21,887 research outputs found
An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization
In this paper we develop an axiomatic setup for algorithmic homological
algebra of Abelian categories. This is done by exhibiting all existential
quantifiers entering the definition of an Abelian category, which for the sake
of computability need to be turned into constructive ones. We do this
explicitly for the often-studied example Abelian category of finitely presented
modules over a so-called computable ring , i.e., a ring with an explicit
algorithm to solve one-sided (in)homogeneous linear systems over . For a
finitely generated maximal ideal in a commutative ring we
show how solving (in)homogeneous linear systems over can be
reduced to solving associated systems over . Hence, the computability of
implies that of . As a corollary we obtain the computability
of the category of finitely presented -modules as an Abelian
category, without the need of a Mora-like algorithm. The reduction also yields,
as a by-product, a complexity estimation for the ideal membership problem over
local polynomial rings. Finally, in the case of localized polynomial rings we
demonstrate the computational advantage of our homologically motivated
alternative approach in comparison to an existing implementation of Mora's
algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu
Obstructions to Genericity in Study of Parametric Problems in Control Theory
We investigate systems of equations, involving parameters from the point of
view of both control theory and computer algebra. The equations might involve
linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference
as well as more complicated ones, which act trivially on the parameters. Such a
system can be identified algebraically with a certain left module over a
non-commutative algebra, where the operators commute with the parameters. We
develop, implement and use in practice the algorithm for revealing all the
expressions in parameters, for which e.g. homological properties of a system
differ from the generic properties. We use Groebner bases and Groebner basics
in rings of solvable type as main tools. In particular, we demonstrate an
optimized algorithm for computing the left inverse of a matrix over a ring of
solvable type. We illustrate the article with interesting examples. In
particular, we provide a complete solution to the "two pendula, mounted on a
cart" problem from the classical book of Polderman and Willems, including the
case, where the friction at the joints is essential . To the best of our
knowledge, the latter example has not been solved before in a complete way.Comment: 20 page
Involutive Bases Algorithm Incorporating F5 Criterion
Faugere's F5 algorithm is the fastest known algorithm to compute Groebner
bases. It has a signature-based and an incremental structure that allow to
apply the F5 criterion for deletion of unnecessary reductions. In this paper,
we present an involutive completion algorithm which outputs a minimal
involutive basis. Our completion algorithm has a nonincremental structure and
in addition to the involutive form of Buchberger's criteria it applies the F5
criterion whenever this criterion is applicable in the course of completion to
involution. In doing so, we use the G2V form of the F5 criterion developed by
Gao, Guan and Volny IV. To compare the proposed algorithm, via a set of
benchmarks, with the Gerdt-Blinkov involutive algorithm (which does not apply
the F5 criterion) we use implementations of both algorithms done on the same
platform in Maple.Comment: 24 pages, 2 figure
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A comparison of fuzzy approaches to e-commerce review rating prediction
This paper presents a comparative analysis of the performance of fuzzy approaches on the task of predicting customer review ratings using a computational intelligence framework based on a genetic algorithm for data dimensionality reduction. The performance of the Fuzzy C-Means (FCM), a neurofuzzy approach combining FCM and the Adaptive Neuro Fuzzy Inference System (ANFIS), and the Simplified Fuzzy ARTMAP (SFAM) was compared on six datasets containing customer reviews. The results revealed that all computational intelligence predictors were suitable for the rating prediction problem, and that the genetic algorithm is effective in reducing the number of dimensions without affecting the prediction performance of each computational intelligence predictor
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