Tootfinder

Explicit lower bound for large gaps between some consecutive primes
Keiju Sono
https://arxiv.org/abs/2404.06951 https://arxiv.org/pdf…

Explicit lower bound for large gaps between some consecutive primes
Let $p_{n}$ denote the $n$th prime and for any fixed positive integer $k$ and $X\geq 2$, put \[ G_{k}(X):=\max _{p _{n+k}\leq X} \min \{ p_{n+1}-p_{n}, \ldots , p_{n+k}-p_{n+k-1} \}. \] Ford, Maynard and Tao proved that there exists an effective abosolute constant $c_{LG}>0$ such that \[ G_{k}(X)\geq \frac{c_{LG}}{k^{2}}\frac{\log X \log \log X \log \log \log \log X}{\log \log \log X} \] holds for any sufficiently large $X$. The main purpose of this paper is to determine the constant $c_{LG}$ a…

Localizations of the category of $A_\infty$ categories and internal Homs over a ring
Alberto Canonaco, Mattia Ornaghi, Paolo Stellari
https://arxiv.org/abs/2404.06610

Localizations of the category of $A_\infty$ categories and internal Homs over a ring
We show that, over an arbitrary commutative ring, the localizations of the categories of dg categories, of unital and of strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. The same result is also proved at the $\infty$-categorical level in the strictly unital case. As an application, we provide a new proof of the existence of internal Homs for the homotopy category of dg categories in terms of the category of unital $A_\inft…

LLM4Decompile: Decompiling Binary Code with Large Language Models
Hanzhuo Tan, Qi Luo, Jing Li, Yuqun Zhang
https://arxiv.org/abs/2403.05286 https://

LLM4Decompile: Decompiling Binary Code with Large Language Models
Decompilation aims to restore compiled code to human-readable source code, but struggles with details like names and structure. Large language models (LLMs) show promise for programming tasks, motivating their application to decompilation. However, there does not exist any open-source LLM for decompilation. Moreover, existing decompilation evaluation systems mainly consider token-level accuracy and largely ignore code executability, which is the most important feature of any program. Therefore,…

Proof complexity of universal algebra in a CSP dichotomy proof
Azza Gaysin
https://arxiv.org/abs/2403.06704 https://arxiv.org/pdf/240…

Proof complexity of universal algebra in a CSP dichotomy proof
The constraint satisfaction problem (CSP) can be formulated as a homomorphism problem between relational structures: given a structure $\mathcal{A}$, for any structure $\mathcal{X}$, whether there exists a homomorphism from $\mathcal{X}$ to $\mathcal{A}$. For years, it has been conjectured that all problems of this type are divided into polynomial-time and NP-complete problems, and the conjecture was proved in 2017 separately by Zhuk (2017) and Bulatov (2017). Zhuk's algorithm solves tractable …

ROUGE-K: Do Your Summaries Have Keywords?
Sotaro Takeshita, Simone Paolo Ponzetto, Kai Eckert
https://arxiv.org/abs/2403.05186 https://

ROUGE-K: Do Your Summaries Have Keywords?
Keywords, that is, content-relevant words in summaries play an important role in efficient information conveyance, making it critical to assess if system-generated summaries contain such informative words during evaluation. However, existing evaluation metrics for extreme summarization models do not pay explicit attention to keywords in summaries, leaving developers ignorant of their presence. To address this issue, we present a keyword-oriented evaluation metric, dubbed ROUGE-K, which provides…

Ring class fields and a result of Hasse
R. Evans, M. van Veen, F. Lemmermeyer
https://arxiv.org/abs/2403.04986 https://arxiv.org/pdf/…

Ring class fields and a result of Hasse
For squarefree $d>1$, let $M$ denote the ring class field for the order $Z[\sqrt{-3d}]$ in $F=Q(\sqrt{-3d})$. Hasse proved that $3$ divides the class number of $F$ if and only if there exists a cubic extension $E$ of $Q$ such that $E$ and $F$ have the same discriminant. Define the real cube roots $v=(a+b\sqrt{d})^{1/3}$ and $v'=(a-b\sqrt{d})^{1/3}$, where $a+b\sqrt{d}$ is the fundamental unit in $Q(\sqrt{d})$. We prove that $E$ can be taken as $Q(v+v')$ if and only if $v \in M$. As byproducts o…

WatChat: Explaining perplexing programs by debugging mental models
Kartik Chandra, Tzu-Mao Li, Rachit Nigam, Joshua Tenenbaum, Jonathan Ragan-Kelley
https://arxiv.org/abs/2403.05334

WatChat: Explaining perplexing programs by debugging mental models
Often, a good explanation for a program's unexpected behavior is a bug in the programmer's code. But sometimes, an even better explanation is a bug in the programmer's mental model of the language they are using. Instead of merely debugging our current code ("giving the programmer a fish"), what if our tools could directly debug our mental models ("teaching the programmer to fish")? In this paper, we apply ideas from computational cognitive science to do exactly that. Given a perplexing program…

ROUGE-K: Do Your Summaries Have Keywords?
Sotaro Takeshita, Simone Paolo Ponzetto, Kai Eckert
https://arxiv.org/abs/2403.05186 https://

ROUGE-K: Do Your Summaries Have Keywords?
Keywords, that is, content-relevant words in summaries play an important role in efficient information conveyance, making it critical to assess if system-generated summaries contain such informative words during evaluation. However, existing evaluation metrics for extreme summarization models do not pay explicit attention to keywords in summaries, leaving developers ignorant of their presence. To address this issue, we present a keyword-oriented evaluation metric, dubbed ROUGE-K, which provides…

A converse of dynamical Mordell--Lang conjecture in positive characteristic
Jungin Lee, Gyeonghyeon Nam
https://arxiv.org/abs/2403.05107 https://

A converse of dynamical Mordell--Lang conjecture in positive characteristic
In this paper, we prove the converse of the dynamical Mordell--Lang conjecture in positive characteristic: For every subset $S \subseteq \mathbb{N}_0$ which is a union of finitely many arithmetic progressions along with finitely many $p$-sets of the form $\left \{ \sum_{j=1}^{m} c_j p^{k_jn_j} : n_j \in \mathbb{N}_0 \right \}$ ($c_j \in \mathbb{Q}$, $k_j \in \mathbb{N}_0$), there exist a split torus $X = \mathbb{G}_m^k$ defined over $K=\overline{\mathbb{F}_p}(t)$, an endomorphism $Φ$ of $X$, $…

The Prime Geodesic Theorem for the Picard Manifold
Ikuya Kaneko
https://arxiv.org/abs/2403.06626 https://arxiv.org/pdf/2403.06626

The Prime Geodesic Theorem for the Picard Manifold
We prove the prime geodesic theorem for the Picard manifold whose error term shrinks as the hybrid subconvex exponent $θ$ for quadratic Dirichlet $L$-functions over Gaussian integers decreases. Our rate of decay is faster than that of Balkanova-Frolenkov (2022) and Kaneko (2022), and leads to the exponent $\frac{202}{139}+\varepsilon$ in the prime geodesic theorem under the generalised Lindelöf hypothesis $θ= 0$, improving upon the existing conditional results. This provides further solid ev…