Periodic Solutions for Circular Restricted 4-body Problems with Newtonian Potentials

We study the existence of non-collision periodic solutions with Newtonian potentials for the following planar restricted 4-body problems: Assume that the given positive masses $m_{1},m_{2},m_{3}$ in a Lagrange configuration move in circular obits around their center of masses, the sufficiently small mass moves around some body. Using variational minimizing methods, we prove the existence of minimizers for the Lagrangian action on anti-T/2 symmetric loop spaces. Moreover, we prove the minimizers are non-collision periodic solutions with some fixed wingding numbers

Advancing Agriculture Sustainability in Rural Maine by Using PFAS-free Paper

Per- and polyfluoroalkyl substances (PFAS) are widely used globally for their properties such as water and oil repellency, chemical resistance, and thermal stability. Their applications include food packaging, non-stick cookware, fabrics and textile, cosmetics, etc. However, PFAS are harmful to human health and environmentally toxic. Their resistant carbon-fluorine bonds cause bioaccumulation in the body, increasing the risk of cancer, lung diseases, and immune system. This persistence also leads to long-term water contamination and ecosystem disruption. Our research aims to replace PFAS with alternative materials that are biologically and environmentally safe. To achieve this, we have developed a PFAS-free omniphobic paper that repels water, oil, and other liquids. To create this omniphobic paper, we used vapor-phase deposition of chlorosilane molecules on release liners to form a nanoscale layer (\u3c 5 nm) of flexible, “liquid like” polymer brushes. This solvent-free strategy prevents paper from conventional damage, such as deformation and wrinkles, caused by immersing in liquids. The obtained paper shows unique properties, such as a low contact angle hysteresis (CAH\u3c60) for liquids with a broad range of surface tension (from 72.8 to 22.4 mN m-1). These unique properties allow for potential applications in various fields such as food packaging containers, oil funnels, microfluidics, and microtiter plates.https://digitalcommons.library.umaine.edu/rural_issues/1102/thumbnail.jp

Sparse Convolution for Approximate Sparse Instance

Computing the convolution $A \star B$ of two vectors of dimension $n$ is one of the most important computational primitives in many fields. For the non-negative convolution scenario, the classical solution is to leverage the Fast Fourier Transform whose time complexity is $O(n \log n)$. However, the vectors $A$ and $B$ could be very sparse and we can exploit such property to accelerate the computation to obtain the result. In this paper, we show that when $\|A \star B\|_{\geq c_1} = k$ and $\|A \star B\|_{\leq c_2} = n-k$ holds, we can approximately recover the all index in $\mathrm{supp}_{\geq c_1}(A \star B)$ with point-wise error of $o(1)$ in $O(k \log (n) \log(k)\log(k/\delta))$ time. We further show that we can iteratively correct the error and recover all index in $\mathrm{supp}_{\geq c_1}(A \star B)$ correctly in $O(k \log(n) \log^2(k) (\log(1/\delta) + \log\log(k)))$ time

Learnable Community-Aware Transformer for Brain Connectome Analysis with Token Clustering

Neuroscientific research has revealed that the complex brain network can be organized into distinct functional communities, each characterized by a cohesive group of regions of interest (ROIs) with strong interconnections. These communities play a crucial role in comprehending the functional organization of the brain and its implications for neurological conditions, including Autism Spectrum Disorder (ASD) and biological differences, such as in gender. Traditional models have been constrained by the necessity of predefined community clusters, limiting their flexibility and adaptability in deciphering the brain's functional organization. Furthermore, these models were restricted by a fixed number of communities, hindering their ability to accurately represent the brain's dynamic nature. In this study, we present a token clustering brain transformer-based model ($\texttt{TC-BrainTF}$) for joint community clustering and classification. Our approach proposes a novel token clustering (TC) module based on the transformer architecture, which utilizes learnable prompt tokens with orthogonal loss where each ROI embedding is projected onto the prompt embedding space, effectively clustering ROIs into communities and reducing the dimensions of the node representation via merging with communities. Our results demonstrate that our learnable community-aware model $\texttt{TC-BrainTF}$ offers improved accuracy in identifying ASD and classifying genders through rigorous testing on ABIDE and HCP datasets. Additionally, the qualitative analysis on $\texttt{TC-BrainTF}$ has demonstrated the effectiveness of the designed TC module and its relevance to neuroscience interpretations