arXiv subjects

Jonathan Leung

Publications and source records attributed to Jonathan Leung.

6 recordsLinked to original sources

arXiv2026

On the revolving structure of the L\'evy dragon and its linear transforms

L\'evy's Dragon Curve is a well-known self-similar fractal, notable for its ability to tile the complex plane. We review a representation of the curve as a set of points given by complex power series satisfying a revolving condition, and study how this representation changes under linear transformations, while preserving its characteristic geometric properties. We introduce a labeled directed graph that encodes these series representations and show that this directed graph remains invariant under linear transformations, with only the labeling subject to change. Furthermore, we provide a geometric characterization of the resulting variations in graph labeling.

math.CA↗

arXiv2026

Shifted L\'evy's Dragon Curve and Directed Graph

It is known that every point on L\'evy's Dragon Curve admits a natural representation as a complex power series. We introduce a directed graph $\mathcal{G}_1$ which characterizes this representation. In this paper, we study the translation of the curve by $s=-1/2+i/2$. We identify another directed graph $\mathcal{G}_2$, that characterizes the translated curve and exhibits a revolving structure analogous to that of $\mathcal{G}_1$.

math.CA↗

arXiv2025

RoleRAG: Enhancing LLM Role-Playing via Graph Guided Retrieval

Large Language Models (LLMs) have shown promise in character imitation, enabling immersive and engaging conversations. However, they often generate content that is irrelevant or inconsistent with a character's background. We attribute these failures to: (1) the inability to accurately recall character-specific knowledge due to entity ambiguity, and (2) a lack of awareness of the character's cognitive boundaries. To address these issues, we propose RoleRAG, a retrieval-based framework that integrates efficient entity disambiguation for knowledge indexing with a boundary-aware retriever for extracting contextually appropriate information from a structured knowledge graph. Experiments on role-playing benchmarks show that RoleRAG's calibrated retrieval helps both general-purpose and role-specific LLMs better align with character knowledge and reduce hallucinated responses.

cs.AI↗

arXiv2025

From Entity-Centric to Goal-Oriented Graphs: Enhancing LLM Knowledge Retrieval in Minecraft

Large Language Models (LLMs) demonstrate impressive general capabilities but often struggle with step-by-step procedural reasoning, a critical challenge in complex interactive environments. While retrieval-augmented methods like GraphRAG attempt to bridge this gap, their fragmented entity-relation graphs hinder the construction of coherent, multi-step plans. In this paper, we propose a novel framework based on Goal-Oriented Graphs (GoGs), where each node represents a goal and edges encode logical dependencies between them. This structure enables the explicit retrieval of causal reasoning paths by identifying a high-level goal and recursively retrieving its prerequisites, forming a coherent chain to guide the LLM. Through extensive experiments on the Minecraft testbed, a domain that demands robust multi-step planning and provides rich procedural knowledge, we demonstrate that GoG substantially improves procedural reasoning and significantly outperforms GraphRAG and other state-of-the-art baselines.

cs.AI↗

arXiv2025

TheoremExplainAgent: Towards Video-based Multimodal Explanations for LLM Theorem Understanding

Understanding domain-specific theorems often requires more than just text-based reasoning; effective communication through structured visual explanations is crucial for deeper comprehension. While large language models (LLMs) demonstrate strong performance in text-based theorem reasoning, their ability to generate coherent and pedagogically meaningful visual explanations remains an open challenge. In this work, we introduce TheoremExplainAgent, an agentic approach for generating long-form theorem explanation videos (over 5 minutes) using Manim animations. To systematically evaluate multimodal theorem explanations, we propose TheoremExplainBench, a benchmark covering 240 theorems across multiple STEM disciplines, along with 5 automated evaluation metrics. Our results reveal that agentic planning is essential for generating detailed long-form videos, and the o3-mini agent achieves a success rate of 93.8% and an overall score of 0.77. However, our quantitative and qualitative studies show that most of the videos produced exhibit minor issues with visual element layout. Furthermore, multimodal explanations expose deeper reasoning flaws that text-based explanations fail to reveal, highlighting the importance of multimodal explanations.

cs.AI↗

arXiv2024

Towards Goal-oriented Prompt Engineering for Large Language Models: A Survey

Large Language Models (LLMs) have shown prominent performance in various downstream tasks and prompt engineering plays a pivotal role in optimizing LLMs' performance. This paper, not only as an overview of current prompt engineering methods, but also aims to highlight the limitation of designing prompts based on an anthropomorphic assumption that expects LLMs to think like humans. From our review of 50 representative studies, we demonstrate that a goal-oriented prompt formulation, which guides LLMs to follow established human logical thinking, significantly improves the performance of LLMs. Furthermore, We introduce a novel taxonomy that categorizes goal-oriented prompting methods into five interconnected stages and we demonstrate the broad applicability of our framework. With four future directions proposed, we hope to further emphasize the power and potential of goal-oriented prompt engineering in all fields.

cs.CL↗