arXiv:2411.17411v2 Announce Type: replace Abstract: Combinatorics studies how discrete objects can be counted, arranged, and combined under specified rules. Motivated by uncertainty in real-world data and decisions, modern set-theoretic formalisms such as fuzzy sets, neutrosophic sets, rough sets, soft sets, and plithogenic sets have been developed. In particular, neutrosophic sets model uncertainty by assigning to each element degrees of truth, indeterminacy, and falsity. In parallel, these uncertainty frameworks are increasingly investigated in graphized and hyperized forms, where generalized graph models encompass classical graphs, hypergraphs, and higher-order "superhyper" structures; related hyper- and superhyper-concepts also arise beyond graph theory. This book (Edition 2.0) surveys and consolidates recent developments at the intersection of combinatorics, uncertain sets, uncertain graphs, and hyper/superhyper frameworks, while introducing several new graph and set concepts. As representative contributions, we extend graph-theoretic notions via Neutrosophic Oversets, Neutrosophic Undersets, Neutrosophic Offsets, and the Nonstandard Real Set. The second edition adds newly introduced concepts, corrects typographical issues, and re-examines mathematical consistency, aiming to serve as a compact reference and a source of inspiration for further research.

arXiv:2411.16085v4 Announce Type: replace-cross Abstract: AdamW has been the default optimizer for transformer pretraining. For many years, our community searched for faster and more stable optimizers with only constrained positive outcomes. In this work, we propose a \textbf{one-line modification in Pytorch} to any momentum-based optimizer, which we rename cautious optimizer, e.g. C-AdamW and C-Lion. Our theoretical result shows that this modification preserves Adam's Hamiltonian function and it does not break the convergence guarantee under the Lyapunov analysis. In addition, a whole new family of optimizers is revealed by our theoretical insight. Among them, we pick the simplest one for empirical experiments, showing not only consistent speed-up on LLM pretraining, but also image classification, with minimum extra tuning on hyperparameters. Code is available at https://github.com/kyleliang919/C-Optim.