一类分数阶脉冲微分方程Dirichlet边值问题非平凡解的存在性

研究了一类带有脉冲的分数阶微分方程Dirichlet边值问题非平凡解的存在性.通过利用变分法和Morse理论证明了此分数阶脉冲微分方程至少存在一个非平凡解.     

Morse函数与非紧流形之加边

设M为连通的非紧的光滑的n维流形带有紧边缘bM.本文定义了M的强Morse函数和强Morse数.M称为有限型的,若M的强Morse数为有限。本文的主要结果是:M可加边的充要条件为M是有限型的.     

Morse函数与非紧流形之加边

设M为连通的非紧的光滑的n维流形带有紧边缘bM.本文定义了M的强Morse函数和强Morse数.M称为有限型的,若M的强Morse数为有限。本文的主要结果是:M可加边的充要条件为M是有限型的.     

具有不定位势Kirchhoff-Schr?dinger-Poisson系统解的存在性

本文研究Kirchhoff-Schr?dinger-Poisson系统.与大多数研究不同,本文允许位势函数是不定的,即位势函数具有有限维负空间.在Morse理论的帮助下,作者得到了Kirchhoff-Schr?dinger-Poisson系统非平凡解的存在性.     

一类具有组合非线性项的分数阶方程的多重解（英文）

本文研究一类具有组合非线性项的分数阶Laplacian方程,在共振与非共振情形下,运用山路理论、Morse理论、Ekeland变分原理,建立了5个非平凡解的存在性结果.     

非线性Hénon方程解的Liouville定理

该文研究了二阶和四阶非线性Henon-Lane-Emden方程有限Morse指标解的Liouville定理.利用一种新方法,即使用单调公式、Pohozaev恒等式和doubling引理等相结合证明了其结果.     

En中齐次多项式芽生成的有限余维理想的判定和应用

本文研究了奇点理论中有限余维理想的一种判定方法,利用Arnold在θn中得出的结论以及Hilbert零点定理,获得C∞实函数芽环En中由齐次多项式芽生成的有限余维理想的特征和判定方法. 其结果是有实用性和有效性的.     

Morse理论及其在一类N-Laplacian方程中的应用

研究了一类次临界增长的N-Laplacian方程.利用Trudinger-Moser不等式和Morse理论,证明了在不具有山路引理的几何性质、不满足Ambrosetti-Rabinowitz条件及整体符号条件下,该方程非平凡解的存在性.     

函数芽的有限V决定性和I决定性

本文定义了函数芽的一种V等价关系,并给出函数芽有限V决定的充要条件。同时得出函数芽I决定和∞决定的条件。并且定义了理想I,使函数芽I决定的判定定理有着十分简明的形式,而且作为一个推论直接给出函数芽∞决定的条件。设E_是全体C~∞函数芽(R~n,0)→R所构成的环。m_={f∈E_|f(0)=0},可以证明它是E_。     

二阶非线性差分方程多重周期解的存在性

本文研究了一类带参数非自治的二阶非线性差分方程多重周期解的存在性问题.利用Morse理论,建立了此类方程多重周期解的存在性准则.     

Morse functions statistics

We answer a question of Arnold concerning the growth rate of the number of Morse functions on the two-sphere. This work was supported in part by the NSF (Grant No. DMS-0303601).     

The Morse–Witten complex via dynamical systems

Given a smooth closed manifold M, the Morse–Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in Weber [Der Morse–Witten Komplex, Diploma Thesis, TU Berlin, 1993] and is based on tools from hyperbolic dynamical systems. For instance, we apply the Grobman–Hartman theorem and the λ-lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.     

Fundamental Theorems of Morse Theory for Optimization on Manifolds with Corners

This note extends the fundamental theorems of Morse theory for stable stationary solutions to optimization problems on manifolds with corners.     

Poincaré-Hopf inequalities

In this article the main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse inequalities to hold. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described.

     

Improved bounds for interatomic distance in Morse clusters

We improve the best known lower bounds on the distance between two points of an optimal Morse cluster, with ρ[4.967,15]. We develop a generalization of a method previously applied to the Lennard-Jones potential, that also leads to improvements of lower bounds for the Morse potential.     

On Functions Whose All Critical Points Are Contained in a Ball

The normal decomposition of operator spaces into inductive scales of locally convex spaces in accordance with the classification of operators by their normal indices is considered. The canonical isomorphisms of operator spaces over Banach space are generalized to operators in locally convex spaces.     

The second closed geodesic on a complex projective plane

We show the existence of at least two geometrically distinct closed geodesics on a complex projective plane with a bumpy and non-reversible Finsler metric.      

R^N中taut子流形的性质与例

给出了紧致taut子流形的一个性质,即它确实是由标准球面堆砌（buildup）而成的．本文也证明了某类近复子流形可taut嵌入到RN．     

ON A BI-HARMONIC EQUATION INVOLVING CRITICAL EXPONENT: EXISTENCE AND MULTIPLICITY RESULTS

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: △2 u = K(x)u p , u > 0 in Ω , △u = u = 0 on Ω , where Ω is a smooth domain in R n , n 5, and p + 1 = 2 n n 4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.     

The stability theorem for smooth pseudoisotopies

The stability theorem states that the suspension map C(M) C(M X I) defined on the pseudoisotopy space C(M)=Diff(M X I rel M X O U M X I) of a compact smooth n-manifold M is n/3-connected. This implies that C(M) has the R~ n/3-homotopy type of the stable pseudoisotopy space P(M) which is related to Waldhausen's algebraic K-theory of spaces by Waldhausen's formula A(X) S(X₊) X B2P(X). This paper gives a detailed proof of the smooth stability theorem following ideas by Hatcher for the proof of a PL stability theorem.Supported by NSF Grant No. MCS-85-02317.