Jiechen Zhang
This note records a common threshold/surplus decomposition for single-threshold stopping rules in the classical prophet inequality. The same decomposition is used to certify several deterministic thresholds, including the median, half-mean, and balanced-surplus thresholds, and to give an averaged certificate for randomized thresholds distributed as the maximum.
Assefa Seyoum Wahd, Banafshe Felfeliyan, Yuyue Zhou et al.
Foundation models like the segment anything model require high-quality manual prompts for medical image segmentation, which is time-consuming and requires expertise. SAM and its variants often fail to segment structures in ultrasound (US) images due to domain shift. We propose Sam2Rad, a prompt learning approach to adapt SAM and its variants for US bone segmentation without human prompts. It introduces a prompt predictor network (PPN) with a cross-attention module to predict prompt embeddings from image encoder features. PPN outputs bounding box and mask prompts, and 256-dimensional embeddings for regions of interest. The framework allows optional manual prompting and can be trained end-to-end using parameter-efficient fine-tuning (PEFT). Sam2Rad was tested on 3 musculoskeletal US datasets: wrist (3822 images), rotator cuff (1605 images), and hip (4849 images). It improved performance across all datasets without manual prompts, increasing Dice scores by 2-7% for hip/wrist and up to 33% for shoulder data. Sam2Rad can be trained with as few as 10 labeled images and is compatible with any SAM architecture for automatic segmentation.
Jiechen Zhang
We prove that, among rectangular grid graphs with a fixed number of vertices, the number of spanning trees increases when the side lengths are made more balanced. In particular, among all rectangular grid graphs with $n^2$ vertices, the square $n\times n$ grid has the largest number of spanning trees. The proof starts with the Laplacian product formula, passes to hyperbolic coordinates, and compares logarithms by separating a discrete-concavity term from a positive decreasing residual term.