51,781 research outputs found
The Ricci flow approach to homogeneous Einstein metrics on flag manifolds
We give the global picture of the normalized Ricci flow on generalized flag
manifolds with two or three isotropy summands. The normalized Ricci flow for
these spaces descents to a parameter depending system of two or three ordinary
differential equations, respectively. We present here the qualitative study of
these system's global phase portrait, by using techniques of Dynamical Systems
theory. This study allows us to draw conclusions about the existence and the
analytical form of invariant Einstein metrics on such manifolds, and seems to
offer a better insight to the classification problem of invariant Einstein
metrics on compact homogeneous spaces.Comment: 17 pages, 2 figure
Spatial Aggregation: Theory and Applications
Visual thinking plays an important role in scientific reasoning. Based on the
research in automating diverse reasoning tasks about dynamical systems,
nonlinear controllers, kinematic mechanisms, and fluid motion, we have
identified a style of visual thinking, imagistic reasoning. Imagistic reasoning
organizes computations around image-like, analogue representations so that
perceptual and symbolic operations can be brought to bear to infer structure
and behavior. Programs incorporating imagistic reasoning have been shown to
perform at an expert level in domains that defy current analytic or numerical
methods. We have developed a computational paradigm, spatial aggregation, to
unify the description of a class of imagistic problem solvers. A program
written in this paradigm has the following properties. It takes a continuous
field and optional objective functions as input, and produces high-level
descriptions of structure, behavior, or control actions. It computes a
multi-layer of intermediate representations, called spatial aggregates, by
forming equivalence classes and adjacency relations. It employs a small set of
generic operators such as aggregation, classification, and localization to
perform bidirectional mapping between the information-rich field and
successively more abstract spatial aggregates. It uses a data structure, the
neighborhood graph, as a common interface to modularize computations. To
illustrate our theory, we describe the computational structure of three
implemented problem solvers -- KAM, MAPS, and HIPAIR --- in terms of the
spatial aggregation generic operators by mixing and matching a library of
commonly used routines.Comment: See http://www.jair.org/ for any accompanying file
Particle-Like Description in Quintessential Cosmology
Assuming equation of state for quintessential matter: , we
analyse dynamical behaviour of the scale factor in FRW cosmologies. It is shown
that its dynamics is formally equivalent to that of a classical particle under
the action of 1D potential . It is shown that Hamiltonian method can be
easily implemented to obtain a classification of all cosmological solutions in
the phase space as well as in the configurational space. Examples taken from
modern cosmology illustrate the effectiveness of the presented approach.
Advantages of representing dynamics as a 1D Hamiltonian flow, in the analysis
of acceleration and horizon problems, are presented. The inverse problem of
reconstructing the Hamiltonian dynamics (i.e. potential function) from the
luminosity distance function for supernovae is also considered.Comment: 35 pages, 26 figures, RevTeX4, some applications of our treatment to
investigation of quintessence models were adde
Anisotropic cosmological models with spinor and scalar fields and viscous fluid in presence of a term: qualitative solutions
The study of a self-consistent system of interacting spinor and scalar fields
within the scope of a Bianchi type I (BI) gravitational field in presence of a
viscous fluid and term has been carried out. The system of equations
defining the evolution of the volume scale of BI universe, energy density and
corresponding Hubble constant has been derived. The system in question has been
thoroughly studied qualitatively. Corresponding solutions are graphically
illustrated. The system in question is also studied from the view point of blow
up. It has been shown that the blow up takes place only in presence of
viscosity.Comment: 18 pages, 14 figures, 12 Tables, section "Basic equations" has been
rewritte
Solutions Classification to the Extended Reduced Ostrovsky Equation
An alternative to the Parkes' approach [SIGMA 4 (2008) 053, arXiv:0806.3155]
is suggested for the solutions categorization to the extended reduced Ostrovsky
equation (the exROE in Parkes' terminology). The approach is based on the
application of the qualitative theory of differential equations which includes
a mechanical analogy with the point particle motion in a potential field, the
phase plane method, analysis of homoclinic trajectories and the like. Such an
approach is seemed more vivid and free of some restrictions contained in the
above mentioned Parkes' paper.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Singularities and qualitative study in LQC
We will perform a detailed analysis of singularities in Einstein Cosmology and in LQC (Loop Quantum Cosmology). We will obtain explicit analytical expressions for the energy density and the Hubble constant for a given set of possible Equations of State. We will also consider the case when the background is driven by a single scalar field, obtaining analytical expressions for the corresponding potential. And, in a given particular case, we will perform a qualitative study of the orbits in the associated phase space of the scalar. eld
Polynomial normal forms of Constrained Differential Equations with three parameters
We study generic constrained differential equations (CDEs) with three
parameters, thereby extending Takens's classification of singularities of such
equations. In this approach, the singularities analyzed are the Swallowtail,
the Hyperbolic, and the Elliptic Umbilics. We provide polynomial local normal
forms of CDEs under topological equivalence. Generic CDEs are important in the
study of slow-fast (SF) systems. Many properties and the characteristic
behavior of the solutions of SF systems can be inferred from the corresponding
CDE. Therefore, the results of this paper show a first approximation of the
flow of generic SF systems with three slow variables.Comment: This is an updated and revised version. Minor modifications mad
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