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31.
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.

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32.
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.

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33.
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   
34.
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   
35.
In this article, we extend a previous work toward presenting a theoretical study of the effects of Fermi resonances and the fundamental anharmonic coupling parameter α between the high‐frequency mode and the H‐bond bridge. The model incorporates (i) both intrinsic anharmonicities of the fast mode (double well potential) and the H‐bond Bridge (Morse potential), (ii) strong anharmonic coupling theory, (iii) Fermi resonances by the aid of an anharmonic coupling between the fast mode and one or several harmonic bending modes, (iv) quadratic modulation of both the angular frequency and the equilibrium position of the X? …Y stretching mode on the intermonomer ? H… motions, and (v) the quantum direct (fast and bending modes) and indirect dampings (slow mode). The IR spectral density is obtained by Fourier transform of the autocorrelation function of the transition dipole moment operator of the X? H bond. The numerical calculation shows that Fermi resonances generate very complicated profiles with multisubstructure and also provide a direct evidence of Fermi resonances which were predicted to be a major feature of H‐bonds. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010  相似文献   
36.
For the first time, the phonon energy per unit volume in a large anharmonic quasi-one-dimensional solid is determined by considering all polarizations of the various modes of phonon propagation and by assuming the solid as a lattice of atoms behaving as the Morse oscillators. In this context the equilibrium phonon occupation number, which is given by the Bose distribution, replaces formally the vibrational quantum number into the expression for the Morse-oscillator energy. In addition, the quasi-harmonic solid is discussed within the above framework so that the phonon energy per unit volume is calculated for a large quasi-harmonic and quasi-one-dimensional solid.  相似文献   
37.
Nano-scale soliton-like supersonic, intrinsic localized excitations in two-dimensional atomic anharmonic lattice layers are here considered. We study the propagation, the velocity and other soliton-like features at head-on collisions of such lattice excitations created by using suitable initial mechanical and thermal conditions. Noteworthy is that narrow, highly-energetic solitons moving along one lattice row are very robust, accompanied by weak anti-phase oscillations in the lateral direction.  相似文献   
38.
We derive the motion equations and the structure equations of neutral and charged test particles, starting from the gravitational field equations. The method consists in the application of conservation laws to singular tensor densities, which represent regions of strong matter concentration. Moreover, a Hamiltonian formulation of the particle equations is given, in the form of implicit differential equations generated by Hamiltonian Morse families.  相似文献   
39.
We consider a Schrödinger–Poisson system in R3R3 with potential indefinite in sign and a general nonlinearity. We use the direct variational method and Morse theory to obtain the existence of multiple nontrivial solutions for this system.  相似文献   
40.
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