A simple proof for folds on both sides in complexes of graph homomorphisms

In this paper we study implications of folds in both parameters of Lovász' Hom complexes. There is an important connection between the topological properties of these complexes and lower bounds for chromatic numbers. We give a very short and conceptual proof of the fact that if is a fold of , then Hom collapses onto Hom, whereas Hom collapses onto Hom.

We also give an easy inductive proof of the only nonelementary fact which we use for our arguments: if is a closure operator on , then collapses onto .

     

Theoretical study of the polarized infrared spectra of the hydrogen bond in 2‐furoic acid crystal dimer

This work presents a theoretical simulation of ν_{O? H} and ν_{O? D} band shapes in the polarized infrared spectra of 2‐furoic acid dimer crystals measured at liquid‐nitrogen temperature. The line shapes are studied theoretically within the framework of the anharmonic couplings between low‐frequency hydrogen‐bond vibrations and degenerate excited states of high‐frequency hydrogen vibrations in hydrogen‐bonded dimers and the anharmonic coupling between the first excited state of the fast mode and the harmonics or band combinations of some low‐frequency bending modes, which lead to Fermi resonances.This approach takes into account the adiabatic approximation, the intrinsic anharmonicity of the low‐frequency mode through a Morse potential, Davydov coupling triggered by resonance exchange between the excited states of the fast modes of the two hydrogen bonds involved in the cyclic dimer, and the direct and indirect damping of the fast‐stretching modes of the hydrogen bonds and of the bending modes. The infrared spectral density was calculated within the linear response theory by Fourier transform of the autocorrelation function of the transition dipole moment operator of the fast mode. Numerical results show that mixing of all these effects allows satisfactory reproduction of the main features of the experimental IR line shapes of crystalline H‐ and D‐bonded 2‐furoic acid at liquid‐nitrogen temperature and for different polarizations. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Topological classification of continuous selections of five linear functions

Continuous selections of linear functions play an important role in Morse theory for piecewise C ²-functions. In this article, the topological properties of continuous selections of linear functions are investigated in detail. These are then utilized to provide a complete classification of all continuous selections of five linear functions. This is done by showing that the first four Betti numbers of a simplicial complex induced by such a function fully determine that function up to topological equivalence. The number of different topological types of continuous selections of linear functions has been known only in the case of four or less selection functions so far. The main result of this article now states that there are exactly 26 different topological types of continuous selections of five linear functions.     

Multiple Solutions for a Fractional p-Laplacian Equation with Concave Nonlinearities

We investigate a fractional $p$-Laplacian equation with right-hand-side nonlinearity which exhibits $(p-1)$-sublinear term of the form $\lambda |u|^{q-2},\, q

Morse theory.     

Critical point theorem for asymptotically quadratic functional without compactness

In this paper we build an abstract critical point theorem for the asymptotically quadratic functional without compactness. Application is devoted to the existence of nontrivial periodic solution of second order Hamiltonian systems.     

Liouville theorem for Choquard equation with finite Morse indices

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation-△u=∫RN|u(y)|p|x-y|αdy|u(x)|p-2u(x) in RN where N ≥ 3, 0 α min{4, N}. Suppose that 2 p (2 N-α)/(N-2),we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=(2 N-α)/(N-2),if i(u) ∞, then we have ∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~α dxdy ∞ and ∫_RN|▽u|~2 dx=∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~αdxdy.     

源自人口动力学的半线性p-Laplace的Dirichlet问题解

研究源自人口动力学的半线性p-Laplace方程的Dirichlet问题,得到了该问题在零点处的能量泛函是平凡的Morse临界群.因而,确定了该问题非平凡解的存在性及其分岔性.     

Complex Valued Bismut-Lott Index Theorem

In this paper, assuming there is a fiberwise Morse function, we extend Bismut-Lott index theorem to the complex valued case.