arXiv subjects

Huimin Zheng

Publications and source records attributed to Huimin Zheng.

8 recordsLinked to original sources

arXiv2026

A two-step supercongruence for an Apéry-like sequence

Let $G_n=\sum_{k=0}^n4^k\binom{2n-2k}{n-k}^{2}\binom{2k}{k}$. Zhi-Hong Sun conjectured that, for primes $p\equiv3\pmod4$, positive odd integers $m$, and $r\ge2$, the two-step congruence $G_{(mp^r-1)/2}\equiv p^2G_{(mp^{r-2}-1)/2}\pmod {p^{2r-1}}$ holds. We prove the stronger valuation statement $G_{(p^2M-1)/2}-p^2G_{(M-1)/2}\in p^{2v_p(M)+3}\mathbb Z_p$ for every positive odd $M$. The proof converts the sum to a terminating ${}_3F_2$, constructs a cancelled digit-transfer operator, and identifies a two-dimensional analytic quotient of its cubic difference operator. The quotient operator has characteristic polynomial $X^2-p^2$; its second trace is evaluated through the Gross--Koblitz formula and Greene's finite-field Dixon identity. A logarithmic-loss Green inverse converts this spectral identity into an actual integral analytic primitive. The exceptional prime $3$ requires a finite exact PARI/GP certificate, while the infinite tail is bounded symbolically. Thus the computation is finite, reproducible, and separated from the uniform part of the proof.

math.NT↗

arXiv2026

Mahler measures at interior CM points: proofs of two conjectures of Samart

We prove two conjectures of Samart as identities of genuine Mahler measures. The first is the case $k=1$ of his evaluations for the family $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k^{1/2}$: $m((x+x^{-1})(y+y^{-1})(z+z^{-1})+1)=4L'(g_7,0)$, where $g_7(τ)=η(τ)^3η(7τ)^3$ is the unique newform of $S_3(Γ_0(7),χ_{-7})$ (LMFDB label 7.3.b.a). The second is the conjugate pair of quadratic CM entries of his 2015 table: $n_2((47\pm 45\sqrt{-7})/2)=(4/7)(54L'(g_7,0)+L'(χ_{-7},-1))$, where $n_2(s):=2m((x+x^{-1})(y+y^{-1})(z+z^{-1})+\sqrt{s})$. In both cases the parameter lies inside (or on the boundary of) the critical locus, so the passage from the holomorphic (modified) Mahler measure---where Samart computed the $L$-value side conditionally, and Fei the real-part level---to the true Mahler measure was open. For the first theorem we close the gap by a differential-comparison continuation of the Mahler differential along an explicitly certified path, followed by a continuity argument at the interior point. For the second theorem no continuation is needed: the second preimage of the parameter under the modular parametrization is the Fricke partner of Samart's CM point and lies in his proved region, and the evaluation is an exact CM lattice-sum computation. The analytic and arithmetic inputs not proved here are stated explicitly with references; the finite numerical inequalities used in the continuation argument are certified by interval arithmetic, and all identities are confirmed numerically to 41--60 digits.

math.NT↗

arXiv2026

Samart's conjecture n_4(81)=40M_7: the exact CM evaluation and the two obstructions. A status report

This note archives the status of Samart's Table-6 conjecture $n_4(81)=40M_7$, $M_7:=L'(g_7,0)$, where $g_7(τ)=η(τ)^3η(7τ)^3$ is the newform of $S_3(Γ_0(7),χ_{-7})$ and $n_4(s):=4m(x^4+y^4+z^4+1+s^{1/4}xyz)$. The conjecture is the cleanest of Samart's open interior-point entries (discriminant $-7$, class number $1$, a single $L$-value), and it was dropped as a theorem target in the companion paper, where the two $n_2$-family conjectures were proved. We record what is proved and precisely where those methods fail. First, a complete proof of the $L$-value side (P1): at the CM point $τ_2=(7+\sqrt{-7})/4$ the Eisenstein--Kronecker expression underlying Samart's formula evaluates exactly to $\mathrm{EK}_4(τ_2)=40M_7$, via lattice sums over the ring of integers of $\mathbb{Q}(\sqrt{-7})$ and the principal ideal $(\bar\varpi)$, with an exact cancellation of the parasitic $ζ_K(2)$-terms. Second, two quantitative obstructions to the remaining half $n_4(81)=\mathrm{EK}_4(τ_2)$: the critical image of the $n_4$-family is a two-dimensional astroid disc containing the parameter $c=3$ in its interior (in contrast to the one-dimensional slit $[0,64]$ of the $n_2$-family), so no continuation path can approach the CM point; and Samart's $U$-series converges on all of the upper half-plane but leaves the geometric sheet of the holomorphic Mahler measure everywhere below $\mathrm{Im}\,τ=1/\sqrt{2}$, so the premise of the differential-comparison continuation fails. A 20-digit direct torus integration then decides the conjecture numerically: $n_4(81)-40M_7=+0.0586706795972872...$, five orders of magnitude above the integration error floor, so the identity as literally stated is refuted; a closed form for the true value $n_4(81)$ remains open and appears to require regulator/monodromy machinery.

math.NT↗

arXiv2024

SleepCoT: A Lightweight Personalized Sleep Health Model via Chain-of-Thought Distillation

We present a novel approach to personalized sleep health management using few-shot Chain-of-Thought (CoT) distillation, enabling small-scale language models (> 2B parameters) to rival the performance of large language models (LLMs) in specialized health domains. Our method simultaneously distills problem-solving strategies, long-tail expert knowledge, and personalized recommendation capabilities from larger models into more efficient, compact models. Unlike existing systems, our approach offers three key functionalities: generating personalized sleep health recommendations, supporting user-specific follow-up inquiries, and providing responses to domain-specific knowledge questions. We focus on sleep health due to its measurability via wearable devices and its impact on overall well-being. Our experimental setup, involving GPT-4o for data synthesis, Qwen-max for instruction set creation, and Qwen2.5 1.5B for model distillation, demonstrates significant improvements over baseline small-scale models in penalization, reasoning, and knowledge application. Experiments using 100 simulated sleep reports and 1,000 domain-specific questions shows our model achieves comparable performance to larger models while maintaining efficiency for real-world deployment. This research not only advances AI-driven health management but also provides a novel approach to leveraging LLM capabilities in resource-constrained environments, potentially enhancing the accessibility of personalized healthcare solutions.

cs.AI↗

arXiv2024

NTIRE 2024 Challenge on Short-form UGC Video Quality Assessment: Methods and Results

This paper reviews the NTIRE 2024 Challenge on Shortform UGC Video Quality Assessment (S-UGC VQA), where various excellent solutions are submitted and evaluated on the collected dataset KVQ from popular short-form video platform, i.e., Kuaishou/Kwai Platform. The KVQ database is divided into three parts, including 2926 videos for training, 420 videos for validation, and 854 videos for testing. The purpose is to build new benchmarks and advance the development of S-UGC VQA. The competition had 200 participants and 13 teams submitted valid solutions for the final testing phase. The proposed solutions achieved state-of-the-art performances for S-UGC VQA. The project can be found at https://github.com/lixinustc/KVQChallenge-CVPR-NTIRE2024.

eess.IV↗

arXiv2023

SoulChat: Improving LLMs' Empathy, Listening, and Comfort Abilities through Fine-tuning with Multi-turn Empathy Conversations

Large language models (LLMs) have been widely applied in various fields due to their excellent capability for memorizing knowledge and chain of thought (CoT). When these language models are applied in the field of psychological counseling, they often rush to provide universal advice. However, when users seek psychological support, they need to gain empathy, trust, understanding and comfort, rather than just reasonable advice. To this end, we constructed a multi-turn empathetic conversation dataset of more than 2 million samples, in which the input is the multi-turn conversation context, and the target is empathetic responses that cover expressions such as questioning, comfort, recognition, listening, trust, emotional support, etc. Experiments have shown that the empathy ability of LLMs can be significantly enhanced when finetuning by using multi-turn dialogue history and responses that are closer to the expression of a psychological consultant.

cs.CL↗

arXiv2023

Pseudorandomness of Sato-Tate Distributions for Elliptic Curves

In this paper we propose conjectures that assert that, the sequence of Frobenius angles of a given elliptic curve over $\mathbf{Q}$ without complex multiplication is pseudorandom, in other words that the Frobenius angles are statistically independently distributed with respect to the Sato-Tate measure. Numerical evidences are presented to support the conjectures.

math.NT↗

arXiv2011

On Measurement and Computation

Inspired by the work of Feynman, Deutsch, We formally propose the theory of physical computability and accordingly, the physical complexity theory. To achieve this, a framework that can evaluate almost all forms of computation using various physical mechanisms is discussed. Here, we focus on using it to review the theory of Quantum Computation. As a preliminary study on more general problems, some examples of other physical mechanism are also given in this paper.

physics.comp-ph↗