166,015 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
Emergent Properties of Tumor Microenvironment in a Real-life Model of Multicell Tumor Spheroids
Multicellular tumor spheroids are an important {\it in vitro} model of the
pre-vascular phase of solid tumors, for sizes well below the diagnostic limit:
therefore a biophysical model of spheroids has the ability to shed light on the
internal workings and organization of tumors at a critical phase of their
development. To this end, we have developed a computer program that integrates
the behavior of individual cells and their interactions with other cells and
the surrounding environment. It is based on a quantitative description of
metabolism, growth, proliferation and death of single tumor cells, and on
equations that model biochemical and mechanical cell-cell and cell-environment
interactions. The program reproduces existing experimental data on spheroids,
and yields unique views of their microenvironment. Simulations show complex
internal flows and motions of nutrients, metabolites and cells, that are
otherwise unobservable with current experimental techniques, and give novel
clues on tumor development and strong hints for future therapies.Comment: 20 pages, 10 figures. Accepted for publication in PLOS One. The
published version contains links to a supplementary text and three video
file
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