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Methods Of Modern Mathematical Physics

Can one hear the shape of a drum

Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete.

The concept of a continuous module is a generalization of that of an injective module, and conditions (), (C) and () are given for this concept in [4]. In this paper, we study modules with properties that are dual to continuity. These will be called discrete and we discuss discrete abelian groups. Throughout R is a ring with identity, M is a module over R, G is an abelian group of finite rank, E is the ring of endomorphisms of G and S is the center of E. Dual to the notion of essential submodules, we define small submodules of a module M over R.(omitted)

On discrete groups

在1954年，科学界庆祝了Poincare的诞辰100周年纪念日，在那个时候，Poincare在数学家们当中的名声并没有达到最高峰，而希尔伯特的精神则主导了很多数学家的思想，Poincare在物理学家们当中的名声也没有达到最高峰，因为在那时候，物理学在本质上是和量子理论相关的。

Henri Poincare，为科学服务的一生

We study Laplacians on graphs and networks via regular Dirichlet forms. We give a sufficient geometric condition for essential selfadjointness and explicitly determine the generators of the associated semigroups on all $\ell^p$, $1\leq p < \infty$, in this case. We characterize stochastic completeness thereby generalizing all earlier corresponding results for graph Laplacians. Finally, we study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph.

Dirichlet forms and stochastic completeness of graphs and subgraphs

We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.

/pdf/unbounded-laplacians-on-graphs-basic-spectral-properties-and-1cbu2po6b3.pdf

Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation

We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally �nite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard formal Laplacian and we can characterize the Dirichlet and Neumann Laplacians as the largest and smallest Markovian restrictions of the standard formal Laplacian. In the case of general graphs, this class contains the Dirichlet and Neumann Laplacians and we describe how these may dier from each other, characterize when they agree, and study connections to essential selfadjointness and stochastic completeness. Finally, we study basic common features of all Laplacians associated to a graph. In particular, we characterize when the associated semigroup is posi- tivity improving and present some basic estimates on its long term behavior. We also characterize boundedness of the Laplacian.

https://www.ems-ph.org/journals/show_pdf.php?issn=1664-039X&vol=2&iss=4&rank=4

Matthias Keller

Papers

Dirichlet forms and stochastic completeness of graphs and subgraphs

Dirichlet forms and stochastic completeness of graphs and subgraphs

Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation

Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions

Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation