We show that for @math and for measurable subsets of the unit square with Lebesgue measure @math there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the boundary and increases the Lebesgue measure of the set to at least @math .
The question considered in this paper has connections to several different problems in analysis, an interested reader is referred to @cite_5 and the references therein for more details.
@cite_5
1786941226
Document authors commonly use tables to support arguments presented in the text. But, because tables are usually separate from the main body text, readers must split their attention between different parts of the document. We present an interactive document reader that automatically links document text with corresponding table cells. Readers can select a sentence (or tables cells) and our reader highlights the relevant table cells (or sentences). We provide an automatic pipeline for extracting such references between sentence text and table cells for existing PDF documents that combines structural analysis of tables with natural language processing and rule-based matching. On a test corpus of 330 (sentence, table) pairs, our pipeline correctly extracts 48.8 of the references. An additional 30.5 contain only false negatives (FN) errors -- the reference is missing table cells. The remaining 20.7 contain false positives (FP) errors -- the reference includes extraneous table cells and could therefore mislead readers. A user study finds that despite such errors, our interactive document reader helps readers match sentences with corresponding table cells more accurately and quickly than a baseline document reader.