142 research outputs found
The H\"older-Poincar\'e Duality for -cohomology
We prove the following version of Poincare duality for reduced
-cohomology: For any , the -cohomology of a
Riemannian manifold is in duality with the interior 1/p+1/p'=11/q+1/q'=1$.Comment: 21 page
Joint characteristic timescales and entropy production analyses for model reduction of combustion systems
The reduction of chemical kinetics describing combustion processes remains one of the major topics in the combustion theory and its applications. Problems concerning the estimation of reaction mechanisms real dimension remain unsolved, this being a critical point in the development of reduction models. In this study, we suggest a combination of local timescale and entropy production analyses to cope with this problem. In particular, the framework of skeletal mechanism is in the focus of the study as a practical and most straightforward implementation strategy for reduced mechanisms. Hydrogen and methane/dimethyl ether reaction mechanisms are considered for illustration and validation purposes. Two skeletal mechanism versions were obtained for methane/dimethyl ether combustion system by varying the tolerance used to identify important reactions in the characteristic timescale analysis of the system. Comparisons of ignition delay times and species profiles calculated with the detailed and the reduced models are presented. The results of the application show transparently the potential of the suggested approach to be automatically implemented for the reduction of large chemical kinetic models
Whirl mappings on generalised annuli and the incompressible symmetric equilibria of the dirichlet energy
In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions.Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation u=(u1,âŠ,uN) :
EL[u,X]=â§â©âšâȘâȘÎu=div(P(x)cofâu)detâu=1uâĄÏinX,inX,onâX,
where X is a finite, open, symmetric N -annulus (with Nâ„2 ), P=P(x) is an unknown hydrostatic pressure field and Ï is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when N=3 , the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when N=2 , the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions Nâ„4 and discuss a number of closely related issues
About Bifurcational Parametric Simplification
A concept of âcriticalâ simplification was proposed by Yablonsky and Lazman (1996) for
the oxidation of carbon monoxide over a platinum catalyst using a Langmuir-Hinshelwood
mechanism. The main observation was a simplification of the mechanism at ignition and
extinction points. The critical simplification is an example of a much more general
phenomenon that we call a bifurcational parametric simplification.
Ignition and extinction points are points of steady state multiplicity bifurcations, i.e.,
they are points of a corresponding bifurcation set for parameters.
Any bifurcation produces a dependence between system parameters. This is a mathematical
explanation and/or justification of the âparametric simplificationâ. It leads us to a
conjecture that âmaximal bifurcational parametric simplificationâ corresponds to the
âmaximal bifurcation complexity.â
This conjecture can have practical applications for experimental study, because at points
of âmaximal bifurcation complexityâ the number of independent system parameters is minimal
and all other parameters can be evaluated analytically or numerically.
We illustrate this method by the case of the simplest possible bifurcation, that is a
multiplicity bifurcation of steady state and we apply this analysis to the Langmuir
mechanism. Our analytical study is based on a coordinate-free version of the method of
invariant manifolds by Bykov, Goldfarb, Golâdshtein (2006). As a result we obtain a more
accurate description of the âcritical (parametric) simplifications.â
With the help of the âbifurcational parametric simplificationâ kinetic mechanisms and
reaction rate parameters may be readily identified from a monoparametric experiment
(reaction rate vs. reaction parameter)
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