arXiv subjects

Hu Tan

Publications and source records attributed to Hu Tan.

6 recordsLinked to original sources

arXiv2026

Sharp Diameter Bounds for Nonnegative Cyclotomic Multiples

Let $N\ge2$ and let $p$ be its least prime divisor. We prove that every nonzero polynomial with nonnegative real coefficients divisible by $\Phi_N$ has support diameter at least $(p-1)N/p$. Equality holds precisely for positive scalar multiples of monomial shifts of the $p$-term geometric sum $\sum_{j=0}^{p-1} X^{jN/p}$, thereby proving a conjecture of Steinberger. The proof turns cyclotomic divisibility into the vanishing of the first $p-1$ Fourier moments of a positive measure on the circle and then applies a classical extremal trigonometric polynomial. As a consequence, we establish the Coven--Meyerowitz diameter bound under their tiling conditions and determine its equality cases. Longer initial intervals of vanishing Fourier coefficients yield stronger diameter bounds, including an explicit refinement in terms of the prime-power divisor sets. The extremal trigonometric polynomial also yields a quantitative concentration estimate for measures and cyclotomic multiples with near-minimal support diameter.

math.NT↗

arXiv2026

Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings

Cyclotomic divisibility imposes strong prime-power structure on integer tiles. We study unsupported cyclotomic divisors: mixed-order divisors for which none of the prime-power components of the order divides the mask, although every prime in the order divides the tile cardinality. Kiss, \L aba, Marshall and Somlai asked whether such a phenomenon can occur in the three-prime setting. We prove that unsupported cyclotomic divisors already occur for periods with three distinct prime factors. For primes $p<q<r$, we characterize the square-period case: an unsupported factor $\Phi_{pqr}$ occurs in a tiling of $\ZZ_{(pqr)^2}$ if and only if $r\in\langle p,q\rangle$, and every such tile lies in a single residue class modulo $r$. Among cyclic tilings with the unsupported order dividing the specified modulus, the smallest modulus is $180$; if the order has three distinct prime factors, it is $900$. An Ap\'ery-set construction gives examples for every triple at period $p^2q^2r^3$.The proof of our results combines Fourier rigidity, a three-cylinder decomposition, and an integer mass obstruction.

math.NT↗

arXiv2026

A Proof of Fraenkel's Conjecture

Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density at least 1/3. After reducing the partition to primitive common-period data, Fourier cancellation produces a finite inverse-sine system. We prove that no such system can exist when every density is below 1/3. The proof combines a divisor-concentration identity with uniform analytic estimates and three exact finite verifications, all carried out with integer or rational arithmetic. The component supplied by the density bound has mean spacing at most three. Deleting it preserves balance, and every surviving periodic balanced set is again a rational Beatty set. Induction determines the surviving binary scales, while a two-sequence disjointness criterion forces the deleted density to be the next binary scale. This yields the asserted density pattern.

math.NT↗

arXiv2026

Beyond Neural Collapse: Task-Intrinsic Geometry Governs Neural Representations in Modular Arithmetic

While neural collapse (NC) predicts that a $K$-class-balanced classifier should organize terminal representations as a $(K-1)$-dimensional simplex equiangular tight frame (ETF), modular addition consistently enters a different regime: networks compress to a two-dimensional cyclic geometry in which both classifier weights and token embeddings lie on circles. We refine the explanation of this phenomenon in three directions. First, we formalize a layerwise non-uniform training mechanism: downstream classifier weights are driven by dense cross-entropy gradients into a rank-2 equiangular configuration before upstream embeddings fully reorganize, and once this classifier plane forms, backpropagated feature gradients constrain embedding motion to the same plane while weight decay suppresses orthogonal components. Second, after this subspace locking, the induced in-plane dynamics admit an entropy-regularized transport interpretation on $S^1$; combined with modular-addition labels, this reduces embedding formation to phase alignment, whose minimizers are single-frequency characters of $\mathbb{Z}/P\mathbb{Z}$ and hence equal-angle points on a circle. Third, we quantify why this solution prevails over NC: a simplex ETF gains only an $O(1)$ advantage in cross-entropy, whereas the cyclic rank-2 solution enjoys a $\Theta(K)$ advantage under Schatten or weight-decay surrogates, yielding a critical threshold $\lambda_{\mathrm{crit}} = \Theta(1/K)$. Our results explain both why classifier weights move first and why embeddings subsequently align with them, showing that grokking on modular arithmetic is governed not by maximal separation alone but by a task-structured trade-off between separation, symmetry, and complexity.

cs.LG↗

arXiv2026

Deciphering Two Training Clocks in Grokking via Deep Linear Network Theory with Conditional ReLU Reduction

Grokking suggests that fitting the training data and learning a simple underlying rule may occur on different time scales. We formalize this phenomenon by separating the fast decay of the classification loss from the slower simplification of the learned representation, and we call the resulting pair of stopping times two training clocks. For deep linear networks, we show that a post-margin gap-growth or one-step tail-contraction condition reduces the cross-entropy loss to level epsilon on a logarithmic time scale. In contrast, when layerwise weight decay is present, the induced regularization on the end-to-end map can be expressed as a Schatten-type penalty; under a sharp late-time Kurdyka-Lojasiewicz tail, this structural energy closes on a polynomial time scale. The two clocks, therefore, separate fitting from representation simplification. We then explain how the same mechanism can appear in ReLU MLPs. In regions where the activation patterns on the training set remain fixed, the network reduces to a linear model in the active coordinates. In a two-layer ReLU embedding model, chain-rule estimates further show that the classifier head can receive larger effective gradients than the embedding block under controlled downstream norms. This supports a two-stage mechanism in which the classifier fits first, while the representation continues to simplify later. We use modular addition as the main experimental setting. The deep linear theory provides the rigorous core of the analysis. But the ReLU results are formulated as conditional reductions that account for empirical behavior without claiming a global proof for nonlinear training dynamics.

cs.LG↗

arXiv2025

Hybrid Time-Domain Behavior Model Based on Neural Differential Equations and RNNs

Nonlinear dynamics system identification is crucial for circuit emulation. Traditional continuous-time domain modeling approaches have limitations in fitting capability and computational efficiency when used for modeling circuit IPs and device behaviors.This paper presents a novel continuous-time domain hybrid modeling paradigm. It integrates neural network differential models with recurrent neural networks (RNNs), creating NODE-RNN and NCDE-RNN models based on neural ordinary differential equations (NODE) and neural controlled differential equations (NCDE), respectively.Theoretical analysis shows that this hybrid model has mathematical advantages in event-driven dynamic mutation response and gradient propagation stability. Validation using real data from PIN diodes in high-power microwave environments shows NCDE-RNN improves fitting accuracy by 33\% over traditional NCDE, and NODE-RNN by 24\% over CTRNN, especially in capturing nonlinear memory effects.The model has been successfully deployed in Verilog-A and validated through circuit emulation, confirming its compatibility with existing platforms and practical value.This hybrid dynamics paradigm, by restructuring the neural differential equation solution path, offers new ideas for high-precision circuit time-domain modeling and is significant for complex nonlinear circuit system modeling.

cs.LG↗