An arc in is a set such that no three points of are collinear. We use the method of hypergraph containers to prove several counting results for arcs. Let denote the family of all arcs in . Our main result is the bound This matches, up to the factor hidden in the (1) notation, the trivial lower bound that comes from considering all subsets of an arc of size . We also give upper bounds for the number of arcs of a fixed (large) size. Let , and let denote the family of all arcs in with cardinality . We prove that This result improves a bound of Roche-Newton and Warren [12]. A nearly matching lower bound follows by considering all subsets of size of an arc of size .
Download full-text PDF |
Source |
|---|---|
| http://www.ncbi.nlm.nih.gov/pmc/articles/PMC11569019 | PMC |
| http://dx.doi.org/10.1007/s00454-023-00622-w | DOI Listing |
Publication Analysis
Top Keywords
Similar Publications
Arctiin alleviates the progression of osteoarthritis by regulating the cholesterol metabolic pathway.
Sci Rep
January 2025
Guangzhou First People's Hospital, the Second Affiliated Hospital, School of Medicine, South China University of Technology; Guangzhou First People's Hospital, Guangzhou Medical University, 1 Panfu Road, Yuexiu District, Guangzhou, 510180, China.
Osteoarthritis (OA) is a multi-factorial degenerative joint disease with unclear pathogenesis. Conservative treatments, primarily aimed at pain relief, fail to halt disease progression. Metabolic syndrome has recently been implicated in OA pathogenesis, underscoring the need for novel therapeutic strategies.
View Article and Find Full Text PDFAn arc in is a set such that no three points of are collinear. We use the method of hypergraph containers to prove several counting results for arcs. Let denote the family of all arcs in .
View Article and Find Full Text PDFFYCO1 regulates autophagy and senescence via PAK1/p21 in cataract.
Arch Biochem Biophys
November 2024
Eye Center, The Second Affiliated Hospital, School of Medicine, Zhejiang University, Zhejiang Provincial Key Laboratory of Ophthalmology, Zhejiang Provincial Clinical Research Center for Eye Diseases, Zhejiang Provincial Engineering Institute on Eye Diseases, Hangzhou, Zhejiang, China. Electronic address:
Article Synopsis
- The study investigates the role of FYCO1, an autophagy adaptor, in the development of age-related cataracts, which cause significant vision impairment.
- Using UVB irradiation to create cataract models in mice and cultured human lens cells, researchers observed structural damage and a decrease in FYCO1 expression associated with impaired autophagy.
- The findings suggest that FYCO1 influences cell senescence and cataract development through its interaction with p21 activated kinase 1 (PAK1), highlighting its potential role in combating cataract-related oxidative stress.
Improvements in hematologic markers and decreases in fatigue with pegcetacoplan for patients with paroxysmal nocturnal hemoglobinuria and mild or moderate anemia (hemoglobin ≥10 g/dL) who had received eculizumab or were naive to complement inhibitors.
PLoS One
July 2024
Sir Y.K. Pao Centre for Cancer & Department of Medicine and Therapeutics, Prince of Wales Hospital, The Chinese University of Hong Kong, Hong Kong, China.
Background: Although complement component 5 inhibitors (C5is) eculizumab and ravulizumab improve paroxysmal nocturnal hemoglobinuria (PNH) outcomes, patients may experience persistent anemia. This post hoc analysis investigated whether the complement component 3-targeted therapy pegcetacoplan also improved hematologic outcomes and reduced fatigue in patients with PNH and mild/moderate anemia.
Methods: Patients with PNH and hemoglobin ≥10.
Comparison of Orchard Networks Using Their Extended μ-Representation.
IEEE/ACM Trans Comput Biol Bioinform
June 2024
Article Synopsis
- Phylogenetic networks are a more complex way to model evolutionary relationships that can include reticulation events, unlike traditional phylogenetic trees.
- The article presents an extended μ-representation for these networks which calculates paths not only to leaves but also to reticulations, enabling better classification of a specific type called orchard networks.
- Orchard networks are significant in biology as they represent coexisting organisms and pose a mathematically interesting challenge, balancing between complexity and the generalizability of their representation.
Want AI Summaries of new PubMed Abstracts delivered to your In-box?
Enter search terms and have AI summaries delivered each week - change queries or unsubscribe any time!