Eigenvalues of collapsing domains and drift Laplacians

By introducing a weight function to the Laplace operator, Bakry and \'Emery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-\'Emery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to generalize \emph{all the results in Riemannian geometry based on gradient estimates to Bakry-\'Emery manifolds}

Contexts of Cultural Diversity Professional Development in Schools

This report is part of a larger MERC study on Professional Development for Success in Culturally Diverse Schools.The goals of the larger study are (1) to understand the landscape of diversity within the schools in the MERC region, (2) to consider the implications for school policy and professional practice, (3) to review the current models of professional development that support teachers and other school professionals in regards to issues of cultural diversity, and (4) to develop and test models of professional development that have positive impacts on teacher practice, student outcomes and overall school success. With these goals in mind, this report contains three parts. Part 1 discusses demographic changes in the MERC region and provides historical, political, social, and economic contexts for understanding these changes. Part 2 describes the federal, state, and local policy contexts relevant to cultural diversity within schools, and professional development (PD) related to this topic. This section also reviews research on how policy contexts shape teachers’ decisions to participate in PD. Part 3 reviews existing studies of cultural diversity PD and describes the types of PD programs currently available in the MERC region. In the conclusion to the report, we present a list of recommendations for policy and and for future development and implementation of PD related to cultural diversity. The final section of the report also describes gaps in the existing body of knowledge and the research needed to better understand PD related to cultural diversity

Generate your neural signals from mine: individual-to-individual EEG converters

Most models in cognitive and computational neuroscience trained on one subject do not generalize to other subjects due to individual differences. An ideal individual-to-individual neural converter is expected to generate real neural signals of one subject from those of another one, which can overcome the problem of individual differences for cognitive and computational models. In this study, we propose a novel individual-to-individual EEG converter, called EEG2EEG, inspired by generative models in computer vision. We applied THINGS EEG2 dataset to train and test 72 independent EEG2EEG models corresponding to 72 pairs across 9 subjects. Our results demonstrate that EEG2EEG is able to effectively learn the mapping of neural representations in EEG signals from one subject to another and achieve high conversion performance. Additionally, the generated EEG signals contain clearer representations of visual information than that can be obtained from real data. This method establishes a novel and state-of-the-art framework for neural conversion of EEG signals, which can realize a flexible and high-performance mapping from individual to individual and provide insight for both neural engineering and cognitive neuroscience.Comment: Proceedings of the 45th Annual Meeting of the Cognitive Science Society (CogSci 2023

The Dirichlet isospectral problem for trapezoids

We show that non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum. This extends our previous result, which was only concerned with the Neumann Laplace spectrum