arXiv subjects
Publications and source records attributed to Lian Duan.
We prove a quantitative non-covering theorem for finite nilpotent groups. As a consequence, every finite covering of $\Z^d$ by affine full-rank sublattices with pairwise distinct indices contains a member of bounded index, uniformly in $d$. We also establish a summability criterion allowing repeated indices, which extends to finite groups containing a normal nilpotent subgroup of bounded index.
We study the adelic Galois representations attached to products of two Drinfeld modules of generic characteristic. As a function-field analogue of Serre's open image theorem for products of elliptic curves, we show that the image of the joint adelic representation is commensurable with the subgroup defined by the natural determinant compatibility conditions if and only if at least one of the two Drinfeld modules has potential complex multiplication, or the two modules are not geometrically isogenous up to any Frobenius twist. Our proof combines Pink's theory of minimal quasi-models, explicit reciprocity laws from global class field theory, and an analysis of residual Galois representations. As an arithmetic application, we prove a torsion finiteness result: if one of the Drinfeld modules has no potential complex multiplication and the two are not geometrically isogenous up to any Frobenius twist, then that Drinfeld module has only finitely many torsion points over the compositum of the maximal abelian extension of the base field and the torsion field of the other module. This is an analogue of the classical results of Ribet and Zarhin for abelian varieties.
We generalize Christol's theorem---which characterizes algebraic power series over a finite field in terms of automatic coefficient sequences---to completions of domains of finite type over a finite field~$\mathbb{F}_{q}$. Let $A$ be such a domain of dimension~$d$ and $\mathfrak{a}$ an ideal of depth~$d$. In the $\mathfrak{a}$-adic completion $\hat{A}$, choose an $\hat{A}$-regular sequence $\boldsymbol{x}=\{x_{1},\dots,x_{d}\}\subset\hat{\mathfrak{a}}$ and a set of representatives $E$ of $\hat{A}/(\boldsymbol{x})$ that is an $\mathbb{F}_{q}$-vector subspace consisting of algebraic elements. Every element $y\in\hat{A}$ then admits a unique expansion $y=\sum_{u\in\mathbb{N}^{d}}a_{u}x^{u}$ with coefficients $a_{u}\in E$. We prove that $y$ is algebraic over $A$ if and only if the coefficient sequence $\{a_{u}\}_{u\in\mathbb{N}^{d}}$ is $q$-automatic. The proof relies on a careful study of henselizations, which are shown to coincide with the algebraic closures inside the completions under certain conditions. As an application, we obtain a complete characterization of the automaticity of $β$-expansions for formal Laurent series over $\mathbb{F}_{q}$, answering questions posed by Hbaib--Mkaouar and Scheicher--Sirvent.
Let $(ρ_λ\colon G_{\mathbb Q}\to \operatorname{GL}_5(\overline{E}_λ))_λ$ be a strictly compatible system of Galois representations such that no Hodge--Tate weight has multiplicity $5$. Under mild assumptions, we show that if $ρ_{λ_0}$ is irreducible for some $λ_0$, then $ρ_λ$ is irreducible for all but finitely many priimes $λ$. More generally, if $(ρ_λ)_λ$ is essentially self-dual, we show that either $ρ_λ$ is irreducible for all but finitely many $λ$, or the compatible system $(ρ_λ)_λ$ 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.
We study the asymptotic density of $\mathbb{F}_q$-point-free linear sections of geometrically irreducible projective varieties over finite fields. We then apply these results to rank-metric codes via determinantal varieties. Our approach recovers the known cases in which the density tends to $0$ or $1$ and determines the limit in the cases where it was previously unknown. To compute these previously unknown limits, we extend the notion of quasireflexivity to higher-dimensional varieties and show that determinantal varieties satisfy this property. This allows us to invoke the Chebotarev density theorem for varieties over finite fields to obtain the desired estimate.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 $q\geq \operatorname{deg}(S)^2$.
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.
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 $δ(S)$ within prime divisors, then $$-\lim_{n\to\infty} \sum_{\substack{1\le °D\le n\\ D\in \mathfrak{D}(K,S)}}\frac{μ(D)}{|D|}=δ(S),$$ where $μ(D)$ is the Möbius 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 Möbius 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.
Using Calegari's result on the Fontaine-Mazur conjecture, we study the irreducibility of pure, regular, rank 3 weakly compatible systems of self-dual l-adic representations. As a consequence, we prove that the Tate conjecture holds for a family of elliptic surfaces defined over Q with geometric genus bigger than 1.