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Lower bounds for nodal sets of Dirichlet and Neumann eigenfunctions

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Sinan Ariturk

Analysis of PDEs · Submitted 31 Oct 2011

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

The authors' main claim We prove lower bounds for the size of the nodal set {\phi=0}.

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arXiv ID1110.6885 (version 1)
Pith IDXKGQCCAE, Pith's permanent record of this paper and its review
CitationSinan Ariturk. "Lower bounds for nodal sets of Dirichlet and Neumann eigenfunctions." arXiv:1110.6885 (2011). https://pith.science/paper/XKGQCCAE
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The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,250 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.40 · compares with prior results 0.05 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {\phi=0}.

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 Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. background 0.99 0.06 0.02 asserted 0.65
s1 We prove lower bounds for the size of the nodal set {\phi=0}. result 0.59 0.96 0.94 asserted 1.00

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

What was examined, and what was not

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not examined
  • full text
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  • references and citing sentences
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  • T body sentences
  • C citations
  • R cross-paper relations

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

Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {\phi=0}.

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2 arXiv papers cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 1 sentence without a confident label

Cites as background 1 paper
  • If $(M,g)$ is a compact real analytic Riemannian manifold with boundary, then Donnelly and Fefferman [ref] proved that [equation] If $(M,g)$ is a compact smooth Riemannian manifold without boundary, then Colding and Minicozzi [ref] proved that

    Lower bounds for nodal sets of eigenfunctions · Introduction · 0.92

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