18,450 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
Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras
The present text surveys some relevant situations and results where basic
Module Theory interacts with computational aspects of operator algebras. We
tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be
published in Lecture Notes in Computer Scienc
Constructive Algebraic Topology
The classical ``computation'' methods in Algebraic Topology most often work
by means of highly infinite objects and in fact +are_not+ constructive. Typical
examples are shown to describe the nature of the problem. The Rubio-Sergeraert
solution for Constructive Algebraic Topology is recalled. This is not only a
theoretical solution: the concrete computer program +Kenzo+ has been written
down which precisely follows this method. This program has been used in various
cases, opening new research subjects and producing in several cases significant
results unreachable by hand. In particular the Kenzo program can compute the
first homotopy groups of a simply connected +arbitrary+ simplicial set.Comment: 24 pages, background paper for a plenary talk at the EACA Congress of
Tenerife, September 199
Singular and non-singular eigenvectors for the Gaudin model
We present a method to construct a basis of singular and non-singular common
eigenvectors for Gaudin Hamiltonians in a tensor product module of the Lie
algebra SL(2). The subset of singular vectors is completely described by
analogy with covariant differential operators. The relation between singular
eigenvectors and the Bethe Ansatz is discussed. In each weight subspace the set
of singular eigenvectors is completed to a basis, by a family of non-singular
eigenvectors. We discuss also the generalization of this method to the case of
an arbitrary Lie algebra.Comment: 19 page
Singular Vectors and Conservation Laws of Quantum KdV type equations
We give a direct proof of the relation between vacuum singular vectors and
conservation laws for the quantum KdV equation or equivalently for
-perturbed conformal field theories. For each degree at which a
classical conservation law exists, we find a quantum conserved quantity for a
specific value of the central charge. Various generalizations (
supersymmetric, Boussinesq) of this result are presented.Comment: 9 page
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