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31.
John Conrad Merkel 《Journal of Dynamics and Differential Equations》2008,20(3):653-668
In the spirit of Palmore and Pacella, Morse Theory is used to obtain a lower bound for the number of central configurations
in the spatial N-body problem. The homology of the configuration ellipsoid with the collision and collinear manifolds removed and the SO(3) symmetry quotiented out is calculated. As intermediate steps, homology calculations are carried out for several additional
manifolds naturally arising in the N-body problem. 相似文献
32.
A simple method is proposed leading to reasonable approximations of the partition function of Morse oscillators in terms of
elementary functions. 相似文献
33.
The Vapnik–Chervonenkis‐dimension is an index of the capacity of a learning machine. It has been computed in several cases, but always in a Euclidean context. This paper extends the notion to classifiers acting in the more general environment of a manifold. General properties are proved, and some examples of simple classifiers on elementary manifolds are given. A large part of the research is directed toward a still open problem on product manifolds. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
34.
D. Visetti 《Journal of Differential Equations》2008,245(9):2397-2439
The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory. 相似文献
35.
Lars Grüne 《Journal of Dynamics and Differential Equations》2000,12(2):435-448
For linear flows on vector bundles we define a uniform exponential spectrum. For a compact invariant set for the projected flow we obtain this spectrum by taking all accumulation points for the time tending to infinity of the union over the finite time exponential growth rates for all initial values in this set. Using direct arguments we show that for a connected compact invariant set this spectrum is a closed interval whose boundary points are Lyapunov exponents. For a compact invariant set on which the flow is chain transitive we show that this spectrum coincides with the Morse spectrum. In particular, this approach admits a straightforward analytic proof for the regularity and continuity properties of the Morse spectrum without using cohomology or ergodicity results. 相似文献
36.
In this paper, we discuss the bifurcation problems for strongly indefinite functional via Morse theory. The generalized topological degree for a class of vector fields is defined. As applications, we study the bifurcation problems for Hamiltonian system and noncooperative elliptic system. 相似文献
37.
Jianxin Zhou. 《Mathematics of Computation》2005,74(251):1391-1411
The objective of this work is to develop some tools for local instability analysis of multiple critical points, which can be computationally carried out. The Morse index can be used to measure local instability of a nondegenerate saddle point. However, it is very expensive to compute numerically and is ineffective for degenerate critical points. A local (weak) linking index can also be defined to measure local instability of a (degenerate) saddle point. But it is still too difficult to compute. In this paper, a local instability index, called a local minimax index, is defined by using a local minimax method. This new instability index is known beforehand and can help in finding a saddle point numerically. Relations between the local minimax index and other local instability indices are established. Those relations also provide ways to numerically compute the Morse, local linking indices. In particular, the local minimax index can be used to define a local instability index of a saddle point relative to a reference (trivial) critical point even in a Banach space while others failed to do so.
38.
Martin Rasmussen 《Proceedings of the American Mathematical Society》2008,136(3):1045-1055
Morse decompositions provide inside information about the global asymptotic behavior of dynamical systems on compact metric spaces. Recently, the existence of Morse decompositions for nonautonomous dynamical systems was proved by restricting attention to the past or the future of the system, but in general, such a construction is not realizable for the entire time. In this article, it is shown that all-time Morse decompositions can be defined for linear systems on the projective space. Moreover, the dynamical properties are discussed and an analogue to the Theorem of Selgrade is proved.
39.
Mireille Bousquet-Mélou Svante Linusson Eran Nevo 《Journal of Algebraic Combinatorics》2008,27(4):423-450
The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution
of hard-particle models. In 2004, it was conjectured by Fendley et al., that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle,
the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the
hard-particle model on these grids at activity −1. This conjecture was recently proved by Jonsson.
Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold
without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, ±1, or, in the
cylindric case, by some power of 2.
We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy
type of these complexes using Forman’s discrete Morse theory. We find that these complexes are either contractible, or homotopic
to a sphere, or, in the cylindric case, to a wedge of spheres.
Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle
model on our graphs at activity −1. These results parallel certain conjectures of Fendley et al., proved by Jonsson in the toric case. 相似文献
40.
《Journal of computational chemistry》2017,38(23):1991-1999
An accurate van der Waals force field (VDW FF) was derived from highly precise quantum mechanical (QM) calculations. Small molecular clusters were used to explore van der Waals interactions between gas molecules and porous materials. The parameters of the accurate van der Waals force field were determined by QM calculations. To validate the force field, the prediction results from the VDW FF were compared with standard FFs, such as UFF, Dreiding, Pcff, and Compass. The results from the VDW FF were in excellent agreement with the experimental measurements. This force field can be applied to the prediction of the gas density (H2, CO2, C2H4, CH4, N2, O2) and adsorption performance inside porous materials, such as covalent organic frameworks (COFs), zeolites and metal organic frameworks (MOFs), consisting of H, B, N, C, O, S, Si, Al, Zn, Mg, Ni, and Co. This work provides a solid basis for studying gas adsorption in porous materials. © 2017 Wiley Periodicals, Inc. 相似文献