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Abstract Let$$p_{1},\ldots ,p_{n}$$ $p_1,\dots ,p_n$be a set of points in the unit square and let$$T_{1},\ldots ,T_{n}$$ $T_1,\dots ,T_n$be a set of$$\delta $$ $\delta$-tubes such that$$T_{j}$$ $T_j$passes through$$p_{j}$$ $p_j$. We prove a lower bound for the number of incidences between the points and tubes under a natural regularity condition (similar to Frostman regularity). As a consequence, we show that in any configuration of points$$p_{1},\ldots , p_{n} \in [0,1]^{2}$$ $p_1,\dots ,p_n\in {[0,1]}^2$along with a line$$\ell _{j}$$ ${\ell }_j$through each point$$p_{j}$$ $p_j$, there exist$$j\neq k$$ $j\ne k$for which$$d(p_{j}, \ell _{k}) \lesssim n^{-2/3+o(1)}$$ $d(p_j,{\ell }_k)\lesssim n^{-2/3+o\left(1\right)}$. It follows from the latter result that any set of$$n$$ $n$points in the unit square contains three points forming a triangle of area at most$$n^{-7/6+o(1)}$$ $n^{-7/6+o\left(1\right)}$. This new upper bound for Heilbronn’s triangle problem attains the high-low limit established in our previous work arXiv:2305.18253. 

Abstract BackgroundWildfire smoke contributes substantially to the global disease burden and is a major cause of air pollution in the US states of Oregon and Washington. Climate change is expected to bring more wildfires to this region. Social media is a popular platform for health promotion and a need exists for effective communication about smoke risks and mitigation measures to educate citizens and safeguard public health. MethodsUsing a sample of 1,287 Tweets from 2022, we aimed to analyze temporal Tweeting patterns in relation to potential smoke exposure and evaluate and compare institutions’ use of social media communication best practices which include (i) encouraging adoption of smoke-protective actions; (ii) leveraging numeric, verbal, and Air Quality Index risk information; and (iii) promoting community-building. Tweets were characterized using keyword searches and the Linguistic Inquiry and Word Count (LIWC) software. Descriptive and inferential statistics were carried out. Results44% of Tweets in our sample were authored between January-August 2022, prior to peak wildfire smoke levels, whereas 54% of Tweets were authored during the two-month peak in smoke (September-October). Institutional accounts used Twitter (or X) to encourage the adoption of smoke-related protective actions (82% of Tweets), more than they used it to disseminate wildfire smoke risk information (25%) or promote community-building (47%). Only 10% of Tweets discussed populations vulnerable to wildfire smoke health effects, and 14% mentioned smoke mitigation measures. Tweets from Washington-based accounts used significantly more verbal and numeric risk information to discuss wildfire smoke than Oregon-based accounts (p = 0.042 andp = 0.003, respectively); however, Tweets from Oregon-based accounts on average contained a higher percentage of words associated with community-building language (p < 0.001). ConclusionsThis research provides practical recommendations for public health practitioners and researchers communicating wildfire smoke risks on social media. As exposures to wildfire smoke rise due to climate change, reducing the environmental disease burden requires health officials to leverage popular communication platforms, distribute necessary health-related messaging rapidly, and get the message right. Timely, evidence-based, and theory-driven messaging is critical for educating and empowering individuals to make informed decisions about protecting themselves from harmful exposures. Thus, proactive and sustained communications about wildfire smoke should be prioritized even during wildfire “off-seasons.” 

Ensuring long-term access to nutrients needed for food production is a growing global challenge. Human urine diversion and recycling is a viable and energy-efficient means of recovering nitrogen, phosphorus, and potassium from wastewater. Before implementation, however, it is critical to understand how communicating differently about human urine-derived fertilizer may influence its public acceptance. This study tests how different strategies of communication (video compared to texts), as well as different amounts of information, impact public acceptance. We also explored how specific characteristics, such as age and education level, may impact the usefulness of the different strategies of communication. The results indicate that short and long videos are the most useful risk communication strategies, and age fully moderates this relationship. This research may serve as a jumping off point for future studies focused on how risk communication strategies may affect consumer acceptance of other emerging food technologies.