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In this study, we conduct parameter estimation analysis on a data assimilation algorithm for two turbulence models: the simplified Bardina model and the Navier–Stokes-α model. Rigorous estimates are presented for the convergence of continuous data assimilation methods when the parameters of the turbulence models are not known a priori. Our approach involves creating an approximate solution for the turbulence models by employing an interpolant operator based on the observational data of the systems. The estimation depends on the parameter alpha in the models. Additionally, numerical simulations are presented to validate our theoretical results. 

The frequency of home spatial activities (e.g., puzzles and blocks) correlates with young children's spatial skills, but causal evidence is limited. We addressed this issue by comparing the effects of a parent‐led intervention aimed at increasing spatial activities to an active control targeting narrative activities (preregistered:https://osf.io/u7qrx). Parents of 80 4‐ and 5‐year‐old children were randomly assigned to either a spatial or narrative condition. Parents learned about the importance and malleability of spatial or narrative skills and engaged their children in spatial or narrative activities provided by the researchers for a month. Unexpectedly, the spatial intervention did not significantly enhance children's spatial skills or parents' motivational beliefs regarding children's spatial abilities. These findings do not support the hypothesis that spatial play causally influences children's skills. However, we note that the families in our sample had high socioeconomic status, and their children may have already benefited from rich spatial environments. 

In this study, we perform a parameter error analysis of the 3D modified Leray-α model using both analytical and numerical approaches. First, we prove the global well-posedness and continuous dependence on the initial data for the assimilated system. Furthermore, under sufficient conditions on the physical parameters and norms of the true solution, we demon- strate that the true solution can be recovered from the approximate solution, with the error determined by the discrepancy between the true and approximate parameters. Numerical simulations are provided to validate the convergence criteria. 

Verbal labels for math concepts influence multiple aspects of math learning. In this study, we examined the influence of point labels (e.g., .42 as “point four two”), decomposed labels (e.g., “four tenths and two hundredths”), and common-unit labels (e.g., “forty-two hundredths”) on children’s processing and representation of decimal magnitudes. We randomly assigned 162 5th- and 6th-graders to briefly learn decomposed, common-unit, or point labels. Children then completed measures of decimal magnitude processing and representation. We found that the place-value labels (i.e., decomposed and common-unit labels) each showed unique advantages in reducing the whole-number bias, and common-unit labels also reduced componential processing. No difference was found in the ratio effect – which served as an index of the precision of decimal magnitude representation - among children from the three conditions. These findings add to our understanding of the role of verbal labels in math learning and have important implications for instructional practices. 

Prior research has shown that the home learning environment (HLE) is critical in the development of spatial skills and that various parental beliefs influence the HLE. However, a comprehensive analysis of the impact of different parental beliefs on the spatial HLE remains lacking, leaving unanswered questions about which specific parental beliefs are most influential and whether inducing a growth mindset can enhance the spatial HLE. To address these gaps, we conducted an online study with parents of 3- to 5-year-olds. We found that parents’ growth mindset about their children’s ability strongly predicted the spatial HLE after controlling for parents’ motivational beliefs about their children, beliefs about their own ability, children’s age, children’s gender, and family SES. Further, reading an article about growth mindset led parents to choose more challenging spatial learning activities for their children. These findings highlight the critical role of parents’ growth mindset in the spatial HLE. Crucially, these findings demonstrate that general growth mindset messages without specific suggestions for parental practices can influence parental behavior intentions. Further, these effects were also observed in the control domain of literacy, underscoring the broad relevance of the growth mindset in the HLE. 

In this paper, we develop parameter-robust numerical algorithms for Biot model and apply the algorithms in brain edema simulations. By introducing an intermediate variable, we derive a multiphysics reformulation of the Biot model. Based on the reformulation, the Biot model is viewed as a generalized Stokes subproblem combining with a reaction–diffusion subproblem. Solving the two subproblems together or separately leads to a coupled or a decoupled algorithm. We conduct extensive numerical experiments to show that the two algorithms are robust with respect to the key physical parameters. The algorithms are applied to study the brain swelling caused by abnormal accumulation of cerebrospinal fluid in injured areas. The effects of the key physical parameters on brain swelling are carefully investigated. It is observed that the permeability has the biggest influence on intracranial pressure (ICP) and tissue deformation; the Young’s modulus and the Poisson ratio do not affect the maximum value of ICP too much but have big influence on the tissue deformation and the developing speed of brain swelling.