2,986 research outputs found
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
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
Modeling Quantum Behavior in the Framework of Permutation Groups
Quantum-mechanical concepts can be formulated in constructive finite terms
without loss of their empirical content if we replace a general unitary group
by a unitary representation of a finite group. Any linear representation of a
finite group can be realized as a subrepresentation of a permutation
representation. Thus, quantum-mechanical problems can be expressed in terms of
permutation groups. This approach allows us to clarify the meaning of a number
of physical concepts. Combining methods of computational group theory with
Monte Carlo simulation we study a model based on representations of permutation
groups.Comment: 8 pages, based on plenary lecture at Mathematical Modeling and
Computational Physics 2017, Dubna, July 3--7, 201
Classification of Multidimensional Darboux Transformations: First Order and Continued Type
We analyze Darboux transformations in very general settings for
multidimensional linear partial differential operators. We consider all known
types of Darboux transformations, and present a new type. We obtain a full
classification of all operators that admit Wronskian type Darboux
transformations of first order and a complete description of all possible
first-order Darboux transformations. We introduce a large class of invertible
Darboux transformations of higher order, which we call Darboux transformations
of continued Type I. This generalizes the class of Darboux transformations of
Type I, which was previously introduced. There is also a modification of this
type of Darboux transformations, continued Wronskian type, which generalize
Wronskian type Darboux transformations
A generalization of Serre's conjecture and some related issues
AbstractSeveral topics concerned with multivariate polynomial matrices like unimodular matrix completion, matrix determinantal or primitive factorization, matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original Serre problem (conjecture) solved nonconstructively in 1976 and constructively, more recently. This generalized Serre conjecture is proved to be equivalent to several other unsettled conjectures and, therfore, all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining
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