On generalized corners and matrix multiplication

Suppose that $S \subseteq [n]^2$ contains no three points of the form $(x,y), (x,y+\delta), (x+\delta,y')$, where $\delta \neq 0$. How big can $S$ be? Trivially, $n \le |S| \le n^2$. Slight improvements on these bounds are obtained from Shkredov's upper bound for the corners problem [Shk06], which shows that $|S| \le O(n^2/(\log \log n)^c)$ for some small $c > 0$, and a construction due to Petrov [Pet23], which shows that $|S| \ge \Omega(n \log n/\sqrt{\log \log n})$. Could it be that for all $\varepsilon > 0$, $|S| \le O(n^{1+\varepsilon})$? We show that if so, this would rule out obtaining $\omega = 2$ using a large family of abelian groups in the group-theoretic framework of Cohn, Kleinberg, Szegedy and Umans [CU03,CKSU05] (which is known to capture the best bounds on $\omega$ to date), for which no barriers are currently known. Furthermore, an upper bound of $O(n^{4/3 - \varepsilon})$ for any fixed $\varepsilon > 0$ would rule out a conjectured approach to obtain $\omega = 2$ of [CKSU05]. Along the way, we encounter several problems that have much stronger constraints and that would already have these implications.Comment: Feedback welcome

Exploring Circle Packing Algorithms

We present an interactive tool for visualizing and experimenting with different circle packing algorithms

The Acute and Residual Effect of a Single Exercise Session on Meal Glucose Tolerance in Sedentary Young Adults

The study goals were to (1) establish the variability in postprandial glucose control in healthy young people consuming a mixed meal and, then (2) determine the acute and residual impact of a single exercise bout on postprandial glucose control. In study 1, 18 people completed two similar mixed meal trials and an intravenous glucose tolerance test (IVGTT). There were strong test-retest correlations for the post-meal area under the curve (AUC) for glucose, insulin, and Cpeptide (r = 0.73–0.83) and the Matsuda insulin sensitivity index (ISI, r = 0.76), and between meal and IVGTT-derived ISI (r = 0.83). In study 2, 11 untrained young adults completed 3 trials. One trial (No Ex) was completed after refraining from vigorous activity for ≥3 days. On the other 2 trials, a 45-min aerobic exercise bout was performed either 17-hours (Prior Day Ex) or 1-hour (Same Day Ex) before consuming the test meal. Compared to No Ex and Prior Day Ex, which did not differ from one another, there were lower AUCs on the Same Day Ex trial for glucose (6%), insulin (20%) and C-peptide (14%). Thus, a single moderate intensity exercise session can acutely improve glycemic control but the effect is modest and short-lived