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Positivity-preserving Hâ model reduction for positive systems
This is the post-print version of the Article - Copyright @ 2011 ElevierThis paper is concerned with the model reduction of positive systems. For a given stable positive system, our attention is focused on the construction of a reduced-order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed Hâ performance. Based upon a system augmentation approach, a novel characterization on the stability with Hâ performance of the error system is first obtained in terms of linear matrix inequality (LMI). Then, a necessary and sufficient condition for the existence of a desired reduced-order model is derived accordingly. Furthermore, iterative LMI approaches with primal and dual forms are developed to solve the positivity-preserving Hâ model reduction problem. Finally, a compartmental network is provided to show the effectiveness of the proposed techniques.The work was partially supported by GRF HKU 7137/09E
Isolated Horizon, Killing Horizon and Event Horizon
We consider space-times which in addition to admitting an isolated horizon
also admit Killing horizons with or without an event horizon. We show that an
isolated horizon is a Killing horizon provided either (1) it admits a
stationary neighbourhood or (2) it admits a neighbourhood with two independent,
commuting Killing vectors. A Killing horizon is always an isolated horizon. For
the case when an event horizon is definable, all conceivable relative locations
of isolated horizon and event horizons are possible. Corresponding conditions
are given.Comment: 14 pages, Latex, no figures. Some arguments tightened. To appear in
Class. Quant. Gra
Genericness of inflation in isotropic loop quantum cosmology
Non-perturbative corrections from loop quantum cosmology (LQC) to the scalar
matter sector is already known to imply inflation. We prove that the LQC
modified scalar field generates exponential inflation in the small scale factor
regime, for all positive definite potentials, independent of initial conditions
and independent of ambiguity parameters. For positive semi-definite potentials
it is always possible to choose, without fine tuning, a value of one of the
ambiguity parameters such that exponential inflation results, provided zeros of
the potential are approached at most as a power law in the scale factor. In
conjunction with generic occurrence of bounce at small volumes, particle
horizon is absent thus eliminating the horizon problem of the standard Big Bang
model.Comment: 4 pages, revtex4, one figure. Only e-print archive numbers correctedi
in the second version. Reference added in the 3rd version. Final version to
appear in Phys. Rev. Lett. Explanations improve
Symplectic Geometries on , Hamiltonian Group Actions and Integrable Systems
Various Hamiltonian actions of loop groups \wt G and of the algebra
of first order differential operators in one variable are
defined on the cotangent bundle T^*\wt G of a Loop Group. The moment maps
generating the actions are shown to factorize through those
generating the loop group actions, thereby defining commuting diagrams of
Poisson maps to the duals of the corresponding centrally extended algebras. The
maps are then used to derive a number of infinite commuting families of
Hamiltonian flows that are nonabelian generalizations of the dispersive water
wave hierarchies. As a further application, sets of pairs of generators of the
nonabelian mKdV hierarchies are shown to give a commuting hierarchy on T^*\wt
G that contain the WZW system as its first element.Comment: 49 page
On Soliton Content of Self Dual Yang-Mills Equations
Exploiting the formulation of the Self Dual Yang-Mills equations as a
Riemann-Hilbert factorization problem, we present a theory of pulling back
soliton hierarchies to the Self Dual Yang-Mills equations. We show that for
each map \C^4 \to \C^{\infty } satisfying a simple system of linear
equations formulated below one can pull back the (generalized) Drinfeld-Sokolov
hierarchies to the Self Dual Yang-Mills equations. This indicates that there is
a class of solutions to the Self Dual Yang-Mills equations which can be
constructed using the soliton techniques like the function method. In
particular this class contains the solutions obtained via the symmetry
reductions of the Self Dual Yang-Mills equations. It also contains genuine 4
dimensional solutions . The method can be used to study the symmetry reductions
and as an example of that we get an equation exibiting breaking solitons,
formulated by O. Bogoyavlenskii, as one of the dimensional reductions
of the Self Dual Yang-Mills equations.Comment: 11 pages, plain Te
Geometric and combinatorial realizations of crystal graphs
For irreducible integrable highest weight modules of the finite and affine
Lie algebras of type A and D, we define an isomorphism between the geometric
realization of the crystal graphs in terms of irreducible components of
Nakajima quiver varieties and the combinatorial realizations in terms of Young
tableaux and Young walls. For affine type A, we extend the Young wall
construction to arbitrary level, describing a combinatorial realization of the
crystals in terms of new objects which we call Young pyramids.Comment: 34 pages, 17 figures; v2: minor typos corrected; v3: corrections to
section 8; v4: minor typos correcte
On obtaining classical mechanics from quantum mechanics
Constructing a classical mechanical system associated with a given quantum
mechanical one, entails construction of a classical phase space and a
corresponding Hamiltonian function from the available quantum structures and a
notion of coarser observations. The Hilbert space of any quantum mechanical
system naturally has the structure of an infinite dimensional symplectic
manifold (`quantum phase space'). There is also a systematic, quotienting
procedure which imparts a bundle structure to the quantum phase space and
extracts a classical phase space as the base space. This works straight
forwardly when the Hilbert space carries weakly continuous representation of
the Heisenberg group and recovers the linear classical phase space
. We report on how the procedure also allows
extraction of non-linear classical phase spaces and illustrate it for Hilbert
spaces being finite dimensional (spin-j systems), infinite dimensional but
separable (particle on a circle) and infinite dimensional but non-separable
(Polymer quantization). To construct a corresponding classical dynamics, one
needs to choose a suitable section and identify an effective Hamiltonian. The
effective dynamics mirrors the quantum dynamics provided the section satisfies
conditions of semiclassicality and tangentiality.Comment: revtex4, 24 pages, no figures. In the version 2 certain technical
errors in section I-B are corrected, the part on WKB (and section II-B) is
removed, discussion of dynamics and semiclassicality is extended and
references are added. Accepted for publication on Classical and Quantum
Gravit
Classical Many-particle Clusters in Two Dimensions
We report on a study of a classical, finite system of confined particles in
two dimensions with a two-body repulsive interaction. We first develop a simple
analytical method to obtain equilibrium configurations and energies for few
particles. When the confinement is harmonic, we prove that the first transition
from a single shell occurs when the number of particles changes from five to
six. The shell structure in the case of an arbitrary number of particles is
shown to be independent of the strength of the interaction but dependent only
on its functional form. It is also independent of the magnetic field strength
when included. We further study the effect of the functional form of the
confinement potential on the shell structure. Finally we report some
interesting results when a three-body interaction is included, albeit in a
particular model.Comment: Minor corrections, a few references added. To appear in J. Phys:
Condensed Matte
Isospectral flow in Loop Algebras and Quasiperiodic Solutions of the Sine-Gordon Equation
The sine-Gordon equation is considered in the hamiltonian framework provided
by the Adler-Kostant-Symes theorem. The phase space, a finite dimensional
coadjoint orbit in the dual space \grg^* of a loop algebra \grg, is
parametrized by a finite dimensional symplectic vector space embedded into
\grg^* by a moment map. Real quasiperiodic solutions are computed in terms of
theta functions using a Liouville generating function which generates a
canonical transformation to linear coordinates on the Jacobi variety of a
suitable hyperelliptic curve.Comment: 12 pg
Discreteness Corrections to the Effective Hamiltonian of Isotropic Loop Quantum Cosmology
One of the qualitatively distinct and robust implication of Loop Quantum
Gravity (LQG) is the underlying discrete structure. In the cosmological context
elucidated by Loop Quantum Cosmology (LQC), this is manifested by the
Hamiltonian constraint equation being a (partial) difference equation. One
obtains an effective Hamiltonian framework by making the continuum
approximation followed by a WKB approximation. In the large volume regime,
these lead to the usual classical Einstein equation which is independent of
both the Barbero-Immirzi parameter as well as . In this work we
present an alternative derivation of the effective Hamiltonian by-passing the
continuum approximation step. As a result, the effective Hamiltonian is
obtained as a close form expression in . These corrections to the
Einstein equation can be thought of as corrections due to the underlying
discrete (spatial) geometry with controlling the size of these
corrections. These corrections imply a bound on the rate of change of the
volume of the isotropic universe. In most cases these are perturbative in
nature but for cosmological constant dominated isotropic universe, there are
significant deviations.Comment: Revtex4, 24 pages, 3 figures. In version 2, one reference and a para
pertaining to it are added. In the version 3, some typos are corrected and
remark 4 in section III is revised. Final version to appear in Class. Quantum
Gra
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