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Xiaoping Yang

Publications and source records attributed to Xiaoping Yang.

At least 19 recordsLinked to original sources
arXiv2026

Near-boundary rigidity and strict maximum modulus for Neumann eigenfunctions on convex planar domains

We develop a near-boundary rigidity theorem for arbitrary nonzero Neumann eigenfunctions on bounded convex planar domains. Precisely, if an eigenfunction $u$ with positive eigenvalue $μ$ has an interior maximum-modulus point $p$ satisfying $\sqrtμ\,\text{dist}(p,\partialΩ)\leπ/\sqrt3$, then $Ω$ must be a rectangle and $u$ must be a cosine of one coordinate, and $μ\ge4μ_2(Ω)$, with the factor $4$ being sharp. This shows that a scale-invariant distance condition at just one maximum-modulus point determines the geometry of the entire domain and the explicit form of the eigenfunction. We further prove that the maximum modulus of every second Neumann eigenfunction on a bounded convex planar domain is attained only on the boundary, without assuming boundary smoothness or eigenvalue simplicity. Moreover, we also prove strict boundary maximum modulus for higher Neumann eigenfunctions under suitable conditions, including all third Neumann eigenfunctions on arbitrary triangles.

math.AP↗
arXiv2026

Symmetry and critical points of second Neumann eigenfunctions on isosceles trapezoids and kites

We classify the symmetry and multiplicity of the second Neumann eigenvalue on isosceles trapezoids and kites as the height varies. Precisely, away from a unique transition height, the second Neumann eigenvalue is simple, and its eigenfunction is either symmetric or antisymmetric. At a transition height, it has multiplicity two, with one symmetric and one antisymmetric basis eigenfunction. We prove that every nonzero real second Neumann eigenfunction has no interior critical points, including arbitrary linear combinations at a transition height, thereby verifying the \textit{hot spots conjecture} for both classes of domains. In the simple-eigenvalue case, we also classify the nonvertex boundary critical points for trapezoids and kites. Reflection reduces the spectral classification to a comparison of a Neumann branch and a mixed Dirichlet--Neumann branch. Energy estimates establish strict transversality at every crossing and hence uniqueness wherever a transition occurs. To treat the two-dimensional eigenspace, we use Jacobian estimates and directional and radial comparison arguments to exclude cancellation of the two basis gradients.

math.AP↗
arXiv2026

Quantitative characterization of deviations from the hot spots conjecture on convex domains in two-dimensional space forms

In this paper, we study critical points of second Neumann eigenfunctions on convex domains in two-dimensional space forms from three complementary perspectives: spectral and geometric criteria for the absence of interior critical points, quantitative localization of possible critical points, and explicit bounds for the hot spots constant. Precisely, we first establish monotonicity-radius localization principles in $\mathbb S^2$ and $\mathbb H^2$. As a consequence, we obtain a unified diameter criterion for convex domains in these two space forms: if $μ_2(Ω)D^2\le j_{1,1}^2,$ then every second Neumann eigenfunction on $Ω$ has no interior critical points. Moreover, when interior critical points may exist, we derive explicit quantitative location restrictions in terms of the domain diameters in $\mathbb{S}^{2}$ and $\mathbb{H}^{2}$. Finally, we develop an analytic approach to study the \emph{hot spots constant} $\mathfrak{C}(Ω)$ on convex domains. For planar convex domains, we improve the Euclidean upper bound to $\mathfrak{C}(Ω)<1.48$. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, $\mathfrak{C}(Ω) < 4$ for $Ω\subset\mathbb{S}^{2}$ contained in a hemisphere; $\mathfrak{C}(Ω) < 11.2$ for $Ω\subset\mathbb{H}^{2}$. Our proofs combine the properties of Bessel and Legendre functions, eigenvalue estimates, and Green's identity. Our results quantitatively measure ``how wrong'' the \emph{hot spots conjecture} can be.

math.AP↗
arXiv2026

Measure upper bounds of nodal sets of solutions to Dirichlet problem of Schrödinger equations

In this paper, we focus on estimating measure upper bounds of nodal sets of solutions to the following boundary value problem \begin{equation*} \left\{ \begin{array}{lll} Δu+Vu=0\quad \mbox{in}\ Ω,\\[2mm] u=0\quad \mbox{on}\ \partialΩ, \end{array}\right. \end{equation*} where $V\in W^{1,\infty}(Ω)$ is a potential function, and $Ω\subset \mathbb{R}^n$ ($n \geq 2$) is a bounded domain whose boundary is of class $C^{1,α}$ for any $0<α<1$. By developing a delicate dividing iteration procedure, we show that upper bound of the $(n-1)$-dimensional Hausdorff measure of the nodal set of $u$ in $Ω$ is $$C\Big(1+\log\left(\|\nabla V\|_{L^{\infty}(Ω)}+1\right)\Big)\cdot\left(\|V\|_{L^{\infty}(Ω)}^{\frac{1}{2}}+\|\nabla V\|_{L^{\infty}(Ω)}^{\frac{1}{2}}+1\right),$$ provided $V$ is analytic, here $C$ is a positive constant depending only on $n$ and $Ω$. In particular, if $\|\nabla V\|_{L^{\infty}(Ω)}$ is small, the upper bound for the measure of the nodal set of $u$ is $C\left(\|V\|^{\frac{1}{2}}_{L^{\infty}(Ω)}+1\right)$, which is sharp in the sense of a famous conjecture of Yau.

math.AP↗
arXiv2025

A Fully Automatic Framework for Intracranial Pressure Grading: Integrating Keyframe Identification, ONSD Measurement and Clinical Data

Intracranial pressure (ICP) elevation poses severe threats to cerebral function, thus necessitating monitoring for timely intervention. While lumbar puncture is the gold standard for ICP measurement, its invasiveness and associated risks drive the need for non-invasive alternatives. Optic nerve sheath diameter (ONSD) has emerged as a promising biomarker, as elevated ICP directly correlates with increased ONSD. However, current clinical practices for ONSD measurement suffer from inconsistency in manual operation, subjectivity in optimal view selection, and variability in thresholding, limiting their reliability. To address these challenges, we introduce a fully automatic two-stage framework for ICP grading, integrating keyframe identification, ONSD measurement and clinical data. Specifically, the fundus ultrasound video processing stage performs frame-level anatomical segmentation, rule-based keyframe identification guided by an international consensus statement, and precise ONSD measurement. The intracranial pressure grading stage then fuses ONSD metrics with clinical features to enable the prediction of ICP grades, thereby demonstrating an innovative blend of interpretable ultrasound analysis and multi-source data integration for objective clinical evaluation. Experimental results demonstrate that our method achieves a validation accuracy of $0.845 \pm 0.071$ (with standard deviation from five-fold cross-validation) and an independent test accuracy of 0.786, significantly outperforming conventional threshold-based method ($0.637 \pm 0.111$ validation accuracy, $0.429$ test accuracy). Through effectively reducing operator variability and integrating multi-source information, our framework establishes a reliable non-invasive approach for clinical ICP evaluation, holding promise for improving patient management in acute neurological conditions.

cs.CV↗
arXiv2024

On the number and geometric location of critical points of solutions to a semilinear elliptic equation in annular domains

In this paper, one of our aims is to investigate the instability of the distribution of the critical point set $\mathcal{C}(u)$ of a solution $u$ to a semilinear equation with Dirichlet boundary condition in the planar annular domains. Precisely, we prove that $\mathcal{C}(u)$ in an eccentric circle annular domain, or a petal-like domain, or an annular domain where the interior and exterior boundaries are equally scaled ellipses contains only finitely many points rather than a Jordan curve. This result indicates that the critical point set $\mathcal{C}(u)$ is unstable when any boundary of planar concentric circle annular domain $Ω$ has some small deformation or minor perturbation. Based on studying the distribution of the nodal sets $u^{-1}_θ(0)(u_θ=\nabla u\cdot θ)$ and $u^{-1}(0)$, we prove that the solution $u$ on each symmetric axis has exactly two critical points under some conditions. Meanwhile, we further obtain that $\mathcal{C}(u)$ only has two critical points in an eccentric circle annular domain, has four critical points in an exterior petal-like domain with the exterior boundary $γ_E$ is an ellipse, and the maximum points are distributed on the long symmetric semi-axis and the saddle points on the short symmetric semi-axis. Moreover, we describe the geometric location of critical points of the solution $u$ by the moving plane method.

math.AP↗
arXiv2023

Morphology Edge Attention Network and Optimal Geometric Matching Connection model for vascular segmentation

There are many unsolved problems in vascular image segmentation, including vascular structural connectivity, scarce branches and missing small vessels. Obtaining vessels that preserve their correct topological structures is currently a crucial research issue, as it provides an overall view of one vascular system. In order to preserve the topology and accuracy of vessel segmentation, we proposed a novel Morphology Edge Attention Network (MEA-Net) for the segmentation of vessel-like structures, and an Optimal Geometric Matching Connection (OGMC) model to connect the broken vessel segments. The MEA-Net has an edge attention module that improves the segmentation of edges and small objects by morphology operation extracting boundary voxels on multi-scale. The OGMC model uses the concept of curve touching from differential geometry to filter out fragmented vessel endpoints, and then employs minimal surfaces to determine the optimal connection order between blood vessels. Finally, we calculate the geodesic to repair missing vessels under a given Riemannian metric. Our method achieves superior or competitive results compared to state-of-the-art methods on four datasets of 3D vascular segmentation tasks, both effectively reducing vessel broken and increasing vessel branch richness, yielding blood vessels with a more precise topological structure.

eess.IV↗
arXiv2023

Upper Measure Bounds of Nodal Sets of Solutions to the Bi-Harmonic Equations on $C^{\infty}$ Riemannian Manifolds

In this paper, we consider the nodal set of a bi-harmonic function $u$ on an $n$ dimensional $C^{\infty}$ Riemannian manifold $M$, that is, $u$ satisfies the equation $\triangle_M^2u=0$ on $M$, where $\triangle_M$ is the Laplacian operator on $M$. We first define the frequency function and the doubling index for the bi-harmonic function $u$, and then establish their monotonicity formulae and doubling conditions. With the help of the smallness propagation and partitions, we show that, for some ball $B_r(x_0)\subseteq M$ with $r$ small enough, an upper bound for the measure of nodal set of the bi-harmonic function $u$ can be controlled by $N^α$, that is, \mathcal{H}^{n-1}\left(\left\{x\in B_{r/2}(x_0)|u(x)=0\right\}\right)\leq CN^αr^{n-1}, where $N=\max\left\{C_0,N(x_0,r)\right\}$, $α$, $C$ and $C_0$ both are positive constants depending only on $n$ and $M$. Here $N(x_0,r)$ is the frequency function of $u$ centered at $x_0$ with radius $r$. Furthermore, we derive that an upper measure for nodal sets of eigenfunctions of the bi-harmonic operator on a $C^{\infty}$ compact Riemannian manifold without boundary can be controlled by $λ^β$ for the corresponding eigenvalue $λ^2$ and some positive constant $β$.

math.AP↗
arXiv2023

Strong unique continuation property for fourth order Baouendi-Grushin type subelliptic operators with strongly singular potential

In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equation \begin{equation*} Δ^2_{X}u=Vu, \end{equation*} where $Δ_{X}=Δ_{x}+|x|^{2α}Δ_{y}$ ($0<α\leq1$), with $x\in\mathbb{R}^{m}, y\in\mathbb{R}^{n}$, denotes the Baouendi-Grushin type subelliptic operators, and the potential $V$ satisfies the strongly singular growth assumption $|V|\leq \frac{c_0}{ρ^4}$, where \begin{equation*} ρ=\left(|x|^{2(α+1)}+(α+1)^2|y|^2\right)^{\frac{1}{2(α+1)}} \end{equation*} is the gauge norm. The main argument is to introduce an Almgren's type frequency function for the solutions, and show its monotonicity to obtain a doubling estimate based on setting up some refined Hardy-Rellich type inequalities on the gauge balls with boundary terms.

math.AP↗
arXiv2023

Quantitative unique continuation property for solutions to a bi-Laplacian equation with a potential

In this paper, we focus on the quantitative unique continuation property of solutions to \begin{equation*} Δ^2u=Vu, \end{equation*} where $V\in W^{1,\infty}$. We show that the maximal vanishing order of the solutions is not large than \begin{equation} C\left(\|V\|^{\frac{1}{4}}_{L^{\infty}}+\|\nabla V\|_{L^{\infty}}+1\right). \end{equation} Our key argument is to lift the original equation to that with a positive potential, then decompose the resulted fourth-order equation into a special system of two second-order equations. Based on the special system, we define a variant frequency function with weights and derive its almost monotonicity to establishing some doubling inequalities with explicit dependence on the Sobolev norm of the potential function.

math.AP↗
arXiv2023

Some Quantitative Properties of Solutions to the Buckling Type Equation

In this paper, we investigate the quantitative unique continuation, propagation of smallness and measure bounds of nodal sets of solutions to the Buckling type equation $\triangle^2u+λ\triangle u-k^2u=0$ in a bounded analytic domain $Ω\subseteq\mathbb{R}^n$ with the homogeneous boundary conditions $u=0$ and $\frac{\partial u}{\partialν}=0$ on $\partialΩ$, where $λ,\ k$ are nonnegative real constants, and $ν$ is the outer unit normal vector on $\partialΩ$. We obtain that, the upper bounds for the maximal vanishing order of $u$ and the $n-1$ dimensional Hausdorff measure of the nodal set of $u$ are both $C(\sqrtλ+\sqrt{k}+1)$, where $C$ is a positive constant only depending on $n$ and $Ω$. Moreover, we also give a quantitative result of the propagation of smallness of $u$.

math.AP↗
arXiv2023

A Weakly Supervised Segmentation Network Embedding Cross-scale Attention Guidance and Noise-sensitive Constraint for Detecting Tertiary Lymphoid Structures of Pancreatic Tumors

The presence of tertiary lymphoid structures (TLSs) on pancreatic pathological images is an important prognostic indicator of pancreatic tumors. Therefore, TLSs detection on pancreatic pathological images plays a crucial role in diagnosis and treatment for patients with pancreatic tumors. However, fully supervised detection algorithms based on deep learning usually require a large number of manual annotations, which is time-consuming and labor-intensive. In this paper, we aim to detect the TLSs in a manner of few-shot learning by proposing a weakly supervised segmentation network. We firstly obtain the lymphocyte density maps by combining a pretrained model for nuclei segmentation and a domain adversarial network for lymphocyte nuclei recognition. Then, we establish a cross-scale attention guidance mechanism by jointly learning the coarse-scale features from the original histopathology images and fine-scale features from our designed lymphocyte density attention. A noise-sensitive constraint is introduced by an embedding signed distance function loss in the training procedure to reduce tiny prediction errors. Experimental results on two collected datasets demonstrate that our proposed method significantly outperforms the state-of-the-art segmentation-based algorithms in terms of TLSs detection accuracy. Additionally, we apply our method to study the congruent relationship between the density of TLSs and peripancreatic vascular invasion and obtain some clinically statistical results.

eess.IV↗
arXiv2023

CTG-Net: An Efficient Cascaded Framework Driven by Terminal Guidance Mechanism for Dilated Pancreatic Duct Segmentation

Pancreatic duct dilation indicates a high risk of various pancreatic diseases. Segmentation of dilated pancreatic ducts on computed tomography (CT) images shows the potential to assist the early diagnosis, surgical planning and prognosis. Because of the ducts' tiny sizes, slender tubular structures and the surrounding distractions, most current researches on pancreatic duct segmentation achieve low accuracy and always have segmentation errors on the terminal parts of the ducts. To address these problems, we propose a terminal guidance mechanism called cascaded terminal guidance network (CTG-Net). Firstly, a terminal attention mechanism is established on the skeletons extracted from the coarse predictions. Then, to get fine terminal segmentation, a subnetwork is designed for jointly learning the local intensity from the original images, feature cues from coarse predictions and global anatomy information from the pancreas distance transform maps. Finally, a terminal distraction attention module which explicitly learns the distribution of the terminal distraction is proposed to reduce the false positive and false negative predictions. We also propose a new metric called tDice to measure the terminal segmentation accuracy for targets with tubular structures and two segmentation metrics for distractions. We collect our dilated pancreatic duct segmentation dataset with 150 CT scans from patients with 5 types of pancreatic tumors. Experimental results on our dataset show that our proposed approach boosts dilated pancreatic duct segmentation accuracy by nearly 20% compared with the existing results, and achieves more than 9% improvement for the terminal segmentation accuracy compared with the state-of-the-art methods.

cs.CV↗
arXiv2023

Automated Peripancreatic Vessel Segmentation and Labeling Based on Iterative Trunk Growth and Weakly Supervised Mechanism

Peripancreatic vessel segmentation and anatomical labeling play extremely important roles to assist the early diagnosis, surgery planning and prognosis for patients with pancreatic tumors. However, most current techniques cannot achieve satisfactory segmentation performance for peripancreatic veins and usually make predictions with poor integrity and connectivity. Besides, unsupervised labeling algorithms cannot deal with complex anatomical variation while fully supervised methods require a large number of voxel-wise annotations for training, which is very labor-intensive and time-consuming. To address these problems, we propose our Automated Peripancreatic vEssel Segmentation and lAbeling (APESA) framework, to not only highly improve the segmentation performance for peripancreatic veins, but also efficiently identify the peripancreatic artery branches. There are two core modules in our proposed APESA framework: iterative trunk growth module (ITGM) for vein segmentation and weakly supervised labeling mechanism (WSLM) for artery branch identification. Our proposed ITGM is composed of a series of trunk growth modules, each of which chooses the most reliable trunk of a basic vessel prediction by the largest connected constraint, and seeks for the possible growth branches by branch proposal network. Our designed iterative process guides the raw trunk to be more complete and fully connected. Our proposed WSLM consists of an unsupervised rule-based preprocessing for generating pseudo branch annotations, and an anatomical labeling network to learn the branch distribution voxel by voxel. We achieve Dice of 94.01% for vein segmentation on our collected dataset, which boosts the accuracy by nearly 10% compared with the state-of-the-art methods. Additionally, we also achieve Dice of 97.01% on segmentation and competitive performance on anatomical labeling for peripancreatic arteries.

eess.IV↗
arXiv2023

Pressure-induced superconductivity in quasi-one-dimensional semimetal $\mathrm{Ta}_2 \mathrm{PdSe}_6$

Here we report the discovery of pressure-induced superconductivity in quasi-one-dimensional $\mathrm{Ta}_2 \mathrm{PdSe}_6$, through a combination of electrical transport, synchrotron x-ray diffraction, and theoretical calculations. Our transport measurements show that the superconductivity appears at a critical pressure $P_{\mathrm{c}} \sim 18.3$ GPa and is robust upon further compression up to $62.6$ GPa. The estimated upper critical field $μ_0 H_{\mathrm{c} 2}(0)$ in the pressurized $\mathrm{Ta}_2 \mathrm{PdSe}_6$ is much lower than the Pauli limiting field, in contrast to the case in its isostructural analogs $M_2 \mathrm{Pd}_{\mathrm{x}} X_5$ $(M=\mathrm{Nb}$, Ta; $X=\mathrm{S}, \mathrm{Se})$. Concomitant with the occurrence of superconductivity, anomalies in pressuredependent transport properties are observed, including sign reversal of Hall coefficient, abnormally enhanced resistance, and dramatically suppressed magnetoresistance. Meanwhile, room-temperature synchrotron x-ray diffraction experiments reveal the stability of the pristine monoclinic structure (space group $C 2 / m$ ) upon compression. Combined with the density functional theory calculations, we argue that a pressure-induced Lifshitz transition could be the electronic origin of the emergent superconductivity in $\mathrm{Ta}_2 \mathrm{PdSe}_6$.

cond-mat.supr-con↗
arXiv2022

The Liver Tumor Segmentation Benchmark (LiTS)

In this work, we report the set-up and results of the Liver Tumor Segmentation Benchmark (LiTS), which was organized in conjunction with the IEEE International Symposium on Biomedical Imaging (ISBI) 2017 and the International Conferences on Medical Image Computing and Computer-Assisted Intervention (MICCAI) 2017 and 2018. The image dataset is diverse and contains primary and secondary tumors with varied sizes and appearances with various lesion-to-background levels (hyper-/hypo-dense), created in collaboration with seven hospitals and research institutions. Seventy-five submitted liver and liver tumor segmentation algorithms were trained on a set of 131 computed tomography (CT) volumes and were tested on 70 unseen test images acquired from different patients. We found that not a single algorithm performed best for both liver and liver tumors in the three events. The best liver segmentation algorithm achieved a Dice score of 0.963, whereas, for tumor segmentation, the best algorithms achieved Dices scores of 0.674 (ISBI 2017), 0.702 (MICCAI 2017), and 0.739 (MICCAI 2018). Retrospectively, we performed additional analysis on liver tumor detection and revealed that not all top-performing segmentation algorithms worked well for tumor detection. The best liver tumor detection method achieved a lesion-wise recall of 0.458 (ISBI 2017), 0.515 (MICCAI 2017), and 0.554 (MICCAI 2018), indicating the need for further research. LiTS remains an active benchmark and resource for research, e.g., contributing the liver-related segmentation tasks in \url{http://medicaldecathlon.com/}. In addition, both data and online evaluation are accessible via \url{www.lits-challenge.com}.

cs.CV↗