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Computational Geometry Column 37
Open problems from the 15th Annual ACM Symposium on Computational Geometry
Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes
In this paper, we establish a lemma in algebraic coding theory that
frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes,
algebraic geometry codes, and affine variety codes. Our lemma corresponds to
the non-systematic encoding of affine variety codes, and can be stated by
giving a canonical linear map as the composition of an extension through linear
feedback shift registers from a Grobner basis and a generalized inverse
discrete Fourier transform. We clarify that our lemma yields the error-value
estimation in the fast erasure-and-error decoding of a class of dual affine
variety codes. Moreover, we show that systematic encoding corresponds to a
special case of erasure-only decoding. The lemma enables us to reduce the
computational complexity of error-evaluation from O(n^3) using Gaussian
elimination to O(qn^2) with some mild conditions on n and q, where n is the
code length and q is the finite-field size.Comment: 37 pages, 1 column, 10 figures, 2 tables, resubmitted to IEEE
Transactions on Information Theory on Jan. 8, 201
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
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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