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Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems. 

Abstract We report that the neutral hydrogen (Hi) mass density of the Universe (ρ_Hi) increases with cosmic time sincez ∼ 5, peaks atz ∼ 3, and then decreases towardz ∼ 0. This is the first result of Qz5, our spectroscopic survey of 63 quasars atz ≳ 5 with VLT/X-SHOOTER and Keck/ESI aimed at characterizing intervening Higas absorbers atz ∼ 5. The main feature of Qz5 is the high resolution (R ∼ 7000–9000) of the spectra, which allows us to (1) accurately detect high column density Higas absorbers in an increasingly neutral intergalactic medium atz ∼ 5 and (2) determine the reliability of previousρ_Himeasurements derived with lower resolution spectroscopy. We find five intervening damped Lyαabsorbers (DLAs) atz > 4.5, which corresponds to the lowest DLA incidence rate ( $0.034_{0.02}^{0.05}$) atz ≳ 2. We also measure the lowestρ_Hiatz ≳ 2 from our sample of DLAs and subDLAs, corresponding toρ_Hi $=0.56_{0.31}^{0.82}\times 10^8{M}_{\odot }$Mpc⁻³atz ∼ 5. Taking into account our measurements atz ∼ 5 and systematic biases in the DLA detection rate at lower spectral resolutions, we conclude thatρ_Hidoubles fromz ∼ 5 toz ∼ 3. From these results emerges a qualitative agreement between how the cosmic densities of Higas mass, molecular gas mass, and star formation rate build up with cosmic time. 

Summary Lignin, a complex heterogenous polymer present in virtually all plant cell walls, plays a critical role in protecting plants from various stresses. However, little is known about how lignin modifications in sorghum will impact plant defense against sugarcane aphids (SCA), a key pest of sorghum.We utilized the sorghumbrown midrib(bmr) mutants, which are impaired in monolignol synthesis, to understand sorghum defense mechanisms against SCA. We found that loss ofBmr12function and overexpression (OE) ofBmr12provided enhanced resistance and susceptibility to SCA, respectively, as compared with wild‐type (WT; RTx430) plants.Monitoring of the aphid feeding behavior indicated that SCA spent more time in reaching the first sieve element phase onbmr12plants compared with RTx430 andBmr12‐OE plants. A combination of transcriptomic and metabolomic analyses revealed thatbmr12plants displayed altered auxin metabolism upon SCA infestation and specifically, elevated levels of auxin conjugate indole‐3‐acetic acid–aspartic acid (IAA–Asp) were observed inbmr12plants compared with RTx430 andBmr12‐OE plants. Furthermore, exogenous application of IAA–Asp restored resistance inBmr12‐OE plants, and artificial diet aphid feeding trial bioassays revealed that IAA–Asp is associated with enhanced resistance to SCA.Our findings highlight the molecular underpinnings that contribute to sorghumbmr12‐mediated resistance to SCA. 

We leverage convex and bilevel optimization techniques to develop a general gradient-based parameter learning framework for neural-symbolic (NeSy) systems. We demonstrate our framework with NeuPSL, a state-of-the-art NeSy architecture. To achieve this, we propose a smooth primal and dual formulation of NeuPSL inference and show learning gradients are functions of the optimal dual variables. Additionally, we develop a dual block coordinate descent algorithm for the new formulation that naturally exploits warm-starts. This leads to over $$100 \times$$ learning runtime improvements over the current best NeuPSL inference method. Finally, we provide extensive empirical evaluations across $$8$$ datasets covering a range of tasks and demonstrate our learning framework achieves up to a $16$ 

This conceptual paper explores language and cultural resources as forms of multicompetence for engaging in engineering epistemologies (what we know) and practices (what we do). The need for a more diverse pool of engineers to tackle the complex challenges facing society is undeniable, but stereotypes about the discipline can create alienation among many students and undermine efforts to build a more inclusive profession. Drawing on scholarship from engineering education, science education, and learning sciences, this paper argues that the resources of Multicompetent Learners (ML), who have acquired valuable experiences and knowledge through social interaction within their communities, are valuable for engineering learning environments. By leveraging the language and cultural resources that students bring with them, engineering education can better prepare learners to develop solutions and knowledge that serve a diverse population. This work underscores the critical role of language and cultural resources in helping students be heard, seen, and understood in engineering and illustrate how these resources can help bridge the gap between students' lives and engineering. The paper further explores the multidimensional nature of language and cultural resources and how students draw on different sets of talk depending on the context, whether near or distal from the activity at hand. It contends that without a deeper understanding of the role of non-dominant ways of speaking in the act of becoming and belonging, efforts to diversify engineering will remain elusive. Ultimately, this paper summarizes these ideas through a conceptual model for engineering learning environments that value and leverage the resources that students bring from their communities. By creating more equitable and socially just solutions, engineering education can better serve the needs of diverse populations and ensure that the profession is truly reflective of the communities it serves. 

The most destructive and deadly wildfires in US history were also fast. Using satellite data, we analyzed the daily growth rates of more than 60,000 fires from 2001 to 2020 across the contiguous US. Nearly half of the ecoregions experienced destructive fast fires that grew more than 1620 hectares in 1 day. These fires accounted for 78% of structures destroyed and 61% of suppression costs ($18.9 billion). From 2001 to 2020, the average peak daily growth rate for these fires more than doubled (+249% relative to 2001) in the Western US. Nearly 3 million structures were within 4 kilometers of a fast fire during this period across the US. Given recent devastating wildfires, understanding fast fires is crucial for improving firefighting strategies and community preparedness.