{"arxiv_id":"2207.11305","incoming":{"papers":0,"sentences":0,"groups":[],"unlabelled_sentences":0},"outgoing":{"papers":3,"sentences":10,"groups":[{"kind":"method_or_tool","label":"Uses the method or tool of","papers":2,"sentences":2,"quotes":[{"arxiv_id":"2004.12522","title":"Foliated corona decompositions","url":"/paper/2004.12522","section":"The case $p<q$","sentence":"The following is an imitation of the argument of [ref] .","intent_p":0.89,"contests_p":null},{"arxiv_id":"2109.05064","title":"Mean Value Inequalities and Characterizations of Sobolev spaces on graded groups","url":"/paper/2109.05064","section":"Proof of Dorronsoro's Theorem [ref] for Carnot groups","sentence":"For $0<α≤1$ we again use characterizations of the fractional Laplacian given by [ref] for $0<α<1$ and [ref] for $α=1$ (subsubsection [ref]).","intent_p":0.95,"contests_p":null}]},{"kind":"data_used","label":"Uses data or values from","papers":1,"sentences":1,"quotes":[{"arxiv_id":"2004.12522","title":"Foliated corona decompositions","url":"/paper/2004.12522","section":"$L^1$-distortion: vertical perimeter versus horizontal perimeter","sentence":"The exponent range is $[4,∞)$ for $G=ℍ^3$ [ref] and is $[2,∞)$ for $G=ℍ^k$, $k≥5$ [ref] .","intent_p":0.88,"contests_p":null}]},{"kind":"background","label":"Cites as background","papers":1,"sentences":1,"quotes":[{"arxiv_id":"2004.12522","title":"Foliated corona decompositions","url":"/paper/2004.12522","section":"Introduction","sentence":"See also Naor and Young [ref] for the endpoint case $p=1$ for the Heisenberg groups.","intent_p":0.83,"contests_p":null}]}],"unlabelled_sentences":6},"claims":[],"thresholds":{"intent_p":0.8,"supports_p":0.95,"contests_p":0.7},"method":"Citing sentences come from each paper's LaTeX source; every \\cite marks the sentence and the works it cites. Pith's classifier typed each sentence. A label is shown at intent probability 0.8 or more (0.95 for support), a contest at 0.7 or more. Measured accuracy against two blind model families is at https://pith.science/accuracy. The quoted sentences are the citing authors' own.","held_up":null,"_links":{"html":"https://pith.science/paper/2207.11305"}}