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Yuchen Ding

Publications and source records attributed to Yuchen Ding.

At least 37 records · Page 2Linked to original sources

Note on shifted primes with large prime factors

We denote by $P^+(n)$ the largest prime factor of the integer $n$. In 1935, Erd\H os studied the quantity $T_c(x)$ defined by $$ T_c(x)=\big|\big\{p\le x: P^+(p-1)\ge p^c\big\}\big|, $$ and he proved $$ \limsup_{x\rightarrow \infty}\frac{T_c(x)}{π(x)}\rightarrow 0, \quad \text{as~}c\rightarrow 1. $$ Recently, Ding gave a quantitative form of Erd\H os' result, showing that $$ \limsup_{x\rightarrow \infty}\frac{T_c(x)}{π(x)}\le 8\big(c^{-1}-1\big). $$ holds for $8/9< c<1$. In this paper, we improve Ding's upper bound to $$ \limsup_{x\rightarrow \infty}\frac{T_c(x)}{π(x)}\le -\frac{7}{2}\log c $$ for $e^{-\frac{2}{7}}< c<1$.

math.NT↗

ERNIE 5.0 Technical Report

In this report, we introduce ERNIE 5.0, a natively autoregressive foundation model desinged for unified multimodal understanding and generation across text, image, video, and audio. All modalities are trained from scratch under a unified next-group-of-tokens prediction objective, based on an ultra-sparse mixture-of-experts (MoE) architecture with modality-agnostic expert routing. To address practical challenges in large-scale deployment under diverse resource constraints, ERNIE 5.0 adopts a novel elastic training paradigm. Within a single pre-training run, the model learns a family of sub-models with varying depths, expert capacities, and routing sparsity, enabling flexible trade-offs among performance, model size, and inference latency in memory- or time-constrained scenarios. Moreover, we systematically address the challenges of scaling reinforcement learning to unified foundation models, thereby guaranteeing efficient and stable post-training under ultra-sparse MoE architectures and diverse multimodal settings. Extensive experiments demonstrate that ERNIE 5.0 achieves strong and balanced performance across multiple modalities. To the best of our knowledge, among publicly disclosed models, ERNIE 5.0 represents the first production-scale realization of a trillion-parameter unified autoregressive model that supports both multimodal understanding and generation. To facilitate further research, we present detailed visualizations of modality-agnostic expert routing in the unified model, alongside comprehensive empirical analysis of elastic training, aiming to offer profound insights to the community.

cs.CL↗

Note on a conjecture of Sárközy on special sequences

Let $α>1$ be an irrational number and $k\ge 2$ a positive integer. Let $f(x)$ be a polynomial with positive integer coefficients. Solving a 2001 problem of Sárközy on special sequences, Hegyvári proved in 2003 that there exists an infinite sequence $A$ with density $\frac{1}{k}-\frac{1}{kα}$ such that $$ \big\{f(a_1)+\ldots+f(a_k): a_i\in A, 1\le i\le k\big\}\cap \big\{\lfloor nα\rfloor: n\in \mathbb{N}\big\}=\emptyset. $$ Hegyvári also proved that the density given by him is optimal for $k=2$. In this article, we show that the density $\frac{1}{k}-\frac{1}{kα}$ given by Hegyvári is actually optimal for all $k\ge 2$.

math.NT↗

Disc growth and vertical heating of lenticular galaxies in the Fornax cluster

We present a detailed analysis of the vertical and radial structure of mono-age stellar populations in three edge-on lenticular galaxies (FCC 153, FCC 170, and FCC 177) in the Fornax cluster, using deep MUSE observations. By measuring the half-mass radius (R$_{50}$) and half-mass height (z$_{50}$) across 1 Gyr-wide age bins, we trace the spatial evolution of stellar populations over cosmic time. All galaxies exhibit a remarkably constant disc thickness for all stars younger than ~6 Gyr, suggesting minimal secular heating and limited impact from environmental processes such as tidal shocking or harassment. Evidence of past mergers (8-10 Gyr ago) is found in the increase of z$_{50}$ for older populations. We find that accreted (metal-poor) stars have been deposited in quite thick configurations, but that the interactions only moderately thickened pre-existing stars in the galaxies, and only caused mild flaring in the outer regions of the discs. The radial structure of the discs varies across galaxies, but in all cases we find that the radial extent of mono-age populations remains constant or grows over the past 8 Gyr. This leads us to argue that within the radial range we consider, strangulation, rather than ram-pressure stripping, is the dominant quenching mechanism in those galaxies. Our results highlight the usefulness of analysing the structure of mono-age population to uncover the mechanisms driving galaxy evolution, and we anticipate broader insights from the GECKOS survey, studying 36 nearby edge-on disc galaxies.

astro-ph.GA↗

Note on a problem of Sárközy on multiplicative representation functions

Motivated by a 2001 problem of Sárközy, we classify all situations of the integers $b,c,e$ and $f$ satisfying \begin{align*} \limsup_{n\rightarrow\infty}|d(\mathcal{A},bn+c)-d(\mathcal{A},en+f)|=\infty \end{align*} for any infinite $\mathcal{A}\subset \mathbb{N}$, where $ d(\mathcal{A},m)=\#\{a\in \mathcal{A}:a|m\}. $

math.NT↗

Primes of the form $ax+by$ in certain intervals with small solutions

Let $1 0.005\cdot \frac{1}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*} Let $π_{\ell,a,b}^{*}$ be the number of primes $p\leq g_{\ell,a,b}$ having at most $\ell$ solutions for (1). For an integer $a\ge 3$ and a large sufficiently integer $b$ with $\gcd(a,b)=1$, we also prove that $$ π^{*}_{\ell,a,b}>\frac{(2\ell+1)a}{2(\ell a+a-1)}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. $$ Moreover, if $\ell \frac{\ell+0.02}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*} These results generalize the previous ones of Chen and Zhu (2025), who established the results for the case $\ell=0$.

math.NT↗

The Diophantine Frobenius Problem revisited

Let $k\ge 2$ and $a_1, a_2, \cdots, a_k$ be positive integers with \[ \gcd(a_1, a_2, \cdots, a_k)=1. \] It is proved that there exists a positive integer $G_{a_1, a_2, \cdots, a_k}$ such that every integer $n$ strictly greater than it can be represented as the form \[ n=a_1x_1+a_2x_2+\cdots+a_kx_k, \quad (x_1, x_2, \cdots, x_k\in\mathbb{Z}_{\ge 0},~\gcd(x_1, x_2, \cdots, x_k)=1). \] We then investigate the size of $G_{a_1, a_2}$ explicitly. Our result strengthens the primality requirement of $x$'s in the classical Diophantine Frobenius Problem.

math.NT↗

Note on a theorem of Birch and Erdős

Let $p,q>1$ be two relatively prime integers and $\mathbb{N}$ the set of nonnegative integers. Let $f_{p,q}(n)$ be the number of different expressions of $n$ written as a sum of distinct terms taken from $\{p^αq^β:α,β\in \mathbb{N}\}$. Erd\H os conjectured and then Birch proved that $f_{p,q}(n)\ge 1$ provided that $n$ is sufficiently large. In this note, for all sufficiently large number $n$ we prove $$ f_{p,q}(n)=2^{\frac{(\log n)^2}{2\log p\log q}\big(1+O(\log\log n/\log n)\big)}. $$ We also show that $\lim_{n\rightarrow\infty}f_{2,q}(n+1)/f_{2,q}(n)=1.$ Additionally, we will point out the relations between $f_{2,q}(n)$ and $m$-ary partitions.

math.NT↗

A Comprehensive Study of Exploitable Patterns in Smart Contracts: From Vulnerability to Defense

With the rapid advancement of blockchain technology, smart contracts have enabled the implementation of increasingly complex functionalities. However, ensuring the security of smart contracts remains a persistent challenge across the stages of development, compilation, and execution. Vulnerabilities within smart contracts not only undermine the security of individual applications but also pose significant risks to the broader blockchain ecosystem, as demonstrated by the growing frequency of attacks since 2016, resulting in substantial financial losses. This paper provides a comprehensive analysis of key security risks in Ethereum smart contracts, specifically those written in Solidity and executed on the Ethereum Virtual Machine (EVM). We focus on two prevalent and critical vulnerability types (reentrancy and integer overflow) by examining their underlying mechanisms, replicating attack scenarios, and assessing effective countermeasures.

cs.CR↗

Note on a conjecture of Ram\'ırez Alfons\'ın and Skałba

Let $2< a<b$ be two relatively prime integers and $g=ab-a-b$. It is proved that there exists at least one prime $p\le g$ of the form $p=ax+by~(x,y\in \mathbb{Z}_{\ge 0})$, which confirms a 2020 conjecture of Ram\'ırez Alfons\'ın and Skałba.

math.NT↗

A generalization of the Romanoff theorem

Let $\mathcal{P}$ be the set of primes and $\mathbb{N}$ the set of positive integers. Let also $r_1,...,r_t$ be positive real numbers and $R_2(r_1,...,r_t)$ the set of odd integers which can be represented as $$ p+2^{\lfloor k_1^{r_1}\rfloor}+\cdot\cdot\cdot+2^{\lfloor k_t^{r_t}\rfloor}, $$ where $p\in \mathcal{P}$ and $k_1,...,k_t\in\mathbb{N}$. Recently, Chen and Xu proved that the set $R_2(r_1,...,r_t)$ has positive lower asymptotic density, provided that $r_1^{-1}+\cdot\cdot\cdot+r_t^{-1}\ge 1$ and at least one of $r_1,...,r_t$ is an integer. Their result reduces to the famous theorem of Romanoff by taking $t=r_1=1.$ In this note, we remove the unnecessary condition that `{\it at least one of $r_1,...,r_t$ is an integer}'.

math.NT↗

Privacy-Preserving Orthogonal Aggregation for Guaranteeing Gender Fairness in Federated Recommendation

Under stringent privacy constraints, whether federated recommendation systems can achieve group fairness remains an inadequately explored question. Taking gender fairness as a representative issue, we identify three phenomena in federated recommendation systems: performance difference, data imbalance, and preference disparity. We discover that the state-of-the-art methods only focus on the first phenomenon. Consequently, their imposition of inappropriate fairness constraints detrimentally affects the model training. Moreover, due to insufficient sensitive attribute protection of existing works, we can infer the gender of all users with 99.90% accuracy even with the addition of maximal noise. In this work, we propose Privacy-Preserving Orthogonal Aggregation (PPOA), which employs the secure aggregation scheme and quantization technique, to prevent the suppression of minority groups by the majority and preserve the distinct preferences for better group fairness. PPOA can assist different groups in obtaining their respective model aggregation results through a designed orthogonal mapping while keeping their attributes private. Experimental results on three real-world datasets demonstrate that PPOA enhances recommendation effectiveness for both females and males by up to 8.25% and 6.36%, respectively, with a maximum overall improvement of 7.30%, and achieves optimal fairness in most cases. Extensive ablation experiments and visualizations indicate that PPOA successfully maintains preferences for different gender groups.

cs.LG↗

On a problem of Nathanson on non-minimal additive complements

Let $C$ and $W$ be two sets of integers. If $C+W=\mathbb{Z}$, then $C$ is called an additive complement to $W$. We further call $C$ a minimal additive complement to $W$ if no proper subset of $C$ is an additive complement to $W$. Answering a problem of Nathanson in part, we give sufficient conditions of $W$ which has no minimal additive complements. Our result also extends the prior result of Chen and Yang.

math.NT↗

The effects of environment on galaxies' dynamical structures: From simulations to observations

We studied the effects of cluster environments on galactic structures by using the TNG50 cosmological simulation and observed galaxies in the Fornax cluster. We focused on galaxies with stellar masses of $10^{8-12}M_{\odot}$ at z=0 that reside in Fornax-like clusters with total masses of $M_{200c} = 10^{13.4-14.3}M_{\odot}$. We characterized the stellar structures by decomposing each galaxy into a dynamically cold disk and a hot non-disk component, and studied the evolution of both the stellar and gaseous constituents. In TNG50, we find that the cold gas is quickly removed when a galaxy falls into a Fornax-mass cluster. About 87\% of the galaxies have lost $80\%$ of their star-forming gas at 4 billion years after infall, with the remaining gas concentrating in the inner regions of the galaxy. The radius of the star-forming gaseous disk decreases to half its original size at 4 billion years after infall for 66\% of the galaxies. As a result, star formation in the extended dynamically cold disk sharply decreases, even though a low level of SF persists at the center for a few additional gigayears. This leads to a tight correlation between the average stellar age in the dynamically cold disk and the infall time of galaxies. Furthermore, the luminosity fraction of the dynamically cold disk in ancient infallers is only about 1/3 of that in recent infallers, controlling for galaxy stellar mass. This quantitatively agrees with what is observed in early-type galaxies in the Fornax cluster. Gas removal stops the possible growth of the disk, with gas removed earlier in galaxies that fell in earlier, and hence the cold-disk fraction is correlated with the infall time. The stellar disk can be significantly disrupted by tidal forces after infall, through a long-term process that enhances the difference among cluster galaxies with different infall times.

astro-ph.GA↗

On a conjecture on shifted primes with large prime factors, II

Let $\mathcal{P}$ be the set of primes and $π(x)$ the number of primes not exceeding $x$. Let also $P^+(n)$ be the largest prime factor of $n$ with convention $P^+(1)=1$ and $$ T_c(x)=\#\left\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\right\}. $$ Motivated by a 2017 conjecture of Chen and Chen, we show that for any $8/9\le c<1$ $$ \limsup_{x\rightarrow\infty}T_c(x)/π(x)\le 8(1/c-1), $$ which clearly means that $$ \limsup_{x\rightarrow\infty}T_c(x)/π(x)\rightarrow 0, \quad \text{as}~c\rightarrow1. $$

math.NT↗

Solutions to a Romanoff type problem

We answer negatively a 2014 problem of Yang and Chen on Romanoff type representations. Sharp results involving their problem were also obtained in this article.

math.NT↗