Showing 1–50 of 349 results for author: Chen, Q

Adaptive Volumetric Parameterization of Simply Connected 3-Manifolds with Applications

Authors: Zhiyuan Lyu, Qiguang Chen, Lok Ming Lui, Gary P. T. Choi

Abstract: Volumetric parameterization, the process of mapping a 3-manifold onto a simplified volumetric domain, is important for many tasks in computer graphics and imaging science. However, most prior volumetric parameterization approaches have only utilized standardized domains such as a solid ball regardless of the overall shape of the given 3-manifolds, which introduces significant geometric distortion… ▽ More Volumetric parameterization, the process of mapping a 3-manifold onto a simplified volumetric domain, is important for many tasks in computer graphics and imaging science. However, most prior volumetric parameterization approaches have only utilized standardized domains such as a solid ball regardless of the overall shape of the given 3-manifolds, which introduces significant geometric distortion and affects the subsequent shape processing and analysis tasks. To overcome this issue, in this work we propose a novel volumetric parameterization framework for simply connected 3-manifolds. Specifically, the proposed framework jointly controls local shape and mass distortions, while adapting the target domain during the optimization process. It enables three progressively more flexible target-domain settings for the parameterization: a prescribed solid ellipsoid, a volume-normalized adaptive ellipsoid with variable radii, and a sea-embedded free-boundary domain. For each setting, the parameterization algorithm consists of a 3D quasi-conformality shape update, a diffusion-based density-equalizing update, and a geometric correction procedure for removing element foldings, thereby allowing for volumetric parameterizations with different desired effects. Experimental results are presented to demonstrate the effectiveness of our proposed framework. Moreover, our framework can be easily applied to multiresolution and localized adaptive volumetric remeshing, volumetric registration, and volumetric morphing. Altogether, our work provides a new way for the representation, processing, and analysis of 3-manifolds. △ Less

Submitted 12 August, 2026; v1 submitted 9 August, 2026; originally announced August 2026.

A new class of irreducible modules over the BMS-Kac-Moody algebra

Authors: Qiu-Fan Chen, Yi-Jun Bai

Abstract: In this paper, we construct a class of non-weight modules over the BMS-Kac-Moody algebra by taking tensor products of finitely many irreducible modules $Φ(λ,\a,\b,\r,h(t))$ with irreducible restricted modules. We obtain the necessary and sufficient conditions for these tensor product modules to be irreducible, and determine the corresponding conditions for two such modules to be isomorphic. Moreov… ▽ More In this paper, we construct a class of non-weight modules over the BMS-Kac-Moody algebra by taking tensor products of finitely many irreducible modules $Φ(λ,\a,\b,\r,h(t))$ with irreducible restricted modules. We obtain the necessary and sufficient conditions for these tensor product modules to be irreducible, and determine the corresponding conditions for two such modules to be isomorphic. Moreover, we compare these modules with other known non-weight modules, showing that these irreducible modules are new. △ Less

Submitted 9 August, 2026; originally announced August 2026.

Comments: 16 pages

MSC Class: 17B10; 17B65; 17B68; 17B70

Debiased Machine Learning: Identification, Estimation, and Shape Constraints

Authors: Qihui Chen, Ka Yan Cheng, Zheng Fang

Abstract: We develop a general framework of identification and estimation for automatic debiased machine learning (DML) where the parameter of interest $θ_0$ is identified by a moment condition involving a nuisance $γ_0$ that may be high dimensional. We establish conditions under which the Riesz representer $α_0$, which is at the core of DML, is identified, and show that the identification occurs precisely… ▽ More We develop a general framework of identification and estimation for automatic debiased machine learning (DML) where the parameter of interest $θ_0$ is identified by a moment condition involving a nuisance $γ_0$ that may be high dimensional. We establish conditions under which the Riesz representer $α_0$, which is at the core of DML, is identified, and show that the identification occurs precisely when $α_0$ uniquely optimizes a quadratic functional. This characterization enables us to develop a general estimation procedure for $α_0$ that allows for generic $γ_0$ including those defined by models with endogeneity and encompasses both classical sieves and modern architectures such as deep neural networks. To improve estimation precision and mitigate the curse of dimensionality, we incorporate shape constraints on $γ_0$ by embedding them into a possibly nonlinear parameter space. We illustrate our estimation procedure through simulations and empirical applications. △ Less

Submitted 29 July, 2026; v1 submitted 27 July, 2026; originally announced July 2026.

Bilinear generating functions of the multivariable Al-Salam-Carlitz polynomials and applications

Authors: Qi Chen, Xinrong Ma, Jin Wang

Abstract: In this paper, by the method of comparing coefficients, we establish a new generating function of multivariable Al-Salam-Carlitz polynomials which contains both Rogers' and Bowman's symmetirc expansion formulas and the classical $q$-Mehler formula as special cases. Some new $q$-series identities related to the multivariable Al-Salam-Carlitz polynomials are also presented. In this paper, by the method of comparing coefficients, we establish a new generating function of multivariable Al-Salam-Carlitz polynomials which contains both Rogers' and Bowman's symmetirc expansion formulas and the classical $q$-Mehler formula as special cases. Some new $q$-series identities related to the multivariable Al-Salam-Carlitz polynomials are also presented. △ Less

Submitted 25 July, 2026; originally announced July 2026.

Comments: 19 pages

MSC Class: 33D15; 05A30

Quantum Weyl Relations arising from Two-Term Complexes

Authors: Qinghua Chen, Yonggang Hu

Abstract: Let $Q$ be a Dynkin quiver over $k=\mathbb F_q$, let $A=kQ$, and let $\Ktwo(\cP)$ be the extriangulated category of two-term complexes of projective $A$-modules. We study the square-root normalized Hall algebra of $\Ktwo(\cP)$. We first establish a PBW-type vector-space factorization into the Ringel--Hall part and the shifted-projective part, and derive an explicit mixed multiplication formula. Fo… ▽ More Let $Q$ be a Dynkin quiver over $k=\mathbb F_q$, let $A=kQ$, and let $\Ktwo(\cP)$ be the extriangulated category of two-term complexes of projective $A$-modules. We study the square-root normalized Hall algebra of $\Ktwo(\cP)$. We first establish a PBW-type vector-space factorization into the Ringel--Hall part and the shifted-projective part, and derive an explicit mixed multiplication formula. For the indecomposable projectives $P_i$, let $p_i$ and $z_i$ be the normalized Hall classes of $(0\to P_i)$ and $(P_i\to0)$, respectively. We prove \[ z_ip_i=q^{-1}p_iz_i+1, \] and determine all off-diagonal products $z_ip_j$ in terms of kernels and cokernels of maps $P_i\to P_j$. Hence each pair $(p_i,z_i)$ generates a rank-one quantum Weyl algebra, while every pairwise Hom-orthogonal family of projectives generates a higher-rank quantum Weyl subalgebra. For these subalgebras we construct explicit Hall--Fock modules, on which the $p_i$ act as creation operators and the $z_i$ act as $q$-annihilation operators. This provides a finite-field Hall model parallel to categorical Hall-type Weyl actions in Donaldson--Thomas theory. Finally, we show that BGP reflection transports the corresponding reflection subalgebras and preserves the mixed Hall coefficients. △ Less

Submitted 24 July, 2026; originally announced July 2026.

Lower bounds on the strength of the determinant

Authors: Qiyuan Chen, Yuhao Zhao

Abstract: We establish new lower bounds for the strength and partition rank of the determinant. For every prime $p$, we prove the exact identity \[ \operatorname{str}(\mathrm{det}_p)=p. \] A weak monotonicity argument, combined with a bound for gaps between consecutive primes, then gives $\operatorname{str}(\mathrm{det}_n)\ge (1-o(1))n^{0.475}$ for sufficiently large $n$. Since the Birch rank of… ▽ More We establish new lower bounds for the strength and partition rank of the determinant. For every prime $p$, we prove the exact identity \[ \operatorname{str}(\mathrm{det}_p)=p. \] A weak monotonicity argument, combined with a bound for gaps between consecutive primes, then gives $\operatorname{str}(\mathrm{det}_n)\ge (1-o(1))n^{0.475}$ for sufficiently large $n$. Since the Birch rank of $\mathrm{det}_n$ is always $4$, this gives the first explicit family showing that the dependence on the degree in bounds for strength in terms of Birch rank is unavoidable. Viewing $\mathrm{det}_n$ as an $n$-linear form in its columns, we also prove that its partition rank is at least the largest prime not exceeding $n$. Consequently, \[ n-n^{0.525}\le \operatorname{prk}(\mathrm{det}_n)\le n \] for all sufficiently large $n$, and hence the partition rank of the determinant is $n-o(n)$. The proof introduces an intersection-theoretic method for lower-bounding strength: a short strength decomposition produces a nowhere-vanishing section of a split vector bundle on the complement of the determinantal hypersurface, while a nonzero top Chern class in the Chow ring of $\mathrm{PGL}_n$ obstructs such a section. △ Less

Submitted 23 July, 2026; originally announced July 2026.

Comments: 16 pages. Comments are welcome!

Norm Inflation for Inviscid and Fully Dissipative Boussinesq Systems in Supercritical Spaces

Authors: Qionglei Chen, Yaowei Xie

Abstract: We prove norm inflation, in the sense of strong ill-posedness, for the two-dimensional Boussinesq system in supercritical Besov spaces. For the inviscid system, norm inflation holds in \(\dot B^β_{p,q}(\mathbb R^2)\times \dot B^β_{p,r}(\mathbb R^2)\) for \(β\neq0\), \(1<p\leq\infty\), \(1\leq q,r\leq\infty\), and \(-2<β-\frac{2}{p}<1\). For the fully dissipative system, the same conclusion holds i… ▽ More We prove norm inflation, in the sense of strong ill-posedness, for the two-dimensional Boussinesq system in supercritical Besov spaces. For the inviscid system, norm inflation holds in \(\dot B^β_{p,q}(\mathbb R^2)\times \dot B^β_{p,r}(\mathbb R^2)\) for \(β\neq0\), \(1<p\leq\infty\), \(1\leq q,r\leq\infty\), and \(-2<β-\frac{2}{p}<1\). For the fully dissipative system, the same conclusion holds in the range \(-2<β-\frac{2}{p}<-1\). In both cases, the results cover almost all supercritical Besov spaces satisfying the local integrability condition. Norm inflation occurs in the density component \(ρ\), while the velocity component \(u\) remains bounded. In the fully dissipative case, the inflation space is supercritical for \(u\), but subcritical for \(ρ\) with respect to its own scaling. This is not a contradiction: the density is transported by a velocity field in a supercritical regime, and this transport mechanism is precisely what produces norm inflation in \(ρ\). △ Less

Submitted 15 July, 2026; originally announced July 2026.

Comments: 35 pages

MSC Class: 35Q35; 35R25; 35A01

The Stability of the Backward Problem for Photoacoustic Imaging in Attenuating Media via Carleman Estimates

Authors: Qihang Chen, Zhiyuan Li, Song Xu

Abstract: This paper investigates the backward problem in time for photoacoustic tomography (PAT) in attenuating media. It is well-established that photoacoustic imaging in attenuating media can be accurately modeled by spatial fractional-order damping. This inverse problem is ill-posed in the sense of Hadamard. In this work, we construct a novel class of Carleman estimates independent of spatial variables,… ▽ More This paper investigates the backward problem in time for photoacoustic tomography (PAT) in attenuating media. It is well-established that photoacoustic imaging in attenuating media can be accurately modeled by spatial fractional-order damping. This inverse problem is ill-posed in the sense of Hadamard. In this work, we construct a novel class of Carleman estimates independent of spatial variables, and by virtue of these estimates, we establish conditional stability estimates for this problem for the first time. Building upon this, we propose a Tikhonov-type regularization functional and derive its associated adjoint system. Furthermore, leveraging the established conditional stability results, we derive the convergence rate of the proposed regularization approach. Finally, we validate the effectiveness of our theoretical findings through extensive numerical experiments. △ Less

Submitted 8 July, 2026; originally announced July 2026.

Spectral Intertwining Operators

Authors: Qiyuan Chen

Abstract: We study spectral intertwining operators between spectral Eisenstein series $\operatorname{Eis}_{P^\vee}$, $\operatorname{Eis}_{Q^\vee}$ for two parabolic subgroups $P, Q$ of a $p$-adic reductive group $G$ with the same Levi subgroup $M$, inspired by the analogy with the classical intertwining operators between parabolic induced representations of $p$-adic reductive groups. In particular, we const… ▽ More We study spectral intertwining operators between spectral Eisenstein series $\operatorname{Eis}_{P^\vee}$, $\operatorname{Eis}_{Q^\vee}$ for two parabolic subgroups $P, Q$ of a $p$-adic reductive group $G$ with the same Levi subgroup $M$, inspired by the analogy with the classical intertwining operators between parabolic induced representations of $p$-adic reductive groups. In particular, we construct the normalized (canonical) intertwining operator that satisfies the transitivity and an unnormalized intertwining operator that is adjoint to a rational section of an analog of the Bruhat-Mackey's filtration. Moreover, the normalized and unnormalized intertwining operators differ by a ratio of L-functions, analogously to the Langlands conjecture about classical ones up to units. Finally we prove that the spectral intertwining operators correspond to classical ones up to units under any conjectural categorical local Langlands correspondence. △ Less

Submitted 16 June, 2026; originally announced June 2026.

Comments: 46 pages

MSC Class: 11S37; 14D24; 22E57

On 3d Quantum Trace Maps

Authors: Qingjing Chen, Andrew Kricker

Abstract: A 3d quantum trace map is a homomorphism from the skein module of an ideally triangulated 3-manifold to its quantum gluing module that quantizes the classical trace map. There are two constructions of such maps, one by Garoufalidis and Yu in [GY], and the other by Panitch and Park in [PP1]. However, the relationship between these two constructions was unknown. We propose a third construction of th… ▽ More A 3d quantum trace map is a homomorphism from the skein module of an ideally triangulated 3-manifold to its quantum gluing module that quantizes the classical trace map. There are two constructions of such maps, one by Garoufalidis and Yu in [GY], and the other by Panitch and Park in [PP1]. However, the relationship between these two constructions was unknown. We propose a third construction of the 3d quantum trace map which agrees with the one given by Garoufalidis and Yu, and extends to certain types of manifolds with ideally triangulated boundaries. Our 3d quantum trace map can be compared with that of [PP1] relatively easily by subdividing the face suspensions of [PP1] and the ideal tetrahedra of our definition into a common subdivision based on face cones. This allows us to give an exact relation between the definitions, which partially addresses the equivalence between the constructions of [GY] and [PP1]. △ Less

Submitted 11 June, 2026; originally announced June 2026.

Comments: 87 pages, 28 figures

MSC Class: 57K31

Lower Bounds on Essential Dimension for Congruence Covers of Mixed Shimura Varieties

Authors: Qi'An Chen

Abstract: We use the fixed point method and toroidal compactifications to establish general lower bounds for the essential dimension of congruence covers $Γ' \backslash \mathcal{X}^0 \rightarrow Γ\backslash \mathcal{X}^0$ of mixed Shimura varieties. Our main result shows that $\mathrm{ed}_{\mathbb{C}}(Γ' \backslash \mathcal{X}^0 \rightarrow Γ\backslash \mathcal{X}^0; p)$ is bounded from below by the dimensi… ▽ More We use the fixed point method and toroidal compactifications to establish general lower bounds for the essential dimension of congruence covers $Γ' \backslash \mathcal{X}^0 \rightarrow Γ\backslash \mathcal{X}^0$ of mixed Shimura varieties. Our main result shows that $\mathrm{ed}_{\mathbb{C}}(Γ' \backslash \mathcal{X}^0 \rightarrow Γ\backslash \mathcal{X}^0; p)$ is bounded from below by the dimension of certain unipotent subgroups associated with the rational boundary components of the given mixed Shimura datum. This generalizes theorems of Brosnan and Fakhruddin to the case of an arbitrary mixed Shimura datum. As a consequence, we obtain incompressibility results for congruence covers of universal families of principally polarized abelian varieties. We also describe explicit fixed points for $(GL_2, \mathcal{H}_2)$ and $(V \rtimes GL_2, \mathcal{Y}_2)$. △ Less

Submitted 25 May, 2026; originally announced May 2026.

Comments: 75 pages

MSC Class: 14G35; 14L30

Manin's conjecture for semi-integral curves and $\mathbb A^1$-connectedness

Authors: Qile Chen, Brian Lehmann, Sho Tanimoto

Abstract: We explore log Manin's conjecture for integral points and its connections to $\mathbb A^1$-connectedness. We prove log Manin's conjecture for Campana rational curves and for $\mathbb A^1$-curves on split toric varieties. Our arguments combine the Cox ring description of the moduli space of rational curves with Batyrev's heuristic-type counting arguments. As our proofs are geometric in nature, they… ▽ More We explore log Manin's conjecture for integral points and its connections to $\mathbb A^1$-connectedness. We prove log Manin's conjecture for Campana rational curves and for $\mathbb A^1$-curves on split toric varieties. Our arguments combine the Cox ring description of the moduli space of rational curves with Batyrev's heuristic-type counting arguments. As our proofs are geometric in nature, they give a geometric explanation of the mysterious leading constant for Campana points proposed by Chow--Loughran--Takloo-Bighash--Tanimoto. △ Less

Submitted 19 May, 2026; originally announced May 2026.

Comments: 47 pages

A new selection problem for degenerate viscous Hamilton-Jacobi equations

Authors: Qinbo Chen, Zhi-Xiang Zhu

Abstract: We study a selection problem for degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians, in which the approximation procedure combines a nonlinear discounted approximation with a small potential perturbation. A key question is how their simultaneous effects influence the asymptotic selection of viscosity solutions of the associated ergodic problem. Based on the nonlinear adjoint me… ▽ More We study a selection problem for degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians, in which the approximation procedure combines a nonlinear discounted approximation with a small potential perturbation. A key question is how their simultaneous effects influence the asymptotic selection of viscosity solutions of the associated ergodic problem. Based on the nonlinear adjoint method, we establish the uniform convergence of the approximating solutions to a distinguished solution of the ergodic problem and derive a formula for the selected limit in terms of generalized Mather measures and the potential. As an application, we show that this selection principle is sufficiently flexible to realize any prescribed solution of the ergodic problem, with an explicit convergence rate. △ Less

Submitted 13 May, 2026; originally announced May 2026.

Newton's method for optimal transport problem on graphs

Authors: Qujiangxue Chen, Jianbo Cui, Luca Dieci, Haomin Zhou

Abstract: In this paper, we study dynamical optimal transport on a connected graph from the perspective of the Benamou-Brenier formulation, where densities are assigned to vertices and velocities to edges. However, directly using Newton's method on the resulting nonlinear systems encounters two potential difficulties: (i) if the graph contains cycles, edge variables are not unique, and (ii) there is no guar… ▽ More In this paper, we study dynamical optimal transport on a connected graph from the perspective of the Benamou-Brenier formulation, where densities are assigned to vertices and velocities to edges. However, directly using Newton's method on the resulting nonlinear systems encounters two potential difficulties: (i) if the graph contains cycles, edge variables are not unique, and (ii) there is no guarantee that the density variables remain positive. To address these challenges, we introduce a finite-difference-type Newton method that eliminates cycle-induced redundancies through a spanning-tree gauge, resulting in a reduced set of independent variables and a well-posed, sparse linear system. For the lattice graph arising from the continuous optimal transport problem, density positivity can also be guaranteed by using an upwind discretization subject to a CFL-type condition. We further demonstrate the versatility of the proposed scheme by applying it to a range of problems, including optimal transport on lattices and random graphs, inverse optimal transport problems, and social network analysis. △ Less

Submitted 8 May, 2026; originally announced May 2026.

MSC Class: 34B45; 49M15; 34A55; 65K10; 65L12; 34A55

Self-normalized tests for multistep conditional predictive ability

Authors: Qitong Chen, Shuwen Lai

Abstract: This paper proposes self-normalized tests for multistep conditional predictive ability in forecast comparison. By normalizing the sample mean of the transformed loss differential using functionals of its cumulative sum (CUSUM) process, specifically an adjusted-range normalizer for scalars and a matrix normalizer for vectors, our approach avoids direct estimation of the long-run covariance matrix.… ▽ More This paper proposes self-normalized tests for multistep conditional predictive ability in forecast comparison. By normalizing the sample mean of the transformed loss differential using functionals of its cumulative sum (CUSUM) process, specifically an adjusted-range normalizer for scalars and a matrix normalizer for vectors, our approach avoids direct estimation of the long-run covariance matrix. Consequently, it eliminates the need for the ad hoc bandwidth, kernel, and lag-truncation choices required by traditional methods. We establish the asymptotic theory for these statistics, deriving pivotal null limiting distributions and proving test consistency. Monte Carlo simulations show that the proposed tests effectively mitigate the finite-sample size distortions associated with traditional heteroskedasticity and autocorrelation consistent (HAC) methods, while retaining strong empirical power against conditional predictability alternatives. △ Less

Submitted 8 May, 2026; originally announced May 2026.

Geometry of multilinear varieties over infinite fields and its applications

Authors: Qiyuan Chen, Ke Ye

Abstract: Multilinear varieties, defined as the sets of rational points of varieties cut out by multilinear functions, were first introduced and studied by Gowers and Milićević[Proc. Edinb. Math. Soc., 2021] for finite $\mathbb{K}$. In this paper, we investigate multilinear varieties over infinite fields from a geometric perspective. We establish two fundamental results: a codimension formula for the Zarisk… ▽ More Multilinear varieties, defined as the sets of rational points of varieties cut out by multilinear functions, were first introduced and studied by Gowers and Milićević[Proc. Edinb. Math. Soc., 2021] for finite $\mathbb{K}$. In this paper, we investigate multilinear varieties over infinite fields from a geometric perspective. We establish two fundamental results: a codimension formula for the Zariski closure of a multilinear variety, and the existence of a high-dimensional irreducible subvariety passing through any given $\mathbb{K}$-rational point. These results serve as a geometric foundation for analyzing various ranks of tensors and homogeneous polynomials, including partition rank, analytic rank, geometric rank, (collective) strength and (collective) Birch rank. As applications, we resolve the Adiprasito-Kazhdan-Ziegler conjecture [arXiv:2102.03659, 2021] on the stability of partition rank for perfect infinite fields. We thereby settle the stability conjecture for collective strength [Selecta Math., 2024], as well as the conjecture on the linear equivalence between strength and Birch rank [arXiv:2410.00248, 2024] for such fields. Moreover, our results immediately yield a strengthening of the theorems of Bik-Draisma-Snowden [arXiv:2401.02067, 2024] and Lampert-Snowden [arXiv:2406.18498, 2024], for multilinear varieties over infinite fields. △ Less

Submitted 6 May, 2026; originally announced May 2026.

Comments: 17 pages.Comments are very welcome!

Ramsey numbers and Gallai--Ramsey numbers of disjoint unions of cherries

Authors: Yanbo Zhang, Qian Chen, Yaojun Chen

Abstract: For graphs $G_1,\ldots,G_k$, the Ramsey number $R(G_1,\ldots,G_k)$ is the smallest positive integer $N$ such that every $k$-edge-coloring of $K_N$ contains a monochromatic copy of $G_i$ in color $i$ for some $i\in[k]$. The Gallai--Ramsey number $GR(G_1,\ldots,G_k)$ is defined analogously, with the colorings restricted to Gallai colorings (i.e., edge-colorings with no rainbow triangle). A copy of… ▽ More For graphs $G_1,\ldots,G_k$, the Ramsey number $R(G_1,\ldots,G_k)$ is the smallest positive integer $N$ such that every $k$-edge-coloring of $K_N$ contains a monochromatic copy of $G_i$ in color $i$ for some $i\in[k]$. The Gallai--Ramsey number $GR(G_1,\ldots,G_k)$ is defined analogously, with the colorings restricted to Gallai colorings (i.e., edge-colorings with no rainbow triangle). A copy of $P_3$ is called a cherry. Let $n_iP_3$ denote the disjoint union of $n_i$ cherries. Wu, Magnant, Nowbandegani, and Xia (Discrete Appl. Math., 2019) proposed two conjectures: \[ R(n_1P_3,\ldots,n_kP_3)=N\ \text{and}\ GR(n_1P_3,\ldots,n_kP_3)=N\,, \] where $N=2\max\{n_1,\ldots,n_k\}+\sum_{i=1}^kn_i-k+1$. We disprove the Ramsey conjecture and provide some sufficient conditions for determining the exact value of $R(n_1P_3,\ldots,n_kP_3)$. In contrast, we confirm the Gallai--Ramsey conjecture. △ Less

Submitted 4 May, 2026; originally announced May 2026.

Parameter estimation for evaporation-driven tear film model in two space dimensions

Authors: Qinying Chen, Tobin Driscoll

Abstract: The tear film (TF) plays a critical role in maintaining ocular surface health, and its disruption through tear breakup (TBU) is closely associated with dry eye disease. Evaporation-driven thinning is a primary mechanism underlying TBU, yet quantitative in vivo estimates of key physical parameters remain limited. In this work, we fit an evaporation-driven TF thinning model, originally developed by… ▽ More The tear film (TF) plays a critical role in maintaining ocular surface health, and its disruption through tear breakup (TBU) is closely associated with dry eye disease. Evaporation-driven thinning is a primary mechanism underlying TBU, yet quantitative in vivo estimates of key physical parameters remain limited. In this work, we fit an evaporation-driven TF thinning model, originally developed by Braun et al. and extended to two dimensions using proper orthogonal decomposition (POD) by Chen et al., to experimental fluorescence (FL) imaging data from normal subjects. The use of dimension reduction enables efficient solution of the governing PDEs and facilitates parameter estimation from imaging data. Our results provide in vivo estimates of evaporation-related and thinning parameters within TBU regions. These findings enhance understanding of TF thinning and dry-spot formation and establish a quantitative baseline for comparison with dry eye patient data. △ Less

Submitted 4 May, 2026; originally announced May 2026.

Gauging the Categorical Connes' $\tildeχ(M)$

Authors: Quan Chen

Abstract: We prove that if a finite group $G$ acts outerly on a McDuff $\rm II_1$ factor $M$, then $\mathsf{Rep}(G/KL)$ is a braided monoidal full subcategory of the categorical Connes' $\tildeχ(M\rtimes G)$ defined in arXiv:2111.06378, where $K$ and $L$ are the centrally trivial and approximately inner parts in $G$ respectively. When $L$ is trivial, we give an explicit formula for the $G/K$-gauging proce… ▽ More We prove that if a finite group $G$ acts outerly on a McDuff $\rm II_1$ factor $M$, then $\mathsf{Rep}(G/KL)$ is a braided monoidal full subcategory of the categorical Connes' $\tildeχ(M\rtimes G)$ defined in arXiv:2111.06378, where $K$ and $L$ are the centrally trivial and approximately inner parts in $G$ respectively. When $L$ is trivial, we give an explicit formula for the $G/K$-gauging procedure on $\tildeχ(M\rtimes G)$. This is the categorical generalization of Connes' short exact sequence on $χ(M\rtimes G)$. Using this machinery, for any finite group $G$, we construct a McDuff $\rm II_1$ factor $M$, whose $\tildeχ(M)$ is braided equivalent to $\mathsf{Rep}(G)$. This is the first example of a braided fusion category which is not modular as $\tildeχ$. △ Less

Submitted 23 April, 2026; originally announced April 2026.

Comments: 36 pages, several tikz figures

MSC Class: 46L37; 18M20; 46M15 primary; 18M15; 18M30 secondary

Compact Runge-Kutta flux reconstruction methods with entropy and/or kinetic energy preserving fluxes

Authors: Arpit Babbar, Qifan Chen, Hendrik Ranocha

Abstract: Compact Runge-Kutta (cRK) methods are a class of high order methods for solving hyperbolic conservation laws characterized by their compact stencil including only immediate neighboring finite elements. A Compact Runge-Kutta flux reconstruction (cRKFR) method for solver hyperbolic conservation laws was introduced in [Babbar, A., Chen, Q., Journal of Scientific Computing, 2025] which uses a time ave… ▽ More Compact Runge-Kutta (cRK) methods are a class of high order methods for solving hyperbolic conservation laws characterized by their compact stencil including only immediate neighboring finite elements. A Compact Runge-Kutta flux reconstruction (cRKFR) method for solver hyperbolic conservation laws was introduced in [Babbar, A., Chen, Q., Journal of Scientific Computing, 2025] which uses a time average flux formulation to perform evolution using a single numerical flux computation at each step, making it a single stage method. Entropy or kinetic energy preserving numerical fluxes are often used for construction of high order entropy stable or kinetic energy preserving methods for hyperbolic conservation laws, and are known to enhance the robustness of numerical methods for under-resolved simulations. In this work, we show how these fluxes can be incorporated into the cRKFR framework for general hyperbolic equations that consist of fluxes and non-conservative products. We test the effectiveness of this new class of methods through numerical experiments for the compressible Euler equations, magnetohydronamics (MHD) equations and multi-ion MHD equations. It is observed that the application of entropy or kinetic energy preserving fluxes enhances the robustness of the cRKFR methods. △ Less

Submitted 2 April, 2026; originally announced April 2026.

MSC Class: 65M60

Some new results on Andrews' and Warnaar's q-identities

Authors: Qi Chen

Abstract: In this paper, by the technique of inverse relations and comparing coefficients, we establish some generalized forms of Andrews' q-series identity and two new Bailey pairs and q-identities closely related to Andrews-Warnaar's sum identity for partial theta functions. In this paper, by the technique of inverse relations and comparing coefficients, we establish some generalized forms of Andrews' q-series identity and two new Bailey pairs and q-identities closely related to Andrews-Warnaar's sum identity for partial theta functions. △ Less

Submitted 30 March, 2026; originally announced March 2026.

Comments: 15 pages

MSC Class: 05A30; 33D15

Towards The Implicit Bias on Multiclass Separable Data Under Norm Constraints

Authors: Shengping Xie, Zekun Wu, Quan Chen, Kaixu Tang

Abstract: Implicit bias induced by gradient-based algorithms is essential to the generalization of overparameterized models, yet its mechanisms can be subtle. This work leverages the Normalized Steepest Descent} (NSD) framework to investigate how optimization geometry shapes solutions on multiclass separable data. We introduce NucGD, a geometry-aware optimizer designed to enforce low rank structures through… ▽ More Implicit bias induced by gradient-based algorithms is essential to the generalization of overparameterized models, yet its mechanisms can be subtle. This work leverages the Normalized Steepest Descent} (NSD) framework to investigate how optimization geometry shapes solutions on multiclass separable data. We introduce NucGD, a geometry-aware optimizer designed to enforce low rank structures through nuclear norm constraints. Beyond the algorithm itself, we connect NucGD with emerging low-rank projection methods, providing a unified perspective. To enable scalable training, we derive an efficient SVD-free update rule via asynchronous power iteration. Furthermore, we empirically dissect the impact of stochastic optimization dynamics, characterizing how varying levels of gradient noise induced by mini-batch sampling and momentum modulate the convergence toward the expected maximum margin solutions.Our code is accessible at: https://github.com/Tsokarsic/observing-the-implicit-bias-on-multiclass-seperable-data. △ Less

Submitted 24 March, 2026; originally announced March 2026.

Quantitative stability of the 2D Monotone shear flow for Boussinesq equation in a finite channel

Authors: Qionglei Chen, Zhen Li

Abstract: Neither natural nor laboratory laminar flows are perfectly steady. Instead, they are frequently highly unsteady, as illustrated by experimental studies on Bénard convection. In the paper, we investigate the transition threshold of the Boussinesq equations around a time-dependent monotone shear flow $(U(t,y),0)$ with a constant background temperature $a\in\mathbb{R}$. The analysis is performed in t… ▽ More Neither natural nor laboratory laminar flows are perfectly steady. Instead, they are frequently highly unsteady, as illustrated by experimental studies on Bénard convection. In the paper, we investigate the transition threshold of the Boussinesq equations around a time-dependent monotone shear flow $(U(t,y),0)$ with a constant background temperature $a\in\mathbb{R}$. The analysis is performed in the finite channel $\mathbb{T}\times[0,1]$ with non-slip boundary condition. By means of the sharp resolvent estimates and space-time estimates, we establish that the Boussinesq system admits a globally stable solution around the monotone shear flow, provided that the initial perturbation satisfies $\|u^{\mathrm{in}}\|_{H^2}\leq cν^{\frac12}, \|\langle D_x\rangle θ^{\mathrm{in}}\|_{L^2} \leq cν^{\frac56}$. Moreover, we derive the enhanced dissipation estimate of the vorticity and inviscid damping estimate of the velocity. △ Less

Submitted 17 March, 2026; originally announced March 2026.

Numerical Approach for On-the-Fly Active Flow Control via Flow Map Learning Method

Authors: Xinyu Liu, Qifan Chen, Dongbin Xiu

Abstract: We present a data-driven numerical approach for on-the-fly active flow control and demonstrate its effectiveness for drag reduction in two-dimensional incompressible flow past a cylinder. The method is based on flow map learning (FML), a recently developed framework for modeling unknown dynamical systems that is particularly effective for partially observed systems. For active flow control, we con… ▽ More We present a data-driven numerical approach for on-the-fly active flow control and demonstrate its effectiveness for drag reduction in two-dimensional incompressible flow past a cylinder. The method is based on flow map learning (FML), a recently developed framework for modeling unknown dynamical systems that is particularly effective for partially observed systems. For active flow control, we construct an FML dynamical model for the quantities of interest (QoIs), namely the drag and lift forces. During offline learning, training data are generated for the responses of drag and lift to the control variable, and a deep neural network (DNN)-based FML model is constructed. The learned FML model enables online optimal flow control without requiring simulations of the flow field. We demonstrate that the FML-based approach can be integrated with existing optimal control strategies, including deep reinforcement learning (DRL) and model predictive control (MPC). Numerical results show that the proposed approach enables on-the-fly flow control and achieves more than $20\%$ drag reduction. By eliminating the need for forward simulations during control optimization, the approach offers the potential for real-time optimal control in other systems. △ Less

Submitted 8 March, 2026; originally announced March 2026.

A global well-posedness result for the three-dimensional inviscid quasi-geostrophic equation over a cylindrical domain

Authors: Qingshan Chen

Abstract: The three-dimensional quasi-geostrophic equation is considered over a cylindrical domain with a multiply connected horizontal cross-section. Homogeneous Neumann boundary conditions, tantamount to homogeneous density fields, are imposed on the top and bottom surfaces, while no-flux boundary conditions combined with constant circulations are imposed on the lateral boundary loops. The global existenc… ▽ More The three-dimensional quasi-geostrophic equation is considered over a cylindrical domain with a multiply connected horizontal cross-section. Homogeneous Neumann boundary conditions, tantamount to homogeneous density fields, are imposed on the top and bottom surfaces, while no-flux boundary conditions combined with constant circulations are imposed on the lateral boundary loops. The global existence and uniqueness of a generalized solution is proven, provided that the initial potential vorticity (PV) field is essentially bounded. If the initial PV field is differentiable, then the solution is shown to satisfy the system in the classical sense. △ Less

Submitted 8 March, 2026; originally announced March 2026.

A base change framework for tensor functions

Authors: Qiyuan Chen

Abstract: The main contribution of this note is to establish a framework to extend results of tensor functions over specific field to general field. As a consequence of this framework, we extend the existing work to more general settings: \emph{(1)} slice rank is linearly bounded by geometric rank for any 3-tensors over any field. \emph{(2)} slice rank of any 3-tensors is quasi-supermultiplicative. As a con… ▽ More The main contribution of this note is to establish a framework to extend results of tensor functions over specific field to general field. As a consequence of this framework, we extend the existing work to more general settings: \emph{(1)} slice rank is linearly bounded by geometric rank for any 3-tensors over any field. \emph{(2)} slice rank of any 3-tensors is quasi-supermultiplicative. As a consequence, the asymptotic slice rank exists for any 3-tensors. △ Less

Submitted 10 March, 2026; v1 submitted 7 March, 2026; originally announced March 2026.

Comments: There is some gap in the proof of Lemma4.3

Accelerating Single-Pass SGD for Generalized Linear Prediction

Authors: Qian Chen, Shihong Ding, Cong Fang

Abstract: We study generalized linear prediction under a streaming setting, where each iteration uses only one fresh data point for a gradient-level update. While momentum is well-established in deterministic optimization, a fundamental open question is whether it can accelerate such single-pass non-quadratic stochastic optimization. We propose the first algorithm that successfully incorporates momentum via… ▽ More We study generalized linear prediction under a streaming setting, where each iteration uses only one fresh data point for a gradient-level update. While momentum is well-established in deterministic optimization, a fundamental open question is whether it can accelerate such single-pass non-quadratic stochastic optimization. We propose the first algorithm that successfully incorporates momentum via a novel data-dependent proximal method, achieving dual-momentum acceleration. Our derived excess risk bound decomposes into three components: an improved optimization error, a minimax optimal statistical error, and a higher-order model-misspecification error. The proof handles mis-specification via a fine-grained stationary analysis of inner updates, while localizing statistical error through a two-phase outer-loop analysis. As a result, we resolve the open problem posed by Jain et al. [2018a] and demonstrate that momentum acceleration is more effective than variance reduction for generalized linear prediction in the streaming setting. △ Less

Submitted 2 March, 2026; originally announced March 2026.

Comments: 50 pages

Turán problems for multilinear maps

Authors: Qiyuan Chen, Zixiang Xu, Ke Ye

Abstract: We study Turán-type extremal problems for alternating and unrestricted multilinear maps. For alternating order-$d$ multilinear maps $T: (\mathbb F^n)^d\to \mathbb{F}^m$, we determine, over algebraically closed fields of arbitrary characteristic, the largest $k$ such that every $T$ vanishes identically on $\mathbb{V}^d$ for some $k$-dimensional subspace $\mathbb{V}$. This extends the bilinear formu… ▽ More We study Turán-type extremal problems for alternating and unrestricted multilinear maps. For alternating order-$d$ multilinear maps $T: (\mathbb F^n)^d\to \mathbb{F}^m$, we determine, over algebraically closed fields of arbitrary characteristic, the largest $k$ such that every $T$ vanishes identically on $\mathbb{V}^d$ for some $k$-dimensional subspace $\mathbb{V}$. This extends the bilinear formula of Buhler, Gupta, and Harris [J. Algebra, 1987] to arbitrary order and resolves a question of Qiao [Discrete Anal., 2023]. We also solve the analogous problem for arbitrary, not necessarily alternating, multilinear maps by determining the largest $k$ such that every $T$ vanishes on $\mathbb{V}_1\times\cdots\times \mathbb{V}_d$ for some $k$-dimensional subspaces $\mathbb{V}_1,\dots,\mathbb{V}_d$. These results yield exact values, over algebraically closed fields, of the Feldman--Propp number [Adv. Math., 1992], the Turán number [Discrete Anal., 2023], and the Gow--Quinlan number [Linear Multilinear Algebra, 2006] associated with alternating multilinear maps. Finally, motivated by the Erdős box problem, we give a purely algebraic derivation of the Conlon--Pohoata--Zakharov lower bound [Discrete Anal., 2021] by combining analytic and partition rank estimates with an incidence count. In the relevant parameter range, we further show that every multilinear map defined over a finite field has many isotropic tuples of $2$-dimensional subspaces over extensions of sufficiently divisible degree. This rules out the natural route to improving the Conlon--Pohoata--Zakharov exponent by selecting multilinear maps with substantially fewer bad isotropic configurations. △ Less

Submitted 23 July, 2026; v1 submitted 28 February, 2026; originally announced March 2026.

Comments: 20 pages. We correct several errors in the original manuscript and extend some of the results

Gale-Robinson Quivers and Principal Coefficients

Authors: Qiyue Chen, Gregg Musiker

Abstract: In this paper, we provide a combinatorial interpretation for Laurent polynomials obtained by iteratively mutating a certain periodic quiver that has been framed with frozen vertices. This yields a family of cluster variables with principal coefficients associated to a family of integer sequences known as Gale-Robinson sequences. The work of this paper completes arguments for preliminary results an… ▽ More In this paper, we provide a combinatorial interpretation for Laurent polynomials obtained by iteratively mutating a certain periodic quiver that has been framed with frozen vertices. This yields a family of cluster variables with principal coefficients associated to a family of integer sequences known as Gale-Robinson sequences. The work of this paper completes arguments for preliminary results announced in earlier work of Jeong-Musiker-Zhang, and relates to works of Bousquet-Mélou-Propp-West, Speyer, Vichitkunakorn, and of Eager-Franco. △ Less

Submitted 22 February, 2026; originally announced February 2026.

Comments: 40 pages, 24 figures. Comments welcome

MSC Class: 13F60 (Primary) 05A15; 05C70 (Secondary)

2-Local derivations on a Block-type Lie algebra

Authors: Qiufan Chen, Xiaohan Guo

Abstract: The present paper is devoted to study 2-local derivations on the Block-type Lie algebra which is an infinite-dimensional Lie algebra with some outer derivations. We prove that every 2-local derivation on the Block-type Lie algebra is a derivation. The present paper is devoted to study 2-local derivations on the Block-type Lie algebra which is an infinite-dimensional Lie algebra with some outer derivations. We prove that every 2-local derivation on the Block-type Lie algebra is a derivation. △ Less

Submitted 10 February, 2026; originally announced February 2026.

Comments: 11 pages

MSC Class: 17B40; 17B65; 17B68

Hodge theory of secant varieties

Authors: Qianyu Chen, Bradley Dirks, Sebastian Olano, Debaditya Raychaudhury

Abstract: We study the local cohomology modules for the secant variety of lines of a smooth projective variety $Y$ and for higher secant varieties of smooth projective curves. We show that the local cohomological defect in the first case is related to the primitive cohomology of $Y$, and in the second case it is $0$. As applications, we compute their (intersection) Hodge-Lyubeznik numbers, the mixed Hodge… ▽ More We study the local cohomology modules for the secant variety of lines of a smooth projective variety $Y$ and for higher secant varieties of smooth projective curves. We show that the local cohomological defect in the first case is related to the primitive cohomology of $Y$, and in the second case it is $0$. As applications, we compute their (intersection) Hodge-Lyubeznik numbers, the mixed Hodge structure on their singular cohomology, the pure Hodge structure on their intersection cohomology, the generating level of the Hodge filtration on their local cohomology modules and their $\mathbf Q$-factoriality defect. As byproducts, we recover and refine various results from the literature by removing restrictive positivity assumptions. △ Less

Submitted 3 February, 2026; originally announced February 2026.

Comments: 101 pages, comments welcome!

Operator learning for models of tear film breakup

Authors: Qinying Chen, Arnab Roy, Tobin A. Driscoll

Abstract: Tear film (TF) breakup is a key driver of understanding dry eye disease, yet estimating TF thickness and osmolarity from fluorescence (FL) imaging typically requires solving computationally expensive inverse problems. We propose an operator learning framework that replaces traditional inverse solvers with neural operators trained on simulated TF dynamics. This approach offers a scalable path towar… ▽ More Tear film (TF) breakup is a key driver of understanding dry eye disease, yet estimating TF thickness and osmolarity from fluorescence (FL) imaging typically requires solving computationally expensive inverse problems. We propose an operator learning framework that replaces traditional inverse solvers with neural operators trained on simulated TF dynamics. This approach offers a scalable path toward rapid, data-driven analysis of tear film dynamics. △ Less

Submitted 27 July, 2026; v1 submitted 12 January, 2026; originally announced January 2026.

MSC Class: 92C35

Circular foliations and shear-radius coordinates on Teichmüller spaces of hyperbolic cone surfaces

Authors: Qiyu Chen, Youliang Zhong

Abstract: We study the Teichmüller space $\mathcal{T}(S,\underline{p})$ of hyperbolic cone-surfaces of fixed topological type with marked cone singularities. Fix a combinatorial triangulation $G$, and let $\mathcal{T}(G)\subset \mathcal{T}(S,\underline{p})$ be the locus where $G$ admits a geodesic realization; varying $G$, these loci form an open cover of $\mathcal{T}(S,\underline{p})$. On $\mathcal{T}(G)$… ▽ More We study the Teichmüller space $\mathcal{T}(S,\underline{p})$ of hyperbolic cone-surfaces of fixed topological type with marked cone singularities. Fix a combinatorial triangulation $G$, and let $\mathcal{T}(G)\subset \mathcal{T}(S,\underline{p})$ be the locus where $G$ admits a geodesic realization; varying $G$, these loci form an open cover of $\mathcal{T}(S,\underline{p})$. On $\mathcal{T}(G)$ we construct a circular foliation adapted to geodesic triangular complementary regions, which is naturally decomposed into interior and peripheral parts. This decomposition defines shear parameters on edges and radius parameters at the singularities, and yields global coordinates on $\mathcal{T}(G)$: the resulting shear-radius map is a homeomorphism onto an explicit open cone in a finite-dimensional real vector space. In the spirit of Thurston, we then introduce partial stretch and anti-stretch deformations by rescaling the transverse measures of the interior or peripheral components. Peripheral stretch rays converge, in the simple-curve length-spectrum topology, to the cusped hyperbolic metric determined by the shear data, while interior anti-stretch rays converge to a circle-packed hyperbolic cone metric determined by the radii. Finally, we give criteria for the realization of prescribed cone angles for fixed $G$ and prove sharp upper bounds for admissible cone angles on the universally triangulable locus. △ Less

Submitted 24 December, 2025; originally announced December 2025.

Comments: 54 pages, 20 figures. Comments and suggestions welcome!

MSC Class: 30F60 (Primary); 57M50; 51M10 (Secondary)

The maximal correlation coefficient associated with the minimum

Authors: Yinshan Chang, Qinwei Chen

Abstract: For independent random variables $(X_i)_{1\leq i\leq n}$, we consider the maximal correlation coefficient $R=R(\min_{i:1\leq i\leq m}X_i,\min_{j:\ell+1\leq j\leq n}X_j)$. If $X_1,X_2,\ldots,X_n$ are identically distributed with the same continuous distribution, we find that $R=(m-\ell)/\sqrt{m(n-\ell)}$. For discrete distributions, we calculate the maximal correlation coefficient $R$ for Bernoulli… ▽ More For independent random variables $(X_i)_{1\leq i\leq n}$, we consider the maximal correlation coefficient $R=R(\min_{i:1\leq i\leq m}X_i,\min_{j:\ell+1\leq j\leq n}X_j)$. If $X_1,X_2,\ldots,X_n$ are identically distributed with the same continuous distribution, we find that $R=(m-\ell)/\sqrt{m(n-\ell)}$. For discrete distributions, we calculate the maximal correlation coefficient $R$ for Bernoulli distributions, geometric distributions, binomial distributions and Poisson distributions. Our paper answers a question in \cite[Section~6]{ChangChen}. △ Less

Submitted 25 March, 2026; v1 submitted 17 December, 2025; originally announced December 2025.

Debiased Bayesian Inference for High-dimensional Regression Models

Authors: Qihui Chen, Zheng Fang, Ruixuan Liu

Abstract: There has been significant progress in Bayesian inference based on sparsity-inducing (e.g., spike-and-slab and horseshoe-type) priors for high-dimensional regression models. The resulting posteriors, however, in general do not possess desirable frequentist properties, and the credible sets thus cannot serve as valid confidence sets even asymptotically. We introduce a novel debiasing approach that… ▽ More There has been significant progress in Bayesian inference based on sparsity-inducing (e.g., spike-and-slab and horseshoe-type) priors for high-dimensional regression models. The resulting posteriors, however, in general do not possess desirable frequentist properties, and the credible sets thus cannot serve as valid confidence sets even asymptotically. We introduce a novel debiasing approach that corrects the bias for the entire Bayesian posterior distribution. We establish a new Bernstein-von Mises theorem that guarantees the frequentist validity of the debiased posterior. We demonstrate the practical performance of our proposal through Monte Carlo simulations and two empirical applications in economics. △ Less

Submitted 9 December, 2025; originally announced December 2025.

Comments: 53 pages

Partial Cohomologically Complete Intersections via Hodge Theory

Authors: Qianyu Chen, Bradley Dirks, Sebastian Olano

Abstract: Using Saito's theory of mixed Hodge modules, we study a generalization of Hellus-Schenzel's "cohomologically complete intersection" property. This property is equivalent to perversity of the shifted constant sheaf. We relate the generalized version to the Hodge filtration on local cohomology, depth of Du Bois complexes, Hodge-Lyubeznik numbers and prove a striking inequality on the codimension of… ▽ More Using Saito's theory of mixed Hodge modules, we study a generalization of Hellus-Schenzel's "cohomologically complete intersection" property. This property is equivalent to perversity of the shifted constant sheaf. We relate the generalized version to the Hodge filtration on local cohomology, depth of Du Bois complexes, Hodge-Lyubeznik numbers and prove a striking inequality on the codimension of the non-perverse locus of the shifted constant sheaf. We study the case of cones over projective rational homology manifolds. We study when such varieties satisfy the weakened condition mentioned above as well as the partial Poincaré duality. To do this, we completely describe their higher local cohomology modules in terms of the Hodge theory of the corresponding projective variety. We apply this to the study of Hodge-Lyubeznik numbers and the intersection cohomology. △ Less

Submitted 4 November, 2025; originally announced November 2025.

Comments: 45 pages

Isotropy and completeness indices of multilinear maps

Authors: Qiyuan Chen, Ke Ye

Abstract: Structures of multilinear maps are characterized by invariants. In this paper we introduce two invariants, named the isotropy index and the completeness index. These invariants capture the tensorial structure of the kernel of a multilinear map. We establish bounds on both indices in terms of the partition rank, geometric rank, analytic rank and height, and present three applications: 1) Using the… ▽ More Structures of multilinear maps are characterized by invariants. In this paper we introduce two invariants, named the isotropy index and the completeness index. These invariants capture the tensorial structure of the kernel of a multilinear map. We establish bounds on both indices in terms of the partition rank, geometric rank, analytic rank and height, and present three applications: 1) Using the completeness index as an interpolator, we establish upper bounds on the aforementioned tensor ranks in terms of the subrank. This settles an open problem raised by Kopparty, Moshkovitz and Zuiddam, and consequently answers a question of Derksen, Makam and Zuiddam. 2) We prove a Ramsey-type theorem for the two indices, generalizing a recent result of Qiao and confirming a conjecture of his. 3) By computing the completeness index, we obtain a polynomial-time probabilistic algorithm to estimate the height of a polynomial ideal. △ Less

Submitted 31 October, 2025; originally announced October 2025.

Comments: 29 pages. Comments welcome

Efficient k-mer Dataset Compression Using Eulerian Covers of de Bruijn Graphs and BWT

Authors: H. Z. Q. Chen, S. Kitaev, X. Lang, A. Pyatkin, R. Tang

Abstract: Transforming an input sequence into its constituent k-mers is a fundamental operation in computational genomics. To reduce storage costs associated with k-mer datasets, we introduce and formally analyze MCTR, a novel two-stage algorithm for lossless compression of the k-mer multiset. Our core method achieves a minimal text representation (W) by computing an optimal Eulerian cover (minimum string c… ▽ More Transforming an input sequence into its constituent k-mers is a fundamental operation in computational genomics. To reduce storage costs associated with k-mer datasets, we introduce and formally analyze MCTR, a novel two-stage algorithm for lossless compression of the k-mer multiset. Our core method achieves a minimal text representation (W) by computing an optimal Eulerian cover (minimum string count) of the dataset's de Bruijn graph, enabled by an efficient local Eulerization technique. The resulting strings are then further compressed losslessly using the Burrows-Wheeler Transform (BWT). Leveraging de Bruijn graph properties, MCTR is proven to achieve linear time and space complexity and guarantees complete reconstruction of the original k-mer multiset, including frequencies. Using simulated and real genomic data, we evaluated MCTR's performance (list and frequency representations) against the state-of-the-art lossy unitigging tool greedytigs (from matchtigs). We measured core execution time and the raw compression ratio cmpr = weight(M)/weight(W), where M is the input sequence data). Benchmarks confirmed MCTR's data fidelity but revealed performance trade-offs inherent to lossless representation. GreedyTigs was significantly faster. Regarding raw compression, GreedyTigs achieved high ratios (cmpr approx 14) on noisy real data for its lossy sequence output. On real data, MCTR (frequency) showed moderate raw compression (cmpr approx 1.5-2.7), while MCTR (list) showed none (cmpr approx 1). Importantly, the full MCTR+BWT pipeline significantly outperforms BWT alone for enhanced lossless compression. Our results establish MCTR as a valuable, theoretically grounded tool for applications demanding efficient, lossless storage and analysis of k-mer multisets, complementing lossy methods optimized for sequence summarization. △ Less

Submitted 25 October, 2025; originally announced October 2025.

Comments: To appear in RAIRO - Theoretical Informatics and Applications

Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition

Authors: Qionglei Chen, Zhen Li, Changxing Miao

Abstract: In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel $\mathbb{R}\times [-1,1]$ with non-slip boundary condition. Compared to the case $\mathbb{T}\times [-1,1]$, we establish the stability in the context of long wave associated with the frequency range $0\leq |k|<1$ by developing the resolvent estimate argument. The new ingredient is to discover t… ▽ More In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel $\mathbb{R}\times [-1,1]$ with non-slip boundary condition. Compared to the case $\mathbb{T}\times [-1,1]$, we establish the stability in the context of long wave associated with the frequency range $0\leq |k|<1$ by developing the resolvent estimate argument. The new ingredient is to discover the key division point at $10ν$ in the frequency interval $(0,1)$ by the sharp Sobolev constant in Wirtinger's inequality together with the refined estimates of the Airy function in the interval $(0,1)$, and then we establish the space-time estimates on the low-frequency $0\leq |k|\leq 10 ν$ and the intermediate-frequency $ 10 ν\leq |k|<1$, respectively. As an application of the space-time estimates, we obtain the nonlinear transition threshold to be $γ\leq\frac{1}{2}$.Meanwhile, we also show that when the frequencies $|k|\geq ν^{1-}$, the enhanced dissipation effect occurs for the linearized Navier-Stokes equations. △ Less

Submitted 21 October, 2025; originally announced October 2025.

Comments: 35 pages. This paper has been submitted in 22 Feb 2025

The optimal transition threshold for the 2D Couette flow in the infinite channel

Authors: Qionglei Chen, Zhen Li, Changxing Miao

Abstract: We investigate the stability of the 2-D Navier-Stokes equations in the infinite channel $\mathbb{R}\times [-1,1]$ with the Navier-slip boundary condition. We show that if the initial perturbations $ω^{in}$ around the Couette flow satisfy $\|ω^{in}\|_{H^3_{x,y}\cap L^1_x H^3_y}\leq cν^{\frac13}$, the solution admits enhanced dissipation at $x$-frequencies $|k|\gg ν$ and inviscid damping effect. The… ▽ More We investigate the stability of the 2-D Navier-Stokes equations in the infinite channel $\mathbb{R}\times [-1,1]$ with the Navier-slip boundary condition. We show that if the initial perturbations $ω^{in}$ around the Couette flow satisfy $\|ω^{in}\|_{H^3_{x,y}\cap L^1_x H^3_y}\leq cν^{\frac13}$, the solution admits enhanced dissipation at $x$-frequencies $|k|\gg ν$ and inviscid damping effect. The key contributions lie in two parts: (1) we adopt the new decomposition of the vorticity $ω=ω_{L}+ω_e$, where $ω_L$ effectively captures a ``weak" enhanced dissipation $(1+ν^{\frac13} t)^{-\frac14}e^{-νt}$ and the corresponding velocity exhibits the inviscid damping effect; (2) we introduce the dyadic decomposition for the long time scale $t\geq ν^{-\frac16}$ and apply the ``infinite superposition principle" to the equation for $ω_e$ in order to control the growth induced by echo cascades, which appears to be novel and may hold independent significance. △ Less

Submitted 21 October, 2025; originally announced October 2025.

Comments: 33 pages. This paper has submitted in July 25, 2025

Asymptotic stability of the symmetric flow via inviscid damping and enhanced dissipation

Authors: Qi Chen, Hao Li, Shunlin Shen, Zhifei Zhang

Abstract: In this paper, we establish the inviscid damping and enhanced dissipation estimates for the linearized Navier-Stokes system around the symmetric flow in a finite channel with the non-slip boundary condition. As an immediate consequence, we prove the asymptotic stability of the symmetric flow in the high Reynolds number regime. Namely, if the initial velocity perturbation $u^{\mathrm{in}}$ satisfie… ▽ More In this paper, we establish the inviscid damping and enhanced dissipation estimates for the linearized Navier-Stokes system around the symmetric flow in a finite channel with the non-slip boundary condition. As an immediate consequence, we prove the asymptotic stability of the symmetric flow in the high Reynolds number regime. Namely, if the initial velocity perturbation $u^{\mathrm{in}}$ satisfies $\Vert u^{\mathrm{in}}-(U(y),0) \Vert_{H^5}\leq c ν^{\frac{2}{3}}$, then inviscid damping and enhanced dissipation estimates also hold for the solution to the Navier-Stokes system. △ Less

Submitted 21 October, 2025; originally announced October 2025.

Asymptotic stability of the Kolmogorov flow at high Reynolds numbers

Authors: Qi Chen, Hao Jia, Dongyi Wei, Zhifei Zhang

Abstract: In this paper we prove the asymptotic stability of the Kolmogorov flow on a non-square torus for perturbations $ω_0$ satisfying $\|ω_0\|_{H^3}\llν^{1/3}$, where $0<ν\ll1$ is the viscosity. Kolmogorov flows are important metastable states to the two dimensional incompressible Navier Stokes equations in the high Reynolds number regime. Our result shows that the perturbed solution will rapidly conver… ▽ More In this paper we prove the asymptotic stability of the Kolmogorov flow on a non-square torus for perturbations $ω_0$ satisfying $\|ω_0\|_{H^3}\llν^{1/3}$, where $0<ν\ll1$ is the viscosity. Kolmogorov flows are important metastable states to the two dimensional incompressible Navier Stokes equations in the high Reynolds number regime. Our result shows that the perturbed solution will rapidly converge to a shear flow close to the Kolmogorov flow, before settling down to the Kolmogorov flow and slowly decaying to $0$ as $t\to\infty$. In fact, our analysis reveals several interesting time scales and rich dynamical behavior of the perturbation in the transition period $0<t\leq 1/ν$. The threshold $ν^{1/3}$, which is the same as that for the Couette flow, is quite surprising since one of the key stability mechanisms, enhanced dissipation, becomes considerably weaker in the case of Kolmogorov flows due to the presence of critical points. To overcome this essential new difficulty, we establish sharp vorticity depletion estimates near the critical points to obtain improved decay rates for the vorticity and velocity fields that are comparable with those for Couette flows, at least for our purposes. We then combine these estimates (enhanced dissipation, inviscid damping and vorticity depletion) with a quasilinear approximation scheme and a multiple-timescale analysis naturally adapted to the dynamics of the perturbation, to obtain the $ν^{1/3}$ threshold for dynamic stability of Kolmogorov flows. The threshold is expected to be sharp when the perturbation is considered in Sobolev spaces. This appears to be the first result that applies vorticity depletion estimates to improve thresholds for nonlinear asymptotic stability in incompressible fluid equations. △ Less

Submitted 15 October, 2025; originally announced October 2025.

Comments: 127 pages

Extremal constructions for apex partite hypergraphs

Authors: Qiyuan Chen, Hong Liu, Ke Ye

Abstract: We establish new lower bounds for the Turán and Zarankiewicz numbers of certain apex partite hypergraphs. Given a $(d-1)$-partite $(d-1)$-uniform hypergraph $\mathcal{H}$, let $\mathcal{H}(k)$ be the $d$-partite $d$-uniform hypergraph whose $d$th part has $k$ vertices that share $\mathcal{ H}$ as a common link. We show that $ex(n,\mathcal{H}(k))=Ω_{\mathcal{ H}}(n^{d-\frac{1}{e(\mathcal{H})}})$ if… ▽ More We establish new lower bounds for the Turán and Zarankiewicz numbers of certain apex partite hypergraphs. Given a $(d-1)$-partite $(d-1)$-uniform hypergraph $\mathcal{H}$, let $\mathcal{H}(k)$ be the $d$-partite $d$-uniform hypergraph whose $d$th part has $k$ vertices that share $\mathcal{ H}$ as a common link. We show that $ex(n,\mathcal{H}(k))=Ω_{\mathcal{ H}}(n^{d-\frac{1}{e(\mathcal{H})}})$ if $k$ is at least exponentially large in $e(\mathcal{H})$. Our bound is optimal for all Sidorenko hypergraphs $\mathcal{H}$ and verifies a conjecture of Lee for such hypergraphs. In particular, for the complete $d$-partite $d$-uniform hypergraphs $\mathcal{K}^{(d)}_{s_1,\dots,s_d}$, our result implies that $ex(n,\mathcal{K}^{(d)}_{s_{1},\cdots,s_{d}})=Θ(n^{d-\frac{1}{s_{1}\cdots s_{d-1}}})$ if $s_{d}$ is at least exponentially large in terms of $s_{1}\cdots s_{d-1}$, improving the factorial condition of Pohoata and Zakharov and answering a question of Mubayi. Our method is a generalization of Bukh's random algebraic method [Duke Math.J. 2024] to hypergraphs, and extends to the sided Zarankiewicz problem. △ Less

Submitted 9 October, 2025; originally announced October 2025.

Comments: 17 pages

Targeted Digital Twin via Flow Map Learning and Its Application to Fluid Dynamics

Authors: Qifan Chen, Zhongshu Xu, Jinjin Zhang, Dongbin Xiu

Abstract: We present a numerical framework for constructing a targeted digital twin (tDT) that directly models the dynamics of quantities of interest (QoIs) in a full digital twin (DT). The proposed approach employs memory-based flow map learning (FML) to develop a data-driven model of the QoIs using short bursts of trajectory data generated through repeated executions of the full DT. This renders the const… ▽ More We present a numerical framework for constructing a targeted digital twin (tDT) that directly models the dynamics of quantities of interest (QoIs) in a full digital twin (DT). The proposed approach employs memory-based flow map learning (FML) to develop a data-driven model of the QoIs using short bursts of trajectory data generated through repeated executions of the full DT. This renders the construction of the FML-based tDT an entirely offline computational process. During online simulation, the learned tDT can efficiently predict and analyze the long-term dynamics of the QoIs without requiring simulations of the full DT system, thereby achieving substantial computational savings. After introducing the general numerical procedure, we demonstrate the construction and predictive capability of the tDT in a computational fluid dynamics (CFD) example: two-dimensional incompressible flow past a cylinder. The QoIs in this problem are the hydrodynamic forces exerted on the cylinder. The resulting tDTs are compact dynamical systems that evolve these forces without explicit knowledge of the underlying flow field. Numerical results show that the tDTs yield accurate long-term predictions of the forces while entirely bypassing full flow simulations. △ Less

Submitted 8 October, 2025; originally announced October 2025.

Feature Augmentation of GNNs for ILPs: Local Uniqueness Suffices

Authors: Qingyu Han, Qian Li, Linxin Yang, Qian Chen, Qingjiang Shi, Ruoyu Sun

Abstract: Integer Linear Programs (ILPs) are central to real-world optimizations but notoriously difficult to solve. Learning to Optimize (L2O) has emerged as a promising paradigm, with Graph Neural Networks (GNNs) serving as the standard backbone. However, standard anonymous GNNs are limited in expressiveness for ILPs, and the common enhancement of augmenting nodes with globally unique identifiers (UIDs) t… ▽ More Integer Linear Programs (ILPs) are central to real-world optimizations but notoriously difficult to solve. Learning to Optimize (L2O) has emerged as a promising paradigm, with Graph Neural Networks (GNNs) serving as the standard backbone. However, standard anonymous GNNs are limited in expressiveness for ILPs, and the common enhancement of augmenting nodes with globally unique identifiers (UIDs) typically introduces spurious correlations that severely harm generalization. To address this tradeoff, we propose a parsimonious Local-UID scheme based on d-hop uniqueness coloring, which ensures identifiers are unique only within each node's d-hop neighborhood. Building on this scheme, we introduce ColorGNN, which incorporates color information via color-conditioned embeddings, and ColorUID, a lightweight feature-level variant. We prove that for d-layer networks, Local-UIDs achieve the expressive power of Global-UIDs while offering stronger generalization. Extensive experiments show that our approach yields substantial and robust gains across ILP benchmarks. △ Less

Submitted 11 May, 2026; v1 submitted 25 September, 2025; originally announced September 2025.

Comments: 19 pages, 9 Tables

Inverse Random Source Problem for the Helmholtz Equation from Statistical Phaseless Data

Authors: Qiao-Ping Chen, Hongyu Liu, Zejun Sun, Li-Li Wang, Guang-Hui Zheng

Abstract: This paper investigates the problem of reconstructing a random source from statistical phaseless data for the two-dimensional Helmholtz equation. The major challenge of this problem is non-uniqueness, which we overcome through a reference source technique. Firstly, we introduce some artificially added point sources into the inverse random source system and derive phase retrieval (PR) formulas for… ▽ More This paper investigates the problem of reconstructing a random source from statistical phaseless data for the two-dimensional Helmholtz equation. The major challenge of this problem is non-uniqueness, which we overcome through a reference source technique. Firstly, we introduce some artificially added point sources into the inverse random source system and derive phase retrieval (PR) formulas for the expectation and variance of the radiated fields. This paper rigorously analyze the uniqueness and stability of the recovered statistics of the radiated fields. Afterwards, since the direct problem has a unique mild solution, by examining the expectation and variance of this solution and combined with the phase retrieval formulas, we derive the Fredholm integral equations to solve the inverse random source problem (IRSP). We prove the stability of the corresponding integral equations. To quantify the uncertainty of the random source, we utilize the Bayesian method to reconstruct the random source and establish the well-posedness of the posterior distribution. Finally, numerical experiments demonstrate the effectiveness of the proposed method and validate the theoretical results. △ Less

Submitted 29 August, 2025; originally announced August 2025.

On the word-representability of $K_m$-$K_n$ graphs

Authors: Herman Z. Q. Chen, Humaira Hameed, Sergey Kitaev

Abstract: Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. Several papers in the literature have explored the word-representability of split graphs, in which the vertices can be partitioned into a clique and an independent set. In this paper, we initiate the study of the word-representabil… ▽ More Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. Several papers in the literature have explored the word-representability of split graphs, in which the vertices can be partitioned into a clique and an independent set. In this paper, we initiate the study of the word-representability of graphs in which the vertices can be partitioned into two cliques. We provide a complete characterization of such word-representable graphs in terms of forbidden subgraphs when one of the cliques has a size of at most four. In particular, if one of the cliques is of size four, we prove that there are seven minimal non-word-representable graphs. △ Less

Submitted 20 August, 2025; originally announced August 2025.

Comments: To appear in the Discussiones Mathematicae Graph Theory

Separable integer partition classes with restrictions on consecutive parts

Authors: Y. Q. Chen, Thomas Y. He, X. M. Huang, T. T. Zou

Abstract: Recently, Andrews introduced separable integer partition classes and studied some well-known theorems. In this article, we will consider the types of partitions with restrictions on consecutive parts. We will show that such partitions are separable integer partition classes and then give the generating functions for such partitions. Recently, Andrews introduced separable integer partition classes and studied some well-known theorems. In this article, we will consider the types of partitions with restrictions on consecutive parts. We will show that such partitions are separable integer partition classes and then give the generating functions for such partitions. △ Less

Submitted 2 October, 2025; v1 submitted 15 August, 2025; originally announced August 2025.

Domino tilings, nonintersecting lattice paths and subclasses of Koutschan-Krattenthaler-Schlosser determinants

Authors: Qipin Chen, Shane Chern, Atsuro Yoshida

Abstract: Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattice paths, thereby partially answering a question of theirs. Furthermore, the determinant evaluations established by Koutschan, Krattenthaler and Schlosser produce… ▽ More Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattice paths, thereby partially answering a question of theirs. Furthermore, the determinant evaluations established by Koutschan, Krattenthaler and Schlosser produce many product formulas for our weighted enumerations of domino tilings and nonintersecting lattice paths. However, there are still two enumerations left corresponding to conjectural formulas made by the three. We hereby prove the two conjectures using the principle of holonomic Ansatz plus the approach of modular reduction for creative telescoping, and hence fill the gap. △ Less

Submitted 17 September, 2025; v1 submitted 21 July, 2025; originally announced July 2025.

Comments: Typo in Thm. 1.4, eq. (1.21) fixed

MSC Class: Primary 15A15; Secondary 05A15; 05B45; 82B20

Inference on Optimal Policy Values and Other Irregular Functionals via Softmax Smoothing

Authors: Justin Whitehouse, Qizhao Chen, Morgane Austern, Vasilis Syrgkanis

Abstract: Constructing confidence intervals for the value of an (unknown) optimal treatment policy is a fundamental problem in causal inference. Insight into the optimal policy value can guide the development of reward-maximizing, individualized treatment regimes. However, because the functional that defines the optimal value is non-differentiable, standard semi-parametric approaches for performing inferenc… ▽ More Constructing confidence intervals for the value of an (unknown) optimal treatment policy is a fundamental problem in causal inference. Insight into the optimal policy value can guide the development of reward-maximizing, individualized treatment regimes. However, because the functional that defines the optimal value is non-differentiable, standard semi-parametric approaches for performing inference fail to be directly applicable. Many existing works circumvent non-differentiability by making the unrealistic assumption of zero probability of treatment non-response, i.e. that every unit responds (either positively or negatively) to an assigned treatment. Further, works that don't circumvent this restriction rely on refitting nuisance models a number of times proportional to the sample size. In this paper, we construct and analyze a simple, softmax smoothing-based estimator for the value of an optimal treatment policy. Our estimator applies in both static and dynamic treatment regimes, only requires fitting a constant number of nuisance models, and is statistically efficient when there is zero probability of non-response to treatment. Also, while our estimator does not require making semi-parametric restrictions, it can exploit them when they exist. We further show how our softmax smoothing approach can be used to estimate general parameters that are specified as a maximum of scores involving nuisance components, and look at conditional Balke and Pearl bounds and $L^1$ calibration error as salient examples. △ Less

Submitted 30 March, 2026; v1 submitted 15 July, 2025; originally announced July 2025.

Comments: 82 pages, 4 figures, 1 table