometrical bisection of the domain, a strategy for the evaluation
of bounding boxes containing the supports of NURBS basis functions is
presented. The versatility and accuracy of the proposed methodology is
demonstrated by convergence studies showing optimal rates and real world
examples in two and three dimensions.Comment: 32 pages, 27 figure
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