arXiv subjects

Qi Zhou

Publications and source records attributed to Qi Zhou.

At least 37 records · Page 2Linked to original sources

arXiv2026

Towards Synergistic Teacher-AI Interactions with Generative Artificial Intelligence

Generative artificial intelligence (GenAI) is increasingly used in education, posing significant challenges for teachers adapting to these changes. GenAI offers unprecedented opportunities for accessibility, scalability and productivity in educational tasks. However, the automation of teaching tasks through GenAI raises concerns about reduced teacher agency, potential cognitive atrophy, and the broader deprofessionalisation of teaching. Drawing findings from prior literature on AI in Education, and refining through a recent systematic literature review, this chapter presents a conceptualisation of five levels of teacher-AI teaming: transactional, situational, operational, praxical and synergistic teaming. The framework aims to capture the nuanced dynamics of teacher-AI interactions, particularly with GenAI, that may lead to the replacement, complementarity, or augmentation of teachers' competences and professional practice. GenAI technological affordances required in supporting teaming, along with empirical studies, are discussed. Drawing on empirical observations, we outline a future vision that moves beyond individual teacher agency toward collaborative decision-making between teachers and AI, in which both agents engage in negotiation, constructive challenge, and co-reasoning that enhance each other's capabilities and enable outcomes neither could realise independently. Further discussion of socio-technical factors beyond teacher-AI teaming is also included to streamline the synergy of teachers and AI in education ethically and practically.

cs.CY↗

arXiv2026

Equidistribution of saddle periodic points for Hénon-like maps

We prove that under a natural assumption on the dynamical degrees, the saddle periodic points of a Hénon-like map in any dimension equidistribute with respect to the equilibrium measure. Our work is a generalization of the results of Bedford-Lyubich-Smillie, Dujardin, and Dinh-Sibony along with improvements of their techniques. We also investigate some fine properties of Green currents associated with the map.

math.DS↗

arXiv2026

Canonical analytic realizations of hyperbolic determinantal processes

Krishnapur asked whether the invariant hyperbolic determinantal point processes on the disk admit a random analytic zero-set interpretation at noninteger parameters. We construct such a realization for every positive real parameter as the full compact-open limit in distribution of normalized finite Blaschke products. The zeros determine the modulus and normalized analytic shape, leaving one independent uniform phase. We prove exact Möbius covariance and classify all realizations with this covariance and square-integrable logarithmic modulus at the origin: they are precisely independent positive random multiples of the canonical function. Within this covariant class, matching the canonical logarithmic mean and variance uniquely determines the canonical function law. The family is weakly continuous in the parameter and agrees at positive integers with determinants of matrix-valued Gaussian power series. An explicit Barnes $G$-function Mellin transform determines the basepoint normalization.

math.PR↗

arXiv2026

The Local Embedding Problem for Hardy Spaces of Dirichlet Series

We solve the local embedding problem for Hardy spaces of Dirichlet series, which is a dimension-free trace problem asking whether the global $\mathscr{H}^p$-norm controls local $L^p$-mass on the critical line $\operatorname{Re}s=1/2$. More precisely, for every $2<p<\infty$, there exists a constant $C_p<\infty$ such that every Dirichlet polynomial $P$ satisfies $$ \sup_{θ\in\mathbb{R}}\int_θ^{θ+1}\left|P\left(\frac12+it\right)\right|^p\,\mathrm{d}t\le C_p\left\lVert P\right\rVert_{\mathscr{H}^p}^{p}, $$ with $C_p$ independent of the number and choice of prime variables on which $P$ depends. Before the present work, the embedding was known at $p=2$ and, by taking integer powers, at the even exponents $p=2k$; it had been conjectured that these exhaust the finite positive cases above $2$. Together with the known failure for $0<p<2$, our theorem gives the sharp finite-exponent classification: the local embedding property holds exactly for $p\ge2$. Thus, the true threshold is $p=2$, rather than even integrality. The proof passes to the dual exponent $q=p/(p-1)\in(1,2)$, where an exact frequency decomposition isolates a single resonant Euler-product term. A covariance-preserving replacement of the shared prime factors reduces the resulting fractional-moment estimate to a log-correlated Gaussian field, and a critical branching-random-walk bound supplies the required multiscale decay. A finite-cyclic square-function estimate assembles the resonant scales, and Hardy-quotient duality converts the resulting vector-valued bound into the critical-line trace. For $1\le p<\infty$, known equivalences give the same sharp threshold in several classical problems, including the conformally invariant half-plane embedding, the reverse local Carleson-measure transfer, and boundedness of all characteristic-zero Gordon--Hedenmalm composition operators.

math.FA↗

arXiv2026

Attractors and Late Time Asymptotics in a Generalized Relativistic Second Order Spin Hydrodynamics

We investigate the attractor of spin density in relativistic spin hydrodynamics using Zubarev's non-equilibrium statistical operator formalism in the spin probe limit. We derive the (0+1)D Bjorken flow equations and the associated attractor equation while retaining second order gradient corrections in the relevant relaxation constitutive equations including couplings associated with nonlinear response and nonlocal memory effects. We analyze the early time fixed point structure and analytically determine the early time attractor solution, thereby clarifying branch selection and the role of different dynamical corrections. We find that source-like driving terms modify the leading correction to the attractor solution without changing the fixed point structure, whereas self feedback terms involving the rotational stress tensor modify the dominant balance and modify the early time fixed point structure. We further analyze the late time asymptotics in the conformal limit and show that the newly added terms modify the algebraic prefactor and logarithmic phase shift, while leaving the leading late time decay unchanged. These results provide a unified picture of early time attractor and late time asymptotics in the conformal limit.

hep-ph↗

arXiv2026

Spectral bipartiteness in generalized odd graphs of diameter three

For a graph $G$ of order $n$, put $σ(G)=(λ_1(G)+λ_n(G))/n$. We determine the first three largest values of this invariant among nonbipartite distance-regular graphs of diameter three and odd girth at least seven. The unique maximizer is the folded $7$-cube, with value $1/32$; the unique second maximizer is the Odd graph $O_4$, with value $1/35$; and the unique third maximizer is $C_7$, with value $2(1-\cos(π/7))/7$. More precisely, every other graph in the class satisfies $σ(G)<1/36$. This answers Problem~11 of Abiad, Taranchuk and van Veluw in \emph{Electronic Journal of Combinatorics} 33(2) (2026), P2.31. The proof combines established local multiplicity and odd-moment bounds: the condition $σ(G)\geq1/36$ forces the valency to be at most $182$. An exhaustive certificate using only integer and rational arithmetic then leaves three intersection arrays. The complete certificate is publicly available, and neither a classification of generalized odd graphs nor the $Q$-polynomial property is assumed. The odd-girth theorem gives the same extremal conclusions for connected $\{C_3,C_5\}$-free graphs with at most four distinct adjacency eigenvalues, without assuming regularity.

math.CO↗

arXiv2026

A Functional Central Limit Theorem for Window Counts of Hardy--Szegő Zeros

The Hardy--Szegő zero process, investigated in the disk by Peres and Virág through the independent identically distributed Gaussian analytic function, is a canonical conformally invariant determinantal point process. This paper studies its upper half-plane realization. Although conformally equivalent to the disk model, this realization has its own natural geometry: real-translation invariance turns the process into a stationary object along the boundary and makes long horizontal windows the natural observables. For every admissible height window, we prove a Donsker-type functional central limit theorem for the centered zero counts in expanding horizontal windows, with an explicit intensity and variance depending on the height window. The proof is based on factorial cumulants and Brillinger mixing. The main technical input is a family of all-order integrability estimates for reduced cumulant densities, obtained by exploiting the determinantal cycle structure before integrating over the height variables. As further consequences, we derive an explicit covariance density, an asymptotic variance formula, and a macroscopic Gaussian white-noise limit for linear statistics.

math.PR↗

arXiv2026

Relocation of compact sets in $\mathbb{R}^n$ by diffeomorphisms and linear separability of datasets in $\mathbb{R}^n$

Relocation of compact sets in an $n$-dimensional manifold by self-diffeomorphism is of its own interest as well as significant potential applications to data classification in data science. This paper presents a theory for relocating a finite number of compact sets in $\mathbb{R}^n$ to be relocated to arbitrary target domains in $\mathbb{R}^n$ by diffeomorphisms of $\mathbb{R}^n$. Furthermore, we prove that for any such collection, there exists a differentiable embedding into $\mathbb{R}^{n+1}$ such that their images become linearly separable. As applications of the established theory, we show that a finite number of compact datasets in $\mathbb{R}^n$ can be made linearly separable by width-$n$ deep neural networks (DNNs) with Leaky-ReLU, ELU, or SELU activation functions, under a mild condition. In addition, we show that any finite number of mutually disjoint compact datasets in $\mathbb{R}^n$ can be made linearly separable in $\mathbb{R}^{n+1}$ by a width-$(n+1)$ DNN.

cs.LG↗

arXiv2026

Power convexity of the torsion function on horo-convex domains in hyperbolic space

In this paper, we study the torsion problem on bounded, smooth, strictly horo-convex domains in hyperbolic space. We prove that if the diameter of the domain is sufficiently small, then $-\sqrt{u}$ is strictly convex, where $u$ denotes the torsion function. The main ingredients of our proof are the constant rank theorem, a boundary convexity estimate, and a deformation argument for the domain. We also show that the exponent $1/2$ is optimal, even among strictly horo-convex domains of arbitrarily small diameter.

math.AP↗

arXiv2026

RBMD 2.0: Random batch molecular dynamics package for large-scale simulations on multi-GPU architectures

Large-scale molecular dynamics simulations of particle systems on multi-GPU architectures are often constrained by the computational and communication costs of nonbonded force evaluation. We present RBMD 2.0, a major new release of the random batch molecular dynamics package designed for cross-node multi-GPU simulations of large-scale systems. It combines the improved random batch Ewald method with three-dimensional domain decomposition and ghost-particle communication to accelerate multi-GPU nonbonded force evaluation, while the DTK CUDA framework facilitates portability across heterogeneous accelerator architectures. Numerical experiments on multiple benchmark systems demonstrate both the accuracy and efficiency of simulations with RBMD 2.0. For simulations involving up to hundreds of millions of particles across multiple accelerator devices, one achieves speedups ranging from severalfold to approximately two orders of magnitude in nonbonded force evaluation while exhibiting over $97.5\%$ weak-scaling behavior. These results demonstrate the promising nature of RBMD 2.0 as a computational engine for future exascale molecular dynamics simulations.

physics.comp-ph↗

arXiv2026

Two Regularity Problems on Analytic Tent Spaces

We study two regularity problems on Hardy-type analytic tent spaces $\mathcal{AT}^p_{q,α}$ on the unit disk: fractional integration and randomization of Taylor coefficients. For fractional integration, we characterize completely the boundedness and compactness of the Hadamard, Flett, and Riemann--Liouville operators between analytic tent spaces, and obtain parallel results for analytic Triebel--Lizorkin spaces. In particular, the case $t=0$ yields a complete solution to the corresponding embedding problem for analytic tent spaces. For randomization, we characterize completely when the random Taylor series $\mathcal{R}f$ belongs almost surely to an analytic tent space whenever $f\in \mathcal{AT}^p_{q,α}$, and we also obtain the Triebel--Lizorkin counterpart. As part of the proof, we identify the random symbol space associated with $\mathcal{AT}^p_{q,α}$ and solve the embedding problem from analytic tent spaces into mixed norm spaces. These results extend classical theorems of Hardy--Littlewood and Littlewood, as well as their later analogues for Bergman and mixed norm spaces.

math.FA↗

arXiv2026

Minimum Block Width for Universal Approximation by Residual Neural Networks with Inner Width One

In this paper, we study the universal approximation property of residual neural networks. For input and output dimensions $d_x$ and $d_y$, and LeakyReLU, ReLU, ReLU-like activation functions, the upper and lower bounds of the minimum block width are established. To achieve $L^p$ approximation $(1\leq p <+\infty)$ on any compact set, we show that the exact minimum block width is $\max\{d_x,d_y\}$ when each residual branch has inner width 1. Furthermore, we show that residual neural networks with block width $\min\{d_x+d_y, \max\{2d_x+1,d_y\}\}$ can achieve uniform approximation on any compact set under the constraint that each residual branch has inner width 1. Besides, for any activation function family, we prove that there exist functions that cannot be approximated by residual neural networks with block width less than $\max\{d_x, d_y\}$, both in the $L^p$ sense and the uniform sense, regardless of inner width. Consequently, for LeakyReLU, ReLU, ReLU-like activation functions and $d_y\geq 2d_x+1$, the exact minimum block width for uniform approximation is $d_y$ when each residual branch has inner width 1.

cs.LG↗

arXiv2026

OpsLLM: Construction of Large Language Model for Software Operations with Multi-stage Learning

In the field of software operations, Large Language Models (LLMs) have attracted increasing attention. However, existing research has not yet achieved efficient and effective endto-end intelligent operations due to low-quality data, fragmented knowledge and insufficient learning. To explore the potential of LLMs in software operations, we propose OpsLLM, a domainspecific LLM that supports both knowledge-based question answering (QA) and root cause analysis (RCA). Moreover, we disclose the detailed workflow for building LLMs specifically in the software operations domain. First, a Human-in-the-Loop mechanism is introduced to curate high-quality data from a large collection of operational data and construct a fine-tuning dataset. Then, based on the data, supervised fine-tuning is conducted to achieve a base model. Furthermore, we introduce a domain process reward model (DPRM) during the reinforcement learning stage to optimize the accuracy and reliability of the fine-tuned model on RCA tasks. Experimental results on the tasks with diverse difficulties demonstrate that OpsLLMs effectively learns and aligns with the operational domain knowledge infused, outperforming existing open-source and closed-source LLMs in accuracy with improvements of 0.2%~11.9% on QA tasks and 8.5%~70.3% on RCA tasks, while exhibiting strong transferability. Moreover, we will open-source three versions of OpsLLM with 7B, 14B and 32B parameters, along with a 15K fine-tuning dataset.

cs.LG↗

arXiv2026

The Ultimate Ishii-Pastur Theorem for Whole-Line Ergodic Block Jacobi Operators

We establish an ultimate Ishii-Pastur theorem for whole-line ergodic block Jacobi operators. We prove that, for almost every realization, the restriction of a maximal spectral measure to the region where the smallest nonnegative Lyapunov exponent is strictly positive is carried by a Borel set of zero capacity. In the scalar case, this settles the whole-line problem formulated by Damanik and Fillman \cite[Problem~4.7.18]{Damanik-Fillman-book}.

math.SP↗

arXiv2026

The log-concavity of eigenfunction to complex Monge-Ampère operator in $\mathbb{C}^2$

Following the authors' recent work \cite{Zhang-Zhou2025}, we further explore the convexity properties of solutions to the Dirichlet problem for the complex Monge-Ampère operator. In this paper, we establish the $\log$-concavity of solutions to the Dirichlet eigenvalue problem for the complex Monge-Ampère operator on bounded, smooth, strictly convex domain in $\mathbb{C}^2$. Our approach combines a constant rank theorem adapted to the study of real convexity in the complex setting with a deformation method.

math.AP↗

arXiv2026

Transfer Operators, Canonical Center Dynamics, and Spectral Applications for Long-Range Operators

We introduce an operator-theoretic framework for long-range operators over general dynamical systems with analytic hopping and small potential. By establishing a partially hyperbolic splitting on the fibered solution bundle, we define the Canonical Center Bundle (CCB) as the center subbundle of this splitting, which is shown to be globally trivial. The center bundle admits a representation via Riesz spectral projections of the transfer operator. Furthermore, we show that, in the local regime, the center bundle arising in this framework essentially coincides, in the sense of gap convergence, with the Intrinsic Center Bundles (ICB) obtained from finite-range approximations in \cite{GJ}. The partially hyperbolic structure thereby reduces the spectral problem to the center bundle, leading to a Johnson-type characterization of the spectrum in terms of the associated center cocycle. We then apply this framework to quasi-periodic Schrödinger operators with analytic hopping, large analytic potentials and Diophantine frequency. In this setting, the center cocycle is analytic and satisfies a Center Thouless formula. As consequences, we establish the absolute continuity of the integrated density of states (IDS), resolving a problem of Eliasson; prove quantitative Hölder continuity of the IDS, partially answering a question of You; and obtain Anderson localization for the original Schrödinger operators.

math.SP↗

arXiv2026

A fast sum-of-Gaussians algorithm for the high-dimensional fractional Fokker-Planck equation

We present a fast, high-order algorithm for the free-space fractional Fokker-Planck equation (FFPE) in arbitrary spatial dimension. Its fundamental solution, corresponding to a Dirac-delta initial condition, is obtained from the explicit Fourier representation by applying a sum-of-Gaussians (SOG) approximation to the nonseparable stretched exponential, using its complete monotonicity as the Laplace transform of a one-sided $α$-stable density. Each Gaussian term is an ordinary heat kernel and therefore factorizes across spatial coordinates. On a tensor-product grid, the separated form can be assembled in $O(MdN)$ work and storage, rather than forming all $O(N^d)$ grid values, where $M$ is the number of Gaussian terms and $N$ is the number of points per dimension. We prove an a~priori error estimate for the pure-fractional fundamental solution and give a parameter-selection procedure for prescribed accuracy over specified ranges of space and time. In numerical experiments the method achieves more than ten digits of relative accuracy, with $M$ growing only logarithmically in the inverse tolerance, and maintains this accuracy in dimensions up to $d=10^{5}$. This exceeds the dimensions reached in comparable radial-quadrature tests, where the integrand becomes increasingly oscillatory as the dimension grows. Because the method represents the fundamental solution as a separated sum of heat kernels, any initial datum given as a finite sum of tensor products can be evolved in closed form using only one-dimensional convolutions. This yields a computable class of high-dimensional solutions that is amenable to error analysis, and tensor neural networks provide one possible way to construct such separated representations for more general data.

math.NA↗

arXiv2026

Gaussian rigidity for infinite exchangeable sequences

We prove a Gaussian rigidity theorem for infinite exchangeable sequences of real-valued random variables: the joint Gaussianity of a single pair of entries already forces the entire sequence to be a Gaussian process. This settles a conjecture raised by Newman (2026). The main analytic ingredient in the proof is Hardy's uncertainty principle. We also obtain a finite-dimensional vector-valued extension.

math.PR↗