arXiv subjects

Li Zhu

Publications and source records attributed to Li Zhu.

At least 19 recordsLinked to original sources

arXiv2026

Cyclotomic polynomials with trinomial irreducible factors over finite fields

We classify the pairs $(q,n)$, where $q$ is a prime power, $n>1$, and $\gcd(n,q)=1$, for which every irreducible factor of $Φ_n(x)$ over $\F_q$ has exactly three nonzero terms. The positive support exponents of these factors have a common greatest common divisor, which gives a unique reduction by a power substitution. The reduced degree is two in odd characteristic and belongs to $\{2,3,4\}$ in characteristic two; the cubic and quartic cases arise by scalar substitution from the layers of orders $7$ and $15$. This yields necessary and sufficient arithmetic conditions on $(q,n)$ and explicit factorizations. Among the classified families, precisely those arising from quartic bases have a divisor layer containing an irreducible factor with more than three terms.

math.NT↗

arXiv2026

Cage-enforced i-wave altermagnetism in boron-carbon clathrates

The spin splitting of an altermagnet arises from a rotation that relates its two magnetic sublattices. In known cases it is fixed by the crystal structure. Here we show that guest-host frameworks separate the two roles. In the boron-carbon clathrates $M$B$_3$C$_3$ the ordered cage removes the translation between the two guest sites and enforces cubic $i$-wave altermagnetism, with six symmetry-enforced nodal planes, for any magnetic guest in the compensated state; strain retunes the class to $d$- or $g$-wave. We demonstrate the platform with GdB$_3$C$_3$: its compensated antiferromagnetic ground state is a narrow-gap semiconductor with spin splitting of 0.19 eV at the conduction-band edge, and its weakly pinned Néel vector switches the helicity-resolved optical response on and off according to the magnetic point group. Thermodynamic stability at high pressure and dynamical stability at ambient pressure indicate that it can be made by the route established for this family.

cond-mat.mtrl-sci↗

arXiv2026

Equal Hamming Weights at Fixed Root Order and Applications to Constacyclic MDS Codes

For a fixed positive integer $n>1$, we study the prime powers $q$ coprime to $n$ for which all monic irreducible factors of $Φ_n$ over $\F_q$ have the same coefficient Hamming weight. Cyclotomic orbit coefficients give a finite, effective classification of these fields. Outside finitely many computable characteristics, every factor has the support of its characteristic-zero orbit polynomial; the uniform condition $p>2^{φ(n)}$ suffices. In each remaining characteristic, extension exponents reduce to divisors of $\ord_n(p)$, and an integer variance identity expressed by polynomial gcd degrees decides equal weight. The same coefficient data determine the full weight distribution. As an application, we prove that an MDS constacyclic code with an irreducible generator forces every irreducible polynomial of the same root order to have full support, including factors associated with different constacyclic constants. This yields a uniform gcd and norm obstruction. We develop its short-multiplier extension, give a prime-order cubic family, and establish an explicit quadratic-residue exclusion without a prime-square alphabet or a near-square length hypothesis.

cs.IT↗

arXiv2026

ARISE: Authentic Reference-Informed Stability Enhancement for Continual Multimodal Forgery Detection

Multimodal forgery detection is commonly studied under a static assumption: training attacks are expected to approximate those encountered after deployment. This assumption becomes increasingly fragile as generators, editing systems, and multimodal large language models continually introduce new forgery mechanisms and forensic evidence. We formulate this problem as attack-incremental continual multimodal forgery detection, where the binary real/fake interface remains fixed while emerging attacks must be learned without forgetting previous ones. We introduce Fable-Bench, which organizes attack evolution through complementary joint-modality, text-side, and image-side protocols. We further propose ARISE (Authentic Reference-Informed Stability Enhancement), motivated by the open-ended forged class and the comparatively reusable geometry of authentic content. ARISE fits a real-only Gaussian mixture model to projected authentic replay representations as a compact historical reference. Meanwhile, the proposed Gaussian-Responsibility-Aware Stability Preservation (GRASP) derives a sample-adaptive anisotropic metric that strongly constrains historically stable representation directions while retaining plasticity elsewhere. Our Vulnerability-Aware Replay (VAR) further estimates stage-wise vulnerability from authentic-density degradation, forged-to-authentic boundary confusion, and authentic-mode occupancy drift, then assigns bounded replay emphasis to fragile stages. All adaptive signals are obtained from training-time replay statistics, while inference uses only the original multimodal backbone and binary classification head. Experiments across three attack-evolution protocols and multiple detector architectures demonstrate improved historical retention and final average performance over representative continual-learning baselines.

cs.CV↗

arXiv2026

Finite Fields: Root Sets, Frobenius, and Algebraic Structure

This monograph develops finite field theory through root sets, Frobenius orbits, trace duality, and cyclic modules, with applications to polynomial constructions and algebraic coding theory. Root-set descent and base extension are expressed through finite reduced algebras, while linearized polynomials connect normal bases, invariant subspaces, and linear recurrences. Coefficient pairings and derivative formulas recover information lost by trace pairings in quotient algebras with repeated factors. Cyclotomic cosets are studied through additive stabilizers, power substitution, and compatible factor lifting, including extensions to Galois rings. A central theme connects low-degree representations of subfields with conjugate root blocks, polynomial composition, and recovery from relative trace data. Return maps on root fibers determine factor degrees and organize constructions of irreducible polynomials, affine symmetries, and permutation polynomials. Applications include cyclic and constacyclic codes, BCH and Goppa distance bounds, rank-metric codes, trace codes, and Fourier support. Character orthogonality and Gauss and Jacobi sums yield counting formulas, a proof of the primitive normal basis theorem, and recurrences for extension-field character sums. Gaussian periods connect these methods with short linear relations in multiplicative subgroups. Proofs of Weil bounds for polynomial character sums combine auxiliary polynomials with generating polynomials and estimates for their reciprocal roots. The emphasis throughout is on explicit arguments connecting field structure, orbit structure, and linear dependence.

math.NT↗

arXiv2026

Support periods of polynomials and support-primitive decomposition of constacyclic codes over Galois rings

Let $A=\GR(p^e,r)$ have residue field $\F_q$, and let $λ\in A^\times$. We relate coefficient compression of monic divisors of $X^N-λ$ to support-primitive decomposition of constacyclic codes. At a simple-root length, the support period of every nonconstant monic divisor is preserved at each coefficient precision and equals the order of the multiplicative stabilizer of its root set. For a cyclic Frobenius-orbit polynomial, the multiple equal-difference structure determines its support period, core degree, and least relative degree of an unramified extension over which all its basic-irreducible factors are binomials. Coordinate projections give every constacyclic code over a finite commutative ring a unique maximal interleaving into copies of one support-primitive code. At simple-root lengths over $A$, the intrinsic period of a code is the greatest common divisor of the periods of its torsion codes. The decomposition yields distance and duality formulas and the arithmetic Singleton bound for free codes. At every length, a nonzero code over $A$ has the same minimum Hamming distance as its highest torsion code, which gives a refined bound. At repeated-root lengths, identical torsion filtrations can occur with different intrinsic periods, while a class of nonzero codes obtained by substitution from simple-root generators retains the same core and minimum distance. Trace matrices recover the coefficients of basic-irreducible cores using unit pivots, and local Hensel corrections lift compressed factors directly. For $X^n-1$ with $p\nmid n$, lifting and output have quasi-linear bit cost in the compressed representation when the arithmetic data and residue factorizations are given.

cs.IT↗

arXiv2026

Acceleration of CTQMC via Denoising Diffusion Models with Auxiliary Spectral Functions

Continuous-time Quantum Monte Carlo (CTQMC) impurity solvers are a numerically exact, but expensive, option for solving the Dynamical Mean Field Theory (DMFT) equations, with statistical noise from undersampling presenting a major challenge to low temperature studies. This noise is especially egregious within the sampled self-energy, which can destabilize the self-consistency loop upon insertion into the Dyson equation. In this work, we introduce a neural denoiser for fermionic imaginary time Green's functions based on the diffusion model architecture trained on synthetic data. The network predicts positive and normalized auxiliary spectral functions consistent with the noisy imaginary axis CTQMC data, which is then refined via autodifferentiation to match the measured self-energy, its first two high-frequency moments, and impurity occupation similar to the Constrained Residual Minimization (CRM) method. This leads to Green's functions and self-energies that are causal by design. Through this denoising framework we are able to reduce the required CTQMC measurement steps by a factor of 10-100. We benchmark our method with a half-filled Hubbard model on a Bethe lattice, a two-patch Dynamical Cluster Approximation (DCA) of the two-dimensional Hubbard model, and the SrVO$_3$ system. We reproduce the Mott transition upon increasing Coulomb interaction for the half-filled Hubbard model with only 5000 CTQMC measurements, the pseudogap transition for the DCA two-dimensional Hubbard model with only 5$\times$10$^5$ measurements, and match the spectral function of SrVO$_3$ with 50 times fewer measurements than the reference. Since we generate auxiliary spectral functions, we recover the approximate spectral functions of the systems for free without performing analytic continuation.

cond-mat.str-el↗

arXiv2026

Affine Symmetries of Polynomial Roots with Multiplicities and Applications to Constacyclic Codes

Let $f(X)$ be a polynomial over a finite field, and let its defining function record the multiplicities of its roots. In this paper, we introduce the affine stabilizer group and the affine stabilizer group scheme of the defining function of $f(X)$. These objects characterize the affine symmetries of the roots of $f(X)$ together with their multiplicities. We establish structural descriptions of these affine stabilizers and relate them to the coefficient structure of $f(X)$. In particular, the affine stabilizer group determines quotient structures of the defining function, which yield canonical descriptions of the corresponding semi-invariant polynomial spaces. The resulting coefficient constraints provide explicit restrictions on the possible supports of polynomials with prescribed affine symmetries. The affine stabilizer group scheme further captures infinitesimal symmetries arising from multiple roots and gives refined coefficient constraints beyond those obtained from ordinary affine transformations. We apply this framework to generator polynomials of constacyclic codes over finite fields. The affine stabilizer structure of a generator polynomial induces subcodes defined by semi-invariant conditions and provides constraints on their dimensions and Hamming distances. In the special case of multiplicative symmetries, the construction specializes to the support-primitive decomposition of constacyclic codes.

cs.IT↗

arXiv2026

Towards Explainable Benchmarking for Data-driven Post-Wildfire Debris Flow Prediction

Post-wildfire debris flows (PFDFs) are destructive sediment-laden hazards triggered when intense rainfall strikes recently burned terrain, destabilizing hillslopes and threatening infrastructure, local economies, and community safety. Data-driven methods have been proposed to learn predictive patterns directly from historical PFDF observations. However, the current research landscape of data-driven PFDF prediction remains highly fragmented across feature spaces, model architectures, and evaluation protocols, making rigorous comparison and the derivation of scientific insights difficult. Moreover, existing studies lack a systematic investigation into the relative importance of heterogeneous factors (e.g., meteorological conditions, terrain characteristics, soil properties, and burn severity) in triggering PFDF. To address these limitations, we present a unified benchmark for data-driven PFDF prediction, enabling fair and comprehensive evaluation across diverse models and feature configurations. Furthermore, to better understand the underlying drivers of PFDF formation, we propose a reinforcement learning-based feature selection framework that identifies factors whose perturbations render positive and negative events indistinguishable, thereby discovering the regional underlying mechanisms of PFDF occurrence across regions. Our code and benchmark are publicly available at https://github.com/KINDLab-Fly/PFDF-Benchmark.

cs.LG↗

arXiv2026

Uncertainty-Aware Consistency Distillation for Few-Step Video Generation

We study few-step video generation, i.e., distilling a multi-step video generator, which typically requires tens of sampling steps, incurring substantial latency and compute, into a few-step student. Consistency distillation is a common recipe, in which a multi-step teacher provides the consistency targets for a few-step student. However, these teacher-guided targets are not equally trustworthy, and the content is harder to learn where it varies rapidly over time, e.g., moving foliage shadows or flowing water. We observe that supervision reliability follows the local difficulty of the content rather than semantic complexity: regions that change little yield consistent endpoint predictions, whereas regions with large temporal variation produce larger discrepancies that coincide with the largest perceptual errors. Motivated by this observation, we propose Uncertainty-Aware Consistency Distillation (UACD), which reweights consistency supervision at each spatiotemporal region using a local, parameter-free uncertainty estimate. Specifically, we construct two independently perturbed teacher-guided consistency paths, whose student endpoint predictions provide a consensus target; the discrepancy between the student's direct prediction and this target is the uncertainty proxy. We then relax the consistency penalty on high-uncertainty regions through an exponential weight, while keeping the full penalty elsewhere, since the student cannot be expected to match targets that are hard to learn. To preserve perceptual quality under aggressive step reduction, we integrate feature-space adversarial training with semantic alignment. With parameter-efficient LoRA adaptation of the 50-step Wan model, our method achieves state-of-the-art 4-step generation on VBench 2.0 (0.556 mean score) and is preferred over competing methods in a user study.

cs.CV↗

arXiv2026

Voxel-based block variational quantum linear solver: a hybrid quantum-classical method for static analysis of solids

In solid mechanics, finite element discretization of large-scale static problems produces large sparse linear systems whose solution requires substantial computation time and memory. The variational quantum linear solver (VQLS) offers a hybrid quantum-classical route, but its use in quantum finite element analysis is limited by the decomposition of nonunitary matrices, barren plateaus, and the measurement cost of expectation values. We propose a voxel-based block variational quantum linear solver (Voxel-BVQLS) that combines structured matrix decomposition, the principle of minimum potential energy, and batched quantum tests. First, we construct an LCU decomposition of the stiffness matrix from the recursive block-banded structure of voxel-grid finite element matrices, with the number of unitary terms bounded independently of the problem size. Second, we optimize the ansatz parameters using a minimum-potential-energy objective in place of a conventional VQLS loss function, thereby mitigating barren plateaus in the studied problems. Third, we introduce a block-Hadamard test whose circuit directly estimates weighted sums of multiple inner products, reducing the number of circuit configurations required per iteration. We assessed the proposed method in noiseless classical simulations using three examples. These examples show that the method reduces both the number of unitary terms in the LCU decomposition and the number of iterations required to converge, while still yielding solutions of finite accuracy. Voxel-BVQLS thus provides a structured hybrid quantum-classical framework for quantum finite element analysis on regular grids.

cs.CE↗

arXiv2026

Vibrational spectroscopy identifies the bond asymmetry of hexagonal diamond

Bulk hexagonal diamond has been synthesized by independent routes, but its structure remains contested: the two recent refinements disagree even on the sign of the difference between its two inequivalent bond lengths, 238~mÅ apart, and both depart from an earlier 2003 refinement. Here we test the competing structures with first-principles lattice dynamics. Relaxed hexagonal diamond has an interlayer bond \emph{longer} than the intralayer bonds by 24~mÅ in both functionals, an effect of its eclipsed conformation that scales with polytype hexagonality. The bright zone-center $A_{1g}$ mode gauges the interlayer bond at $\approx\!-2{,}100$~\icm~Å$^{-1}$, and neither refined coordinate reproduces the full pattern of measured modes. The only structure matching the twinned sample's three bands requires tens-of-gigapascals confining stress and lattice constants excluded by its own diffraction. Raman spectroscopy and diffraction jointly select a small positive bond asymmetry: inverting the spectrum of the phase-pure sample gives $\OB-\OA=24\pm3$~mÅ (95\% interval), and two determinations on separate samples give $+33\pm8$ and $+60\pm45$~mÅ. The 1{,}529~\icm{} feature cannot be assigned to homogeneous ideal 2H diamond, and the local HRTEM observation remains an open puzzle.

cond-mat.mtrl-sci↗

arXiv2026

Partition Scores Are Not System Scores: Deployment-Fidelity Gaps in Decomposed Algorithm Selection

Oracle-style quantities, including virtual best solvers, selected-portfolio VBS, virtual-best encodings, and best-in-family summaries, are widely reported as upper bounds on what a deployable selector could achieve. In decomposed algorithm selection, an analogous partition-level score grants an oracle choice of the best algorithm within the selected family; once the family selector is fixed, the deployable system must replace that within-family oracle with a learned within-family selector. We define the deployment-fidelity gap G(R) as the difference between partition-level and deployable end-to-end utility and derive two accounting consequences: a per-instance margin-regret stability condition that tells us when a partition-time family choice is deployment-optimal, and a sharp partition-only identification interval that, when it strictly crosses zero, prevents the partition-level report from certifying the deployable winner. Across five public algorithm-selection benchmarks spanning tabular AutoML and combinatorial CSP/SAT, every decomposed pipeline has positive G(R), ranging from 0.012 on TabZilla to 0.13 on PROTEUS-2014. Four of ten decomposed-versus-flat decisions have sign-changing point estimates; on PROTEUS-2014, a 33-point partition advantage shrinks to a 20-point end-to-end advantage. A training-side validation gap-correction diagnostic recovers the point-estimate deployable sign on all four sign-changing cells; it is a reporting aid, not a substitute for direct end-to-end evaluation. Partition and end-to-end scores should be reported side by side.

cs.AI↗

arXiv2026

The Optimal Coefficients-Based Criterion for Primitive Quadratic Polynomials over Finite Fields

Let $\mathbb F_q$ be a finite field and consider quadratic polynomials $f(X)=X^2+bX+c\in\mathbb F_q[X]$ with primitive constant term $c$. We construct an optimal coefficients-based determining polynomial for primitive quadratic polynomials over every finite field. More precisely, for every primitive $c\in\mathbb F_q$, its specialization is the unique monic square-free polynomial whose roots are exactly the coefficients $b$ for which $X^2+bX+c$ is primitive. The construction is based on Lucas polynomials and Lucas atoms over finite fields. In odd characteristic, the optimal determining polynomial is the $(q+1)$-st Lucas atom. In characteristic $2$, this Lucas atom has multiplicity $2$ in the variable $B$, and the optimal polynomial is obtained by removing this multiplicity through the inverse Frobenius over $A=\mathbb F_q[C]/(Φ_{q-1}(C))$. We prove that the determining polynomial is unique for each fixed primitive constant term and, globally, unique as an element of $A[B]$. We also give equivalent criteria involving only recursively computable Lucas polynomials and, as an application, a first-zero coefficient description of the binomial order and order of an irreducible quadratic polynomial.

math.NT↗

arXiv2026

Learning and retrieval for warm-starting charge-self-consistent DFT+DMFT

Charge-self-consistent (CSC) DFT+DMFT delivers quantitative correlated-electron physics one configuration at a time, making ensemble sampling dependent on reliable warm starts for its expensive fixed-point iteration. Here we compare retrieval from the most similar converged structure with an E(3)-equivariant model that predicts a physics-structured self-energy and Fermi level. Because the full CSC loop refines both initializations, every production result remains a converged DFT+DMFT solution. Across metallic Fe, correlated FeO, and Mott-insulating NiO, learning reduces the typical iterations to sustained convergence from 8 to 3, 8 to 3, and 6 to 1. Retrieval matches this median speed when a dense same-state archive is available, but several poor transplants reveal that structural similarity does not predict transplant quality. In a pre-registered volume window excluded from training and donor pools, learning retains its speed, whereas retrieval starts $3.4\times$ farther from the fixed point and approaches cold-start cost. Either initialization can select a distinct near-degenerate branch on a rugged CSC landscape. Applied end to end, the workflow generates over one thousand correlated energy and force labels for iron at Earth's-core conditions and trains an equivariant interatomic potential. Solid--liquid coexistence with 9216 atoms gives $T_m = 6225 \pm 42\,\mathrm{K}_{\mathrm{stat}}$ at 330~GPa, consistent with recent experiments. A 50-configuration DFT+DMFT audit resolves the potential's energy calibration but does not justify a corrected melting temperature, because four-atom cells cannot realize a liquid. The resulting regime map favors retrieval within dense coverage, amortized learning at and beyond its boundary, and solver refinement throughout.

cond-mat.mtrl-sci↗

arXiv2026

Explicit Representatives and Sizes of Cyclotomic Cosets II: Cyclotomic Systems and Applications to Constacyclic Codes

Let $q=p^e$ be a prime power, let $\ell$ be a prime with $\gcd(\ell,qn)=1$, and let $n$ be a positive integer. While $q$-cyclotomic cosets are usually considered modulo a fixed integer, we consider their behavior along the sequence $n,\ell n,\ell^2n,\ldots$ of moduli. To describe the resulting compatibility among cyclotomic cosets, we introduce the $\ell$-adic $q$-cyclotomic system with base module $n$, defined as the projective limit of the spaces of $q$-cyclotomic cosets modulo $\ell^i n$. We associate to each $q$-cyclotomic coset modulo $n$ a cyclotomic $\ell$-adic integer and use its $\ell$-adic expansion to describe all compatible liftings through the successive levels. This leads to a classification of the corresponding sequences according to their splitting and stable behaviors, together with explicit formulas for the representatives and sizes of their components. We treat separately the cases of odd $\ell$ and $\ell=2$. These results are then used to determine representatives and sizes of $q$-cyclotomic cosets for arbitrary admissible parameters. As an application, we obtain explicit irreducible factorizations of binomials over finite fields and use them to describe the corresponding constacyclic codes.

math.NT↗

arXiv2026

Can Video Diffusion Models Predict Past Frames? Bidirectional Cycle Consistency for Reversible Interpolation

Video frame interpolation aims to synthesize realistic intermediate frames between given endpoints while adhering to specific motion semantics. While recent generative models have improved visual fidelity, they predominantly operate in a unidirectional manner, lacking mechanisms to self-verify temporal consistency. This often leads to motion drift, directional ambiguity, and boundary misalignment, especially in long-range sequences. Inspired by the principle of temporal cycle-consistency in self-supervised learning, we propose a novel bidirectional framework that enforces symmetry between forward and backward generation trajectories. Our approach introduces learnable directional tokens to explicitly condition a shared backbone on temporal orientation, enabling the model to jointly optimize forward synthesis and backward reconstruction within a single unified architecture. This cycle-consistent supervision acts as a powerful regularizer, ensuring that generated motion paths are logically reversible. Furthermore, we employ a curriculum learning strategy that progressively trains the model from short to long sequences, stabilizing dynamics across varying durations. Crucially, our cyclic constraints are applied only during training; inference requires a single forward pass, maintaining the high efficiency of the base model. Extensive experiments show that our method achieves state-of-the-art performance in imaging quality, motion smoothness, and dynamic control on both 37-frame and 73-frame tasks, outperforming strong baselines while incurring no additional computational overhead.

cs.CV↗

arXiv2026

Before Thinking, Learn to Decide: Proactive Routing for Efficient Visual Reasoning

Large multimodal models have achieved strong reasoning on complex visual tasks, but their inference efficiency is often restricted by long chains of thought. A promising solution is to pair a small draft model with a large target model, enabling cooperative inference employing a routing signal that adaptively routes queries to either the draft or target model based on their difficulties for optimal efficiency and accuracy. Yet, the remaining bottleneck is to establish a reliable query difficulty signal under multimodal settings. Existing approaches designed for language models either rely on post-hoc token probabilities, which fall short in multimodal scenarios, or depend on supervised fine-tuning, which is a data-sensitive strategy. Both paradigms perform routing only after a complete output, and ignore whether the target model can actually solve the routed instances. To address this, we propose PRP, a Proactive Routing Paradigm that enables early decision-making by jointly evaluating the competence of both the draft and target models. Our Draft Rating Learning (DRL) equips the draft model with an internal confidence estimator, while Joint Rating Learning (JRL) predicts how well the target model can handle a given query, thereby prioritizing the allocation of samples it excels at rather than the hardest ones. These ratings enable fine-grained, instance-level \textbf{Proactive Routing} and substantially accelerate inference without compromising overall performance. Extensive experiments across multiple multimodal reasoning benchmarks validate our effectiveness and efficiency.

cs.CL↗