@article{aslam_ardanza-trevijano_xiong_arsuaga_sazdanovic_2022, title={TAaCGH Suite for Detecting Cancer-Specific Copy Number Changes Using Topological Signatures}, volume={24}, ISSN={["1099-4300"]}, url={https://www.mdpi.com/1099-4300/24/7/896}, DOI={10.3390/e24070896}, abstractNote={Copy number changes play an important role in the development of cancer and are commonly associated with changes in gene expression. Persistence curves, such as Betti curves, have been used to detect copy number changes; however, it is known these curves are unstable with respect to small perturbations in the data. We address the stability of lifespan and Betti curves by providing bounds on the distance between persistence curves of Vietoris–Rips filtrations built on data and slightly perturbed data in terms of the bottleneck distance. Next, we perform simulations to compare the predictive ability of Betti curves, lifespan curves (conditionally stable) and stable persistent landscapes to detect copy number aberrations. We use these methods to identify significant chromosome regions associated with the four major molecular subtypes of breast cancer: Luminal A, Luminal B, Basal and HER2 positive. Identified segments are then used as predictor variables to build machine learning models which classify patients as one of the four subtypes. We find that no single persistence curve outperforms the others and instead suggest a complementary approach using a suite of persistence curves. In this study, we identified new cytobands associated with three of the subtypes: 1q21.1-q25.2, 2p23.2-p16.3, 23q26.2-q28 with the Basal subtype, 8p22-p11.1 with Luminal B and 2q12.1-q21.1 and 5p14.3-p12 with Luminal A. These segments are validated by the TCGA BRCA cohort dataset except for those found for Luminal A.}, number={7}, journal={ENTROPY}, author={Aslam, Jai and Ardanza-Trevijano, Sergio and Xiong, Jingwei and Arsuaga, Javier and Sazdanovic, Radmila}, year={2022}, month={Jul} }
@article{aslam_chen_frick_saloff-coste_setiabrata_thomas_2020, title={SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM}, volume={8}, ISSN={["2050-5094"]}, DOI={10.1017/fms.2019.51}, abstractNote={Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz’s square peg problem. We prove Hadwiger’s 1971 conjecture that any simple loop in $3$ -space inscribes a parallelogram. We show that any simple planar loop inscribes sufficiently many rectangles that their vertices are dense in the loop. If the loop is rectifiable, there is a rectangle that cuts the loop into four pieces which can be rearranged to form two loops of equal length. (The previous two results are independently due to Schwartz.) A rectifiable loop in $d$ -space can be cut into $(r-1)(d+1)+1$ pieces that can be rearranged by translations to form $r$ loops of equal length. We relate our results to fair divisions of necklaces in the sense of Alon and to Tverberg-type results. This provides a new approach and a common framework to obtain inscribability results for the class of all continuous curves.}, journal={FORUM OF MATHEMATICS SIGMA}, author={Aslam, Jai and Chen, Shujian and Frick, Florian and Saloff-Coste, Sam and Setiabrata, Linus and Thomas, Hugh}, year={2020}, month={Jan} }
@article{aslam_chen_coldren_frick_setiabrata_2019, title={On the generalized Erdos-Kneser conjecture: Proofs and reductions}, volume={135}, ISSN={["1096-0902"]}, DOI={10.1016/j.jctb.2018.08.004}, abstractNote={Abstract Alon, Frankl, and Lovasz proved a conjecture of Erdős that one needs at least ⌈ n − r ( k − 1 ) r − 1 ⌉ colors to color the k-subsets of { 1 , … , n } such that any r of the k -subsets that have the same color are not pairwise disjoint . A generalization of this problem where one requires s-wise instead of pairwise intersections was considered by Sarkaria. He claimed a proof of a generalized Erdős–Kneser conjecture establishing a lower bound for the number of colors that reduces to Erdős' original conjecture for s = 2 . Lange and Ziegler pointed out that his proof fails whenever r is not a prime. Here we establish this generalized Erdős–Kneser conjecture for every r, as long as s is not too close to r. Our result encompasses earlier results but is significantly more general. We discuss relations of our results to conjectures of Ziegler and of Abyazi Sani and Alishahi, and prove the latter in several cases.}, journal={JOURNAL OF COMBINATORIAL THEORY SERIES B}, author={Aslam, Jai and Chen, Shuli and Coldren, Ethan and Frick, Florian and Setiabrata, Linus}, year={2019}, month={Mar}, pages={227–237} }