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604 recordsLinked to original sources
arXiv2025

Turing complete Navier-Stokes steady states via cosymplectic geometry

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

math.DG↗
arXiv2026

SG-UMP: Sequence-Guided Universal Multimodal Prioritization Calculation Framework

Multimodal sequential recommendation (MSR) improves recommendation by incorporating heterogeneous information such as text, images, and user interactions. However, existing MSR methods often fail to capture user-level preference heterogeneity and dataset-level modality bias, limiting their adaptability across users and datasets. To address this issue, we propose \textbf{S}equence-\textbf{G}uided \textbf{U}niversal \textbf{M}ultimodal \textbf{P}rioritization Calculation Framework (\textbf{SG-UMP}), a plug-and-play plugin for enhancing multimodal information processing in MSR. SG-UMP includes a Module Combiner for flexible multimodal processing and a Module Router for dynamic module ordering, enabling adaptation to both user preferences and dataset characteristics. Experiments on four real-world datasets show that SG-UMP consistently improves recommendation performance across different backbones and multimodal settings. The code is available at https://github.com/esemsc-xz524/SG-UMP .

cs.IR↗
arXiv2026

SG-AMP: Scene-Graph-Guided Active Perception and Semantics-Aware Motion Planning for Pepper Plants

We present SG-AMP, integrating robust depth completion with input-conditioned uncertainty, persistent panoptic mapping, plant scene-graph reasoning, and semantics-aware active view-motion planning. Beyond inspecting uncertain observed regions, the scene graph explicitly hypothesizes unobserved pepper--peduncle attachments and directs close-range sensing toward them. Candidate views are selected according to expected information gain, while class-dependent motion costs distinguish protected peppers, peduncles, and stems from conditionally traversable foliage. On pepper data, the perception network achieves $55.27\%$ semantic mIoU, $38.67\%$ PQ, and $40.62\,\mathrm{mm}$ depth RMSE, while input-conditioned uncertainty improves NYUv2 NLL from $-1.6518$ to $-1.6925$ and AUSE from $0.0102$ to $0.0087$.

cs.RO↗
arXiv2026

OVIP-SG: Open-Vocabulary Instance-Preserving Scene Graphs for Mapping and Retrieval of Small, Fine-Grained Objects

Integrating open-vocabulary perception into object-level 3D scene graphs is a double-edged sword. While vision-language detectors recover long-tail categories and small, fine-grained objects overlooked by closed-set models, they also tend to fragment large surfaces and merge small objects into larger neighboring objects, compromising instance-level consistency and undermining mapping fidelity. Moreover, existing methods struggle to retrieve previously unmapped targets or determine whether a queried object is absent, hindering robust embodied open-world navigation and exploration. We present OVIP-SG, a unified framework for instance-preserving semantic mapping, functional scene partitioning, and language-guided small, fine-grained object retrieval. OVIP-SG uses a vision-language model (VLM) to enumerate scene-specific categories for robust open-world detection. Symmetric 3D Intersection over Union (IoU) association and area-weighted feature fusion preserve small independent instances, while VLM-inferred object functions partition scenes into compact functional search regions. A four-stage cascaded retrieval pipeline further incorporates voxel voting and determines target absence from exploration coverage. Under a unified evaluation protocol on Replica, OVIP-SG outperforms ConceptGraphs by 6.31 points in class-mean accuracy (mAcc) and 5.15 points in frequency-weighted mIoU (F-mIoU) while achieving a class-agnostic native-instance Panoptic Quality (PQ) of 0.398. It reduces the search area to 21.8% of the indoor floor space and reaches 0.773 balanced accuracy for object-presence classification. Real-world robotic experiments further demonstrate its practical effectiveness. Code is available at https://github.com/Agibot-Spatial-Intelligence/OVIP-SG.

cs.RO↗
arXiv2026

X-SG$^2$S: Safe and Generalizable Gaussian Splatting with X-dimensional Watermarks

3D Gaussian Splatting (3DGS) has been widely used in 3D reconstruction and 3D generation. However, the rapid adoption of 3D Gaussian Splatting raises growing concerns about information leakage and unauthorized use, urging the exploration of effective watermarking techniques. However, existing methods are limited by low capacity, fragility under geometric perturbations, and the infeasible requirement for costly fine-tuning or pipeline modifications, motivating the need for a generalizable, feed-forward framework capable of robust multi-modal embedding with minimal intrusion. In this paper, we propose a new framework X-SG$^2$S which can simultaneously inject 1D to 3D watermarks for copyright protection, while keeping the high fidelity of original 3DGS scenes. Specifically, we first split the watermarks into message patches. A self-adaptive gate is developed to select the injection positions of the watermark messages. Then, we use an XD (multi-dimensional) injection head to inject multi-modal messages into sorted 3DGS points. To restore watermarking messages, a learnable gate is developed to recognize the watermarked locations, from which our XD-extraction heads are used to restore hidden messages. X-SG$^2$S is the first framework to unify 1D-to-3D watermarking and enable simultaneous multi-modal watermark embedding in 3DGS, achieving this with minimal rendering interference and zero modifications to parameters or pipelines. Extensive experiments demonstrate that X-SG$^2$S effectively preserves consistency between the watermark and the original 3DGS, exhibits robustness against model degradation, and maintains accurate judgment capabilities.

cs.CR↗
arXiv2026

Accelerating Reinforcement Learning via MPC Solver-Gradient Guidance for Weights-varying MPC

In Model Predictive Control (MPC), cost-function weights shape closed-loop behavior, yet changing conditions often make fixed parametrizations suboptimal and motivate context-dependent online adaptation. Learning such policies is difficult because behavior depends implicitly on numerical MPC solutions, producing nonlinear, potentially nonsmooth, long-horizon dependencies on policy parameters. This creates a bias-variance tradeoff: Reinforcement Learning (RL) optimizes realized closed-loop return from environment samples but is sample-inefficient, whereas Gradient-Based Policy Learning (GB-PL) uses low-variance solver gradients from differentiable MPC to optimize surrogate losses on predicted trajectories but can be biased under model mismatch. We propose Solver-Gradient Guided Reinforcement Learning (SG-RL), a solver-sensitivity augmentation for RL-based online MPC cost-weight adaptation. SG-RL keeps sampled closed-loop return as the objective and uses bounded solver-derived gradients as auxiliary guidance to improve stability and sample efficiency. We instantiate SG-RL in Proximal Policy Optimization (PPO) with four modular algorithms that inject solver-gradient guidance into actor-update scaling, policy loss, advantage estimation, and value-function learning. On two full-scale autonomous racing platforms with intentional model mismatch, SG-RL reaches PPO's best closed-loop return with up to 70.6% fewer samples, outperforms GB-PL baselines by at least 54% in closed-loop return, and generalizes zero-shot to unseen environments.

cs.RO↗
arXiv2026

Geometric mean and Lebesgue-type decomposition of completely positive maps

We introduce the geometric mean and the parallel sum of completely positive (CP) maps between von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM--GM--HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi--Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo--Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando's decomposition of bounded positive operators and Kosaki's decomposition of normal positive functionals on von Neumann algebras.

math.OA↗
arXiv2026

Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.

math.NA↗
arXiv2026

Multiplicative comparisons of Rényi entropies for weighted Bernoulli sums

We establish improved multiplicative bounds relating the Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic bound between the zeroth-order and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain, Sah, and Sawhney. Additionally, we obtain explicit constant-factor bounds for comparisons among Rényi entropies of nonzero orders.

math.PR↗
arXiv2026

Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

math.CO↗
arXiv2026

Shannon's problem on the monotonicity of entropy and a Conjecture of Tao

Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.

math.PR↗
arXiv2026

Geometric integrators for adiabatically closed simple thermodynamic systems

A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples.

math-ph↗
arXiv2026

Edge codes constructed from unicyclic graphs

Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.

math.CO↗
arXiv2026

Difference equations of average entropies

Exact cumulants of entanglement entropies of random state ensembles have traditionally been studied within the random matrix framework. In this work, we propose an alternative approach based on the intrinsic connection to integrable systems. The central idea is to embed entropic quantities into tau functions satisfying Toda-type lattice equations, which in turn yield linear difference equations for their averages. Directly solving the difference equations recovers exact entropy formulas in the literature. The integrable systems approach bypasses the case-by-case, ensemble-dependent derivations required by random matrix methods. The approach also suggests a possible route towards unified and more efficient higher-order cumulant calculations by exploring integrable hierarchies.

math-ph↗
arXiv2026

A Complete Characterization of Tensorizable $f$-divergences

Csiszar's formulation of the $f$-divergence introduced a vast family of functionals for quantifying dissimilarity between probability distributions. However, many applications in statistics and information theory rely only on a few $f$-divergences, such as the Kullback-Leibler divergence, the $χ^2$-divergence, and the squared Hellinger distance. These divergences are especially useful because they admit simple compositional formulas under product measures, a property sometimes referred to as tensorization. In this work, we refine a formalism of tensorization previously introduced in the literature. Then, we show that any possible tensorization formula has a multi-affine form characterized by a single parameter, and identify all tensorizable $f$-divergences under our adopted notion of tensorization.

cs.IT↗
arXiv2026

Spatial symmetry invariance of solution of Kolmogorov flow

We prove a mathematical theorem that solution for all $t > 0$ of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition keeps the same spatial symmetry as its smooth initial condition. The proof of a similar theorem for the three-dimensional NS equations is given in the appendix. These mathematical theorems can be used to check the correctness and reliability of numerical simulations of NS turbulence. For example, they support the corresponding CNS (clean numerical simulation) results of the 2D and 3D turbulent Kolmogorov flows [1-3] that remain the same spatial symmetry in the whole time interval of simulation, but do not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate these mathematical theorems. Thus, these mathematical theorems rigorously confirm that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. All of these indicate that CNS can indeed provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.

physics.flu-dyn↗
arXiv2026

A simple derivation of the Kalman filter

In this lecture note, we present a concise and self-contained derivation of the discrete-time Kalman filter equations that requires only a basic understanding of least squares estimation. The treatment is designed to minimize mathematical overhead while preserving both rigor and generality.

math.OC↗