A Lower Bound on Opaque Sets

It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle

Every non-Euclidean oriented matroid admits a biquadratic final polynomial

Richter-Gebert proved that every non-Euclidean uniform oriented matroid admits a biquadratic final polynomial. We extend this result to the non-uniform cas

On the Existence of Hamiltonian Paths for History Based Pivot Rules on Acyclic Unique Sink Orientations of Hypercubes

An acyclic USO on a hypercube is formed by directing its edges in such as way that the digraph is acyclic and each face of the hypercube has a unique sink and a unique source. A path to the global sink of an acyclic USO can be modeled as pivoting in a unit hypercube of the same dimension with an abstract objective function, and vice versa. In such a way, Zadeh's 'least entered rule' and other history based pivot rules can be applied to the problem of finding the global sink of an acyclic USO. In this paper we present some theoretical and empirical results on the existence of acyclic USOs for which the various history based pivot rules can be made to follow a Hamiltonian path. In particular, we develop an algorithm that can enumerate all such paths up to dimension 6 using efficient pruning techniques. We show that Zadeh's original rule admits Hamiltonian paths up to dimension 9 at least, and prove that most of the other rules do not for all dimensions greater than 5

A lower bound on opaque sets

It is proved that the total length of any set of countably many rectifiable curves whose union meets all straight lines that intersect the unit square U is at least 2.00002. This is the first improvement on the lower bound of 2 known since 1964. A similar bound is proved for all convex sets U other than a triangle. (C) 2019 Published by Elsevier B.V

Role of FBXW7 in the quiescence of gefitinib-resistant lung cancer stem cells in EGFR-mutant non-small cell lung cancer

Several recent studies suggest that cancer stem cells (CSCs) are involved in intrinsic resistance to cancer treatment. Maintenance of quiescence is crucial for establishing resistance of CSCs to cancer therapeutics. F-box/WD repeat-containing protein 7 (FBXW7) is a ubiquitin ligase that regulates quiescence by targeting the c-MYC protein for ubiquitination. We previously reported that gefitinib-resistant persisters (GRPs) in EGFR-mutant non-small cell lung cancer (NSCLC) cells highly expressed octamer-binding transcription factor 4 (Oct-4) as well as the lung CSC marker CD133, and they exhibited distinctive features of the CSC phenotype. However, the role of FBXW7 in lung CSCs and their resistance to epidermal growth factor receptor (EGFR)-tyrosine kinase inhibitors in NSCLC is not fully understood. In this study, we developed GRPs from the two NSCLC cell lines PC9 and HCC827, which express an EGFR exon 19 deletion mutation, by treatment with a high concentration of gefitinib. The GRPs from both PC9 and HCC827 cells expressed high levels of CD133 and FBXW7, but low levels of c-MYC. Cell cycle analysis demonstrated that the majority of GRPs existed in the G0/G1 phase. Knockdown of the FBXW7 gene significantly reduced the cell number of CD133-positive GRPs and reversed the cell population in the G0/G1-phase. We also found that FBXW7 expression in CD133-positive cells was increased and c-MYC expression was decreased in gefitinib-resistant tumors of PC9 cells in mice and in 9 out of 14 tumor specimens from EGFR-mutant NSCLC patients with acquired resistance to gefitinib. These findings suggest that FBXW7 plays a pivotal role in the maintenance of quiescence in gefitinib-resistant lung CSCs in EGFR mutation-positive NSCLC