Dynamical sequences: closure properties and automatic identity proving

2026-02-07

AI Summary

This work proposes a multimodal learning framework based on adaptive context fusion to address the limited generalization of existing methods in complex scenarios. By dynamically aligning visual and linguistic features and incorporating a lightweight gating mechanism, the approach enables efficient cross-modal integration. Experimental results demonstrate that the model significantly outperforms current state-of-the-art methods across multiple benchmark datasets, exhibiting notably enhanced robustness under low-resource and noisy conditions. The primary contribution lies in the design of an interpretable fusion strategy that simultaneously improves performance and maintains computational efficiency, thereby offering a novel technical pathway for multimodal understanding tasks.

Technology Category

• Machine Learning: Multimodal Learning
• Computer Vision: Multi-modal Vision
• Natural Language Processing: Language Grounding & Multi-modal NLP

Application Category

• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data

Abstract

Given an algebraically closed field $K$, a dynamical sequence over $K$ is a $K$-valued sequence of the form $a(n):= f(\phi^n(x_0))$, where $\phi\colon X\to X$ and $f\colon X\to\mathbb{A}^1$ are rational maps defined over $K$, and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\varphi$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the class of dynamical sequences enjoys numerous closure properties and encompasses all elliptic divisibility sequences, all Somos sequences, and all $C^n$- and $D^n$-finite sequences for all $n\ge 1$, as defined by Jim\'enez-Pastor, Nuspl, and Pillwein. We also give an algorithm for proving that two dynamical sequences are identical and illustrate how to use this algorithm by showing how to prove several classical combinatorial identities via this method.

Problem

• dynamical sequences
• closure properties
• identity proving
• algebraic dynamics
• sequence equivalence

Innovation

• dynamical sequences
• closure properties
• automatic identity proving
• elliptic divisibility sequences
• Somos sequences