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Lu Hao

Publications and source records attributed to Lu Hao.

10 recordsLinked to original sources
arXiv2026

Flow Decomposition and Sharp Integral Fujita Criteria on Weighted Graphs

We study the Fujita phenomenon for semilinear heat inequalities generated by variable-speed Laplacians on infinite weighted graphs. Assuming that the graph carries a proper adapted path metric, we establish an integral volume-growth criterion forcing every nonnegative global classical supersolution on the open cylinder $(0,\infty)\times V$ to vanish, without assuming an initial value or trace. We prove that a nontrivial supersolution of this kind exists if and only if the equation has a positive global Cauchy solution for some nonzero point-source datum. A complementary heat-kernel construction gives global Cauchy solutions for all sufficiently small point-source data when the same volume integral converges and a matching anchored heat-kernel upper bound is available. This proves sharpness on integer lattices and on a family of logarithmically perturbed weighted half-lines; in the latter examples, even the exponent of an iterated logarithm can determine the existence--nonexistence alternative. A finer nonexistence criterion couples intrinsic volume growth with the capacity of intrinsic annuli. Its proof combines parabolic testing, a Laplace--resolvent reduction, and a pathwise decomposition of resolvent currents. The nonexistence results require no volume-doubling property, Poincar\'e inequality, heat-kernel bound, or stochastic completeness.

math.AP↗
arXiv2026

A sharp integral criterion for the Lane--Emden system of inequalities on weighted graphs

We establish a sharp integral nonexistence criterion for the Lane--Emden system of inequalities \[ -\Delta u\ge v^p,\qquad -\Delta v\ge u^q, \qquad p,q>0,\quad pq>1, \] on arbitrary infinite, connected, locally finite weighted graphs. In the asymmetric case $p\ne q$, set $P=\max\{p,q\}$. If, for some root $o\in V$, \[ \sum_{n=2}^{\infty} \frac{n^{2pq+2P-1}}{\mu(B(o,n))^{pq-1}}=\infty, \] then every nonnegative solution $(u,v)$ satisfies $u\equiv v\equiv0$. The proof combines flow decomposition of the finite Green current with nonlinear testing. In the symmetric case $p=q>1$, the Liouville problem reduces, via the sum $u+v$, to the scalar criterion \[ \sum_{n=2}^{\infty} \frac{n^{2p-1}}{\mu(B(o,n))^{p-1}}=\infty. \] Weighted half-line examples show that the critical logarithmic endpoint in the asymmetric result is sharp.

math.AP↗
arXiv2026

Beyond Retrieval: Learning Compact User Representations for Scalable LLM Personalization

Personalizing large language models requires adapting model behavior to individual users while preserving robustness and deployment-scale efficiency. Existing approaches typically personalize LLMs either at the input level, by retrieving user histories or constructing profile prompts, or at the parameter level, by maintaining user-specific parameter-efficient modules. The former makes personalization sensitive to retrieval quality and prompt design, whereas the latter incurs storage and maintenance costs that grow with the user population. To address these limitations, we propose TAP-PER (Temporal Attentive Prefix for PERsonalization), a prefix-based framework that encodes user preferences as learnable representations, avoiding the serialization of user histories into prompts and replacing heavy per-user adapters with lightweight user-state prefix embeddings. Inspired by personalized recommendation systems, TAP-PER decomposes user modeling into user-state and query-conditioned components, and incorporates temporal signals to capture the evolving nature of user interests. Experiments on six LaMP tasks show that TAP-PER consistently outperforms prompt-based and model-based baselines across classification, rating, and generation settings. Moreover, TAP-PER uses 130x fewer per-user parameters than OPPU and roughly half the total parameter footprint of PER-PCS at the 1,000-user scale, demonstrating scalable personalization without prompt-serialized histories or heavy per-user adapters.

cs.IR↗
arXiv2026

A Volume-Growth Criterion for the p-Laplace Inequality on Weighted Graphs

We prove a nonexistence result for nonnegative solutions of the quasi-linear elliptic inequality \[ -\Delta_p u\ge \sigma(x)u^q \] on infinite locally finite connected weighted graphs, where $1 p-1$, $\sigma$ is a nonnegative Radon measure. Under the non-$p$-parabolic setting, we show that every nonnegative solution is identically zero, provided the volume of intrinsic balls satisfy \[ \int_1^\infty \frac{r^{\frac{pq}{p-1}-1}} {\nu(B_\rho(o,r))^{\frac{q-p+1}{p-1}}} \dd r =\infty, \] This criterion recovers the known sharp pointwise critical volume-growth threshold and is strictly more flexible, since it allows irregular growth and does not require uniform upper bounds at every large radius. The proof adapts the finite-network current method to the $p$-Laplace setting, combining a path decomposition with one-dimensional Hardy estimates, $p$-parallel-sum bounds across metric cuts, and the global $p$-Green function furnished by non-$p$-parabolicity.

math.AP↗
arXiv2026

Flow Decomposition, Green Testing, and Lane--Emden Inequalities on Weighted Graphs

We study positive solutions of the superlinear Lane--Emden inequality \[ -\Delta u\ge \sigma u^q,\qquad q>1, \] on infinite locally finite weighted graphs and connected domains. When the Dirichlet Green function is finite, the existence of a positive solution is equivalent to \[ G_\Omega\bigl(\sigma g_\Omega(o,\cdot)^q\bigr)(x) \le C g_\Omega(o,x) \] for some pole \(o\in\Omega\). Under Green function estimates, this yields sharp existence criteria and the Serrin-type exponents on \(\mathbb Z^d\) and orthant domains. For nonexistence, the principal method is flow decomposition. Its basic estimate bounds Green energy from below in terms of the relative capacities of intrinsic balls. %It yields annular conductance, capacity-to-infinity, and Nash--Williams cut-resistance criteria. For \(\sigma>0\), set \(\nu=\sigma\mu\). We show that if \(d_\rho\) is a complete \(\nu\)-adapted path metric and \[ \int_1^\infty \frac{r^{2q-1}}{\nu(B_{d_\rho}(o,r))^{q-1}}\,dr=\infty, \] then every nonnegative solution is identically zero. The proof combines a flow decomposition of the acyclic Green current, a pathwise Hardy estimate, and a relative capacity estimate. It requires none of (VD), (PI), (P$_0$), or the (3G) condition.

math.AP↗
arXiv2026

A Wiener criterion at infinity for $p$-massiveness on weighted graphs

We study boundary value problems at infinity for the graph $p$-Laplacian on infinite, connected, locally finite weighted graphs. Our main result is a Wiener criterion for $p$-massiveness. Assuming volume doubling and a weak $(1,p)$-Poincar\'e inequality, we show that every infinite connected $p$-massive set satisfies a dyadic capacitary condition expressed through relative $p$-capacities in nested balls; under the additional $(p_0)$ condition, the converse also holds. This yields a nonlinear criterion at the point at infinity in a rough weighted-graph setting and extends the Wiener viewpoint to a nonlinear discrete framework. We also prove, without these geometric assumptions, that $p$-massiveness is equivalent to a strengthened nonuniqueness property for exterior Dirichlet problems. As a further consequence, bounded nonconstant $p$-harmonic functions are characterized by the existence of two disjoint massive sets. In this way, the Wiener criterion is placed in a broader and more flexible picture of exterior boundary behavior and Liouville-type phenomena on weighted graphs.

math.AP↗
arXiv2026

QARM V2: Quantitative Alignment Multi-Modal Recommendation for Reasoning User Sequence Modeling

With the evolution of large language models (LLMs), there is growing interest in leveraging their rich semantic understanding to enhance industrial recommendation systems (RecSys). Traditional RecSys relies on ID-based embeddings for user sequence modeling in the General Search Unit (GSU) and Exact Search Unit (ESU) paradigm, which suffers from low information density, knowledge isolation, and weak generalization ability. While LLMs offer complementary strengths with dense semantic representations and strong generalization, directly applying LLM embeddings to RecSys faces critical challenges: representation unmatch with business objectives and representation unlearning end-to-end with downstream tasks. In this paper, we present QARM V2, a unified framework that bridges LLM semantic understanding with RecSys business requirements for user sequence modeling.

cs.IR↗
arXiv2025

Poster: Long PHP webshell files detection based on sliding window attention

Webshell is a type of backdoor, and web applications are widely exposed to webshell injection attacks. Therefore, it is important to study webshell detection techniques. In this study, we propose a webshell detection method. We first convert PHP source code to opcodes and then extract Opcode Double-Tuples (ODTs). Next, we combine CodeBert and FastText models for feature representation and classification. To address the challenge that deep learning methods have difficulty detecting long webshell files, we introduce a sliding window attention mechanism. This approach effectively captures malicious behavior within long files. Experimental results show that our method reaches high accuracy in webshell detection, solving the problem of traditional methods that struggle to address new webshell variants and anti-detection techniques.

cs.CR↗
arXiv2024

On the equivalence of Lp-parabolicity and Lq-liouville property on weighted graphs

We study the equivalence between the $L^p$-parabolicity, the $L^q$-Liouville property of positive super-harmonic functions, and the existence of nonharmonic positive solutions to the following elliptic differential system \begin{equation*} \left\{ \begin{array}{lr} -\Delta u\geq 0, \Delta(|\Delta u|^{p-2}\Delta u)\geq 0, \end{array} \right. \end{equation*} on weighted graphs, where $1\leq p< \infty$, and $(p, q)$ are H\"{o}lder conjugate exponent pair. Furthermore, by refining a new technique on estimate of heat kernel, we can establish two-sided estimates of Green function on graph, and find the sharp volume growth criteria for the $L^q$-Liouville property on a large class of graphs. As an application, many non-trivial interesting examples are presented.

math.AP↗