arXiv subjects

Qinfeng Li

Publications and source records attributed to Qinfeng Li.

At least 19 recordsLinked to original sources

arXiv2026

On hot spots of the torsion function and the ground state of the Laplacian

We compare the maximum points $P_Ω$ and $Q_Ω$ of the torsion function and the positive first Dirichlet eigenfunction. We begin with a systematic analytic description of both hot spots over the full range of opening angles of circular sectors: uniqueness, strict monotonicity with respect to the angle, distinct small-angle asymptotics, and explicit limiting equations in the slit-disk limit. We also prove the existence of a coincidence angle and establish arithmetic properties of the limiting coordinates. We then consider bounded convex planar domains with a line of symmetry, and we prove the sharp diameter-normalized identity \[ \sup_Ω \frac{|P_Ω-Q_Ω|}{\operatorname{diam}(Ω)}=\frac12. \] Its upper-bound argument uses standard localization in the heart and an elementary geometric enclosure. Sharpness follows from thin convex domains with a vanishing relative width perturbation peaking near one endpoint: the torsion hot spot approaches that endpoint, whereas the ground-state hot spot approaches the midpoint.

math.AP↗

arXiv2026

DTCA: Denoising Token Credit Assignment for Reinforcement Learning in Diffusion Language Models

Diffusion language models (DLMs) generate text through iterative denoising, enabling bidirectional context modeling and parallel token refinement. Recent studies have applied reinforcement learning (RL) to DLMs and improved their reasoning performance. However, the iterative denoising process makes credit assignment more challenging, as final outcome rewards provide limited information about how different tokens contribute to the final output. In this paper, we propose Denoising Token Credit Assignment (DTCA), a framework for token-level credit assignment in reinforcement learning for diffusion language models. DTCA estimates token contributions by measuring the improvement in prediction confidence of remaining masked positions after token commitment and groups these signals according to part-of-speech (POS) categories. These contribution weights are used to select informative denoising timesteps and assign token-level learning signals during RL training. DTCA further calibrates these learning signals according to token-level entropy changes induced by policy updates. Experiments on LLaDA-8B-Instruct show that DTCA consistently improves mathematical reasoning performance across different generation budgets, with gains of up to 9.0 points on MATH500 and 5.3 points on GSM8K over the base model.

cs.CL↗

arXiv2026

Concave entire solutions of the Liouville equation

We prove that every concave entire solution of the Liouville equation $-Δu=e^{2u}$ in $\R^n$ is one-dimensional, and we extend the result to a class of general semilinear equations under suitable conditions on the nonlinearity. In dimension two, we also provide a PDE-based proof of the known classification of quasiconcave solutions to planar Liouville solutions, without relying on complex-analytic methods developed in the previous work by Bergweiler-Eremenko-Langley \cite{BEL}

math.AP↗

arXiv2026

A quantitative Robin Pólya--Szegő theorem for tangential polygons

We extend the quantitative Dirichlet theorem of Gui, Hu, and Li \cite{GuiHuLi2026} to Robin torsional rigidity. For every fixed $β>0$, we prove that the regular $N$-gon uniquely maximizes Robin torsional rigidity among tangential $N$-gons of prescribed area. Our explicit perimeter deficit bound contains the Dirichlet deficit obtained by Gui, Hu, and Li \cite{GuiHuLi2026} and an additional Robin term that is positive for nonregular tangential polygons. For triangles, we answer the fixed-area optimization problem recorded as open by Keady and Wiwatanapataphee \cite[Appendix A.1.2]{KeadyWiwatanapataphee2016}. We also establish strict monotonicity along equal-area regular polygons as the number of sides increases, with an explicit bound for each consecutive gap. The proof combines mixed Robin--Neumann triangular cells, spectral estimates for angular concavity, and a quantitative scaling estimate.

math.AP↗

arXiv2026

Pressure estimates and classifications for steady $H^1_{\rm loc}$-solutions to two-dimensional incompressible Euler equations

We study two-dimensional steady incompressible Euler flows with finite total curvature. A distinguished family of such flows arises from finite-Morse-index solutions to semilinear elliptic equations in two dimensions. We establish three main properties of finite-total-curvature flows in the $L^\infty\cap H^1_{\rm loc}$ class. First, in the plane and the half-plane, every such flow without stagnation points is a parallel shear flow. Second, in the plane, the half-plane, and the infinite strip, the pressure converges uniformly to a constant at each end at infinity. Third, in the plane, the half-plane, the infinite strip, and the periodic strip, we establish sharp lower bounds for the total curvature in terms of the pressure oscillation. The rigidity result in fact extends to a broader class of flows: in the plane and the half-plane, any steady Euler flow without stagnation points is a parallel shear flow provided that $\nabla P\in L^q$ for some $1\le q\le 2$. Combined with standard elliptic estimates for the pressure, this result yields a concise new proof of Hamel and Nadirashvili's rigidity theorems, with the regularity assumption sharpened optimally to $H^1_{\rm loc}$. The pressure serves as a unifying quantity throughout the analysis.

math.AP↗

arXiv2026

Scaling Properties of Same-Family On-Policy Distillation

*Reinforcement learning (RL)* can induce substantial reasoning capabilities in large language models (LLMs), but how much of this capability transfers across model scales, and how quickly, remains unclear. We study the scaling properties of *on-policy distillation (OPD)* across *weak-to-strong*, *same-base*, and *strong-to-weak* teacher--student setups. We find that early OPD training dynamics uniformly exhibit a regular *useful-transfer* regime, in which held-out accuracy (the *gold score*, $G$) rises approximately linearly in $d=\sqrt{\mathrm{KL}(π_θ\Vert π_{\mathrm{ref}})}$, the square root of token-level reverse KL divergence from the student initialization. In every observed weak-to-strong pair, the student's peak gold score exceeds its teacher's own, so a compact RL expert can transfer capability to a much larger student via OPD. To estimate OPD outcomes, we fit *power laws* for how $G_{\mathrm{peak}}$ and the slope of the useful-transfer regime scale with student and teacher parameter counts and with teacher gold score. These laws show that peak gold score improves with teacher scale only up to roughly the student's scale, and that at a matched gold score smaller teachers transfer better, so a teacher's score alone does not define its supervision value. We also study the scaling effects of two OPD variants, bootstrapping weak-to-strong OPD, and the degree of on-policy supervision.

cs.LG↗

arXiv2026

Monotonicity properties of the Robin torsion function in a class of symmetric planar domains

We prove the monotonicity property of the Robin torsion function in a smooth planar domain $Ω$ with a line of symmetry, provided that the Robin coefficient $β$ is greater than or equal to the negative of the boundary curvature $κ$, i.e., $β\geq - κ$ on $\partialΩ$. We also construct a counterexample showing that the curvature condition cannot, in general, be removed.

math.AP↗

arXiv2026

EmbodiedMemory-Bench: Benchmarking Embodied Memory for Long-Horizon Embodied Tasks

Long-horizon embodied interaction requires agents to retain and continually update information about the environment as they observe, act, and encounter change. Yet current agents struggle to maintain such memory reliably. Our analysis traces this limitation to four key deficiencies: weak fine-grained visual memory, unreliable dynamic world-state tracking, failing to record world state revealed by interaction outcomes, and limited generalization from prior experience. However, existing benchmarks do not directly assess these memory capabilities during long-horizon embodied interaction. To address this gap, we introduce EmbodiedMemory-Bench (EMem-Bench), comprising 2,554 interactive episodes across four task families. EMem-Bench requires agents to build and update memory from interaction history, then use it to complete a later task by acting in the environment. We further present Embodied-Memorizer (EMem), an external memory system that organizes embodied experience into spatial, event, and scene memories. We also train EMem-8B, an 8B policy that manages and uses these memories. We evaluate a diverse range of open-source and proprietary MLLMs and representative multimodal memory systems. Results show that current models remain weak and uneven across the four challenges. Under matched backbones, EMem achieves the best overall performance among the evaluated memory systems and improves both open-source and proprietary models, while EMem-8B further improves over its backbone. Project page: https://zju-omniai.github.io/EmbodiedMemoryBench/

cs.CV↗

arXiv2026

Global Strict Monotonicity and Asymptotics of the First Steklov Eigenvalue for Regular Polygons

For a bounded Lipschitz planar domain $Ω$, let $σ_1(Ω)$ denote its first nonzero Steklov eigenvalue. Let $Ω_N$ be the regular $N$-gon normalized to have perimeter $2π$. We prove that \[ σ_1(Ω_{N+1})>σ_1(Ω_N),\qquad N\ge3. \] Consequently, $σ_1(Ω_N)$ increases strictly to the disk value $1$. We also show that the real first eigenspace has dimension two and derive the asymptotic expansion \[ σ_1(Ω_N) =1-\frac{2ζ(3)}{N^3} -\frac{8ζ(4)}{N^4} -\frac{26ζ(5)}{N^5} +O(N^{-6}). \] The proof combines an equivariant Schwarz--Christoffel pullback, a nodal selection of the two critical Fourier residue classes, and a positivity-improving Perron comparison for the associated reciprocal operators. The asymptotic expansion follows independently from a Schur reduction and the evaluation of Euler-type sums.

math.AP↗

arXiv2026

The Pólya--Szegő conjecture for convex polygons with many sides

For every sufficiently large N, we prove that the regular N-gon uniquely minimizes the first Dirichlet eigenvalue among convex polygons of prescribed area with at most N sides, up to rigid motions. Quantitative Faber--Krahn stability localizes minimizers near the disk but leaves their discrete geometry undetermined. We encode that geometry by a centered measure of exterior angles and prove an inverse-cubic Fourier inequality whose equality case is the uniform root configuration. Its defect gives a lower bound for the quadratic spectral excess over the regular polygon. Cyclic cancellation and a common material-coordinate comparison make the nonlinear difference small relative to the same defect. Exact constraint coordinates extend this comparison to arbitrarily small positive defects. Minimality then forces zero defect, which characterizes the regular polygon.

math.AP↗

arXiv2026

A Quantitative Pólya--Szegő Theorem for Tangential Polygons

For a bounded Lipschitz domain $Ω\subset\mathbb R^2$, let $T(Ω)=\int_Ωu_Ω\, dx$ denote its torsional rigidity, where $-Δu_Ω=1$ in $Ω$ and $u_Ω=0$ on $\partialΩ$. We prove a quantitative Pólya--Szegő inequality for tangential polygons. Let $N\ge3$, let $P$ be a tangential $N$-gon, set $A=|P|$, and let $R_N$ be the regular $N$-gon of area $A$. Writing $L(\cdot)$ for perimeter, we obtain the explicit deficit estimate \[ T(R_N)-T(P)\ge \frac{A^2}{8N\tan(π/N)} \left(1-\frac{L(R_N)^2}{L(P)^2}\right)^2.\]Thus, at fixed area, the torsional deficit is controlled from below purely by the perimeter ratio. In particular, the regular $N$-gon is the unique maximizer of torsional rigidity among tangential $N$-gons of prescribed area; for $N=3$ this gives the classical triangular Pólya--Szegő theorem with a quantitative estimate. The same perimeter estimate yields an explicit positive lower bound for $T(R_{N+1})-T(R_N)$ for equal-area regular polygons, and hence a rather short alternative proof of the strict monotonicity of torsional rigidity in $N$. Combined with the Kohler--Jobin inequality, it also gives an explicit sufficient condition for the polygonal Faber--Krahn inequality within the tangential class. Our full quantitative inequality is stronger: it contains, in addition, a nonnegative angular Jensen deficit, which yields quantitative angular stability away from degenerate configurations.

math.AP↗

arXiv2026

AttriMem: Attribution-Guided Process Feedback for Agent Memory Construction

Effective memory is crucial for LLM agents, yet constructing it effectively remains challenging. A memory-construction policy decides what information to extract, store, update, compress, or discard as interactions accumulate. Heuristic memory methods rely on subjective, task-specific rules, which can misalign with downstream objectives and limit cross-task adaptability. RL-based methods, by contrast, learn from task feedback but mainly use outcome- or module-level rewards. These coarse signals indicate task success but cannot identify which intermediate memory contents support the final answer, creating a fine-grained credit-assignment bottleneck. However, constructing such process feedback is prohibitively difficult because intermediate memory decisions lack unique ground-truth targets, while the appropriate credit varies with the agent's uncertain reasoning trajectory and therefore cannot be specified in advance. We propose AttriMem, an attribution-guided process-feedback framework for learning memory-construction policies with RL. AttriMem augments the global outcome reward with local rewards derived from token-level contributions to the final answer. Experiments on long-horizon dialogue question answering show that AttriMem outperforms retrieval-based, heuristic, and RL-based baselines, generalizes across benchmarks and answer models, stabilizes RL optimization.

cs.AI↗

arXiv2026

SkillAligner: Treating Retrieved Skills as Adaptable Drafts at Execution Time

General-purpose skills promise reusable procedural knowledge for language agents, yet semantic relevance does not guarantee execution utility: a retrieved skill may encode assumptions that conflict with the current task, execution environment, or other retrieved skills. We formalize this problem as the skill--execution misfit. To address it, we propose SkillAligner, a training-free execution-time skill adaptation framework that treats retrieved skills as adaptable drafts rather than fixed instructions. Before execution, SkillAligner performs a one-time joint adaptation that specializes useful skill fragments to task requirements, aligns their procedural assumptions with the available execution interface, and composes the resulting guidance by resolving dependencies, conflicts, and redundancy across skills. The adapted content is consolidated into a compact execution guide and reused throughout the subsequent trajectory. Extensive experiments across diverse agent benchmarks and model backbones show that SkillAligner substantially improves task performance over existing skill-use baselines, reduces skill-induced regressions at the instance level, and lowers total inference cost.

cs.LG↗

arXiv2026

Serrin's Problem under Dirichlet Perturbations: Geometric Compactness and Sharp Planar Stability

In earlier work [21], we posed a stability question for Serrin's overdetermined problem under Dirichlet perturbations and proved that the answer is negative in dimensions $n\ge3$. Here we resolve the question in the planar convex class and obtain a sharp quantitative theory without any a priori geometric nondegeneracy. Let $u_Ω$ solve \[ -Δu_Ω=1\ \text{in }Ω,\qquad \partial_νu_Ω=-\frac{|Ω|}{P(Ω)}\ \text{on }\partialΩ, \qquad \int_{\partialΩ}u_Ω\,dσ=0, \] and set $O(Ω):=\text{osc}_{\partial Ω}u_Ω$. We construct fixed-area annuli with $O(Ω_k)\to0$ that remain far from every disk, showing that convexity is essential in dimension two. By contrast, if $Ω_k\subset\mathbb R^2$ are convex, $|Ω_k|=π$, and $O(Ω_k)\to0$, then, up to translations, $Ω_k$ converges in Hausdorff distance to the unit disk. Moreover, \[ R_Ω-r_Ω+\inf_{z\in\mathbb R^2}d_H(Ω,B_1(z)) \le C\,O(Ω) \] for all planar convex $Ω$ with $|Ω|=π$ and sufficiently small $O(Ω)$, and the linear order is optimal. The proof combines a new mechanism excluding long-thin degeneration, the rough-domain Serrin rigidity theorem of Figalli--Zhang, new tangential-gradient and linear boundary-growth estimates, a boundary $P$-function estimate, and the reverse-Serrin identity of Magnanini--Molinarolo--Poggesi. We also study the weaker deficit \[ A(Ω):=\frac1{P(Ω)}\int_{\partialΩ}u_Ω,dσ-\min_{\partialΩ}u_Ω. \] In the planar convex class, $A(Ω_k)\to0$ still forces convergence to a disk, and \[ R_Ω-r_Ω+\inf_z d_H(Ω,B_1(z)) \le C A(Ω)^{2/3} \] for $|Ω|=π$ and sufficiently small $A(Ω)$.

math.AP↗

arXiv2026

Towards Steering without Sacrifice: Principled Training of Steering Vectors for Prompt-only Interventions

Recently, steering vectors (SVs) have emerged as an effective and lightweight approach to steer behaviors of large language models (LLMs), among which fine-tuned SVs are more effective than optimization-free ones. However, current approaches to fine-tuned SVs suffer from two limitations. First, they require careful selection of steering factors on a per-SV basis to balance steering effectiveness and generation quality at inference time. Second, they operate as full-sequence SVs (FSSVs), which can sacrifice generation quality regardless of factor selection due to excessive intervention on the model generation process. To address the first limitation, we propose joint training of steering factors and directions, such that post-hoc factor selection is no longer required. Using neural network scaling theory, we find that moderately large initialization sizes and learning rates for steering factors are essential for stability and efficiency of joint training. To tackle the second limitation, we draw inspiration from representation fine-tuning and introduce Prompt-only SV (PrOSV), an SV that intervenes only on a few prompt tokens. Our empirical results show that PrOSV outperforms traditional FSSVs on AxBench when using our joint training scheme. We also find that PrOSV achieves a better tradeoff between general model utility and adversarial robustness than FSSV.

cs.LG↗

arXiv2026

Mixed Torsion on Right Triangles and the Pólya--Szegő Monotonicity Problem for Regular Polygons

Motivated by the polygonal Pólya--Szegő conjecture for torsional rigidity, we study two monotonicity problems for torsional rigidity. The first concerns a mixed torsion problem on fixed-area right triangles, with a Dirichlet condition on one leg and Neumann conditions on the other leg and on the hypotenuse. We prove that the mixed torsional rigidity strictly increases as the ratio of the Neumann leg to the Dirichlet leg increases. The proof uses a Hadamard shape derivative, a Pohozaev-type identity, and a monotonicity result for the mixed torsion function. We also prove a similar result for the mixed ground state of Laplacian. The second concerns regular polygons. If $P_N$ denotes the regular $N$-gon of area $π$, we prove, by a purely analytic Schwarz--Christoffel/Bergman analytic-content argument, that \[ T^D(P_{N+1})>T^D(P_N),\qquad N\ge3, \] where $T^D$ is the Dirichlet torsional rigidity. We also obtain the asymptotic expansion \[ T^D(P_N)=\fracπ{8}-\frac{πζ(3)}{N^3} +\frac{π^5}{45N^4}+O(N^{-5}). \]

math.AP↗

arXiv2026

PragLocker: Protecting Agent Intellectual Property in Untrusted Deployments via Non-Portable Prompts

LLM agents rely on prompts to implement task-specific capabilities based on foundation LLMs, making agent prompts valuable intellectual property. However, in untrusted deployments, adversaries can copy and reuse these prompts with other proprietary LLMs, causing economic losses. To protect these prompts, we identify four key challenges: proactivity, runtime protection, usability, and non-portability that existing approaches fail to address. We present PragLocker, a prompt protection scheme that satisfies these requirements. PragLocker constructs function-preserving obfuscated prompts by anchoring semantics with code symbols and then using target-model feedback to inject noise, yielding prompts that only work on the target LLM. Experiments across multiple agent systems, datasets, and foundation LLMs show that PragLocker substantially reduces cross-LLM portability, maintains target performance, and remains robust against adaptive attackers.

cs.CR↗

arXiv2026

CoreGuard: Safeguarding Foundational Capabilities of LLMs Against Model Stealing in Edge Deployment

Proprietary large language models (LLMs) exhibit strong generalization capabilities across diverse tasks and are increasingly deployed on edge devices for efficiency and privacy reasons. However, deploying proprietary LLMs at the edge without adequate protection introduces critical security threats. Attackers can extract model weights and architectures, enabling unauthorized copying and misuse. Even when protective measures prevent full extraction of model weights, attackers may still perform advanced attacks, such as fine-tuning, to further exploit the model. Existing defenses against these threats typically incur significant computational and communication overhead, making them impractical for edge deployment. To safeguard the edge-deployed LLMs, we introduce CoreGuard, a computation- and communication-efficient protection method. CoreGuard employs an efficient protection protocol to reduce computational overhead and minimize communication overhead via a propagation protocol. Extensive experiments show that CoreGuard achieves upper-bound security protection with negligible overhead.

cs.CR↗