arXiv subjects

Lu Lu

Publications and source records attributed to Lu Lu.

At least 37 records · Page 2Linked to original sources

arXiv2026

The Second Largest Eigenvalue of Stiffness Matrices of Normalized Complete Frameworks

Let $R(G,p)$ be the normalized rigidity matrix of a framework $(G,p)$ in $\mathbb R^d$, and let \[ L(G,p)=R(G,p)R(G,p)^{T} \] be the associated stiffness matrix. We study the extremal eigenvalues of $L(K_n,p)$ for complete frameworks whose vertices lie on the unit sphere and have centroid at the origin. Our main result shows that, whenever $d\ge2$ and the image of $p$ contains at least three distinct points, the second largest eigenvalue of $L(K_n,p)$ is exactly $n/2$. This settles the eigenvalue part of a conjecture of Lew et al. [Israel J. Math. 256, 2023]. We further construct an infinite family of examples, given by regular polygons embedded in a two-dimensional subspace, for which the eigenvalue $n/2$ has multiplicity $2n-4$. Consequently, the multiplicity predicted in the conjecture is not correct in general. Our results reveal a dichotomy: the value of the second largest eigenvalue is universal, while its multiplicity is sensitive to the geometry of the underlying point configuration.

math.CO↗

arXiv2026

Building a Scalable, Reproducible, Evaluatable, and Closed-Loop Simulation Environment Foundation for Embodied Intelligence

This paper presents a cloud-native simulation infrastructure framework for embodied intelligence that supports large-scale training, standardized evaluation, and simulation-based data collection. The framework unifies simulation environment generation, task execution, trajectory collection, model evaluation, data management, and cloud services into a scalable and reproducible platform. To address the high cost, limited scalability, and poor reproducibility of real-world robotic data collection, the framework adopts cloud-native technologies including elastic resource scheduling, containerized simulation, unified data management, and service-oriented system design, enabling efficient large-scale simulation for multi-model and multi-task workloads. Built on a four-layer architecture, the framework provides standardized environment assets, automated task generation, trajectory collection, benchmark evaluation, and closed-loop data optimization. It further integrates representative systems including D-VLA, RL-VLA3, Sword, and Pre-VLA to support scalable simulation, dynamic scheduling, visual augmentation, and real-time data filtering. We argue that cloud-native simulation infrastructure provides a unified foundation for data generation, model training, standardized evaluation, and real-world deployment, and will play a key role in the future development of embodied intelligence.

cs.RO↗

arXiv2026

Thresholds for the Frankl-Wang $3/7$ conjecture on maximum-degree ratios

Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family, $Δ(\mathcal{F})=\max_{x\in[n]}|\{F\in\mathcal{F}:x\in F\}|$, and $\varrho(\mathcal{F})=Δ(\mathcal{F})/|\mathcal{F}|$. Frankl and Wang conjectured that if $n>100k$ and $|\mathcal{F}|>\binom{n-3}{k-3}$, then $\varrho(\mathcal{F})\ge 3/7$; the constant $3/7$ is sharp because of the Fano-plane construction. In this note we obtain three results. First, we show that no linear threshold $n>Ck$ can be sufficient: using a truncated Fano-plane construction we exhibit, for every constant $C$ and all large $k$, an intersecting family with $n>Ck$, $|\mathcal{F}|>\binom{n-3}{k-3}$, yet $\varrho(\mathcal{F})<3/7$. In particular, the original condition $n>100k$ does not guarantee the conclusion. Second, for $k=3$ we prove that $\varrho(\mathcal{F})\ge 3/7$ holds for every nonempty intersecting $3$-uniform family; the proof is nontrivial and does not rely on any assumption on $n$ or $|\mathcal{F}|$. Third, using the classical pseudo-sunflower bound $|\mathcal{F}|\le t^k$ (for families containing no pseudo-sunflower of size $t+1$), we obtain a completely explicit polynomial threshold for all $k\ge4$: if $n>(k-3)(7k^4+k)+3$ and $|\mathcal{F}|>\binom{n-3}{k-3}$, then $\varrho(\mathcal{F})\ge 3/7$. In particular, the simplified bound $n>7k^5$ is sufficient for every $k\ge4$.

math.CO↗

arXiv2026

On the sum of the two largest eigenvalues of the curl-curl operator on graphs

The Grone--Merris conjecture, proved by Bai in~2011, states that the spectrum of the graph Laplacian $Δ_0 = -\operatorname{div}\operatorname{grad}$ is majorized by the conjugate of the vertex degree sequence. Duval and Reiner proposed a simplicial complex analogue of this statement. On a graph, where triangles serve as $2$-simplices, their conjecture reduces to the assertion that the spectrum of $\operatorname{curl}^*\operatorname{curl}$ is majorized by the conjugate of the second-order degree sequence, which records the number of triangles containing each vertex. We prove that the sum of the two largest eigenvalues of $\operatorname{curl}^*\operatorname{curl}$ does not exceed the sum of the first two entries of that conjugate sequence. This confirms the first two majorization inequalities predicted by Duval and Reiner for $\operatorname{curl}^*\operatorname{curl}$. As a corollary, we obtain upper bounds for the two largest eigenvalues of the full graph Helmholtzian $Δ_1 = -\operatorname{grad}\operatorname{div} + \operatorname{curl}^*\operatorname{curl}$. The same result extends to the up-Laplacian of any $3$-family, yielding a concrete step towards the Duval--Reiner conjecture in dimension~$1$.

math.CO↗

arXiv2026

Data-driven discovery of governing differential equations across physical systems

Differential equations play a critical role in scientific discovery because they provide a mathematical framework to describe the behaviour of physical phenomena. As a promising alternative to traditional first principles, data-driven differential equation discovery has attracted increasing attention for its ability to infer governing laws directly from experimental or simulated data, especially when the underlying physics is unclear. However, the field has expanded rapidly along diverse methodological directions, particularly with the emergence of AI-based approaches, and still lacks a clear organizing perspective. In this Review, we propose a problem-oriented perspective on data-driven differential equation discovery. We first introduce a two-dimensional phase diagram of equation discoverability, where discovery problems are organized according to structural complexity and coefficient complexity. This phase diagram shows how the field has moved from the discovery of sparse equations with simple coefficients toward more complex governing laws with richer structures and more flexible parameterizations. It also clarifies why different methodological families succeed or fail in different problem settings. We then present the representation-evaluation-optimization (REO) framework as a fundamental abstraction of the discovery process. By identifying the core problems of equation discovery that persist across algorithmic variations, REO shifts the discussion from individual algorithms to the fundamental principles that determine discoverability. We connect these perspectives to applications across physics and adjacent sciences, and argue that the next challenge is not merely recovering equations, but using them to revise existing theories, distil mechanisms and form new scientific concepts.

cs.LG↗

arXiv2026

EarlyTom: Early Token Compression Completes Fast Video Understanding

Video large language models (Video-LLMs) have demonstrated strong capabilities in video understanding tasks. However, their practical deployment is still hindered by the inefficiency introduced by processing massive amounts of visual tokens. Although recent approaches achieve extremely low token retention ratios while maintaining accuracy comparable to full-token baselines, most of them perform compression only at the late stage of prefilling, leaving the efficiency of the vision encoder unoptimized. In this paper, we first show that vision encoding contributes a large portion to the time-to-first-token (TTFT). Therefore, instead of compressing visual tokens only after the vision encoder, performing compression inside the encoder still leaves substantial room for exploration. Based on this insight, we propose EarlyTom, a training-free token compression framework that performs early-stage visual token compression inside the vision encoder, enabling significantly better TTFT reduction and higher throughput. In addition, we introduce a decoupled spatial token selection strategy that improves the overall compression effectiveness. EarlyTom reduces TTFT by up to 2.65x and FLOPs by up to 61% on a single NVIDIA A100 GPU for the LLaVA-OneVision-7B model, while maintaining accuracy comparable to the full-token baseline. These improvements substantially enhance the practicality of deploying Video-LLMs in real-world production scenarios.

cs.CV↗

arXiv2026

Helmholzian Spectra of Graphs: Novel Properties

Let $\grad$, $\curl$, and $\dv$ be the graph-theoretic analogues of the gradient, curl, and divergence operators from multivariate calculus. The graph Laplacian $-\dv \grad$ gives rise to the celebrated Laplacian matrix, while the matrix representation of the graph Helmholtzian $\grad \grad^* + \curl^* \curl$ is called the Helmholtzian matrix. In this paper, we present a new graph-theoretic proof that the Helmholtzian matrix indeed represents the graph Helmholtzian. We then investigate the spectral properties of this matrix. Our main results are as follows: (i) a classification of graphs having exactly two distinct Helmholtzian eigenvalues; (ii) the nullity of the Helmholtzian matrix; and (iii) a combinatorial interpretation of the coefficients of the Helmholtzian polynomial. Furthermore, we determine the Helmholtzian spectrum for certain graph products and characterize Helmholtzian integral graphs, as well as derive bounds for the smallest Helmholtzian eigenvalue. Meanwhile, we pose some open problems for future research.

math.CO↗

arXiv2026

Helmholzian spectra of graphs: basic properties

The Helmholtzian matrix of a graph $G=(V(G),E(G))$ is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) [S. Li, L. Lu, J.F. Wang, A graph discretization of vector Laplacian, 379 (2026) 446--460]. Motivated by the applications of graph Helmholtzian in simplicial networks, we will investiagte its basic spectral properties. As the first graph matrix indexed by edge set, we find that Helmholtzian matrix is positive semi-definite and its non-negativity correlates with the odd cycles in $G$ and the orientation on $E(G)$, while its irreducibility relates to the signed graphs with loops. We show that the eigenvalues of Helmholtzian matrix are independent of the orientation and further investigate the eigenvalue interlacing under edge additions. One of striking findings is that the non-zero eigenvalues of the Laplacian matrix are those of Helmholtzian matrix of every graph. All these discoveries reveal that the Helmholtzian spectrum of $G$ balances and bridges the oriented graphs, weighted graphs and signed graphs as well as their adjacency or Laplacian spectra.

math.CO↗

arXiv2026

End-to-end Listen, Look, Speak and Act

Human interaction is inherently multimodal and full-duplex: we listen while watching, speak while acting, and fluidly adapt to turn-taking and interruptions. Realizing these capabilities is essential for building models simulating humans. We present ELLSA (End-to-end Listen, Look, Speak and Act), which, to our knowledge, is the first full-duplex, end-to-end model that simultaneously perceives and generates across vision, text, speech, and action within a single architecture, enabling interaction patterns previously out of reach, yielding more natural, human-like behaviors. At its core is a novel SA-MoE architecture (Self-Attention Mixture-of-Experts) that routes each modality to specialized experts and fuses them through a unified attention backbone. This provides a generalizable solution for joint multimodal perception and concurrent generation, leveraging strong pre-trained components while enabling efficient modality integration and mitigating modality interference. On speech-interaction and robot-manipulation benchmarks, ELLSA matches modality-specific baselines, while uniquely supporting advanced multimodal and full-duplex behaviors such as dialogue and action turn-taking, defective instruction rejection, speaking-while-acting, context-grounded visual question answering, and action barge-ins. We contend that ELLSA represents a step toward more natural and general interactive intelligence, contributing to the broader pursuit of artificial general intelligence. All data, code and model checkpoints will be released at https://github.com/bytedance/SALMONN/tree/ELLSA.

cs.AI↗

arXiv2026

Nested Fourier-enhanced neural operator for efficient modeling of radiation transfer in fires

Computational fluid dynamics (CFD) has become an essential tool for predicting fire behavior, yet maintaining both efficiency and accuracy remains challenging. A major source of computational cost in fire simulations is the modeling of radiation transfer, which is usually the dominant heat transfer mechanism in fires. Solving the high-dimensional radiative transfer equation (RTE) with traditional numerical methods can be a performance bottleneck. Here, we present a machine learning framework based on Fourier-enhanced multiple-input neural operators (Fourier-MIONet) as an efficient alternative to direct numerical integration of the RTE. We first investigate the performance of neural operator architectures for a small-scale 2D pool fire and find that Fourier-MIONet provides the most accurate radiative solution predictions. The approach is then extended to 3D CFD fire simulations, where the computational mesh is locally refined across multiple levels. In these high-resolution settings, monolithic surrogate models for direct field-to-field mapping become difficult to train and computationally inefficient. To address this issue, a nested Fourier-MIONet is proposed to predict radiation solutions across multiple mesh-refinement levels. We validate the approach on 3D McCaffrey pool fires simulated with FireFOAM, including fixed fire sizes and a unified model trained over a continuous range of heat release rates (HRRs). The proposed method achieves global relative errors of 2-4% for 3D varying-HRR scenarios while providing faster inference than the estimated cost of one finite-volume radiation solve in FireFOAM for the 16-solid-angle case. With fast and accurate inference, the surrogate makes higher-fidelity radiation treatments practical and enables the incorporation of more spectrally resolved radiation models into CFD fire simulations for engineering applications.

physics.flu-dyn↗

arXiv2026

Optimization of Magnetic Milli-Spinner for Robotic Endovascular Intervention

Vascular diseases such as atherosclerosis, thrombosis, and aneurysms can lead to life-threatening medical events. Conventional catheter- or guidewire-based interventional devices often struggle to navigate through highly tortuous vasculature. The recently developed multifunctional magnetic milli-spinner offers a promising wireless solution by integrating a central through-hole and side slits into a cylindrical body with helical fins, enabling rapid and stable navigation for clot debulking, targeted drug delivery, and aneurysm treatment. Here, we combine computational fluid dynamics simulations with experimental validation to optimize the milli-spinner's structural design for high-velocity propulsion and high-efficiency clot debulking in tubular flow environments. By systematically investigating the effects of through-hole radius, fin number, fin helical angle, and slit dimension on propulsion performance, the optimized milli-spinner achieves swimming velocities of 55 cm/s (175 body lengths per second) in saline water and 44 cm/s (140 body lengths per second) in a fluid with viscosity (3.5 mPa.s) comparable to that of arterial blood at high shear rates, far exceeding existing untethered magnetic robots in tubular environments (less than 80 body lengths per second). This exceptional velocity enables stable upstream operation against strong physiological flows representative of major arteries and veins, establishing the milli-spinner as a robust untethered navigation platform for operation in high-flow, tortuous vasculature.

physics.med-ph↗

arXiv2026

ParaS2S: Benchmarking and Aligning Spoken Language Models for Paralinguistic-aware Speech-to-Speech Interaction

Speech-to-Speech (S2S) models have shown promising dialogue capabilities, but their ability to handle paralinguistic cues - such as emotion, tone, and speaker attributes - and to respond appropriately in both content and style remains under-explored. Progress is further hindered by the scarcity of high-quality and expressive demonstrations. To address this, we introduce a new reinforcement learning (RL) framework for paralinguistic-aware S2S, ParaS2S, which evaluates and optimizes both response content and speaking style directly at the waveform level. We first construct ParaS2SBench, a benchmark that evaluates the naturalness of input-output pairs in terms of content and speaking style using expressive and challenging queries. For the automatic judge, we propose a PolyTone training strategy and a multi-stage framework, preventing the style hallucination of end-to-end audio LLM judging. Our judge correlates well with human preferences and is scalable, enabling the model to interact and learn from unlabeled speech via RL. Experiments show that existing S2S models fail to respond appropriately to paralinguistic attributes, performing no better than pipeline-based baselines. Our RL approach (ParaS2SAlign) achieves a 10% relative improvement in the appropriateness of response content and speaking style on ParaS2SBench over supervised fine-tuning (SFT), surpassing all prior models while requiring substantially fewer paired demonstrations than pure SFT. Our findings highlight the need for a scalable and accurate automatic evaluator for speech-to-speech interaction.

eess.AS↗

arXiv2026

Subspace variations of the weighted skew Bollobás theorem

Let $V$ be a finite-dimensional real vector space. A collection $\mathcal{P} = \{(A_i,B_i)\}_{i=1}^m$ of pairs of subspaces of $V$ is called a skew Bollobás system if $\dim(A_i\cap B_i)=0$ for each $i\in [m]$ and $\dim(A_i\cap B_j)>0$ for all $1\leq i<j \leq m$. Assume that $V = V^{(1)}\oplus \cdots \oplus V^{(r)}$ and $\mathcal{P}= \{(A_i,B_i)\}_{i=1}^m$ is a skew Bollobás system of subspaces of $V$ satisfying $ A_i = \bigoplus_{k=1}^r (A_i \cap V^{(k)})$ and $ B_i = \bigoplus_{k=1}^r (B_i \cap V^{(k)})$ for each $i\in [m]$. Denote $a_{i,k} = \dim(A_i \cap V^{(k)})$ and $b_{i,k} = \dim(B_i \cap V^{(k)})$. Suppose that $a_{1,k} \le \cdots \le a_{m,k}$ and $b_{1,k} \ge \cdots \ge b_{m,k}$ for each $k\in [r]$. Using the exterior algebraic method developed by Lovász and Scott--Wilmer, we prove that $$ \sum_{i=1}^{m} \frac{1}{\prod_{k=1}^{r} \binom{a_{i,k}+b_{i,k}}{a_{i,k}}} \le 1 . $$ This generalizes the results of Alon (JCTA, 1985) and Scott--Wilmer (JLMS, 2021) to multipart weighted setting. Secondly, we solve a conjecture of Hegedüs (AJC, 2015) concerning projective subspaces, showing that any skew Bollobás system of projective subspaces in an $n$-dimensional projective space contains at most $2^{n+1} - 2$ pairs. Thirdly, we prove that if $\mathcal{P}= \{(A_i,B_i)\}_{i=1}^m$ is a skew Bollobás system of subspaces of $V$ with $a_i=\dim (A_i)$ and $b_i=\dim (B_i)$, then $$ \sum_{i=1}^m \frac{1}{(a_i+ b_i+1)\binom{a_i+b_i}{a_i}} \le 1. $$ This gives an extension to the subspace setting of the results of Hegedüs--Frankl (EUJC, 2024) and Yue (DM, 2026). Finally, we extend the above inequality to systems of $d$-tuples of subspaces, giving a unified bound that implies the corresponding results for $d$-tuples of subsets.

math.CO↗

arXiv2026

RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion

Partial differential equation (PDE)-governed inverse problems are fundamental across various scientific and engineering applications; yet they face significant challenges due to nonlinearity, ill-posedness, and sensitivity to noise. Here, we introduce a new computational framework, RED-DiffEq, by integrating physics-driven inversion and data-driven learning. RED-DiffEq leverages pretrained diffusion models as a regularization mechanism for PDE-governed inverse problems. We apply RED-DiffEq to solve the full waveform inversion problem in geophysics, a challenging seismic imaging technique that seeks to reconstruct high-resolution subsurface velocity models from seismic measurement data. Our method shows enhanced accuracy and robustness compared to conventional methods. Additionally, it exhibits strong generalization ability to more complex velocity models that the diffusion model is not trained on. Our framework can also be directly applied to diverse PDE-governed inverse problems.

cs.LG↗

arXiv2026

Active operator learning with predictive uncertainty quantification for partial differential equations

With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. We propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework's uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.

cs.LG↗

arXiv2026

FlowPortrait: Reinforcement Learning for Audio-Driven Portrait Video Generation

Generating realistic talking-head videos remains challenging due to persistent issues such as imperfect lip synchronization, unnatural motion, and evaluation metrics that correlate poorly with human perception. We propose FlowPortrait, a reinforcement-learning framework for audio-driven portrait animation built on a multimodal backbone for autoregressive audio-to-video generation. FlowPortrait introduces a human-aligned evaluation system based on Multimodal Large Language Models (MLLMs) to assess lip-sync accuracy, expressiveness, and motion quality. These signals are combined with perceptual and temporal consistency regularizers to form a stable composite reward, which is used to post-train the generator via Group Relative Policy Optimization (GRPO). Extensive experiments, including both automatic evaluations and human preference studies, demonstrate that FlowPortrait consistently produces higher-quality talking-head videos, highlighting the effectiveness of reinforcement learning for portrait animation.

cs.CV↗

arXiv2026

Probabilistic modeling of Cherenkov emission from particle showers

Subatomic particles can interact with target nuclei in matter or decay in flight, and an individual high-energy particle can induce a particle shower composed of numerous, lower-energy secondaries. These particle showers broadly exhibit universality across diverse media, including air, water, ice, and other materials, with their development governed by the Standard Model. Full Monte Carlo simulation of particle showers, where each secondary is individually tracked and propagated, can be a computational challenge to perform at scale. Experiments thus resort to parametrized approximations when efficient simulation becomes necessary. Here, we construct distributions of parameters capable of describing the Cherenkov light yield from particle showers in ice or water. Sampling from the distributions allows for a much improved description of event-to-event fluctuations, in amplitude and shape, along the shower axis. Including these effects is essential for a more accurate simulation of signal and background events in current and next-generation neutrino telescopes.

astro-ph.HE↗

arXiv2026

What Quality Engineers Need to Know about Degradation Models

Degradation models play a critical role in quality engineering by enabling the assessment and prediction of system reliability based on data. The objective of this paper is to provide an accessible introduction to degradation models. We explore commonly used degradation data types, including repeated measures degradation data and accelerated destructive degradation test data, and review modeling approaches such as general path models and stochastic process models. Key inference problems, including reliability estimation and prediction, are addressed. Applications across diverse fields, including material science, renewable energy, civil engineering, aerospace, and pharmaceuticals, illustrate the broad impact of degradation models in industry. We also discuss best practices for quality engineers, software implementations, and challenges in applying these models. This paper aims to provide quality engineers with a foundational understanding of degradation models, equipping them with the knowledge necessary to apply these techniques effectively in real-world scenarios.

stat.AP↗