Index Reduction for Differential-Algebraic Equations with Mixed Matrices

Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. The difficulty in solving numerically a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is important to convert high-index DAEs into low-index DAEs. Most of existing simulation software packages for dynamical systems are equipped with an index-reduction algorithm given by Mattsson and S\"{o}derlind. Unfortunately, this algorithm fails if there are numerical cancellations. These numerical cancellations are often caused by accurate constants in structural equations. Distinguishing those accurate constants from generic parameters that represent physical quantities, Murota and Iri introduced the notion of a mixed matrix as a mathematical tool for faithful model description in structural approach to systems analysis. For DAEs described with the use of mixed matrices, efficient algorithms to compute the index have been developed by exploiting matroid theory. This paper presents an index-reduction algorithm for linear DAEs whose coefficient matrices are mixed matrices, i.e., linear DAEs containing physical quantities as parameters. Our algorithm detects numerical cancellations between accurate constants, and transforms a DAE into an equivalent DAE to which Mattsson--S\"{o}derlind's index-reduction algorithm is applicable. Our algorithm is based on the combinatorial relaxation approach, which is a framework to solve a linear algebraic problem by iteratively relaxing it into an efficiently solvable combinatorial optimization problem. The algorithm does not rely on symbolic manipulations but on fast combinatorial algorithms on graphs and matroids. Furthermore, we provide an improved algorithm under an assumption based on dimensional analysis of dynamical systems.Comment: A preliminary version of this paper is to appear in Proceedings of the Eighth SIAM Workshop on Combinatorial Scientific Computing, Bergen, Norway, June 201

On the Reduction of Singularly-Perturbed Linear Differential Systems

In this article, we recover singularly-perturbed linear differential systems from their turning points and reduce the rank of the singularity in the parameter to its minimal integer value. Our treatment is Moser-based; that is to say it is based on the reduction criterion introduced for linear singular differential systems by Moser. Such algorithms have proved their utility in the symbolic resolution of the systems of linear functional equations, giving rise to the package ISOLDE, as well as in the perturbed algebraic eigenvalue problem. Our algorithm, implemented in the computer algebra system Maple, paves the way for efficient symbolic resolution of singularly-perturbed linear differential systems as well as further applications of Moser-based reduction over bivariate (differential) fields.Comment: Keywords: Moser-based Reduction, Perturbed linear Differential systems, turning points, Computer algebr

Non-unimodular reductions and N = 4 gauged supergravities

We analyze the class of four-dimensional N = 4 supergravities obtained by gauging the axionic shift and axionic rescaling symmetries. These theories are formulated with the machinery of embedding tensors and shown to be deducible from higher dimensions using a Scherk--Schwarz reduction with a twist by a non-compact symmetry. This allows to evade the usual unimodularity requirement and completes the dictionary between heterotic gaugings and fluxes, at least for the "geometric sector".Comment: 15 page

Duality Twists on a Group Manifold

We study duality-twisted dimensional reductions on a group manifold G, where the twist is in a group \tilde{G} and examine the conditions for consistency. We find that if the duality twist is introduced through a group element \tilde{g} in \tilde{G}, then the flat \tilde{G}-connection A =\tilde{g}^{-1} d\tilde{g} must have constant components M_n with respect to the basis 1-forms on G, so that the dependence on the internal coordinates cancels out in the lower dimensional theory. This condition can be satisfied if and only if M_n forms a representation of the Lie algebra of G, which then ensures that the lower dimensional gauge algebra closes. We find the form of this gauge algebra and compare it to that arising from flux compactifications on twisted tori. As an example of our construction, we find a new five dimensional gauged, massive supergravity theory by dimensionally reducing the eight dimensional Type II supergravity on a three dimensional unimodular, non-semi-simple, non-abelian group manifold with an SL(3,R) twist.Comment: 22 page

Poisson sigma model on the sphere

We evaluate the path integral of the Poisson sigma model on sphere and study the correlators of quantum observables. We argue that for the path integral to be well-defined the corresponding Poisson structure should be unimodular. The construction of the finite dimensional BV theory is presented and we argue that it is responsible for the leading semiclassical contribution. For a (twisted) generalized Kahler manifold we discuss the gauge fixed action for the Poisson sigma model. Using the localization we prove that for the holomorphic Poisson structure the semiclassical result for the correlators is indeed the full quantum result.Comment: 38 page

Generators of split extensions of Abelian groups by cyclic groups

Let $G \simeq M \rtimes C$ be an $n$-generator group with $M$ Abelian and $C$ cyclic. We study the Nielsen equivalence classes and T-systems of generating $n$-tuples of $G$. The subgroup $M$ can be turned into a finitely generated faithful module over a suitable quotient $R$ of the integral group ring of $C$. When $C$ is infinite, we show that the Nielsen equivalence classes of the generating $n$-tuples of $G$ correspond bijectively to the orbits of unimodular rows in $M^{n -1}$ under the action of a subgroup of $GL_{n - 1}(R)$. Making no assumption on the cardinality of $C$, we exhibit a complete invariant of Nielsen equivalence in the case $M \simeq R$. As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag-Solitar groups, lamplighter groups and split metacyclic groups.Comment: 36 pages, The former Theorem F.ii has been retracted because the proof was wrong and couldn't be repaired. To appear in Groups, Geometry and Dynamic