arXiv subjects

Wei Ji

Publications and source records attributed to Wei Ji.

At least 19 recordsLinked to original sources

arXiv2026

Finitely many solutions for the prescribed scalar curvature problem at the critical flatness order

We study the following prescribed scalar curvature problem $$ -Δu=K(y)u^{2^*-1},\ \ u>0 \,\ \ \text{in}\ \ \mathbb{R}^N, $$ where $2^*=\frac{2N}{N-2}$. For $N\geq4$, we construct a positive $K(y)$ with maximum set $\{|y|=1\}$ and critical order of flatness $m=N-2$ at $|y|=1$, such that the equation admits only finitely many positive solutions in $D^{1,2}(\mathbb{R}^N)$. To this end, we prove the compactness of the solution set for this equation by blow-up analysis and then apply Sard's theorem to choose $K$ for which each solution is non-degenerate and hence isolated, so that the solution set is finite. Our results also give a definite answer to Wei--Yan's conjecture in [J. Wei and S. Yan, JFA, 2010].

math.AP↗

arXiv2026

Sequential convergence in the lower topology and co-sobriety of Scott spaces

We investigate how order structure controls sequential convergence in the lower topology of a poset and how this, in turn, governs topological duality and co-sobriety of Scott spaces. We first give a purely order-theoretic characterization of lower convergence. For complete lattices this yields a lower-limsup formula, and on complete chains the lower limsup coincides with the usual lattice limsup. We also characterize first countability of the lower topology at a point by the existence of a countable coinitial subset outside its principal ideal. We prove that every compact saturated subset of a Scott space is sequentially closed in the lower topology and deduce that, for every poset, the lower topology and the co-compact (de Groot dual) topology of its Scott space have exactly the same convergent sequences and limits. For coherent Scott spaces we identify the lower closure of a $k$-irreducible compact saturated set with its double-cone closure and obtain an infimum criterion for co-sobriety. Consequently, every bounded-complete dcpo with sequential lower topology has a co-sober Scott space. In particular, every countable bounded-complete dcpo, and hence every countable complete lattice, has a co-sober Scott space.

math.GN↗

arXiv2026

A co-sober Alexandrov space whose Scott power space is not co-sober

We show that the Scott power construction does not preserve co-sobriety. For a dcpo $P$, let $\operatorname{Fin}(P)$ be the poset of nonempty finitely generated upper sets ordered by reverse inclusion. We prove that co-sobriety of $Σ\operatorname{Fin}(P)$ implies co-sobriety of $ΣP$. The proof adapts the point-saturation and union argument used for full compact saturated power posets; the finite step follows from a result of Heckmann and Keimel. Applying this transfer to a dcpo with a non-co-sober Scott space yields a co-sober Alexandrov space whose Scott power space is not co-sober. We also construct a countable example for which the Smyth power poset is countable. In the countable example, the same poset is co-sober under the upper Vietoris topology but not under the Scott topology.

math.GN↗

arXiv2026

Lie Routing Vision Language Models for View-Robust 3D Understanding

Multimodal Large Language Models (MLLMs) have made substantial progress in 3D scene understanding from multi-view RGB observations. Yet a fixed object and query can elicit inconsistent target-related responses across camera viewpoints, a phenomenon we characterize as viewpoint-induced semantic drift. We introduce LieVLM, a Lie routing framework for view-robust 3D understanding. Motivated by the rotational component of viewpoint change, Lie routing adapts visual representations by applying local $\mathrm{SO}(3)$ transformations to projected features within learned three-dimensional subspaces. Geometry tokens extracted by VGGT predict local rotation parameters and channel-wise routing coefficients. A closed-form Padé--Cayley map transforms projected visual features, whose identity-relative corrections are projected back and gated into the original visual and geometry features. This design retains the pretrained semantic and direct spatial context while learning view-dependent feature adjustments. Experiments on 3D captioning, grounding, detection, and spatial reasoning show consistent improvements over additive fusion across training scales and backbones. Controlled ablations examine the roles of geometric conditioning and transformation structure. Viewpoint analyses further reveal systematic variation in the predicted routing states alongside more localized target responses and stronger task performance across viewpoint changes.

cs.CV↗

arXiv2026

Frequency-Domain Decomposition and Recomposition for Robust Audio-Visual Segmentation

Audio-visual segmentation (AVS) plays a critical role in multimodal machine learning by effectively integrating audio and visual cues to precisely segment objects or regions within visual scenes. Recent AVS methods have demonstrated significant improvements. However, they overlook the inherent frequency-domain contradictions between audio and visual modalities--the substantial noise often present in audio high-frequency signals vs. the structurally rich details in visual high-frequency signals. Ignoring these differences can result in suboptimal performance. In this paper, we rethink the AVS task from a deeper perspective by reformulating AVS task as a frequency-domain decomposition and recomposition problem. To this end, we introduce a novel Frequency-Aware Audio-Visual Segmentation (FAVS) framework consisting of two key modules: Frequency-Domain Enhanced Decomposer (FDED) module and Synergistic Cross-Modal Consistency (SCMC) module. FDED module employs a residual-based iterative frequency decomposition to discriminate modality-specific semantics and structural features, and SCMC module leverages a mixture-of-experts architecture to reinforce semantic consistency and modality-specific feature preservation through dynamic expert routing. Extensive experiments demonstrate that our FAVS framework achieves state-of-the-art performance on three benchmark datasets, and abundant qualitative visualizations further verify the effectiveness of the proposed FDED and SCMC modules. The project repository is available at https://github.com/happylife-pk/FAVS.

cs.CV↗

arXiv2026

Products of co-sober spaces need not be co-sober

We answer negatively the question whether the product of two co-sober spaces must be co-sober. For a nonempty closed subset $A$ of a $T_1$-space $(X,τ)$, we consider the topology consisting of the co-finite open sets together with the $τ$-open sets disjoint from $A$. We determine its $k$-irreducible compact subsets and derive a criterion that produces two compact co-sober spaces whose product contains a non-singleton $k$-irreducible compact saturated subset of the diagonal. Applying the criterion to a specific Kovář--van Douwen space gives two countable compact Fréchet--Urysohn $T_1$ anti-Hausdorff co-sober spaces whose product is not co-sober. A further construction yields a countable compact Fréchet--Urysohn $T_1$ anti-Hausdorff co-sober space whose square is not co-sober. The failure of product closure also gives an alternative proof of the known non-reflectivity of co-sober spaces in the category of $T_0$-spaces.

math.GN↗

arXiv2026

Compact localization and co-sobriety of Smyth power spaces

We establish a compact-localization theorem for finite iterates of the Smyth power space. Successive union maps assign to every finite-level Smyth object a canonical least compact saturated carrier in the original space, and co-sobriety at each positive finite level is equivalent to co-sobriety of the corresponding iterates over all compact saturated carriers. Consequently, if every compact saturated subspace of a $T_0$-space $X$ is stably compact---in particular, if every such subspace is Hausdorff---then $\PS^n(X)$ is co-sober for every finite $n\geq 1$. This applies to every $k_1$-Hausdorff space and hence to every Hausdorff or metrizable space. For the first iterate, we also give a direct full-Vietoris argument, independent of stable compactness, which shows that each $k$-irreducible compact saturated family is generated by its union.

math.GN↗

arXiv2026

Uniqueness of Bound States for Double-Power Scalar Field Equations

We study the uniqueness of bound states for the following scalar field equation $$ -Δu+u=-\varepsilon |u|^{q-1}u+|u|^{p-1}u \,\ \ \mbox{in}\,\ \mathbb R^N, \,\ \varepsilon>0,\tag*{$(*)$} $$ where $ N\ge3,\ 1 1$ satisfies $q \neq p$. For each integer $k \geq 1$, we prove that the equation $(*)$ has, up to translation and reflection, a unique bound state with precisely $k$ nodes in a natural coupling regime, where $\varepsilon>0$ is sufficiently small. Our results address a conjecture of [H. Berestycki and P. L. Lions, ARMA, 1983][M. X. Tang, Invent. Math., 2026] on the uniqueness of fixed-node bound states for the equation $(*)$.

math.AP↗

arXiv2026

Seeing Through the Glare: A Multi-Source Benchmark and Ocular-Adaptive Pixel MeanFlow for Eyeglass Reflection Removal

Eyeglass reflection removal is important across smartphone imaging, video conferencing, and other face-centric visual applications. The task is challenging because reflections range from mild photometric contamination to severe ocular occlusion, requiring selective correction and plausible reconstruction without altering identity or natural appearance. Existing datasets cover limited reflection conditions, constraining generalization to complex real-world scenes and systematic evaluation. We introduce \textbf{OcuBench}, a multi-source benchmark comprising 10,280 controllable synthetic pairs, 732 real-input pseudo-pairs, and 458 independent real-world test images, supporting both paired evaluation and assessment beyond generated supervision. We further propose \textbf{OcuFlow}, an ocular-adaptive pixel MeanFlow (pMF) framework for efficient, detail-preserving restoration. It combines geometry-adaptive representation with one-step pMF to focus reconstruction on reflection-obscured ocular regions, together with native-resolution frequency-preserving synthesis to retain reliable observed details. Experiments across diverse reflection conditions demonstrate that OcuFlow achieves consistent advantages in reflection removal quality, ocular fidelity, and efficiency. In a blind user study, it receives $67.32\%$ of selections, $6.2\times$ the next-best share. Both the code and dataset will be released.

cs.CV↗

arXiv2026

A countable metric space whose Hoare power space is not co-sober

Xu asked whether the Hoare power space of a co-sober space, and in particular of a $T_2$-space or a metric space, must be co-sober. We give a negative answer. He and Zhao recently constructed a countable $T_{1}$ space $X$ whose lattice of open sets has a non-sober Scott topology. We observe that their topology has a countable subbase. The classical Ponomarev construction then yields an open continuous surjection from a countable zero-dimensional metrizable space $M$ onto $X$. Direct and inverse images then make the Scott space $Σ\mathcal{O}(X)$ a retract of $Σ\mathcal{O}(M)$, so $Σ\mathcal{O}(M)$ is not sober. Finally, we show that, for every $T_{0}$-space $Y$, the Hoare power space $\PH(Y)$ is co-sober if and only if the Scott space $Σ\mathcal{O}(Y)$ is sober. Consequently, $\PH(M)$ is not co-sober. Therefore, the same example answers negatively a question of He and Zhao about Scott sobriety of the lattices of open sets of countable Hausdorff spaces.

math.GN↗

arXiv2026

Scott products and sobriety of countable meet-continuous dcpos

We study Scott-product compatibility and sobriety for countable meet-continuous directed-complete partial orders (dcpos). We construct a countable meet-continuous dcpo $\mathcal R$ whose Scott space is well-filtered but not sober, although the Scott topology on every finite power is the product of the factor Scott topologies. The ideals of $\mathcal R$ admit a unique finite-parameter description. We also construct a countable meet-continuous dcpo $\mathbb{H}$ with a greatest element whose Scott space is coherent and well-filtered, but whose Scott square has a topology strictly finer than the ordinary product topology. These constructions answer negatively two questions about countable meet-continuous dcpos. On the positive side, we prove that a meet-continuous $L$-dcpo, meaning a dcpo whose principal ideals are complete lattices, is Scott sober whenever its Scott square carries the ordinary product topology. This answers Jia's core-compactness question affirmatively for $L$-dcpos without any countability assumption. Finally, we characterize well-filteredness of $L$-dcpos by boundedness of closed Rudin sets and formulate related open questions.

math.LO↗

arXiv2026

Distribution-Conditioned Task Routing for Class-Incremental Learning

Parameter-efficient adaptation enables continual learners to acquire task-specific knowledge through compact model updates while maintaining strong within-task performance. However, class-incremental inference requires each input to be classified among all classes seen so far without access to its task identity. For learners equipped with task-specific parameter-efficient modules, this introduces a critical task-routing challenge beyond catastrophic forgetting. We study post-hoc task routing without retraining the learner or introducing a separately trained router. Such training-free inference-time calibration remains comparatively underexplored in parameter-efficient class-incremental learning. We identify three sources of routing error (feature-level, task-level, and class-level misalignment) and propose Feature Distribution Calibration (FDC). Its three components address these misalignments: Task Subspace Filtering (TSF) suppresses feature components outside each task's principal subspace, Residual Likelihood Calibration (RLC) evaluates the typicality of its subspace residual, and Prototype Affinity Calibration (PAC) measures compatibility with the task's class prototypes. Experiments demonstrate plug-and-play applicability to eight parameter-efficient class-incremental methods using a shared encoder. With one component configuration selected per method across all five benchmarks, FDC improves final accuracy in all 40 method-dataset pairs by 4.39 percentage points on average. Enabling all components improves 35 of the 40 pairs, with an average gain of 4.45 points. When applied to a simple baseline, FDC achieves strong overall performance.

cs.LG↗

arXiv2026

SAMatcher: Dense Co-Visibility Modeling via Cross-View Fusion for Scale-Imbalance Image Matching

Reliable correspondence estimation supports image processing and 3D vision tasks, including Structure from Motion, visual localization, and image registration. Wide-baseline matching is difficult under cross-view scale imbalance. The same content may appear at different resolutions and spatial extents, creating unequal observation quality. Such pairs constitute low-quality multi-view data, even when each image is clear. Image information that is distinctive in one view may be weak or unavailable in the other. Existing methods rely on pixel- or patch-level appearance and do not explicitly identify where reliable evidence is shared across views. We propose SAMatcher, a modular co-visibility framework for scale-imbalanced image matching. It uses symmetric cross-view interaction to fuse the two feature representations and identify their shared spatial support. From the fused features, SAMatcher predicts dense co-visible masks and bounding boxes. These predictions define view-specific cropping windows for downstream point-level matching. Built upon the Segment Anything Model (SAM), SAMatcher extends monocular region modeling to cross-view co-visibility reasoning. Point-sampled mask learning, box regression, and mask--box consistency jointly supervise its multi-granularity predictions. Extensive experiments follow an evaluation protocol centered on cross-view scale imbalance. The results show that SAMatcher consistently improves matching robustness across diverse matching pipelines. When integrated with RoMa, SAMatcher improves AUC@5 and mAA@5 by 5.58 and 4.97 points, respectively. The results demonstrate that cross-view fusion and dense co-visibility modeling provide reliable priors for matching under scale-induced quality imbalance. Code and project page are available at https://xupan.top/projects/samatcher

cs.CV↗

arXiv2026

Discrete Tokenization for Multimodal LLMs: A Comprehensive Survey

The rapid advancement of large language models (LLMs) has intensified the need for effective mechanisms to transform continuous multimodal data into discrete representations suitable for language-based processing. Discrete tokenization, with vector quantization (VQ) as a central approach, offers both computational efficiency and compatibility with LLM architectures. Despite its growing importance, there is a lack of a comprehensive survey that systematically examines VQ techniques in the context of LLM-based systems. This work fills this gap by presenting the first structured taxonomy and analysis of discrete tokenization methods designed for LLMs. We categorize 8 representative VQ variants that span classical and modern paradigms and analyze their algorithmic principles, training dynamics, and integration challenges with LLM pipelines. Beyond algorithm-level investigation, we discuss existing research in terms of classical applications without LLMs, LLM-based single-modality systems, and LLM-based multimodal systems, highlighting how quantization strategies influence alignment, reasoning, and generation performance. In addition, we identify key challenges including codebook collapse, unstable gradient estimation, and modality-specific encoding constraints. Finally, we discuss emerging research directions such as dynamic and task-adaptive quantization, unified tokenization frameworks, and biologically inspired codebook learning. This survey bridges the gap between traditional vector quantization and modern LLM applications, serving as a foundational reference for the development of efficient and generalizable multimodal systems. A continuously updated version is available at: https://github.com/jindongli-Ai/LLM-Discrete-Tokenization-Survey.

cs.CL↗

arXiv2026

Every countable meet-continuous lattice is Scott sober

We prove that every countable meet-continuous lattice is Scott sober. For any family of such lattices, the Scott topology on the product equals the product of the factor Scott topologies, and the product Scott space is sober. These results answer affirmatively Questions 7.1 and 7.2 of Xu (arXiv:2609.18032v1), extending the countable-frame results beyond finite distributivity. A diagonal argument on meet coordinates extracts directed punctured intervals from nonprincipal ideals. Join translates of these intervals form a countable family detecting Scott openness and yield Scott open rectangles by successive finite choices. We also show that the cardinal spectrum of Scott non-sober meet-continuous lattices is upward closed and, under the Continuum Hypothesis, consists of all uncountable cardinals.

math.LO↗

arXiv2026

Equation Recast for Canonical Operator Learning Across Parametric PDEs

Learning solution operators across broad parameter ranges can require substantial coverage of both input functions and physical parameters, particularly for purely data-driven parametric models. In addition, the resulting models may fail silently outside the training distribution. We introduce equation recast, which reformulates parametric operator learning as the learning of a single canonical operator. Parameter-induced operator variations are derived analytically from the governing equation and absorbed into effective sources, enabling zero-shot prediction across new parameter regimes. Across multi-parameter, nonlinear, and singular PDE settings, equation recast supports extrapolation, integrates sparse heterogeneous datasets in a shared canonical representation, and uses loss of convergence as an internal warning signal for failure of the recast iteration. In high-fidelity tokamak simulations for nuclear fusion, the framework unifies electron-temperature data across four device geometries through canonical-domain mapping within one jointly trained operator. Equation recast provides a route toward reusable neural PDE solvers combining equation-guided transfer, data efficiency, and monitorable inference.

cs.LG↗

arXiv2026

Orbital occupation selects structural dimensionality in binary transition-metal oxides

Orbital-lattice coupling in transition-metal oxides is usually discussed within a given bonding framework, where orbital occupation is intertwined with local coordination, strain, or symmetry breaking. Here, we show that orbital occupation can also select the bonding framework itself, thereby determining structural dimensionality. Using first-principles calculations, we identify CrO as a prototype in which the single active $3d\,e_g$ electron of high-spin Cr$^{2+}$ gives rise to two competing orbital-structure states. The $d_{x^2-y^2}$ occupation favors a three-dimensionally connected covalent phase, whereas the $d_{z^2}$ occupation stabilizes a weakly coupled layered phase. Constrained-occupation calculations show that increasing the $d_{z^2}$ filling continuously contracts the in-plane lattice while expanding the structure along the layer normal. The two phases exhibit distinct magnetic ground states and ferroelastic responses. Moreover, the layered phase is robust against exchange-correlation functional and on-site ($U$) variations, remains dynamically stable down to the monolayer limit, and has a low exfoliation energy of 46 meV/Angstrom^2. Extending the analysis across related $3d$ binary oxides reveals a filling-dependence relation between accessible orbital filling and the preference for 2D or 3D connected bonding motifs, providing a microscopic basis for exploring low-dimensional oxide materials.

cond-mat.mtrl-sci↗

arXiv2026

Decoupled domain-texture switching from magnetic easy axis in kagome ferromagnet EuTi3Bi4

Magnetic anisotropy defines the easy axis of a magnetic material and governs the spatial arrangement of its domains. To date, anisotropy engineering has focused on reorienting the easy axis or tuning the anisotropy energy, both of which demand substantial energy input. Here, we demonstrate that magnetic domain textures can be switched without reorienting the easy axis, as observed in a kagome ferromagnet EuTi3Bi4 crystal. Using low-temperature magnetic force microscopy, we observe that the preferred orientation of magnetic domains switches from the a-axis to the b-axis upon temperature variation, and that this switching can also be triggered by an out-of-plane magnetic-field reset. Magnetization measurements and density functional theory calculations confirm a robust c-axis easy magnetization, ruling out a conventional spin-reorientation transition. Instead, the texture switching is governed by the temperature dependence of the in-plane variation of the Magnetic anisotropy energy landscape, which arises from two competing interactions with different decay rates: single-ion anisotropy favors a-oriented spin components, while nearest-neighbor anisotropic exchange favors b-oriented ones. Furthermore, the critical switching temperature is substantially elevated in a mechanically exfoliated EuTi3Bi4 flake. Our findings establish that macroscopic magnetic textures can be effectively manipulated by tuning the competition between in-plane anisotropic interactions, without the energy cost of reorienting the easy axis.

cond-mat.mtrl-sci↗