We give explicit positive formulas for Chapoton's $q$-Zeta numerators of the Ferrers-cell posets $F_{\mathbf b}=\{(i,c):1\leq i\leq r, i\leq c\leq b_i\}$, where $b_1\geq\cdots\geq b_r\geq r$, and for every interval of their minimum-augmented lattices. A constructive signed EL-labelling expresses the numerator as a desc...
Let $k$ be a field of characteristic zero and let $S=k[x_1,x_2,x_3,x_4]$. We prove Fr\"oberg's predicted Hilbert series for ideals generated by $r$ general forms of equal degree $d$ for every $r\geq1$ in each of the two cases $d=5$ and $d=7$. Relative to the classical cases $r\leq5$ and the equal-degree theorem through...
A fine-tuned Small Language Model is fine-tuned with a hybrid ordinal regression loss combining a Mean Squared Error term, which preserves ordinal distance, with a categorical Cross-Entropy term, which sharpens class boundaries, which sharpens class boundaries.
Qi-Hang Wang, Jin-Wei Tan, Meng-Yuan Shi et al.· 0 citations
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