13,864 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
Gradings, Braidings, Representations, Paraparticles: some open problems
A long-term research proposal on the algebraic structure, the representations
and the possible applications of paraparticle algebras is structured in three
modules: The first part stems from an attempt to classify the inequivalent
gradings and braided group structures present in the various parastatistical
algebraic models. The second part of the proposal aims at refining and
utilizing a previously published methodology for the study of the Fock-like
representations of the parabosonic algebra, in such a way that it can also be
directly applied to the other parastatistics algebras. Finally, in the third
part, a couple of Hamiltonians is proposed, and their sutability for modeling
the radiation matter interaction via a parastatistical algebraic model is
discussed.Comment: 25 pages, some typos correcte
Stronger computational modelling of signalling pathways using both continuous and discrete-state methods
Starting from a biochemical signalling pathway model expresses in a process algebra enriched with quantitative information, we automatically derive both continuous-space and discrete-space representations suitable for numerical evaluation. We compare results obtained using approximate stochastic simulation thereby exposing a flaw in the use of the differentiation procedure producing misleading results
Yang-Baxter Equations, Computational Methods and Applications
Computational methods are an important tool for solving the Yang-Baxter
equations(in small dimensions), for classifying (unifying) structures, and for
solving related problems. This paper is an account of some of the latest
developments on the Yang-Baxter equation, its set-theoretical version, and its
applications. We construct new set-theoretical solutions for the Yang-Baxter
equation. Unification theories and other results are proposed or proved.Comment: 12 page
The Classification of All Crossed Products
Using the computational approach introduced in [Agore A.L., Bontea C.G.,
Militaru G., J. Algebra Appl. 12 (2013), 1250227, 24 pages, arXiv:1207.0411] we
classify all coalgebra split extensions of by , where is
the cyclic group of order and is Sweedler's -dimensional Hopf
algebra. Equivalently, we classify all crossed products of Hopf algebras by explicitly computing two classifying objects: the cohomological
'group' and
the set of types of isomorphisms of all crossed products .
More precisely, all crossed products are described by
generators and relations and classified: they are -dimensional quantum
groups , parameterized by the set of all pairs consisting of an arbitrary unitary map and an -th root
of . As an application, the group of Hopf algebra
automorphisms of is explicitly described
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