7,536 research outputs found
An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier-Stokes Equations
We discuss model reduction for a particular class of quadratic-bilinear (QB)
descriptor systems. The main goal of this article is to extend the recently
studied interpolation-based optimal model reduction framework for QBODEs
[Benner et al. '16] to a class of descriptor systems in an efficient and
reliable way. Recently, it has been shown in the case of linear or bilinear
systems that a direct extension of interpolation-based model reduction
techniques to descriptor systems, without any modifications, may lead to poor
reduced-order systems. Therefore, for the analysis, we aim at transforming the
considered QB descriptor system into an equivalent QBODE system by means of
projectors for which standard model reduction techniques for QBODEs can be
employed, including aforementioned interpolation scheme. Subsequently, we
discuss related computational issues, thus resulting in a modified algorithm
that allows us to construct \emph{near}--optimal reduced-order systems without
explicitly computing the projectors used in the analysis. The efficiency of the
proposed algorithm is illustrated by means of a numerical example, obtained via
semi-discretization of the Navier-Stokes equations
On the algebraic structure of rational discrete dynamical systems
We show how singularities shape the evolution of rational discrete dynamical
systems. The stabilisation of the form of the iterates suggests a description
providing among other things generalised Hirota form, exact evaluation of the
algebraic entropy as well as remarkable polynomial factorisation properties. We
illustrate the phenomenon explicitly with examples covering a wide range of
models
Spin dynamics of the bilinear-biquadratic Heisenberg model on the triangular lattice: a quantum Monte Carlo study
We study thermodynamic properties as well as the dynamical spin and
quadrupolar structure factors of the O(3)-symmetric spin-1 Heisenberg model
with bilinear-biquadratic exchange interactions on the triangular lattice.
Based on a sign-problem-free quantum Monte Carlo approach, we access both the
ferromagnetic and the ferroquadrupolar ordered, spin nematic phase as well as
the SU(3)-symmetric point which separates these phases. Signatures of Goldstone
soft-modes in the dynamical spin and the quadrupolar structure factors are
identified, and the properties of the low-energy excitations are compared to
the thermodynamic behavior observed at finite temperatures as well as to
Schwinger-boson flavor-wave theory.Comment: 7 pages, 8 figure
Symmetric designs on Lie algebras and interactions of hamiltonian systems.
Nonhamiltonian interaction of hamiltonian systems is considered. Dynamical
equations are constructed by use of symmetric designs on Lie algebras. The
results of analysis of these equations show that some class of symmetric
designs on Lie algebras beyond Jordan ones may be useful for a description of
almost periodic, asymptotically periodic, almost asymptotically periodic, and,
possibly, more chaotic systems. However, the behaviour of systems related to
symmetric designs with additional identities is simpler than for general ones
from different points of view. These facts confirm a general thesis that
various algebraic structures beyond Lie algebras may be regarded as certain
characteristics for a wide class of dynamical systems
Clifford geometric parameterization of inequivalent vacua
We propose a geometric method to parameterize inequivalent vacua by dynamical
data. Introducing quantum Clifford algebras with arbitrary bilinear forms we
distinguish isomorphic algebras --as Clifford algebras-- by different
filtrations resp. induced gradings. The idea of a vacuum is introduced as the
unique algebraic projection on the base field embedded in the Clifford algebra,
which is however equivalent to the term vacuum in axiomatic quantum field
theory and the GNS construction in C^*-algebras. This approach is shown to be
equivalent to the usual picture which fixes one product but employs a variety
of GNS states. The most striking novelty of the geometric approach is the fact
that dynamical data fix uniquely the vacuum and that positivity is not
required. The usual concept of a statistical quantum state can be generalized
to geometric meaningful but non-statistical, non-definite, situations.
Furthermore, an algebraization of states takes place. An application to physics
is provided by an U(2)-symmetry producing a gap-equation which governs a phase
transition. The parameterization of all vacua is explicitly calculated from
propagator matrix elements. A discussion of the relation to BCS theory and
Bogoliubov-Valatin transformations is given.Comment: Major update, new chapters, 30 pages one Fig. (prev. 15p, no Fig.
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