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Fei Xue

Publications and source records attributed to Fei Xue.

At least 19 recordsLinked to original sources
arXiv2026

The lattice covering density of the regular octahedron

We prove that the lattice covering density of the regular octahedron is $9/8$, establishing the optimality of the classical body-centered cubic covering. A hypothetical lattice covering of smaller density would yield a lattice basis whose six Kuhn tetrahedra each fit in a translate of the octahedron. A sharp determinant bound then gives a contradiction. As a consequence, the maximum order of an undirected Abelian Cayley graph with three generators and their inverses and diameter at most $k$ is $\frac{32}{27}k^3+O(k^2)$ as $k\to\infty$.

math.MG↗
arXiv2026

Bounds for lattice points in terms of volume and successive minima

We prove, in every dimension, the upper bound conjectured by Betke, Henk and Wills for the lattice-point enumerator in terms of volume and successive minima. For every convex body $K\subset R^n$ and full-rank lattice $Λ$, we establish $$ \#(K\capΛ)\le\frac{\mathrm{vol}_n(K)}{\detΛ}\prod_{i=1}^n\left(1+\frac{iλ_i}{2}\right), $$ where $λ_i$ are the successive minima of $(K-K)/2$ with respect to $Λ$. Equality holds precisely for lattice translates of positive integral dilates of unimodular simplices. We also prove $$ \#(\mathrm{int}K\capΛ)\ge\frac{\mathrm{vol}_n(K)}{\detΛ} \left(1-\frac{n}{4}\sum_{i=1}^{n-1}λ_i-\frac{n}{2}λ_n\right), $$ which implies the lower estimate with factor $1-\frac{1}{2}\sum_{i=1}^n iλ_i$. In dimension two we prove the lower bound with factor $(1-λ_1/2)(1-λ_2)$ when $λ_2\le1$. We also observe that a conjectured product lower bound for origin-symmetric bodies fails in dimension three. The proofs combine geometric compression, lattice-cell counts in coordinate sections, and an elementary coefficient comparison.

math.MG↗
arXiv2026

Quantum-Geometric Subgap Second-Harmonic Generation in PT-Symmetric Antiferromagnetic Insulators

Although quantum geometry plays an established role in electronic transport and optical responses, the contribution of the time-reversal-odd Riemannian connection to fully subgap responses remains less explored. Here we identify intrinsic second-harmonic generation in PT-symmetric antiferromagnets, predominantly governed by the real sector of the interband Hermitian connection and arising entirely from virtual interband transitions. In bilayer MnBi$_2$Te$_4$, the response persists in trivial phases but is enhanced in the layer-Chern phase and reverses sign with Néel order, offering a probe of quantum geometry and phase-sensitive readout of antiferromagnetic order without mobile carriers or real excitations.

cond-mat.mes-hall↗
arXiv2026

The Discrete Monotonicity Property in Rank Two

The Betke--Henk--Wills conjecture predicts a bound for lattice-point counts in terms of successive minima. Malikiosis introduced the discrete monotonicity property for normalized residue counts as a sufficient condition for this bound and its extension to nonsymmetric convex bodies. We prove that every compact convex set satisfies this property for lattice subgroups of rank at most two, in arbitrary ambient dimension.

math.MG↗
arXiv2026

On $κ$-covering minima

We introduce $κ$-covering minima by requiring the permutation simplices of one independent lattice frame to fit into translates of a common dilate of a convex body. We prove a lower bound for their normalized product, with coefficient $(n+1)/2^n$ for centrally symmetric bodies and an explicit dependence on Minkowski asymmetry in general. On Euclidean balls, the supremum over full-rank lattices of the ratio of the resulting fitting radius to the covering radius has order $\sqrt n$. We also establish basic comparisons with classical lattice minima and bounds under products, projections, Minkowski addition, and free direct sums. We extend the definition by allowing projected periodic lattice triangulations. These extended minima agree with the classical covering minima on ellipsoids.

math.MG↗
arXiv2026

Heterogeneity and Subgrouping in Modern Statistics and AI

Heterogeneity is a defining feature of modern data: individuals, subpopulations, sites, modalities, tasks, and time points may differ in distributions, associations, treatment effects, or decision objectives. This review develops a unified perspective in which heterogeneous analysis is a problem of structured information borrowing: deciding what should be shared, what should remain distinct, and how uncertainty about that structure should enter inference and decisions. We connect classical stratification, interaction models, finite mixtures, mixed-effects models, and empirical Bayes with subgroup discovery, individualized learning, synthetic data generation, large language models, federated learning, privacy, security, and mobile health. Precision medicine provides a central example: subgroup rules offer interpretable approximations to individualized response surfaces, whereas partially pooled models stabilize patient-specific estimation. Across applications, the same tensions recur: population averages can conceal consequential variation, unrestricted personalization is unstable, adaptive subgroup discovery creates selection bias, and borrowing across incompatible sources causes negative transfer. These difficulties are amplified in contemporary AI, where sparse user histories, entangled representations, heterogeneous preferences, feedback loops, distribution shift, and privacy constraints make reliable personalization and evaluation especially challenging. We conclude with an agenda for calibrated personalization that integrates multilevel representations, selective borrowing, subgroup-aware evaluation, and valid uncertainty quantification.

stat.OT↗
arXiv2026

A note on consequences of Mahler's inequalities

We derive Makai's inequality $\vol_n(K)\vol_n((K-K)^\circ)\ge(n+1)/n!$ in every dimension from the symmetric Mahler theorem in one higher dimension, with equality precisely for simplices. The proof uses a classical two-layer construction, an identity for its polar volume, and a Rogers--Shephard volume estimate. We then record basic consequences of the Mahler and Makai inequalities for polar successive minima, successive minima at the Santaló point, critical determinants, first covering minima, and non-separable lattice arrangements. We also combine Alexandrov--Fenchel with the volume-product inequalities to obtain sharp bounds for paired mixed volumes.

math.MG↗
arXiv2026

Foundation4D: Towards a unified perception and reconstruction model for autonomous vehicles

We introduce Foundation4D, a unified feed-forward model for geometry estimation, 3D detection, semantic occupancy, and Gaussian reconstruction from multi-camera driving video. A shared backbone uses factorized spatiotemporal attention to efficiently combine observations across cameras and time. Our video inference pipeline uses pose graph optimization to refine camera trajectories and assembles object tracks across input windows. We also develop a data annotation pipeline and release PAI-Foundation4D, a benchmark with dense depth and semantic occupancy labels for 1,253 driving clips. With benchmark-specific fine-tuning, Foundation4D achieves the lowest depth AbsRel on Waymo and nuScenes, the highest detection mAP on nuScenes, and the highest occupancy mIoU on OpenOccupancy among the compared methods. Our task-combination study shows benefits for both detection and geometry: co-training detection with geometry and occupancy improves top-class mAP by 1.2 points over detection-only training, while co-training geometry with detection reduces camera-to-rig translation error by $13.4\%$ relative to geometry-only training. We further use the predicted geometry and trajectories as priors for lidarless NuRec, supporting reconstruction of driving scenes for simulation.

cs.CV↗
arXiv2026

MAEM: Mitigating Recency Bias via a Replay-Free, Capacity-Controlled Dynamic Mixture of Experts for Class-Incremental Graph Learning

Graph Neural Networks (GNNs) excel on graph-structured data but suffer catastrophic forgetting when classes arrive sequentially. In class-incremental graph learning, predictions are often dominated by recently learned classes, causing severe recency bias. Dynamic expansion can isolate knowledge across sessions, but many methods assume task-incremental settings and require task identities at test time to select branches or heads---information unavailable in class-incremental learning. Learnable routers offer an alternative, yet continual fine-tuning the router on new data can bias routing toward recent components. We propose MAEM, a replay-free, capacity-controlled dynamic mixture-of-experts framework for class-incremental graph learning. MAEM builds representations with Simplified Graph Convolution (SGC) and trains a dedicated expert for each new session, then freezes it to protect prior knowledge. For routing without task IDs or a shared trainable router, each expert uses a masked autoencoder (MAE); test nodes are assigned to the expert with the lowest reconstruction error, reducing routing-level recency bias. When the expert pool reaches a fixed capacity, MAEM merges the least-used experts via Git Re-Basin alignment and further distills them into a fused expert using topology-aware pseudo data synthesized from MAE distributions (DeepInversion-style), without storing or revisiting historical graph data. On four class-incremental graph benchmarks, MAEM mitigates recency bias, outperforms strong continual graph learning baselines, and keeps model capacity bounded under strict replay-free learning.

cs.LG↗
arXiv2026

Convergence of repeated-pole rational approximation of the exponential on infinite sectors

For $h>0$, degree $m\ge0$, $0\leβ<π$, and $q>0$, let $Σ_β=\{re^{iθ}:r\ge0,\ |θ|\leβ/2\}$ and let $E_{m,β}(q)$ denote the best uniform error for approximating $e^{-hz}$ on $Σ_β$ by rational functions whose only possible finite pole is $s=-q/h$, with multiplicity at most $m$. For the linear scaling $q=αm$ with a fixed $α>0$, a Möbius map reduces the problem to polynomial approximation on a compact lens. Write $γ_β=2-β/π$ and let $α_c(γ_β)$ denote the saddle-coalescence threshold. For $0<α<α_c(γ_β)$, let $H_β(α)\in(0,1)$ be the exponential of the real saddle action. An exact boundary Faber integral, a global fractional-power contour deformation, and a two-saddle expansion yield the sharp two-sided order $E_{m,β}(αm)\asymp m^{-1/2}H_β(α)^m$, uniformly on compact subsets of $0<α<α_c(γ_β)$. On this noncoalescing interval, the exponential factor $H_β(α)$ has the unique minimizer $α_β^*=1/[γ_β\sin(π/(2γ_β))]$ and minimum $\tan(π/(4γ_β))$, recovering the classical half-line factor $\tan(π/8)=\sqrt{2}-1$ at $β=0$. Numerical-range estimates give matrix dimension-independent upper bounds for projected shift-and-invert Arnoldi, while finite-degree pole searches and nonnormal experiments illustrate the distinction between the sector-uniform law and individual matrix computations.

math.NA↗
arXiv2026

Shifted tapered-pole approximation on finite sectors: an exact asymptotic and dimension control

We study explicit lightning-plus-polynomial (LP) approximation of $g(z)z^α(\log z)^m$ on a closed unit-radius sector, where $0<α<1$, $m$ is fixed, and $g$ is analytic nearby. Since the basic sector root-exponential rate is already known, we address three finer questions. First, a one-sided Abel--Poisson formula and a pole-free notched contour give a branchwise error bound that is uniform for grid shifts in compact subsets of $(0,1)$ and valid for every fixed logarithmic power. Second, for $z^α$ in the dense and balanced regimes, an exact vertex-scale rescaling identifies a positive finite leading constant for a specified shifted sequence. Third, a rate-matched band of additional representation poles followed by polynomial compression gives the explicit representation dimension $N+2\sqrt{2αλ_σN}+O(1)$ while leaving the first $N$ tapered poles unchanged. A slightly overresolved hybrid sequence also preserves the exact leading constant. Matched-dimension experiments, including reentrant sectors, illustrate the allocation. These exact-asymptotic and dimension statements concern the stated constructions, not sector minimax approximation or total computational complexity.

math.NA↗
arXiv2026

Fixed-Grid Reversibility of High-Order Splittings for Nonlinear Schrödinger Equations with Arbitrary-Axis 3D Rotation

High-order splittings for rotational nonlinear Schrödinger equations may use an exact continuous factorization of the Laplace--rotation flow, but its fixed-grid Fourier realization need not retain the symmetry required by standard symmetric compositions. For the specified realization of the 3D arbitrary-axis explicit exact integrator (EEI), we prove that the complete map has a nonzero quadratic coefficient in its local matrix logarithm for every nonzero rotation on centered even grids. The defect persists under every real consistent composition of the same unrepaired nonlinear sandwich through a positive sum-of-squares factor. We construct two self-adjoint remedies: an adjoint-symmetrized EEI and a palindromic shear alternative. Fourier-tail estimates illustrate the representation mechanism, while complete-map diagnostics and a three-dimensional dipolar benchmark are consistent with the predicted order behavior. An implementation audit separates floating-point coefficient cancellation from the exact-arithmetic symmetry defect.

math.NA↗
arXiv2026

Length--Velocity Gauge Equivalence of Quantum Geometric Nonlinear Conductivity in the Adiabatic DC Limit

Nonlinear transport has emerged as a sensitive probe of quantum geometry beyond the Berry-curvature physics of linear response. However, the intrinsic second-order dc response remains conceptually subtle: different quantum and semiclassical formulations can appear to give different static limits, with different assignments of Fermi sea and Fermi surface contributions. Here we resolve this ambiguity by developing a gauge-consistent density-matrix theory of intrinsic nonlinear conductivity in both the length gauge, where the electric field couples through the position operator, and the velocity gauge, where it enters through the vector potential. We show that the two gauges give the same adiabatic dc response when the same retarded continuation is used for all external frequencies and when the velocity gauge current includes all field-dependent vertices. The apparent Fermi sea terms cancel in the full expression, leaving a Fermi surface quantum geometric contribution determined by the band-normalized quantum metric. This result implies that a fully gapped insulator has no residual dc nonlinear Hall current in the adiabatic clean limit. The reactive part of the Fermi surface term agrees with the original semiclassical Berry-connection-polarizability response, while the dissipative Ohmic sector requires a more careful treatment of relaxation and impurity scattering. Our work establishes the length-velocity gauge equivalence for quantum geometric nonlinear response and provides a foundation for using nonlinear transport to probe magnetic quantum geometry, especially in PT-symmetric antiferromagnets.

cond-mat.mes-hall↗
arXiv2026

Physics-Informed Neural Networks for Static Black-Hole Exterior Metrics: Charge and Cosmological-Constant Sweeps

We apply physics-informed neural networks (PINNs) to recover the time-time component of static, spherically symmetric black-hole exterior metrics from a reduced ordinary differential equation (ODE). Inspired by recent work on solving Einstein field equations with deep learning~\cite{Li2023}, we encode the vacuum/charged exterior through a residual loss and an asymptotic boundary constraint---not by embedding analytic metric terms such as $2M/r$ directly into the network output. Unlike the distributed PINN (DPINN) strategy of Ref.~\cite{Li2023}, which partitions the radial domain into subdomains with separate networks, we employ a \emph{unified} fully connected network over the entire interval $[r_{\min},r_{\max}]$, avoiding spurious jumps at subdomain interfaces. Moreover, whereas Ref.~\cite{Li2023} restricts training to $r\in(10,300)$---well outside the steep $1/r$ and $1/r^2$ curvature of the inner exterior---we begin at $r_{\min}=10^{-2}M$, spanning three decades in radius and covering the strongly varying region that DPINNs sidestep by domain truncation. Holding the mass fixed at $M=1$, we sweep electric charge $Q\in\{0,0.5,1.0,1.1\}$ at $Λ=0$ and sweep $Λ\in\{0,0.1,-0.1\}$ at $Q=0.5$. All configurations achieve relative $\mathcal{L}_2$ errors below $6\%$ against the analytic reference. For the representative case $Q=0.5$, $Λ=0.1$, three independent trainings with fixed seeds yield relative errors of $3.89\%$, $1.08\%$, and $2.76\%$ (mean $2.58\%$, standard deviation $1.41\%$), demonstrating robustness of the mesh-free approach without labeled field data.

gr-qc↗
arXiv2026

Uniform Chebyshev asymptotics for repeated-pole rational approximation of the exponential

We study uniform approximation of $\exp(tz)$, $t>0$, on $(-\infty,0]$ by rational functions $P_m(z)/(q_m-z)^m$ with a prescribed repeated pole $q_m>0$. A Möbius transformation reduces the problem to polynomial approximation of $F_λ(x)=\exp\!\left(-λ\frac{1-x}{1+x}\right)$ on $[-1,1]$, where $λ=tq_m$. We derive a uniform two-saddle asymptotic formula for the Chebyshev coefficients, including explicit amplitude and phase and a relative remainder for each localized complex saddle contribution, covering fixed, sublinear, and linear pole scalings away from saddle coalescence. Because the two saddle contributions can cancel in a single coefficient, we pass to a growing block of neighboring coefficients and prove that the whole block cannot cancel. This transfers the coefficient asymptotics to approximation errors. For $q_m=(α/t)m$, $0<α<3\sqrt3/2$, the best uniform error has two-sided order $m^{-1/2}H_e(α)^m$; at the optimal ratio $α=1/\sqrt2$ this becomes $m^{-1/2}(\sqrt2-1)^m$. The normalized Chebyshev-weighted $L^2$ projection error has an explicit bounded oscillatory profile. These prefactor-resolved estimates yield a two-term precision-to-work law and quantify mismatch between pole-design and stopping degrees. Finally, the scalar error gives an exact worst-case matrix-action benchmark for self-adjoint negative semidefinite matrices and dimension-independent shift-and-invert Krylov bounds.

math.NA↗
arXiv2026

Minimal Covering Bodies and a Minkowski-Type Criterion for Lattice Coverings

We study convex bodies whose translates by a fixed lattice cover space and for which every proper convex subbody loses this property. In three dimensions, the convex hull of independent translates of the six Kuhn tetrahedra always gives a lattice covering. We give a geometric proof using auxiliary tetrahedra and the parity of the covering multiplicity. We also prove polytopality and boundary restrictions for minimal covering bodies, and show that every three-dimensional parallelohedron admits a Kuhn representation with respect to its face-to-face tiling lattice. Constructions from Reeve tetrahedra give non-symmetric minimal covering bodies with eight vertices in dimension three and centrally symmetric ones with sixteen vertices in dimension four, with unbounded volumes for the integer covering lattice. They have pairwise distinct arithmetic contact types, which record lattice contacts and the faces containing them. The covering theorem also gives a finite intersection criterion for a prescribed lattice basis, with three intersection tests in the centrally symmetric case.

math.MG↗
arXiv2026

Non-Separable Homothetic Triangles, Part I: Constructions and Lower Bounds

A finite family of planar convex bodies is called a non-separable family if no line disjoint from its union has at least one member in each open half-plane. In this paper, we prove that there exist finite non-separable families of positive homothetic triangles with covering factors strictly exceeding the sharp three-member bound $μ= \frac{2}{3} + \frac{2}{3\sqrt{3}}$. This resolves negatively a question posed by K. Bezdek and Z. Lángi, who originally established this three-member bound after proving that the classic factor 1 covering theorem by A. W. Goodman and R. E. Goodman for disks fails for arbitrary positive homothets. Besides giving explicit algebraic examples with four, five, and six members having factors of approximately 1.0533161, 1.0551900, and 1.0572061 respectively, we provide a common cyclic recurrence that yields a 303-member family with the exact factor 250000000/235141779. Finally, we derive a continuous model suggested by increasingly fine recurrences, yielding a numerical candidate of 1.0633083; its attainability and optimality remain open.

math.MG↗
arXiv2026

Post-Training Recipe, More Than Model Family, Shapes Multi-Agent LLM Conversational Behavior

Multi-LLM systems use multiple language models to deliberate, judge each other's outputs, or coordinate as agents. Their value depends on the models producing measurably different conversational behaviors when given the same input. Prior offline studies recommend drawing one model per family for behavioral diversity, because LLMs prefer outputs from their own family when rating one another in isolation. Whether the same family label predicts behavior in interactive multi-LLM systems, the setting that real deployed systems use, has not been tested. We study this with a 940,000-chain 11-checkpoint corpus and a 1.6M-chain same-base Llama factorial. On our validated headline metric, hedging, a reasoning-distilled Llama checkpoint shifts by 18% depending on which same-base partner it replies to, more than any cross-family hedging gap in the controlled subset. Qwen, closed-API, and runtime checks suggest the pattern is not isolated, while repair and challenge analyses remain exploratory because their surface-cue detectors are weaker. Overall, the results identify post-training recipe as a first-class axis for multi-LLM panel composition and show that model family alone is an incomplete proxy for conversational diversity.

cs.CL↗