Cheeger and Yau Estimates on the Minimum Positive Eigenvalue

2017 ◽  
pp. 53-82
Keyword(s):  
Positive Eigenvalue

scholarly journals Instability of standing waves for non-linear Schrödinger-type equations

1988 ◽  
Vol 8 (8) ◽  
pp. 119-138 ◽  
Keyword(s):  
Dynamical Systems ◽  
Standing Wave ◽  
Optical Waveguides ◽  
Standing Waves ◽  
Positive Eigenvalue ◽  
Shooting Argument ◽  
Wave Solutions ◽  
Non Linear ◽  
Linearized Operator

AbstractA theorem is proved giving a condition under which certain standing wave solutions of non-linear Schrödinger-type equations are linearly unstable. The eigenvalue equations for the linearized operator at the standing wave can be analysed by dynamical systems methods. A positive eigenvalue is then shown to exist by means of a shooting argument in the space of Lagrangian planes. The theorem is applied to a situation arising in optical waveguides.

Signed graphs with small positive index of inertia

2016 ◽  
Vol 31 ◽  
pp. 232-243 ◽  
Author(s):  
Guihai Yu ◽  
Lihua Feng ◽  
Hui Qu
Keyword(s):  
Positive Eigenvalue ◽  
Signed Graphs ◽  
Unicyclic Graphs ◽  
Bicyclic Graphs ◽  
Positive Index

In this paper, the signed graphs with one positive eigenvalue are characterized, and the signed graphs with pendant vertices having exactly two positive eigenvalues are determined. As a consequence, the signed trees, the signed unicyclic graphs and the signed bicyclic graphs having one or two positive eigenvalues are characterized.

scholarly journals A partial converse to the Andreotti–Grauert theorem

2018 ◽  
Vol 155 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Xiaokui Yang
Keyword(s):  
Line Bundle ◽  
Ricci Curvature ◽  
Positive Eigenvalue ◽  
Projective Manifold ◽  
Hermitian Metric ◽  
Partial Converse

Let $X$ be a smooth projective manifold with $\dim _{\mathbb{C}}X=n$. We show that if a line bundle $L$ is $(n-1)$-ample, then it is $(n-1)$-positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold $X$ is uniruled if and only if there exists a Hermitian metric $\unicode[STIX]{x1D714}$ on $X$ such that its Ricci curvature $\text{Ric}(\unicode[STIX]{x1D714})$ has at least one positive eigenvalue everywhere.

The Eigenvalue Problem and Saint-Venant Decay Rate for a Nonhomogeneous Semi-Infinite Strip in Plane Strain

2013 ◽  
Vol 706-708 ◽  
pp. 1822-1826
Author(s):  
Qing Yang ◽  
Kai Zhang ◽  
Bai Lin Zheng ◽  
Jian Xin Zhu
Keyword(s):  
Numerical Calculation ◽  
Eigenvalue Problem ◽  
Decay Rate ◽  
Plane Strain ◽  
Plane Problem ◽  
Significant Influence ◽  
Positive Eigenvalue ◽  
Infinite Strip ◽  
Homogeneous Plane

The eigenvalue problem referring to a nonhomogeneous semi-infinite strip in plane strain is investigated here, by using the analogous methodology proposed by Papkovich and Fadle in homogeneous plane problem. Two types of nonhomogeneity are considered: (i) the modulus varies with the thickness coordinate exponentially, (ii) it varies with the length coordinate exponentially. The eigenvalues for these two cases are obtained by numerical calculation. By considering the smallest positive eigenvalue, the Saint-Venant Decay rate of the problem is estimated, which indicates material nonhomogeneity can have a significant influence on the Saint-Venant decay rate.

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