论文标题
常数和图表上的热流
Constants and heat flow on graphs
论文作者
论文摘要
在本文中，我们首先介绍了向量场及其分歧的概念，我们回想起渐变，拉普拉斯运营商，Cheeger常数，特征值和热核在本地有限的图形$ v $上的概念。我们给出了特征值的投影特征。我们还提供了Barta定理的扩展。然后，我们在局部有限的图表上介绍了函数的迷你最大值。我们表明，对于在本地有限和本地连接的图上的强制函数，只要它具有两个严格的局部最小值，就有一个函数的迷你最大值。我们考虑在本地有限图$ v $中的有限图上的热流的离散摩尔斯流量。我们表明，在图表上的合适假设下，在任何时间间隔内，$ s $上的热流的离散摩尔斯流量很弱。我们还以时变电位及其离散的摩尔斯流量来研究热流。我们提出了谐波图的概念，从图到riemannian歧管，并提出了一些开放的问题。
In this article, we first introduce the concepts of vector fields and their divergence, and we recall the concepts of the gradient, Laplacian operator, Cheeger constants, eigenvalues, and heat kernels on a locally finite graph $V$. We give a projective characteristic of the eigenvalues. We also give an extension of Barta Theorem. Then we introduce the mini-max value of a function on a locally finite and locally connected graph. We show that for a coercive function on on a locally finite and locally connected graph, there is a mini-max value of the function provided it has two strict local minima values. We consider the discrete Morse flow for the heat flow on a finite graph in the locally finite graph $V$. We show that under suitable assumptions on the graph one has a weak discrete Morse flow for the heat flow on $S$ on any time interval. We also study the heat flow with time-variable potential and its discrete Morse flow. We propose the concepts of harmonic maps from a graph to a Riemannian manifold and pose some open questions.
