主化格上的超模性和次可加性的主化前驱
Majorization precursors to supermodularity and subadditivity on the majorization lattice
AI总结：
本文建立了两种结构性的主化关系（称为前驱），它们支撑了主化诱导格上的超模性和次可加性，并证明了Tsallis熵和Rényi熵在这些格上的超模性和次可加性，以及严格次可加性和严格超模性。
摘要：
我们在主化诱导的格上建立了两种结构性的主化关系，称为前驱，它们支撑了超模性和次可加性的性质。这些前驱关系直接蕴含所有凹函数的和（我们称之为和凹函数）在主化格上是超模和次可加的。利用这些主化关系，我们随后展示了Tsallis熵（对所有$α$）和Rényi熵（对所有$α>1$）的超模性和次可加性（在格论意义上），同时在此过程中恢复了Shannon熵的这些性质。我们进一步强化了这些不等式，证明：(i) 所有这些熵泛函在主化格上是严格次可加的；(ii) Tsallis熵（因此也包括Shannon熵）在主化格上是严格超模的。
英文摘要：
We establish two structural majorization relations, which we call precursors, underlying the properties of supermodularity and subadditivity on the lattice induced by majorization. These are precursors in that they immediately imply that all sums of concave functions, which we dub sum-concave functions, are supermodular and subadditive on the majorization lattice. Using these majorization relations, we then show the supermodularity and subadditivity (in the lattice-theoretic sense) of Tsallis entropies (for all $α$) and Rényi entropies (for all $α> 1$), also recovering these properties for the Shannon entropy in the process. We further strengthen these inequalities, showing that: (i) all these entropic functionals are strictly subadditive on the majorization lattice; (ii) Tsallis entropies (and therefore the Shannon entropy as well) are strictly supermodular on the majorization lattice.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
