arXiv subjects

Xiang Fan

Publications and source records attributed to Xiang Fan.

At least 19 recordsLinked to original sources

arXiv2026

Rational functions over finite fields with Galois closure of genus zero

Let $k=\mathbb F_q$. We classify, up to pre- and post-composition by $k$-M\"obius transformations, all separable $k$-indecomposable rational functions $f\in k(X)$ of degree greater than one whose Galois closure has genus zero. The classification is valid in arbitrary characteristic and includes exact arithmetic conditions and class counts over the prescribed field. Semilinear Frobenius descent determines the finite-field forms and which geometric decompositions descend to $k$. For every separable $f\in k(X)$ of degree greater than one with Galois closure of genus zero and every $m\geqslant1$, we prove that $f$ permutes $\mathbf P^1(\mathbb F_{q^m})$ if and only if it is exceptional over $\mathbb F_{q^m}$, meaning that it permutes $\mathbf P^1(L)$ for infinitely many finite extensions $L/\mathbb F_{q^m}$. No indecomposability assumption or lower bound on $q$ is needed. The argument combines fixed-point averaging with ramification on the Galois-closure curve. If the full constant field is $\mathbb F_{q^d}$, these properties depend only on $\gcd(m,d)$. For each classified family we determine the permutation extension degrees explicitly and characterize polynomial representatives.

math.NT↗

arXiv2026

Exceptional covers of the projective line with genus-one Galois closure: arithmetic forms over finite fields

Let $k=\mathbb F_q$. We classify, up to two-sided $k$-M\"obius equivalence, all separable indecomposable tame exceptional maps $\mathbf P^1_k\to\mathbf P^1_k$ whose geometric Galois closure has genus one. The resulting arithmetic fixed-field forms are encoded by Frobenius-stable affine elliptic quotient data recovered from the cover; we prove a converse and an exact equivalence criterion. The same data determine branch arithmetic, geometric and arithmetic monodromy, the constant field, and behavior over every finite extension. In particular, over each $\mathbb F_{q^r}$, permutation is equivalent to exceptionality. We obtain explicit rank-two support formulas, necessary and sufficient occurrence criteria, and exact two-sided class counts in every tame signature, including the surviving cases in characteristics $2$ and $3$. In characteristic greater than $3$, tameness is automatic. A sharp $3$-adic obstruction arises only for a stronger cubic self-endomorphism realization problem, not for the fixed-field classification itself.

math.NT↗

arXiv2026

Indecomposable rational functions over finite fields with Galois closure of genus one: Reconstruction and exceptionality

Let $k=\mathbb F_q$. We reconstruct every $k$-indecomposable $g\in k(X)$ of degree greater than one whose normal closure has genus one. Such a map is automatically separable and, after independent degree-one changes of the source and target coordinates over $k$, arises from a separable equivariant isogeny between elliptic curves equipped with compatible finite group actions stable under Frobenius conjugation. In particular, $\deg g=\ell$ or $\ell^2$ for a prime $\ell$, with $\ell\ne\operatorname{char}k$ in the latter case. More generally, every separable rational function with genus-one Galois closure admits a canonical factorization class, modulo degree-one changes of the intermediate coordinates over $k$, determined by the intrinsic translation subgroup of its geometric monodromy group; every $k$-indecomposable factor of the remaining map has Galois closure of genus zero. For every equivariant-isogeny quotient and every finite extension $k_r/k$, the same exact finite-kernel condition characterizes both permutation of $\mathbf P^1(k_r)$ and exceptionality over $k_r$. The finite kernel and its induced Frobenius and linear symmetry actions also determine the arithmetic and geometric monodromy permutation groups, decomposition classes, and the exact periodic set of permutation extension degrees, including its least period and limiting proportion. Together with the corresponding genus-zero theorem, these results yield permutation if and only if exceptionality over every finite extension for every separable rational function whose Galois closure has genus at most one; under $k$-decomposition, the common extension-degree set is the intersection of the corresponding sets for the factors.

math.NT↗

arXiv2026

Shifted dihedral isogeny quotients and the classification of exceptional rational functions of degree five

We give a complete classification, up to $k$-M\"obius equivalence, of exceptional rational functions of degree five over every finite field. The classification gives explicit normal forms in every characteristic, exact parameter identifications, and the resulting class counts. The main structural input is an arithmetic theory of shifted dihedral isogeny quotients valid in every odd degree. For each odd $n\geqslant3$, the separable degree-$n$ rational maps whose geometric monodromy group is isomorphic to $D_n$ and whose nontrivial inertia groups are generated by reflections are precisely the maps induced by cyclic $n$-isogenies on shifted Kummer quotients. We determine the exact ambiguity of the isogeny data under two-sided $k$-M\"obius equivalence by means of a signed Frobenius-descent invariant. The theory allows arbitrary odd $n$, reduced kernels when the characteristic divides $n$, and wild reflection inertia. If Frobenius acts on the cyclic kernel by $\lambda\in(\mathbb Z/n\mathbb Z)^\times$, the induced map is exceptional exactly when both $\lambda-1$ and $\lambda+1$ are units modulo $n$.

math.NT↗

arXiv2026

Hyperon polarization in isobaric Zr+Zr collisions at $\sqrt{s_{NN}}=200$ GeV: TRENRo3D + CLVisc with an initial longitudinal flow gradient

We present a theoretical study of global and azimuthal-angle-dependent $\Lambda$ hyperon polarization in isobaric $^{96}_{40}$Zr+$^{96}_{40}$Zr collisions at $\sqrt{s_{NN}}=200$~GeV using the TRENTo3D initial condition model coupled to the (3+1)-D viscous hydrodynamic model CLVisc. A longitudinal flow velocity gradient, controlled by $f_v$, is introduced into TRENTo3D for the first time, providing an essential source of initial vorticity in this symmetric isobaric system. Within the isothermal polarization framework, the model provides a simultaneous description of STAR measurements of the global polarization $-P^{y}$ (centrality, $p_T$, and $\eta$ dependences) and the azimuthal modulation coefficients $P_{y,\mathrm{c0}}$ and $P_{y,\mathrm{c2}}$. The $p_T$ dependence reflects the competition between thermal vorticity and shear contributions: the thermal term decreases with $p_T$, while the shear term rises and increasingly shapes the curvature of the total polarization. In this decomposition, $P_{y,\mathrm{c2}}$ is dominantly shear-driven and serves as a clean probe of shear-induced polarization. Scans of $f_v$, $k_T$, and nuclear structure provide complementary constraints on the initial state, while the bulk-viscosity dependence is also examined; the five nuclear structure configurations from the STAR isobar blind analysis yield nearly indistinguishable polarization. For $P_z$, the isothermal scenario captures the azimuthal modulation but overpredicts the high-$p_T$ modulation amplitude, and comparison with the standard thermal treatment shows that neither scenario achieves a unified description of all observables.

nucl-th↗

arXiv2026

RefDecoder: Enhancing Visual Generation with Conditional Video Decoding

Video generation powers a vast array of downstream applications. However, while the de facto standard, i.e., latent diffusion models, typically employ heavily conditioned denoising networks, their decoders often remain unconditional. We observe that this architectural asymmetry leads to significant loss of detail and inconsistency relative to the input image. To address this, we argue that the decoder requires equal conditioning to preserve structural integrity. We introduce RefDecoder, a reference-conditioned video VAE decoder by injecting high-fidelity reference image signal directly into the decoding process via reference attention. Specifically, a lightweight image encoder maps the reference frame into the detail-rich high-dimensional tokens, which are co-processed with the denoised video latent tokens at each decoder up-sampling stage. We demonstrate consistent improvements across several distinct decoder backbones (e.g., Wan 2.1 and VideoVAE+), achieving up to +2.1dB PSNR over the unconditional baselines on the Inter4K, WebVid, and Large Motion reconstruction benchmarks. Notably, RefDecoder can be directly swapped into existing video generation systems without additional fine-tuning, and we report across-the-board improvements in subject consistency, background consistency, and overall quality scores on the VBench I2V benchmark. Beyond I2V, RefDecoder generalizes well to a wide range of visual generation tasks such as style transfer and video editing refinement.

cs.CV↗

arXiv2026

MolmoAct2: Action Reasoning Models for Real-world Deployment

Vision-Language-Action (VLA) models aim to provide a single generalist controller for robots, but today's systems fall short on the criteria that matter for real-world deployment. Frontier models are closed, open-weight alternatives are tied to expensive hardware, reasoning-augmented policies pay prohibitive latency for their grounding, and fine-tuned success rates remain below the threshold for dependable use. We present MolmoAct2, a fully open action reasoning model built for practical deployment, advancing its predecessor along five axes. We introduce MolmoER, a VLM backbone specialized for spatial and embodied reasoning, trained on a 3.3M-sample corpus with a specialize-then-rehearse recipe. We release three new datasets spanning low-to-medium cost platforms, including MolmoAct2-BimanualYAM, 720 hours of teleoperated bimanual trajectories that constitute the largest open bimanual dataset to date, together with quality-filtered Franka (DROID) and SO100/101 subsets. We provide OpenFAST, an open-weight, open-data action tokenizer trained on millions of trajectories across five embodiments. We redesign the architecture to graft a flow-matching continuous-action expert onto a discrete-token VLM via per-layer KV-cache conditioning. Finally, we propose MolmoThink, an adaptive-depth reasoning variant that re-predicts depth tokens only for scene regions that change between timesteps, retaining geometric grounding at a fraction of prior latency. In the most extensive empirical study of any open VLA to date, spanning 7 simulation and real-world benchmarks, MolmoAct2 outperforms strong baselines including Pi-05, while MolmoER surpasses GPT-5 and Gemini Robotics ER-1.5 across 13 embodied-reasoning benchmarks. We release model weights, training code, and complete training data. Project page: https://allenai.org/blog/molmoact2

cs.RO↗

arXiv2026

Deep learning approaches to extract nuclear deformation parameters from initial-state information in heavy-ion collisions

The deformation of heavy nuclei leaves characteristic imprints on the initial conditions of relativistic heavy-ion collisions. However, event-by-event fluctuations make the quantitative extraction of this information challenging. This study examines the identifiability of the quadrupole ($\beta_2$) and hexadecapole ($\beta_4$) deformation parameters from nucleon configurations sampled from a deformed Woods-Saxon distribution commonly used in initial-state modeling of heavy-ion collisions. As a baseline, we first establish an upper bound on the "intrinsic identifiability" of deformation information at the most microscopic level by constructing permutation-invariant point-cloud networks under controlled multi-event grouping. We then extend the analysis to the more realistic initial entropy-density profiles generated by the TRENTo model, where both standard regression and simulation-based inference (SBI) with conditional normalizing flows are employed to reconstruct the deformation parameters from ensembles of event images supplemented with global attributes. Multi-event averaging is found to be essential in this setting for suppressing stochastic fluctuations and revealing the underlying deformation information. While standard regression efficiently captures the central trends of deformation through point estimates, SBI provides calibrated posterior distributions, offering a more complete and robust characterization of uncertainty. Collectively, our results demonstrate that deformation information is effectively encoded in the initial state and becomes increasingly identifiable with sufficient ensemble averaging, laying a solid foundation for future extensions toward more complete dynamical modeling and final-state observables.

nucl-th↗

arXiv2025

Initial spin fluctuations as a probe of cluster spin structure in $^{16}\mathrm{O}$ and $^{20}\mathrm{Ne}$ nuclei

We investigate the imprint of $\alpha$ clustering on initial spin fluctuations in relativistic $^{16}\mathrm{O}+{}^{16}\mathrm{O}$ and $^{20}\mathrm{Ne}+{}^{20}\mathrm{Ne}$ collisions at $\sqrt{s_{\mathrm{NN}}}=5.36$~TeV. Utilizing \textit{ab initio} configurations from Nuclear Lattice Effective Field Theory (NLEFT) and phenomenological $\alpha$-cluster models within a Monte-Carlo Glauber framework, we compute the event-by-event variance of the initial net spin polarization. We find that the strong short-range spin--isospin correlations characteristic of $\alpha$ clusters lead to a significant suppression of spin fluctuations compared to a spherical Woods--Saxon baseline with uncorrelated spins. By constructing a scaled fluctuation observable that accounts for trivial finite-size effects, we demonstrate that this suppression exhibits a non-monotonic centrality dependence sensitive to the detailed cluster geometry. Furthermore, we propose the ratio of scaled spin fluctuations between $^{20}\mathrm{Ne}$ and $^{16}\mathrm{O}$ systems as a robust probe. Our results predict distinct percent-level deviations from the baseline for clustered nuclei, suggesting that measurements of final-state $\Lambda$-hyperon spin correlations can provide novel constraints on the ground-state spin structure of light nuclei.

nucl-th↗

arXiv2025

Revisiting finite Abelian hidden subgroup problem and its distributed exact quantum algorithm

We revisit the finite Abelian hidden subgroup problem (AHSP) from a mathematical perspective and make the following contributions. First, by employing amplitude amplification, we present an exact quantum algorithm for the finite AHSP, our algorithm is more concise than the previous exact algorithm and applies to any finite Abelian group. Second, utilizing the Chinese Remainder Theorem, we propose a distributed exact quantum algorithm for finite AHSP, which requires fewer qudits, lower quantum query complexity, and no quantum communication. We further show that our distributed approach can be extended to certain classes of non-Abelian groups. Finally, we develop a parallel exact classical algorithm for finite AHSP with reduced query complexity; even without parallel execution, the total number of queries across all nodes does not exceed that of the original centralized algorithm under mild conditions.

quant-ph↗

arXiv2025

OmniView: An All-Seeing Diffusion Model for 3D and 4D View Synthesis

Prior approaches injecting camera control into diffusion models have focused on specific subsets of 4D consistency tasks: novel view synthesis, text-to-video with camera control, image-to-video, amongst others. Therefore, these fragmented approaches are trained on disjoint slices of available 3D/4D data. We introduce OmniView, a unified framework that generalizes across a wide range of 4D consistency tasks. Our method separately represents space, time, and view conditions, enabling flexible combinations of these inputs. For example, OmniView can synthesize novel views from static, dynamic, and multiview inputs, extrapolate trajectories forward and backward in time, and create videos from text or image prompts with full camera control. OmniView is competitive with task-specific models across diverse benchmarks and metrics, improving image quality scores among camera-conditioned diffusion models by up to 33\% in multiview NVS LLFF dataset, 60\% in dynamic NVS Neural 3D Video benchmark, 20\% in static camera control on RE-10K, and reducing camera trajectory errors by 4x in text-conditioned video generation. With strong generalizability in one model, OmniView demonstrates the feasibility of a generalist 4D video model. Project page is available at https://snap-research.github.io/OmniView/

cs.CV↗

arXiv2025

Probabilistic Bounds on the Number of Elements to Generate Finite Nilpotent Groups and Their Applications

This work establishes a new probabilistic bound on the number of elements to generate finite nilpotent groups. Let $\varphi_k(G)$ denote the probability that $k$ random elements generate a finite nilpotent group $G$. For any $0 < \epsilon < 1$, we prove that $\varphi_k(G) \ge 1 - \epsilon$ if $k \ge \operatorname{rank}(G) + \lceil \log_2(2/\epsilon) \rceil$ (a bound based on the group rank) or if $k \ge \operatorname{len}(G) + \lceil \log_2(1/\epsilon) \rceil$ (a bound based on the group chain length). Moreover, these bounds are shown to be nearly tight. Both bounds sharpen the previously known requirement of $k \ge \lceil \log_2 |G| + \log_2(1/\epsilon) \rceil + 2$. Our results provide a foundational tool for analyzing probabilistic algorithms, enabling a better estimation of the iteration count for the finite Abelian hidden subgroup problem (AHSP) standard quantum algorithm and a reduction in the circuit repetitions required by Regev's factoring algorithm.

quant-ph↗

arXiv2025

A deep learning approach for predicting multiple observables in Au+Au collisions at RHIC

We develop a neural network model, based on the processes of high-energy heavy-ion collisions, to study and predict several experimental observables in Au+Au collisions. We present a data-driven deep learning framework for predicting multiple bulk observables in Au+Au collisions at RHIC energies. A single neural network is trained exclusively on experimental measurements of charged-particle pseudorapidity density distributions, transverse-momentum spectra and elliptic flow coefficients over a broad range of collision energies and centralities. The network architecture is inspired by the stages of a heavy-ion collision, from the quark-gluon plasma to chemical and kinetic freeze-out, and employs locally connected hidden layers and a structured input design that encodes basic geometric and kinematic features of the system. We demonstrate that these physics-motivated choices significantly improve test performance compared to purely fully connected baselines. The trained model is then used to predict the above observables at collision energies not yet explored experimentally at RHIC, and the results are validated using the energy dependence of the total charged-particle multiplicity per participant pair as well as comparisons to a CLVisc hydrodynamic calculation with TRENTo initial conditions. Our findings indicate that such physics-guided neural networks can serve as efficient surrogates to fill critical data gaps at RHIC and to support further phenomenological studies of QGP properties.

nucl-th↗

arXiv2025

Thermal photon emission from quark-gluon plasma: 1+1D magnetohydrodynamics results

We investigate thermal photon production in the quark-gluon plasma (QGP) under strong magnetic fields using a magnetohydrodynamic (MHD) framework. Adopting the Bjorken flow model with power-law decaying magnetic fields $\mathbf{B}(\tau) = \mathbf{B}_0 (\tau_0/\tau)^a$ (where $a$ controls the decay rate, $B_0 = \sqrt{\sigma} T_0^2$, and $\sigma$ characterizes the initial field strength), we employ relativistic ideal fluid dynamics under the non-resistive approximation. The resulting QGP temperature evolution exhibits distinct $a$- and $\sigma$-dependent behaviors. Thermal photon production rates are calculated for three dominant processes: Compton scattering with $q\bar{q}$ annihilation (C+A), bremsstrahlung (Brems), and $q\bar{q}$ annihilation with additional scattering (A+S). These rates are integrated over the space-time volume to obtain the photon transverse momentum $(p_T)$ spectrum. Our results demonstrate that increasing $a$ enhances photon yields across all $p_T$, with $a \to \infty$ (super-fast decay) providing an upper bound. For $a = 2/3$, larger $\sigma$ suppresses yields through accelerated cooling, whereas for $a \to \infty$, larger $\sigma$ enhances yields via prolonged thermal emission. Low-$p_T$ photons receive significant contributions from all QGP evolution stages, while high-$p_T$ photons originate predominantly from early times. The central rapidity region $(y=0)$ dominates the total yield. This work extends photon yield studies to the MHD regime under strong magnetic fields, elucidating magnetic field effects on QGP electromagnetic signatures and establishing foundations for future investigations of magnetization and dissipative phenomena.

hep-ph↗

arXiv2025

RefTok: Reference-Based Tokenization for Video Generation

Effectively handling temporal redundancy remains a key challenge in learning video models. Prevailing approaches often treat each set of frames independently, failing to effectively capture the temporal dependencies and redundancies inherent in videos. To address this limitation, we introduce RefTok, a novel reference-based tokenization method capable of capturing complex temporal dynamics and contextual information. Our method encodes and decodes sets of frames conditioned on an unquantized reference frame. When decoded, RefTok preserves the continuity of motion and the appearance of objects across frames. For example, RefTok retains facial details despite head motion, reconstructs text correctly, preserves small patterns, and maintains the legibility of handwriting from the context. Across 4 video datasets (K600, UCF-101, BAIR Robot Pushing, and DAVIS), RefTok significantly outperforms current state-of-the-art tokenizers (Cosmos and MAGVIT) and improves all evaluated metrics (PSNR, SSIM, LPIPS) by an average of 36.7% at the same or higher compression ratios. When a video generation model is trained using RefTok's latents on the BAIR Robot Pushing task, the generations not only outperform MAGVIT-B but the larger MAGVIT-L, which has 4x more parameters, across all generation metrics by an average of 27.9%.

cs.CV↗

arXiv2025

Contrastive Flow Matching

Unconditional flow-matching trains diffusion models to transport samples from a source distribution to a target distribution by enforcing that the flows between sample pairs are unique. However, in conditional settings (e.g., class-conditioned models), this uniqueness is no longer guaranteed--flows from different conditions may overlap, leading to more ambiguous generations. We introduce Contrastive Flow Matching, an extension to the flow matching objective that explicitly enforces uniqueness across all conditional flows, enhancing condition separation. Our approach adds a contrastive objective that maximizes dissimilarities between predicted flows from arbitrary sample pairs. We validate Contrastive Flow Matching by conducting extensive experiments across varying model architectures on both class-conditioned (ImageNet-1k) and text-to-image (CC3M) benchmarks. Notably, we find that training models with Contrastive Flow Matching (1) improves training speed by a factor of up to 9x, (2) requires up to 5x fewer de-noising steps and (3) lowers FID by up to 8.9 compared to training the same models with flow matching. We release our code at: https://github.com/gstoica27/DeltaFM.git.

cs.CV↗

arXiv2025

Left-right splitting of elliptic flow in heavy ion collisions: TRENTo-3D initialization and CLVisc hydrodynamic simulations

Using the TRENTo-3D initial condition model coupled with (3+1)-dimensional CLVisc hydrodynamic simulations, we systematically investigate the left-right splitting of elliptic flow ($\Delta v_{2}$) for soft particles in relativistic heavy-ion collisions. Our study reveals that the final distribution characteristics of $\Delta v_{2}$ are primarily depend on the odd flow harmonics and $v_{2}$ itself. We find that the parton transverse momentum scale $k_\mathrm{T}$ not only determines the geometric tilt of the QGP fireball but also significantly affects the rapidity dependence of both $v_1$ and $\Delta v_{2}$, providing new insights into the splitting mechanism of $\Delta v_{2}$. Furthermore, our results demonstrate that $\Delta v_{2} (p_\mathrm{T})$ exhibits significant sensitivity to influences such as the sub-nucleonic degrees of freedom (or `hotspots'), transverse momentum scale, and fragmentation region profile. By analyzing the $\Delta v_{2}$ and $\Delta v_{2}/v_{2}$ ratio, our findings provide new constraints on the uncertainties of the QGP initial state and provide additional constraints for refining model parameters.

nucl-th↗

arXiv2024

Videoshop: Localized Semantic Video Editing with Noise-Extrapolated Diffusion Inversion

We introduce Videoshop, a training-free video editing algorithm for localized semantic edits. Videoshop allows users to use any editing software, including Photoshop and generative inpainting, to modify the first frame; it automatically propagates those changes, with semantic, spatial, and temporally consistent motion, to the remaining frames. Unlike existing methods that enable edits only through imprecise textual instructions, Videoshop allows users to add or remove objects, semantically change objects, insert stock photos into videos, etc. with fine-grained control over locations and appearance. We achieve this through image-based video editing by inverting latents with noise extrapolation, from which we generate videos conditioned on the edited image. Videoshop produces higher quality edits against 6 baselines on 2 editing benchmarks using 10 evaluation metrics.

cs.CV↗