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Graph-Directed Sprays and Their Tube Volumes via Functional Equations

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Derya Çeli̇k, Şahin Koçak, Yunus Özdemir, A. Ersin Üreyen

Metric Geometry · Submitted 27 Jul 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Using a simple functional equation satisfied by the volume of the inner $\varepsilon$-neighborhood of such a generalized spray, we establish a tube formula for them.

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arXiv ID1507.07461 (version 1)
Pith IDO57RNANU, Pith's permanent record of this paper and its review
CitationA. Ersin \"Ureyen, \c{S}ahin Ko\c{c}ak, Derya \c{C}el\.ik, Yunus \"Ozdemir. "Graph-Directed Sprays and Their Tube Volumes via Functional Equations." arXiv:1507.07461 (2015). https://pith.science/paper/O57RNANU
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,944 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.30 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

The notion of sprays introduced by Lapidus and his co-workers has proved useful in the context of fractal tube formulas. In the present note, we propose a more general concept of sprays, where we allow several generators to make them more convenient for applications to graph-directed fractals. Using a simple functional equation satisfied by the volume of the inner $\varepsilon$-neighborhood of such a generalized spray, we establish a tube formula for them.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The notion of sprays introduced by Lapidus and his co-workers has proved useful in the context of fractal tube formulas. background 1.00 0.04 0.03 asserted 1.00
s1 In the present note, we propose a more general concept of sprays, where we allow several generators to make them more convenient for applications to graph-directed fractals. contribution 1.00 0.89 0.09 asserted 0.96
s2 Using a simple functional equation satisfied by the volume of the inner $\varepsilon$-neighborhood of such a generalized spray, we establish a tube formula for them. result 0.49 0.95 0.86 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.82 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

The notion of sprays introduced by Lapidus and his co-workers has proved useful in the context of fractal tube formulas. In the present note, we propose a more general concept of sprays, where we allow several generators to make them more convenient for applications to graph-directed fractals. Using a simple functional equation satisfied by the volume of the inner $\varepsilon$-neighborhood of such a generalized spray, we establish a tube formula for them.

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