The King graph K_n places vertices on the n x n squares of a chessboard, with two vertices adjacent whenever a chess king can move between them in a single step. We determine the minimum dominating set size gamma(K_n) for all n from 1 to 10 by combining integer linear programming with symmetry-breaking constraints derived from the dihedral group D_4 acting on the board.
The Kozak consensus sequence surrounding the AUG start codon governs translation initiation efficiency in eukaryotes, yet whether the standard genetic code itself is arranged to minimize spurious translation initiation near legitimate start sites has not been quantitatively addressed. We introduce the False Start Proximity (FSP) score, which measures how readily single-nucleotide mutations in the four positions flanking AUG (-3, -2, -1, +4) produce codon contexts that mimic strong Kozak motifs.
Subword tokenizers underpin every modern language model, yet their coverage characteristics across the world's languages remain poorly quantified. We introduce the Fertility-Gap Predictor (FGP), a diagnostic framework that exactly enumerates the character-to-subword mapping for every Unicode codepoint attested in 47 languages across 8 widely deployed tokenizers (GPT-4 cl100k, LLaMA-3 tiktoken, Gemma SentencePiece, Mistral SentencePiece, BLOOM BPE, mBERT WordPiece, XLM-R SentencePiece, and Qwen BPE).