A new matroid lift construction and an application to group-labeled graphs

A well-known result of Brylawski constructs an elementary lift of a matroid M from a linear class of circuits of M. We generalize this result by constructing a rank-k lift of M from a rank-k matroid on the set of circuits of M. We conjecture that every lift of M arises via this construction. We then apply this result to group-labeled graphs, generalizing a construction of Zaslavsky. Given a graph G with edges labeled by a group, Zaslavsky\u27s lift matroid K is an elementary lift of the graphic matroid M(G) that respects the group-labeling; specifically, the cycles of G that are circuits of K coincide with the cycles that are balanced with respect to the group-labeling. For k 2, when does there exist a rank-k lift of M(G) that respects the group-labeling in this same sense? For abelian groups, we show that such a matroid exists if and only if the group is isomorphic to the additive group of a non-prime finite field

Matroid lifts and representability

A 1965 result of Crapo shows that every elementary lift of a matroid $M$ can be constructed from a linear class of circuits of $M$. In a recent paper, Walsh generalized this construction by defining a rank-$k$ lift of a matroid $M$ given a rank-$k$ matroid $N$ on the set of circuits of $M$, and conjectured that all matroid lifts can be obtained in this way. In this sequel paper we simplify Walsh's construction and show that this conjecture is true for representable matroids but is false in general. This gives a new way to certify that a particular matroid is non-representable, which we use to construct new classes of non-representable matroids. Walsh also applied the new matroid lift construction to gain graphs over the additive group of a non-prime finite field, generalizing a construction of Zaslavsky for these special groups. He conjectured that this construction is possible on three or more vertices only for the additive group of a non-prime finite field. We show that this conjecture holds for four or more vertices, but fails for exactly three

Planar Tur\'an number of the 7-cycle

The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_{\ell}$ denote the cycle of length $\ell$. The planar Tur\'an number $\textrm{ex}_{\mathcal P}(n,C_{\ell})$ behaves differently for $\ell\le 10$ and for $\ell\ge 11$, and it is known when $\ell \in \{3,4,5,6\}$. We prove that $\textrm{ex}_{\mathcal P}(n,C_7) \le \frac{18n}{7} - \frac{48}{7}$ for all $n > 38$, and show that equality holds for infinitely many integers $n$

Extended formulations for a class of polyhedra with bimodular cographic constraint matrices

We are motivated by integer linear programs (ILPs) defined by constraint matrices with bounded determinants. Such matrices generalize the notion of totally-unimodular matrices. When the determinants are bounded by $2$, the matrix is called bimodular. Artmann et al. give a polynomial-time algorithm for solving any ILP defined by a bimodular constraint matrix. Complementing this result, Conforti et al. give a compact extended formulation for a particular class of bimodular-constrained ILPs, namely those that model the stable set polytope of a graph with odd cycle packing number $1$. We demonstrate that their compact extended formulation can be modified to hold for polyhedra such that (1) the constraint matrix is bimodular, (2) the row-matroid generated by the constraint matrix is cographic and (3) the right-hand side is a linear combination of the columns of the constraint matrix. This generalizes the important special case from Conforti et al. concerning 4-connected graphs with odd cycle transversal number at least four. Moreover, our results yield compact extended formulations for a new class of polyhedra

Dense circuit graphs and the planar Tur\'an number of a cycle

The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_k$ denote the cycle of length $k$. The planar Tur\'an number $\textrm{ex}_{\mathcal P}(n,C_k)$ is known for $k\le 7$. We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Tur\'an numbers. In particular, we prove that there is a constant $D$ so that $\textrm{ex}_{\mathcal P}(n,C_k) \le 3n - 6 - Dn/k^{\log_2^3}$ for all $k, n\ge 4$. When $k \ge 11$ this bound is tight up to the constant $D$ and proves a conjecture of Cranston, Lidick\'y, Liu, and Shantanam

Frequency of cannabis and illicit opioid use among people who use drugs and report chronic pain: A longitudinal analysis.

BACKGROUND:Ecological research suggests that increased access to cannabis may facilitate reductions in opioid use and harms, and medical cannabis patients describe the substitution of opioids with cannabis for pain management. However, there is a lack of research using individual-level data to explore this question. We aimed to investigate the longitudinal association between frequency of cannabis use and illicit opioid use among people who use drugs (PWUD) experiencing chronic pain. METHODS AND FINDINGS:This study included data from people in 2 prospective cohorts of PWUD in Vancouver, Canada, who reported major or persistent pain from June 1, 2014, to December 1, 2017 (n = 1,152). We used descriptive statistics to examine reasons for cannabis use and a multivariable generalized linear mixed-effects model to estimate the relationship between daily (once or more per day) cannabis use and daily illicit opioid use. There were 424 (36.8%) women in the study, and the median age at baseline was 49.3 years (IQR 42.3-54.9). In total, 455 (40%) reported daily illicit opioid use, and 410 (36%) reported daily cannabis use during at least one 6-month follow-up period. The most commonly reported therapeutic reasons for cannabis use were pain (36%), sleep (35%), stress (31%), and nausea (30%). After adjusting for demographic characteristics, substance use, and health-related factors, daily cannabis use was associated with significantly lower odds of daily illicit opioid use (adjusted odds ratio 0.50, 95% CI 0.34-0.74, p < 0.001). Limitations of the study included self-reported measures of substance use and chronic pain, and a lack of data for cannabis preparations, dosages, and modes of administration. CONCLUSIONS:We observed an independent negative association between frequent cannabis use and frequent illicit opioid use among PWUD with chronic pain. These findings provide longitudinal observational evidence that cannabis may serve as an adjunct to or substitute for illicit opioid use among PWUD with chronic pain

The Grizzly, February 18, 2010

Every Ending Starts with a Beginning • Record-Breaking Blizzard Evokes Varied Reactions • Could Watching the Super Bowl Damage Your Heart? • Snow Storm Photos • Senior Class Gift Drive • SPINTfest \u2710 Brings New Themes for Houses • UC Goes Red to Raise Awareness About the Risks of Heart Disease • Opinion: Teenage Pregnancy TV Shows are a Big Hit, But What\u27s the Effect? • Tragedy Strikes in Early Hours of Winter Olympics • Men\u27s Basketball Shuts Down McDanielhttps://digitalcommons.ursinus.edu/grizzlynews/1806/thumbnail.jp

The Grizzly, February 12, 2010

Main Street Renovation Improves Safety • Community Brings Down Crime at Ursinus College • Annual Scottish Irish Festival This Weekend • Bonner Leaders Hold Fair in Lower Wismer • What You Should Really Expect from Study Abroad • New Member Education Starts Up Again and Looks Forward to Positive Change • UC Gymnastics is Flipping Through 2010 Season • Moliken Named New Athletic Directorhttps://digitalcommons.ursinus.edu/grizzlynews/1805/thumbnail.jp

The Grizzly, April 8, 2010

First Annual Backwards Beauty Pageant Held • Ursinus Senior to Travel Abroad for a Year with Watson • Kappa Alpha Psi and Seismic Step Emerge on Campus • UC Welcomes New Field Hockey Coach • Sex on Wheels Documentary Screening • Volunteering with St. Christopher\u27s Children\u27s Hospital • Feeling Good in The Skin We\u27re In • The Sacrifice Your Body Makes for [Better] Grades • Opinion: Kyleigh\u27s Law Profiles Drivers by Age in New Jersey • Upper Classmen Off-Campus Living • UC Gymnastics Closes Season with Successhttps://digitalcommons.ursinus.edu/grizzlynews/1810/thumbnail.jp