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Lian Duan

Publications and source records attributed to Lian Duan.

At least 19 recordsLinked to original sources
arXiv2026

Galois representation of the product of two Drinfeld modules of generic characteristic

In this paper, we study the Galois representations attached to products of Drinfeld modules. As a function-field analogue of Serre's classical open image theorem for products of elliptic curves, we prove that if the two Drinfeld modules are not geometrically isogenous up to any Frobenius twist, then for any finite set of primes, the image of the associated product representation is sufficiently large. That is, the image group is commensurable with a subgroup defined by natural determinant compatibility conditions. Our approach combines Pink's minimal quasi-model theory for compact subgroups of linear algebraic groups over local fields with explicit reciprocity laws from global class field theory. As an arithmetic application of our main theorem, we establish a mutual torsion finiteness property for non-isogenous Drinfeld modules, mirroring the classical results of Ribet and Zarhin for abelian varieties.

math.NT↗
arXiv2026

Safety-Potential Pruning for Enhancing Safety Prompts Against VLM Jailbreaking Without Retraining

Safety prompts constitute an interpretable layer of defense against jailbreak attacks in vision-language models (VLMs); however, their efficacy is constrained by the models' latent structural responsiveness. We observe that such prompts consistently engage a sparse set of parameters that remain largely quiescent during benign use. This finding motivates the Safety Subnetwork Hypothesis: VLMs embed structurally distinct pathways capable of enforcing safety, but these pathways remain dormant without explicit stimulation. To expose and amplify these pathways, we introduce Safety-Potential Pruning, a one-shot pruning framework that amplifies safety-relevant activations by removing weights that are less responsive to safety prompts without additional retraining. Across three representative VLM architectures and three jailbreak benchmarks, our method reduces attack success rates by up to 22% relative to prompting alone, all while maintaining strong benign performance. These findings frame pruning not only as a model compression technique, but as a structural intervention to emerge alignment-relevant subnets, offering a new path to robust jailbreak resistance.

cs.CV↗
arXiv2025

Static Batching of Irregular Workloads on GPUs: Framework and Application to Efficient MoE Model Inference

It has long been a problem to arrange and execute irregular workloads on massively parallel devices. We propose a general framework for statically batching irregular workloads into a single kernel with a runtime task mapping mechanism on GPUs. We further apply this framework to Mixture-of-Experts (MoE) model inference and implement an optimized and efficient CUDA kernel. Our MoE kernel achieves up to 91% of the peak Tensor Core throughput on NVIDIA H800 GPU and 95% on NVIDIA H20 GPU.

cs.DC↗
arXiv2024

Interpretable Hierarchical Attention Network for Medical Condition Identification

Accurate prediction of medical conditions with straight past clinical evidence is a long-sought topic in the medical management and health insurance field. Although great progress has been made with machine learning algorithms, the medical community is still skeptical about the model accuracy and interpretability. This paper presents an innovative hierarchical attention deep learning model to achieve better prediction and clear interpretability that can be easily understood by medical professionals. This paper developed an Interpretable Hierarchical Attention Network (IHAN). IHAN uses a hierarchical attention structure that matches naturally with the medical history data structure and reflects patients encounter (date of service) sequence. The model attention structure consists of 3 levels: (1) attention on the medical code types (diagnosis codes, procedure codes, lab test results, and prescription drugs), (2) attention on the sequential medical encounters within a type, (3) attention on the individual medical codes within an encounter and type. This model is applied to predict the occurrence of stage 3 chronic kidney disease (CKD), using three years medical history of Medicare Advantage (MA) members from an American nationwide health insurance company. The model takes members medical events, both claims and Electronic Medical Records (EMR) data, as input, makes a prediction of stage 3 CKD and calculates contribution from individual events to the predicted outcome.

cs.LG↗
arXiv2024

Five-dimensional compatible systems and the Tate conjecture for elliptic surfaces

Let $(\rho_\lambda\colon G_{\mathbb Q}\to \operatorname{GL}_5(\overline{E}_\lambda))_\lambda$ be a strictly compatible system of Galois representations such that no Hodge--Tate weight has multiplicity $5$. Under mild assumptions, we show that if $\rho_{\lambda_0}$ is irreducible for some $\lambda_0$, then $\rho_\lambda$ is irreducible for all but finitely many priimes $\lambda$. More generally, if $(\rho_\lambda)_\lambda$ is essentially self-dual, we show that either $\rho_\lambda$ is irreducible for all but finitely many $\lambda$, or the compatible system $(\rho_\lambda)_\lambda$ decomposes as a direct sum of lower-dimensional compatible systems. We apply our results to study the Tate conjecture for elliptic surfaces. For example, if $X_0\colon y^2 + (t+3)xy + y= x^3$, we prove the codimension one $\ell$-adic Tate conjecture for all but finitely many $\ell$, for all but finitely many general, degree $3$, genus $2$ branched multiplicative covers of $X_0$. To prove this result, we classify the elliptic surfaces into four one-dimensional families and two isolated classes. For each of the four families, we prove, using perverse sheaf theory and a result of Cadoret--Tamagawa, that if the relevant $5$-dimensional Galois representation is irreducible for one surface in a family, then it is irreducible for all but finitely many surfaces in that family. We then verify this irreducibility for one representative of each family by making our irreducibility result explicit: for the compatible system arising from the transcendental part of $H^2_{\mathrm{et}}(X_{\overline{\mathbb Q}}, \mathbb{Q}_\ell(1))$ for a representative $X$, we formulate an algorithm that takes as input the characteristic polynomials of Frobenius, and terminates if and only if the compatible system is irreducible.

math.NT↗
arXiv2024

Transverse linear subspaces to hypersurfaces over finite fields

Ballico proved that a smooth projective variety $X$ of degree $d$ over a finite field of $q$ elements admits a smooth hyperplane section if $q\geq d(d-1)^{\dim X}$. In this paper, we refine this criterion for higher codimensional linear sections on smooth hypersurfaces and for hyperplane sections on Frobenius classical hypersurfaces. We also prove a similar result for the existence of reduced hyperplane sections on reduced hypersurfaces.

math.AG↗
arXiv2023

Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation

Turbulent flows have historically presented formidable challenges to predictive computational modeling. Traditional numerical simulations often require vast computational resources, making them infeasible for numerous engineering applications. As an alternative, deep learning-based surrogate models have emerged, offering data-drive solutions. However, these are typically constructed within deterministic settings, leading to shortfall in capturing the innate chaotic and stochastic behaviors of turbulent dynamics. We introduce a novel generative framework grounded in probabilistic diffusion models for versatile generation of spatiotemporal turbulence. Our method unifies both unconditional and conditional sampling strategies within a Bayesian framework, which can accommodate diverse conditioning scenarios, including those with a direct differentiable link between specified conditions and generated unsteady flow outcomes, and scenarios lacking such explicit correlations. A notable feature of our approach is the method proposed for long-span flow sequence generation, which is based on autoregressive gradient-based conditional sampling, eliminating the need for cumbersome retraining processes. We showcase the versatile turbulence generation capability of our framework through a suite of numerical experiments, including: 1) the synthesis of LES simulated instantaneous flow sequences from URANS inputs; 2) holistic generation of inhomogeneous, anisotropic wall-bounded turbulence, whether from given initial conditions, prescribed turbulence statistics, or entirely from scratch; 3) super-resolved generation of high-speed turbulent boundary layer flows from low-resolution data across a range of input resolutions. Collectively, our numerical experiments highlight the merit and transformative potential of the proposed methods, making a significant advance in the field of turbulence generation.

physics.flu-dyn↗
arXiv2023

On the essential torsion finiteness of abelian varieties over torsion fields

The classical Mordell-Weil theorem implies that an abelian variety $A$ over a number field $K$ has only finitely many $K$-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension $K^{\rm cyc}=K\mathbb{Q}^{\mathrm{ab}}$ by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety $A$ over the infinite algebraic extension $K_B$ obtained by adjoining the coordinates of all torsion points of an abelian variety $B$. Assuming the Mumford-Tate conjecture, and up to a finite extension of the base field $K$, we give a necessary and sufficient condition for the finiteness of $A(K_B)_{\rm tors}$ in terms of Mumford--Tate groups. We give a complete answer when both abelian varieties have dimension both three, or when both have complex multiplication.

math.NT↗
arXiv2022

Frobenius nonclassical hypersurfaces

A smooth hypersurface over a finite field $\mathbb{F}_q$ is called Frobenius nonclassical if the image of every geometric point under the $q$-th Frobenius endomorphism remains in the unique hyperplane tangent to the point. In this paper, we establish sharp lower and upper bounds for the degrees of such hypersurfaces, give characterizations for those achieving the maximal degrees, and prove in the surface case that they are Hermitian when their degrees attain the minimum. We also prove that the set of $\mathbb{F}_q$-rational points on a Frobenius nonclassical hypersurface form a blocking set with respect to lines, which indicates the existence of many $\mathbb{F}_q$-points.

math.AG↗
arXiv2022

Generalizations of Alladi's formula for arithmetical semigroups

In this article, we prove that a general version of Alladi's formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom $A$ or Axiom $A^{\#}$. As applications, we apply our main results to certain semigroups coming from algebraic number theory, arithmetical geometry and graph theory, particularly generalizing the results of Wang 2021, Kural et al. 2020 and Duan et al. 2020.

math.NT↗
arXiv2021

Nonsplitting of the Hilbert exact sequence and the principal Chebotarev density theorem

Let $K/k$ be a finite Galois extension of number fields, and let $H_K$ be the Hilbert class field of $K$. We find a way to verify the nonsplitting of the short exact sequence $$1\to Cl_K\to \text{Gal}(H_K/k)\to\text{Gal}(K/k)\to 1$$ by finite calculation. Our method is based on the study of the principal version of the Chebotarev density theorem, which represents the density of the prime ideals of $k$ that factor into the product of principal prime ideals in $K$. We also find explicit equations to express the principal density in terms of the invariants of $K/k$. In particular, we prove that the group structure of the ideal class group of $K$ can be determined by reading the principal densities.

math.NT↗
arXiv2021

Transverse lines to surfaces over finite fields

We prove that if $S$ is a smooth reflexive surface in $\mathbb{P}^3$ defined over a finite field $\mathbb{F}_q$, then there exists an $\mathbb{F}_q$-line meeting $S$ transversely provided that $q\geq c\operatorname{deg}(S)$, where $c=\frac{3+\sqrt{17}}{4}\approx 1.7808$. Without the reflexivity hypothesis, we prove the existence of a transverse $\mathbb{F}_q$-line for $q\geq \operatorname{deg}(S)^2$.

math.AG↗
arXiv2021

Faltings Serre method on three dimensional selfdual representations

We prove that a selfdual $GL_3$-Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to $3$-dimensional Galois representations with the ground field not equal to $\mathbb{Q}$. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.

math.NT↗
arXiv2020

Analogues of Alladi's formula over global function fields

In this paper, we show an analogue of Kural, McDonald and Sah's result on Alladi's formula for global function fields. Explicitly, we show that for a global function field $K$, if a set $S$ of prime divisors has a natural density $\delta(S)$ within prime divisors, then $$-\lim_{n\to\infty} \sum_{\substack{1\le \deg D\le n\\ D\in \mathfrak{D}(K,S)}}\frac{\mu(D)}{|D|}=\delta(S),$$ where $\mu(D)$ is the M\"{o}bius function on divisors and $\mathfrak{D}(K,S)$ is the set of all effective distinguishable divisors whose smallest prime factors are in $S$. As applications, we get the analogue of Dawsey's and Sweeting and Woo's results to the Chebotarev Density Theorem for function fields, and the analogue of Alladi's result to the Prime Polynomial Theorem for arithmetic progressions. We also display a connection between the M\"obius function and the Fourier coefficients of modular form associated to elliptic curves. The proof of our main theorem is similar to the approach in Kural et al.'s article.

math.NT↗
arXiv2019

Flows Over Periodic Hills of Parameterized Geometries: A Dataset for Data-Driven Turbulence Modeling From Direct Simulations

Computational fluid dynamics models based on Reynolds-averaged Navier--Stokes equations with turbulence closures still play important roles in engineering design and analysis. However, the development of turbulence models has been stagnant for decades. With recent advances in machine learning, data-driven turbulence models have become attractive alternatives worth further explorations. However, a major obstacle in the development of data-driven turbulence models is the lack of training data. In this work, we survey currently available public turbulent flow databases and conclude that they are inadequate for developing and validating data-driven models. Rather, we need more benchmark data from systematically and continuously varied flow conditions (e.g., Reynolds number and geometry) with maximum coverage in the parameter space for this purpose. To this end, we perform direct numerical simulations of flows over periodic hills with varying slopes, resulting in a family of flows over periodic hills which ranges from incipient to mild and massive separations. We further demonstrate the use of such a dataset by training a machine learning model that predicts Reynolds stress anisotropy based on a set of mean flow features. We expect the generated dataset, along with its design methodology and the example application presented herein, will facilitate development and comparison of future data-driven turbulence models.

physics.flu-dyn↗
arXiv2019

GRID: a Student Project to Monitor the Transient Gamma-Ray Sky in the Multi-Messenger Astronomy Era

The Gamma-Ray Integrated Detectors (GRID) is a space mission concept dedicated to monitoring the transient gamma-ray sky in the energy range from 10 keV to 2 MeV using scintillation detectors onboard CubeSats in low Earth orbits. The primary targets of GRID are the gamma-ray bursts (GRBs) in the local universe. The scientific goal of GRID is, in synergy with ground-based gravitational wave (GW) detectors such as LIGO and VIRGO, to accumulate a sample of GRBs associated with the merger of two compact stars and study jets and related physics of those objects. It also involves observing and studying other gamma-ray transients such as long GRBs, soft gamma-ray repeaters, terrestrial gamma-ray flashes, and solar flares. With multiple CubeSats in various orbits, GRID is unaffected by the Earth occultation and serves as a full-time and all-sky monitor. Assuming a horizon of 200 Mpc for ground-based GW detectors, we expect to see a few associated GW-GRB events per year. With about 10 CubeSats in operation, GRID is capable of localizing a faint GRB like 170817A with a 90% error radius of about 10 degrees, through triangulation and flux modulation. GRID is proposed and developed by students, with considerable contribution from undergraduate students, and will remain operated as a student project in the future. The current GRID collaboration involves more than 20 institutes and keeps growing. On August 29th, the first GRID detector onboard a CubeSat was launched into a Sun-synchronous orbit and is currently under test.

astro-ph.IM↗
arXiv2018

Galois action on the Neron-Severi group of Dwork surfaces

We study the Galois action attached to the Dwrok surfaces $X_λ:X_0^4+X_1^4+X_2^4+X_3^4-4λX_0X_1X_2X_3=0$ with parameter $λ$ in a number field $F$. We show that when $X_λ$ has geometric Picard number $19$, its Néron-Severi group $NS(\overline{X}_λ)\otimes \mathbb{Q}$ is a direct sum of quadratic characters. We provide two proofs to this conclusion in our article. In particular, the geometrically proof determines the conductor of each of quadratic characters. Our result matches the one in \cite{Voight2}. With this decomposition, we give another proof to a result of Wan.

math.NT↗