Functions of finite Dirichlet sums and compactifications of infinite graphs

Hattori, Tae; Kasue, Atsushi

doi:10.2969/aspm/05710141

Cited by 5 publications

(4 citation statements)

References 3 publications

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“…The arguments there are valid for a proof of Proposition 4.7 and further it is possible to show the proposition for functions of finite Dirichlet sums of order p > 1. We refer the reader to [2] for some related results to Proposition 4.3, [7], [19] for those to Proposition 4.6, and [25] for some extensions of Proposition 4.1, Theorem 4.2 and Proposition 4.7 to the case of Dirichlet sums of order p > 1.…”

Section: Infinite Networkmentioning

confidence: 99%

Convergence of metric graphs and energy forms

Kasue¹

2010

Rev. Mat. Iberoam.

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In this paper, we begin with clarifying spaces obtained as limits of sequences of finite networks from an analytic point of view, and we discuss convergence of finite networks with respect to the topology of both the Gromov-Hausdorff distance and variational convergence called Γ-convergence. Relevantly to convergence of finite networks to infinite ones, we investigate the space of harmonic functions of finite Dirichlet sums on infinite networks and their Kuramochi compactifications.

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Section: Infinite Networkmentioning

confidence: 99%

Convergence of metric graphs and energy forms

Kasue¹

2010

Rev. Mat. Iberoam.

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“…To make this paper accessible to diverse audiences, we have included a number of definitions we shall need from the theory of (i) infinite networks, and (ii) the use of unbounded operators on Hilbert space in discrete contexts. Some useful background references for the first are [Soa] and [Woe2] and the multifarious references cited therein; see also [Yam,Zem,HK,KY3,KY2,KY1,MYY,vBL,DJ2]. For the second, see [DS] (especially Ch.…”

Section: Introductionmentioning

confidence: 99%

Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian

Jorgensen,

Pearse

2015

Preprint

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We compare the space of square-summable functions on an infinite graph (denoted 2 (G)) with the space of functions of finite energy (denoted H E ). There is a notion of inclusion that allows 2 (G) to be embedded into H E , but the required inclusion operator is unbounded in most interesting cases. These observations assist in the construction of the Krein extension of the Laplace operator on H E . We investigate the Krein extension and compare it to the Friedrichs extension developed by the authors in a previous paper.

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“…G (x, y) is called the effective resistance between x and y; the quantity R ( p) G (x, y) for p > 1 has been considered in [9,10,14,15]. We note that the reciprocal of R ( p) G (x, y) is the p-capacity of the pair (x, y).…”

mentioning

confidence: 98%

“…Obviously, (R ( p) G ) 1/ p induces a distance on V and we note that for x, y ∈ V , R [14,15]). For p = 2, R…”

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Expansion Constants and Hyperbolic Embeddings of Finite Graphs

Hattori

Kasue²

2014

Mathematika

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In this paper, we study a finite connected graph which admits a quasimonomorphism to hyperbolic spaces and give a geometric bound for the Cheeger constants in terms of the volume, an upper bound of the degree, and the quasimonomorphism. §1. Introduction. We consider a graph G = (V, E) with the set of vertices V and the set of edges E that consists of pairs of vertices. In this paper, a graph admits no loops and multiple edges, and the set of vertices is finite or countably infinite. A graph is considered as a metric space with its graph distance d G .There are various results known on geometric bounds for the Cheeger constant, or the expansion constant h(G) and the second eigenvalue, or the first positive eigenvalue λ 2 (G) of the combinatorial Laplacian of a connected finite graph G = (V, E), in terms of an upper bound of the degree, d, the cardinality of V , |V |, the diameter of G, diam(G), and other characteristic properties.

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Functions of finite Dirichlet sums and compactifications of infinite graphs

Cited by 5 publications

References 3 publications

Convergence of metric graphs and energy forms

Convergence of metric graphs and energy forms

Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian

Expansion Constants and Hyperbolic Embeddings of Finite Graphs

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