arXiv subjects

Zhijie Chen

Publications and source records attributed to Zhijie Chen.

At least 19 recordsLinked to original sources

arXiv2026

Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schrödinger equations

We study positive solutions of the partially confined stationary nonlinear Schrödinger equation $$-Δu+|y|^2u+λu=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d -d$, we establish the existence of positive solutions under some standard assumptions. Furthermore, every positive solution decaying at infinity is radially symmetric and strictly decreasing in the confined variables and, up to one common translation, radially symmetric and strictly decreasing in the free variables. \vskip 0.2in Dedicated to our supervisor Prof. Wenming Zou on the occasion of his 60th birthday.

math.AP↗

arXiv2026

Exact counting of spherical metrics with one conical singularity on rectangular tori

We prove that for every integer $n\geq 2$ and $8π(n-1)<ρ<8πn$, the singular Liouville equation $Δu+\e^u=ρδ_0$ on a rectangular torus $E_{\mathrm{i}b}=\mathbb{C}/(\mathbb Z+\mathrm{i} b\mathbb Z)$ has exactly $n$ solutions, which are all axisymmetric. Together with previous results by Chen-Lin and Lin-Wang, this yields that \begin{itemize} \item $E_{\mathrm{i} b}$ admits no spherical metrics with a conical singularity of angle $2π\vartheta$ as long as $\vartheta$ is a positive odd integer. \item For every integer $n\geq 1$, $E_{\mathrm{i} b}$ admits exactly $n$ spherical metrics with a conical singularity of angle $2π\vartheta$ for each $\vartheta\in (2n-1, 2n+1)$. \end{itemize} The basic idea is to prove that the linearized equation has only trivial solutions in the space of axisymmetric functions. The previous method of analysing nodal domains via Bol's isoperimetric inequality only works for $ρ\leq 8π$. We develop a unified approach for all $ρ\in (0,+\infty)\setminus 8π\mathbb{N}_{\geq 1}$ by exploring the deep connection with the monodromy of the classical Lamé equation.

math.AP↗

arXiv2026

Qualitative analysis of positive singular solutions for a critical elliptic system in a punctured ball

We study qualitative properties of positive singular solutions to a weakly coupled elliptic system with critical exponents in a punctured ball. We give a sharp criterion on the removablity of the isolated singularity. We prove that semi-singular solutions (i.e., solutions with one component bounded near the singularity and the other component unbounded near the singularity) do not exist for the dimensions $N\geq 4$ but do exist for $N=3$. Asymptotic symmetry and sharp pointwise estimates are also proved for singular solutions. These generalizes some classical results of (Caffarelli, Gidas and Spruck, Comm. Pure Appl. Math, 1989) to the weakly coupled system. Moreover, for the weakly coupled system, we demonstrate a novel phenomenon that does not arise in the scalar equation.

math.AP↗

arXiv2026

Geometric analysis on rhombus torus: Green function with two singularities

Let $G(z)$ be the Green function on the flat torus $E_τ=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has at most one pair of nontrivial critical points. This is the third of a series of papers to study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. We study how the geometry of the torus and the location of singularities $\pm p$ affect the structure of critical points of $G_p(z)$. In Part I \cite{CFL}, we proved that $G_p(z)$ has at most three pairs of nontrivial critical points for all tori. In Part II \cite{CFL-II} (Proc. Lond. Math. Soc. 2026), we studied the important case that $E_τ$ is a rectangular torus. In this paper, first we prove that if $G_p(z)$ has three pairs of nontrivial critical points, then critical points are all non-degenerate. Secondly, we study the other important but more challenging case that $E_τ$ is a rhombus torus, by developing different approaches from \cite{CFL, CFL-II}. As applications, we show that the curvature equation $Δu+e^{u}=4π(δ_p+δ_{-p})$ on $E_τ$ has exactly either $0$, $1$ or $2$ even axisymmetric solutions and each number really occurs.

math.AP↗

arXiv2026

Sharp weighted Carleman and Huber isoperimetric inequalities on the unit ball in higher dimensions

In this paper, using a limiting approach, we establish a new type of weighted Carleman inequality in all dimensions $n\geq 2$ and classify all extremal functions. In particular, when $n=2$, we prove that our inequality is equivalent to a sharp norm inequality in the Bergman space. In even dimensions, we further establish a sharp weighted Huber isoperimetric inequality on the unit ball, which generalizes Huber's original result \cite[Ann. Math., 1954]{Huber} and may be regarded as a sharp counterpart of Y. Wang's isoperimetric inequality in the unit ball \cite[Adv. Math., 2015]{Wang}.

math.DG↗

arXiv2026

Never Too Late for Force: Accelerating VLA Post-Training with Reactive Force Injection

Pretrained vision-language-action (VLA) policies provide strong language-conditioned manipulation knowledge, but they remain largely vision-driven and can struggle once manipulation enters contact states where the scene is occluded, depth is ambiguous, or small force errors push execution off the offline demonstration distribution. We present LIFT (Late Reactive Injection of Force for VLA Post-Training), a force-aware post-training framework that adds contact reactivity to a pretrained VLA policy while preserving its general manipulation knowledge. LIFT grafts a reactive action expert beside the original action expert, initializes it from pretrained action weights, and injects recent 6D end-effector force through causal force memory and zero-initialized cross attention, enabling actions to be refreshed during execution. To address the policy-dependent distribution shift of contact feedback, LIFT further couples reactive force injection with an online DAgger loop that trains on a mixture of offline task-alignment data and human-corrected online rollouts. Across towel folding, book insertion, and Hanoi ring placement, LIFT learns faster and reaches higher performance than vision-only post-training, while ablations show that reactive force memory and online corrective data are both important for robust contact-rich manipulation. Our code and data will be publicly available.

cs.RO↗

arXiv2026

Towards UAV Detection in the Real World: A New Multispectral Dataset UAVNet-MS and a New Method

The proliferation of unmanned aerial vehicles (UAVs) has created urgent demand for precise UAV monitoring. Existing RGB-based systems rely on spatial cues that degrade at small scales, particularly with high inter-type similarity, target-clutter ambiguity, and low contrast. Multispectral imaging (MSI) encodes material-aware spectral signatures, yet MSI-based fine-grained small-UAV detection remains underexplored due to lack of dedicated datasets. We introduce UAVNet-MS, the first multispectral dataset for fine-grained small-UAV detection, comprising 15,618 temporally synchronized RGB-MSI data cubes (1440x1080) with bounding box annotations. The dataset features challenging small objects (93.7% <= 32^2 pixels, average 18^2 pixels, ~0.02% image area) under low contrast. We propose MFDNet, a dual-stream baseline addressing array-induced parallax and spatial-spectral fusion. Extensive evaluation under RGB-only, MSI-only, and RGB+MSI protocols against 20 detectors shows MFDNet achieves +6.2% AP50 improvement over best RGB-only methods, demonstrating spectral cues provide complementary material evidence beyond spatial cues. This work provides foundational dataset, strong baseline, and benchmark for multispectral UAV monitoring research.

cs.CV↗

arXiv2026

On the Classification of blow-up solutions of a singular Liouville equation on the disk

We study the blow-up behavior of solutions to the singular Liouville equation \[ Δ\tilde u+λe^{\tilde u}=4παδ_0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, \] where $α>0$, $λ>0$ and $B\subset\mathbb R^2$ is the unit disk. Our main results give a complete classification of all blow-up solutions and determine the exact number of solutions to the above equation. More precisely, for fixed $α>0$ and $λ\in(0,λ_α)$, the singular Liouville equation has exactly $\lceil α\rceil+2$ solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each $1\le m\le\lceil α\rceil$, a unique $m$-peak sequence whose blow-up points are the vertices of a regular $m$-gon centered at the origin. This result answers the questions raised in Bartolucci-Montefusco \cite{Bartolucci-Montefusco06} and Bartolucci \cite{Bartolucci10}. We also prove the non-degeneracy of these solutions. Thus we provide a full description of the blow-up structure for the singular Liouville equation on the disk.

math.AP↗

arXiv2026

Architectural Scaling Surpass Basis Complexity? Efficient KANs with Single-Parameter Design

The landscape of Kolmogorov-Arnold Networks (KANs) is rapidly expanding, yet lacks a unified theoretical framework and a clear principle for efficient architecture design. This paper addresses these gaps with three core contributions. First, we introduce the Universal KAN (Uni-KAN) framework, a novel abstraction that formally unifies all KAN-style networks through dense and sparse representations. We prove their interchangeability and provide an open-source library for this framework, facilitating future research. Second, we propose the Efficient KAN Expansion (EKE) Hypothesis, a design philosophy positing that allocating parameters to architectural scaling rather than basis function complexity yields superior performance. Third, we present Single-Parameter KANs (SKANs), a family of ultra-lightweight networks that embody the EKE Hypothesis. Our comprehensive experiments provide the first strong empirical validation for the theoretical necessity of basis function smoothness for stable training. Furthermore, SKANs demonstrate state-of-the-art performance, improving F1 scores by up to 6.51\% and reducing test loss by 93.1\%, while achieving up to 6x faster training speeds compared to existing KAN variants. These results establish a robust framework, a guiding hypothesis, and a practical methodology for designing the next generation of efficient and powerful neural networks. The code is accessible at https://anonymous.4open.science/r/SKAN-EBBB/.

cs.AI↗

arXiv2026

LArctan-SKAN: Simple and Efficient Single-Parameterized Kolmogorov-Arnold Networks using Learnable Trigonometric Function

This paper proposes a novel approach for designing Single-Parameterized Kolmogorov-Arnold Networks (SKAN) by utilizing a Single-Parameterized Function (SFunc) constructed from trigonometric functions. Three new SKAN variants are developed: LSin-SKAN, LCos-SKAN, and LArctan-SKAN. Experimental validation on the MNIST dataset demonstrates that LArctan-SKAN excels in both accuracy and computational efficiency. Specifically, LArctan-SKAN significantly improves test set accuracy over existing models, outperforming all pure KAN variants compared, including FourierKAN, LSS-SKAN, and Spl-KAN. It also surpasses mixed MLP-based models such as MLP+rKAN and MLP+fKAN in accuracy. Furthermore, LArctan-SKAN exhibits remarkable computational efficiency, with a training speed increase of 535.01% and 49.55% compared to MLP+rKAN and MLP+fKAN, respectively. These results confirm the effectiveness and potential of SKANs constructed with trigonometric functions. The experiment code is available at https://github.com/chikkkit/LArctan-SKAN .

cs.AI↗

arXiv2025

Green functions, Hitchin's formula and curvature equations on tori II: Rectangular torus

Let $G(z)$ be the Green function on the flat torus $E_τ=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has either $3$ or $5$ critical points (depending on the choice of $τ$). Here we study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. In Part I \cite{CFL}, we proved that for any $p$ satisfying $p\neq -p$ in $E_τ$, the number of critical points of $G_p(z)$ belongs to $\{4,6,8,10\}$ (depending on the choice of $(τ, p)$) and each number really occurs. In the Part II of this series, we study the important case $τ=ib$ with $b>0$, i.e. $E_τ$ is a rectangular torus. By developing a completely different approach from Part I, we show the existence of $8$ real values $d_1<d_2<\cdots<d_7<d_8$ such that if $$\wp(p)\in (-\infty, d_1]\cup [d_2, d_3]\cup [d_4, d_5]\cup [d_6, d_7]\cup [d_8,+\infty),$$ then $G_p(z)$ has no nontrivial critical points; if $$\wp(p)\in (d_1, d_2)\cup (d_3, d_4)\cup (d_5, d_6)\cup (d_7, d_8),$$ then $G_p(z)$ has a unique pair of nontrivial critical points that are always non-degenerate saddle points. This allows us to study the possible distribution of the numbers of critical points of $G_p(z)$ for generic $p$. Applications to the Painlevé VI equation and the curvature equation are also given.

math.AP↗

arXiv2025

Sharp stability on the second Robin eigenvalue with negative boundary parameters

In this paper, we prove a quantitative refinement of the isoperimetric type inequality for the second Robin eigenvalue with negative boundary parameters established by Freitas and Laugesen [Amer.J.Math.143 (2021), no.3, 969-994].Such new stability estimate is proved when the boundary parameter is not too far from 0.By constructing a suitable family of nearly spherical domains, we prove that the exponent for the Fraenkel asymmetry in this quantitative type inequality is sharp.

math.AP↗

arXiv2025

ReVul-CoT: Towards Effective Software Vulnerability Assessment with Retrieval-Augmented Generation and Chain-of-Thought Prompting

Context: Software Vulnerability Assessment (SVA) plays a vital role in evaluating and ranking vulnerabilities in software systems to ensure their security and reliability. Objective: Although Large Language Models (LLMs) have recently shown remarkable potential in SVA, they still face two major limitations. First, most LLMs are trained on general-purpose corpora and thus lack domain-specific knowledge essential for effective SVA. Second, they tend to rely on shallow pattern matching instead of deep contextual reasoning, making it challenging to fully comprehend complex code semantics and their security implications. Method: To alleviate these limitations, we propose a novel framework ReVul-CoT that integrates Retrieval-Augmented Generation (RAG) with Chain-of-Thought (COT) prompting. In ReVul-CoT, the RAG module dynamically retrieves contextually relevant information from a constructed local knowledge base that consolidates vulnerability data from authoritative sources (such as NVD and CWE), along with corresponding code snippets and descriptive information. Building on DeepSeek-V3.1, CoT prompting guides the LLM to perform step-by-step reasoning over exploitability, impact scope, and related factors Results: We evaluate ReVul-CoT on a dataset of 12,070 vulnerabilities. Experimental results show that ReVul-CoT outperforms state-of-the-art SVA baselines by 16.50%-42.26% in terms of MCC, and outperforms the best baseline by 10.43%, 15.86%, and 16.50% in Accuracy, F1-score, and MCC, respectively. Our ablation studies further validate the contributions of considering dynamic retrieval, knowledge integration, and CoT-based reasoning. Conclusion: Our results demonstrate that combining RAG with CoT prompting significantly enhances LLM-based SVA and points out promising directions for future research.

cs.SE↗

arXiv2025

Green functions, Hitchin's formula and curvature equations on tori

Let $G(z)=G(z;τ)$ be the Green function on the flat torus $E_τ=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has either $3$ or $5$ critical points (depending on the choice of $τ$). Later, Bergweiler and Eremenko (Proc. Amer. Math. Soc. 2016) gave a new proof of this remarkable result by using anti-holomorphic dynamics. In this paper, firstly, we prove that once $G(z)$ has $5$ critical points, then these $5$ critical points are all non-degenerate. Secondly, we study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. We prove that for any $p$ satisfying $p\neq -p$ in $E_τ$, the number of critical points of $G_p(z)$ belongs to $\{4,6,8,10\}$ (depending on the choice of $(τ, p)$) and each number really occurs. We apply Hitchin's formula (J. Differ. Geom. 1995) in a surprising way to prove the generic non-degeneracy of critical points. This allows us to study the distribution of the numbers of critical points of $G_p(z)$ as $p$ varies. Applications to the curvature equation $Δu+e^{u}=4π(δ_{p}+δ_{-p})$ on $E_τ$ are also given, and how the geometry of the torus affects the solution structure is studied.

math.AP↗

arXiv2025

Sketched Gaussian Mechanism for Private Federated Learning

Communication cost and privacy are two major considerations in federated learning (FL). For communication cost, gradient compression by sketching the clients' transmitted model updates is often used for reducing per-round communication. For privacy, the Gaussian mechanism (GM), which consists of clipping updates and adding Gaussian noise, is commonly used to guarantee client-level differential privacy. Existing literature on private FL analyzes privacy of sketching and GM in an isolated manner, illustrating that sketching provides privacy determined by the sketching dimension and that GM has to supply any additional desired privacy. In this paper, we introduce the Sketched Gaussian Mechanism (SGM), which directly combines sketching and the Gaussian mechanism for privacy. Using Rényi-DP tools, we present a joint analysis of SGM's overall privacy guarantee, which is significantly more flexible and sharper compared to isolated analysis of sketching and GM privacy. In particular, we prove that the privacy level of SGM for a fixed noise magnitude is proportional to $1/\sqrt{b}$, where $b$ is the sketching dimension, indicating that (for moderate $b$) SGM can provide much stronger privacy guarantees than the original GM under the same noise budget. We demonstrate the application of SGM to FL with either gradient descent or adaptive server optimizers, and establish theoretical results on optimization convergence, which exhibits only a logarithmic dependence on the number of parameters $d$. Experimental results confirm that at the same privacy level, SGM based FL is at least competitive with non-sketching private FL variants and outperforms them in some settings. Moreover, using adaptive optimization at the server improves empirical performance while maintaining the privacy guarantees.

cs.LG↗

arXiv2025

Show-o: One Single Transformer to Unify Multimodal Understanding and Generation

We present a unified transformer, i.e., Show-o, that unifies multimodal understanding and generation. Unlike fully autoregressive models, Show-o unifies autoregressive and (discrete) diffusion modeling to adaptively handle inputs and outputs of various and mixed modalities. The unified model flexibly supports a wide range of vision-language tasks including visual question-answering, text-to-image generation, text-guided inpainting/extrapolation, and mixed-modality generation. Across various benchmarks, it demonstrates comparable or superior performance to existing individual models with an equivalent or larger number of parameters tailored for understanding or generation. This significantly highlights its potential as a next-generation foundation model. Code and models are released at https://github.com/showlab/Show-o.

cs.CV↗

arXiv2025

Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations

We study the curvature equation with multiple singular sources on a torus \[Δu+e^{u}=8π\sum_{k=0}^{3}n_{k}δ_{\frac{ω_{k}}{2}}% +4π\left( δ_{p}+δ_{-p}\right) \quad \text{ on }\;E_τ:=\mathbb{C}/(\mathbb Z+\mathbb{Z}τ),\] where $n_k\in\mathbb N$ and $δ_a$ denotes the Dirac measure at $a$. This is known as a critical case for which the apriori estimate does not hold, and the existence of solutions has been a long-standing problem. In this paper, by establishing a deep connection with Painlevé VI equations, we show that the existence of even solutions (i.e. $u(z)=u(-z)$) depends on the location of the singular point $p$, and we give a sharp criterion of $p$ in terms of Painlevé VI equations.

math.AP↗

arXiv2025

On the distribution of critical points of the Eisenstein series $E_6$ and monodromy interpretation

In previous works joint with Lin, we proved that the Eisenstein series $E_4$ (resp. $E_2$) has at most one critical point in every fundamental domain $γ(F_0)$ of $Γ_{0}(2)$, where $γ(F_0)$ are translates of the basic fundamental domain $F_0$ via the Möbius transformation of $γ\inΓ_{0}(2)$. But the method can not work for the Eisenstein series $E_6$. In this paper, we develop a new approach to show that $E_6'(τ)$ has exactly either $1$ or $2$ zeros in every fundamental domain $γ(F_0)$ of $Γ_{0}(2)$. A criterion for $γ(F_0)$ containing exactly $2$ zeros is also given. Furthermore, by mapping all zeros of $E_6'(τ)$ into $F_0$ via the Möbius transformations of $Γ_{0}(2)$ action, the images give rise to a dense subset of the union of three disjoint smooth curves in $F_0$. A monodromy interpretation of these curves from a complex linear ODE is also given. As a consequence, we give a complete description of the distribution of the zeros of $E_6'(τ)$ in fundamental domains of $SL(2,\mathbb{Z})$.

math.NT↗