End compactifications in non-locally-finite graphs

There are different definitions of ends in non-locally-finite graphs which are all equivalent in the locally finite case. We prove the compactness of the end-topology that is based on the principle of removing finite sets of vertices and give a proof of the compactness of the end-topology that is constructed by the principle of removing finite sets of edges. For the latter case there exists already a proof in \cite{cartwright93martin}, which only works on graphs with countably infinite vertex sets and in contrast to which we do not use the Theorem of Tychonoff. We also construct a new topology of ends that arises from the principle of removing sets of vertices with finite diameter and give applications that underline the advantages of this new definition.Comment: 17 pages, to appear in Math. Proc. Cambridge Philos. So

From Euclidean Geometry to Knots and Nets

This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe

Physical Consequences of Complex Dimensions of Fractals

It has recently been realized that fractals may be characterized by complex dimensions, arising from complex poles of the corresponding zeta function, and we show here that these lead to oscillatory behavior in various physical quantities. We identify the physical origin of these complex poles as the exponentially large degeneracy of the iterated eigenvalues of the Laplacian, and discuss applications in quantum mesoscopic systems such as oscillations in the fluctuation $\Sigma^2 (E)$ of the number of levels, as a correction to results obtained in Random Matrix Theory. We present explicit expressions for these oscillations for families of diamond fractals, also studied as hierarchical lattices.Comment: 4 pages, 3 figures; v2: references added, as published in Europhysics Letter

Spectral analysis on infinite Sierpinski fractafolds

A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpinski gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blowups of a Sierpinski gasket, where spectrum is pure point of infinite multiplicity. Both works used the method of spectral decimations to obtain explicit description of the eigenvalues and eigenfunctions. In this paper we combine the ideas from these earlier works to obtain a description of the spectral resolution of the Laplacian for noncompact fractafolds. Our main abstract results enable us to obtain a completely explicit description of the spectral resolution of the fractafold Laplacian. For some specific examples we turn the spectral resolution into a "Plancherel formula". We also present such a formula for the graph Laplacian on the 3-regular tree, which appears to be a new result of independent interest. In the end we discuss periodic fractafolds and fractal fields

Green-Functions on Self-Similar Graphs and BOUNDS FOR THE SPECTRUM OF THE LAPLACIAN

Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method of spectral analysis on self-similar graphs. We give an axiomatic definition of self-similar graphs which correspond to general nested but not necessarily finitely ramified fractals. For this class of graphs a graph theoretic analogue to the Banach fixed point theorem is proved. Functional equations and a decomposition algorithm for the Green functions of self-similar graphs with some more symmetric structure are obtained. Their analytic continuations are given by rapidly converging expressions. We study the dynamics of a certain complex rational Green function d on finite directed subgraphs. If the Julia set J of d is a Cantor set, then the reciprocal spectrum spec −1 P = {1/z | z ∈ spec P} of the Markov transition operator P can be identified with the set of singularities of any Green function of the whole graph. Finally we get explicit upper and lower bounds for the reciprocal spectrum, where D is a countable set of the d-backwards iterates of a certain finite set of real numbers

VERTEX CUTS

We generalise structure tree theory, which is based on removing finitely many edges, to removing finitely many vertices. This gives a significant generalization of Tutte’s tree decomposition of 2-connected graphs into 3-connected blocks. For a finite graph there is a structure tree that contains information about k-connectivity for any k. The theory can also be applied to infinite graphs that have more than one vertex end, i.e. ends that can be separated by removing a finite number of vertices. This gives a generalization of Stallings’ structure theorem for groups with more than one end