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Assessment of the glomerular basement membrane (GBM) in transmission electron microscopy (TEM) is crucial for diagnosing chronic kidney disease (CKD). The lack of domain-independent automatic segmentation tools for the GBM necessitates an AI-based solution to automate the process. In this study, we introduce GBMSeg, a training-free framework designed to automatically segment the GBM in TEM images guided only by a one-shot annotated reference. Specifically, GBMSeg first exploits the robust feature matching capabilities of the pretrained foundation model to generate initial prompt points, then introduces a series of novel automatic prompt engineering techniques across the feature and physical space to optimize the prompt scheme. Finally, GBMSeg employs a class-agnostic foundation segmentation model with the generated prompt scheme to obtain accurate segmentation results. Experimental results on our collected 2538 TEM images confirm that GBMSeg achieves superior segmentation performance with a Dice similarity coefficient (DSC) of 87.27% using only one labeled reference image in a training-free manner, outperforming recently proposed one-shot or few-shot methods. In summary, GBMSeg introduces a distinctive automatic prompt framework that facilitates robust domain-independent segmentation performance without training, particularly advancing the automatic prompting of foundation segmentation models for medical images. Future work involves automating the thickness measurement of segmented GBM and quantifying pathological indicators, holding significant potential for advancing pathology assessments in clinical applications. The source code is available on https://github.com/SnowRain510/GBMSeg, Comment: Accepted for MICCAI2024

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s+s'$, $s, s'\in S$, $s

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$ denote the counting function of $A$.Bell and Shallit recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{\frac{k-2}{k}-\epsilon})$ for some $\epsilon>0$, then $R_{\mathbb{N}\setminus A,k}(n)$ is eventually strictly increasing. In this paper, we improve this result to $A(n)=O(n^{\frac{k-2}{k-1}})$. We also give an example to show that this bound is best possible., Comment: 11 pages

For integer $h\geq2$ and $A\subseteq\mathbb{N}$, we define $hA$ to be all integers which can be written as a sum of $h$ elements of $A$. The set $A$ is called an asymptotic basis of order $h$ if $n\in hA$ for all sufficiently large integers $n$. An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$. For $W\subseteq\mathbb{N}$, denote by $\mathcal{F}^*(W)$ the set of all finite, nonempty subsets of $W$. Let $A(W)$ be the set of all numbers of the form $\sum_{f \in F} 2^f$, where $F \in \mathcal{F}^*(W)$. In this paper, we give some characterizations of the partitions $\mathbb{N}=W_1\cup\cdots \cup W_h$ with the property that $A=A(W_1)\cup\cdots \cup A(W_{h})$ is a minimal asymptotic basis of order $h$. This generalizes a result of Chen and Chen, recent result of Ling and Tang, and also recent result of Sun., Comment: 8 pages

Let $\varphi (x_{1}, \ldots, x_{h})=c_{1} x_{1}+\cdots+c_{h} x_{h}$ be a linear form with coefficients in a field $\mathbf{F}$, and let $V$ be a vector space over $\mathbf{F}$. A nonempty subset $A$ of $V$ is a $\varphi$-Sidon set if $\varphi\left(a_{1}, \ldots, a_{h}\right)=\varphi\left(a_{1}^{\prime}, \ldots, a_{h}^{\prime}\right)$ implies $\left(a_{1}, \ldots, a_{h}\right)=$ $\left(a_{1}^{\prime}, \ldots, a_{h}^{\prime}\right)$ for all $h$-tuples $\left(a_{1}, \ldots, a_{h}\right) \in A^{h}$ and $\left(a_{1}^{\prime}, \ldots, a_{h}^{\prime}\right) \in A^{h}$. We call $A$ a polynomial perturbation of $B$ if for some $r>0$ and positive integer $k_0$, $|a_k-b_k|< k^r$ holds for all integers $k \geq k_0$. In this paper, for a given set $B$, we prove that there exists a $\varphi$-Sidon set $A$ of integers that is a polynomial perturbation of $B$. This gives an affirmative answer to a recent problem of Nathanson. Some other results are also proved., Comment: 6 pages. arXiv admin note: text overlap with arXiv:2101.01034

Let $\Delta(n)$ denote the smallest positive integer $m$ such that $a^3+a(1\le a\le n)$ are pairwise distinct modulo $m$. The purpose of this paper is to determine $\Delta(n)$ for all positive integers $n$., Comment: Correct a print mistake in Theorem 1.1

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s_1+s_2$, $s_1,s_2\in S$ and $s_1

Given a positive integer $n\ge 2$, let $D(n)$ denote the smallest positive integer $m$ such that $a^3+a(1\le a\le n)$ are pairwise distinct modulo $m^2$. A conjecture of Z.-W. Sun states that $D(n)=3^k$, where $3^k$ is the least power of $3$ no less than $\sqrt{n}$. The purpose of this paper is to confirm this conjecture.