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David Gao

Publications and source records attributed to David Gao.

25 records · Page 2Linked to original sources
arXiv2024

Sofic actions on graphs

We develop a theory of soficity for actions on graphs and obtain new applications to the study of sofic groups. We establish various examples, stability and permanence properties of sofic actions on graphs, in particular soficity is preserved by taking several natural graph join operations. We prove that an action of a group on its Cayley graph is sofic if and only if the group is sofic. We show that arbitrary actions of amenable groups on graphs are sofic. Using a graph theoretic result of E. Hrushovski, we also show that arbitrary actions of free groups on graphs are sofic. Notably we show that arbitrary actions of sofic groups on graphs, with amenable stabilizers, are sofic, settling completely an open problem from \cite{gao2024soficity}. We also show that soficity is preserved by taking limits under a natural Gromov-Hausdorff topology, generalizing prior work of the first author \cite{gao2024actionslerfgroupssets}. Our work sheds light on a family of groups called graph wreath products, simultaneously generalizing graph products and generalized wreath products. Extending various prior results in this direction including soficity of generalized wreath products \cite{gao2024soficity}, B. Hayes and A. Sale \cite{HayesSale}, and soficity of graph products \cite{CHR, charlesworth2021matrix}, we show that graph wreath products are sofic if the action and acting groups are sofic. These results provide several new examples of sofic groups in a systematic manner.

math.GR↗
arXiv2024

On soficity for certain fundamental groups of graphs of groups

In this note we study a family of graphs of groups over arbitrary base graphs where all vertex groups are isomorphic to a fixed countable sofic group $G$, and all edge groups $H<G$ are such that the embeddings of $H$ into $G$ are identical everywhere. We prove soficity for this family of groups under a flexible technical hypothesis for $H$ called $σ$-co-sofic. This proves soficity for group doubles $*_H G$, where $H<G$ is an arbitrary separable subgroup and $G$ is countable and sofic. This includes arbitrary finite index group doubles of sofic groups among various other examples.

math.GR↗
arXiv2024

Internal sequential commutation and single generation

We extract a precise internal description of the sequential commutation equivalence relation introduced in [KEP23] for tracial von Neumann algebras. As an application we prove that if a tracial von Neumann algebra $N$ is generated by unitaries $\{u_i\}_{i\in \mathbb{N}}$ such that $u_i\sim u_j$ (i.e, there exists a finite set of Haar unitaries $\{w_i\}_{i=1}^{n}$ in $N^\mathcal{U}$ such that $[u_i, w_1]= [w_k, w_{k+1}]=[w_n,u_j]=0$ for all $1\leq k< n$) then $N$ is singly generated. This generalizes and recovers several known single generation phenomena for II$_1$ factors in the literature with a unified proof.

math.OA↗
arXiv2023

Testing LLMs on Code Generation with Varying Levels of Prompt Specificity

Large language models (LLMs) have demonstrated unparalleled prowess in mimicking human-like text generation and processing. Among the myriad of applications that benefit from LLMs, automated code generation is increasingly promising. The potential to transform natural language prompts into executable code promises a major shift in software development practices and paves the way for significant reductions in manual coding efforts and the likelihood of human-induced errors. This paper reports the results of a study that evaluates the performance of various LLMs, such as Bard, ChatGPT-3.5, ChatGPT-4, and Claude-2, in generating Python for coding problems. We focus on how levels of prompt specificity impact the accuracy, time efficiency, and space efficiency of the generated code. A benchmark of 104 coding problems, each with four types of prompts with varying degrees of tests and specificity, was employed to examine these aspects comprehensively. Our results indicate significant variations in performance across different LLMs and prompt types, and its key contribution is to reveal the ideal prompting strategy for creating accurate Python functions. This study lays the groundwork for further research in LLM capabilities and suggests practical implications for utilizing LLMs in automated code generation tasks and test-driven development.

cs.SE↗
arXiv2023

On Topics Related to Tensor Products with Abelian von Neumann Algebras

In this note, we developed several results concerning abelian von Neumann algebras, their spectrums, and their tensor products with other von Neumann algebras. In particular, we developed a theory connecting elements of the spectrum of $L^\infty(Ω, μ)$ with ultrafilters on a Boolean algebra associated with $(Ω, μ)$, generalizing the case where $Ω$ is a discrete space and its spectrum is identified with the space of ultrafilters on $Ω$. These ultrafilters were also used to define a new kind of tracial ultrapowers. We also developed a theory representing elements of $L^\infty(Ω, μ) \bar{\otimes} M$ as functions that are continuous from the spectrum to $M$ equipped with the weak* topology, analogous to the classical direct integral theory, but which can apply to non-separable von Neumann algebras. We then applied this theory to prove several structural results of isomorphisms $L^\infty(Ω, μ) \bar{\otimes} M \rightarrow L^\infty(Ω, μ) \bar{\otimes} N$ where $M$ and $N$ are $\mathrm{II}_1$ factors and placed several constraints on $M$ and $N$. The ultrafilter representation of the spectrum was then used to perform certain calculations to provide specific examples regarding the spectrum representations of tensor products as well as the structures of isomorphisms $L^\infty(Ω, μ) \bar{\otimes} M \rightarrow L^\infty(Ω, μ) \bar{\otimes} N$.

math.OA↗
arXiv2012

On the minimal distance between two surfaces

This article revisits previous results presented in Optimization which were challenged later by Voisei and Zalinescu (V-Z) in the same journal. We aim to use the points of view of V-Z to modify the original results and highlight that the consideration of the so called Gao-Strang total complementary function is indeed quite useful for establishing necessary conditions for solving this problem.

math.OC↗